Signatures of a Deconfined Phase Transition on the Shastry-Sutherland Lattice:Applications to Quantum Critical SrCu2(BO3)2

Jong Yeon Lee,1 Yi-Zhuang You,1, 2 Subir Sachdev,1 and Ashvin Vishwanath1

1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA2Department of Physics, University of California, San Diego, California 92093, USA

(Dated: September 13, 2019)

We study a possible deconfined quantum phase transition in a realistic model of a two-dimensionalShastry-Sutherland quantum magnet, using both numerical and field theoretic techniques. Usingthe infinite density matrix renormalization group (iDMRG) method, we verify the existence ofan intermediate plaquette valence bond solid (pVBS) order, with two fold degeneracy, betweenthe dimer and Neel ordered phases. We argue that the quantum phase transition between theNeel and pVBS orders may be described by a deconfined quantum critical point (DQCP) with anemergent O(4) symmetry. By analyzing the correlation length spectrum obtained from iDMRG, weprovide evidence for the DQCP and emergent O(4) symmetry in the lattice model. Such a phasetransition has been reported in the recent pressure tuned experiments in the Shastry-Sutherlandlattice material SrCu2(BO3)2 [1]. The non-symmorphic lattice structure of the Shastry-Sutherlandcompound leads to extinction points in the scattering, where we predict sharp signatures of a DQCPin both the phonon and magnon spectra associated with the spinon continuum. The effect of weakinterlayer couplings present in the three dimensional material is also discussed. Our results shouldhelp guide the experimental study of DQCP in quantum magnets.

Contents

I. Introduction 1

II. Numerical Study 3A. Detection of the pVBS Phase 3B. Correlation Length Spectra, Monopole

Fluctuations, and Emergent O(4) Symmetry4

III. Symmetry Analysis 5

IV. Dangerously Irrelevant Scaling and ItsAbsence 7

V. Spectral Signatures of DQCP 10

VI. Effects of Interlayer Coupling 12

VII. Predictions for Experiment 14

VIII. Conclusions 15

Acknowledgments 15

A. Monopole Transformation 15

B. Comparison with a DMRG simulationresult of the J1-J2 model in the squarelattice 17

C. VBS-Phonon Coupling and PhononSpectrum 19

D. Detailed Numerical Data 21

References 22

I. INTRODUCTION

Quantum magnets can host some of the most exoticphenomena in condensed matter physics, due to thestrong quantum fluctuations of the microscopic spin de-grees of freedom. Notably, deconfined gauge fluctuationsand fractionalized spinon excitations can emerge in quan-tum magnets, which bear no analog in classical spin sys-tems. Such behavior can exist either in a quantum spinliquid, which is a stable phase of matter with topolog-ical order [2–4]; or by tuning a single parameter to acritical point known as the deconfined quantum criticalpoint (DQCP) [5, 6]. The DQCP describes the possiblecontinuous phase transition between two distinct sym-metry breaking phases, which is beyond the conventionalLandau-Ginzburg paradigm. While the search for quan-tum spin liquids is still an ongoing research effort in con-densed matter physics [7], the possibility of observing theDQCP in materials could provide us with an alternativeopportunity to study the properties of deconfined spinonsand emergent gauge fields in quantum magnets, as wellas in interacting fermion systems that realizes the DQCP[8–13].

In a recent experiment [1], a phase transition be-tween Neel antiferromagnet and plaquette valence bondsolid (crystal) [14] was observed in a single crystal ofSrCu2(BO3)2 under pressure. The material is a layeredquantum magnet. Within each two-dimensional layer,the copper ions carry the spin-1/2 degrees of freedom andare arranged on a Shastry-Sutherland lattice as shown inFig. 1(a). The spin system was proposed to be effectivelydescribed by the Shastry-Sutherland model [15, 16]

H = J1

∑ij∈n.n.

Si · Sj + J2

∑ij∈dimer

Si · Sj , (1)

where the J1 and J2 bonds are specified according to

arX

iv:1

904.

0726

6v3

[con

d-m

at.st

r-el

] 12

Sep

201

9

2

σ xy

σxy

Gx

Gy

J1

J2

(a)

kx

ky

0 π 2π

π

2π

Γ X

Y M

(b)

dVBS pVBS Néel0.675 0.77 J1 / J2

1storder DQCP

(c)

1

FIG. 1: (a) The Shastry-Sutherland lattice of copper sites(small circles) in SrCu2(BO3)2, on which the spins reside.The spins are coupled across nearest neighbor bonds (J1, inblue) and dimer bonds (J2, in red). Each unit cell containsfour sites, as shaded in yellow. The glide reflection Gx, Gyand the diagonal reflection σxy, σxy symmetries are indicatedon the lattice. (b) Diffraction peaks from copper sites. Thedarker dot indicates a higher intensity. The extinction pointsare marked out by red circles. The first Brillouin zone isshaded in yellow, corresponding to the unit cell in (a). Spe-cial momentum points Γ, X, Y,M are defined as labeled. (c)The phase diagram of the spin model Eq. (1). The Neel anti-ferromagnetic and dimer valence bond solid (dVBS) phasesare separated by the intermediate plaquette valence bondsolid (pVBS) phase upon tuning the J1/J2 ratio. The criticalpoints are determined in Tab. I based on our iDMRG result.The transition between pVBS and Neel phases is likely to bea DQCP (or weakly first-order proximate to a DQCP).

Fig. 1(a). The ratio J1/J2 between the coupling con-stants is tunable by pressure in experiments within cer-tain range. In the large J1 (or large J2) limit, the modelreduces to the square lattice Heisenberg model (or thedecoupled dimerized model), which stabilizes the Neelphase (or the dimer valence bond solid (dVBS) phase).Between these two limits, numerical [14, 17–20] and the-oretical [21] analysis of the model have revealed an in-termediate plaquette valence bond solid (pVBS) phase,as illustrated in Fig. 1(c). Remarkably, the experimentin Ref.1 seems to confirm this phase diagram. Since thepVBS and Neel phases separately break two distinct sym-metries, the lattice and the spin rotation symmetry, adirect second-order transition between them would nec-essarily go beyond the Landau-Ginzburg paradigm andpoint to the possibility of the DQCP. Although the na-ture of the pVBS-Neel transition remains unresolved byexperiments, there are promising signs for the excitingopportunity that SrCu2(BO3)2 might provide the firstexperimental platform to realize DQCP.

Recent studies on different models with the same sym-metry class showed that the transition between pVBS

and Neel phases could be first-order[22, 23]. However,despite being first-order, the transition is accompaniedwith an extended region of quantum-critical-like scalingand an emergent O(4) symmetry, implying that the tran-sition could be close to a DQCP (possibly as an avoidedcriticality). Thus the DQCP is still the best theory to ac-count for these anomalous features in the critical region,even though it may eventually break down at longestscales. Note that the J-Q model or loop model stud-ied in Monte Carlo simulations[22, 23] are designed dif-ferently from the original Shastry-Sutherland model toavoid the sign problem. Given that the first- or second-order nature of the transition can be tuned by modelparameters [24–27] and is therefore a model-dependentproperty, the fate of the pVBS-Neel transition in theShastry-Sutherland model remains to be fully resolvedyet.

The goal of this work is to investigate the pVBS-Neel transition in the Shastry-Sutherland model Eq. (1)in more detail using both field theory and the densitymatrix renormalization group (DMRG) approach, andto identify the unique signatures of DQCP that can beprobed by inelastic neutron scattering (INS) or resonantinelastic X-ray scattering (RIXS) experiments. We usethe infinite DMRG technique to overcome the sign prob-lem. Our numerical simulation indicates (i) that the tran-sition between pVBS and Neel phases appears continuousup to the largest available system size (infinite cylinderwith the circumference of 10 lattice sites), although wecan not rule out the possibility of a weakly first-ordertransition due to our limited system size. (ii) We alsoobserve the asymptotic degeneracy between spin-tripletand spin-singlet excitations over a large length scale,demonstrating an approximate emergent O(4) symmetrywhich rotates among the Neel and pVBS order parame-ters. Our theoretical analysis further suggests that (iii)in the Shastry-Sutherland lattice, in contrast to previ-ous realizations of DQCP, a dangerously irrelevant oper-ator is absent which has consequences for numerics andthat (iv) critical spinon continua appear at the extinctionpoints of lattice diffraction peaks (c.f. Fig. 1(b)) in boththe magnon and phonon channels at low-temperaturearound the DQCP. The universal critical behaviors ofthese continua are examined as well, which could guidethe experimental study of the candidate DQCP in theSrCu2(BO3)2 material.

The rest of the paper is organized as follows. InSec. II, we perform an infinite DMRG simulation on theShastry-Sutherland spin model and discuss the natureof the phase transition between Neel and VBS phasesbased on a correlation length spectra. In Sec. III, weanalyze symmetry quantum numbers of a monopole op-erator whose proliferation induces the transition to theVBS phase. By investigating the transformation prop-erty of the monopole, we show that the single-monopoleterm is suppressed while the double-monopole term canappear in the action describing the Neel order in theShastry-Sutherland lattice. We compare the differences

3

among various microscopic models – easy-plane, rectan-gular, and Shastry-Sutherland – whose possible emer-gent symmetry is all O(4). One distinct feature of theShastry-Sutherland lattice is the presence of the relevantanisotropy operator that breaks the four-fold lattice rota-tion symmetry, which stabilizes the VBS order and givesrise to a fast-growing spectral gap in the spin-0 chan-nel as the system enters the VBS phase. In Sec. V, wepropose the spectral signatures of DQCP in the Shastry-Sutherland model, including the SO(4) conserved currentfluctuation in the magnon spectrum and the VBS fluc-tuation in the phonon spectrum. Both of these featuresappear at the extinction point of the Shastry-Sutherlandlattice, which is detectable at low energy without over-whelmed by the elastic scattering signals. We concludeour discussion in Sec. VII.

II. NUMERICAL STUDY

In this section, we study the model in Eq. (1) using theinfinite density matrix renormalization group (iDMRG)method [28, 29]. We wrap the 2D system onto a cylin-der which is infinite along x-direction but compact alongy-direction, with a finite circumference L. In the simu-lation, spin-1/2s are mapped into the hard-core bosonswith density 〈n〉 = 1/2 per site; an antiferromagnetic spininteraction would then be translated into hopping anddensity-density interaction terms for bosons. As the bo-son number is conserved in the simulation, we have an ex-plicit U(1)z symmetry which allows us to extract correla-tion functions for an operator with a specific U(1)z quan-tum number. During the simulation, we fixed the valueof J2 = 1 and tuned the value of J1 across the phase tran-sition between the pVBS and Neel order phases. Sincethe unit cell of Shastry-Sutherland lattice contains 2× 2square unit cells, the iDMRG unit cell becomes 2× (2m)slice of the infinite cylinder, where the circumference ofthe cylinder L = 2m.

In the iDMRG simulation, we have two limiting factorsto describe the exact two-dimensional state: the circum-ference length L and the bond dimension χ. Due to lim-ited computational capacity, it is difficult to conclusivelyidentify whether the phase transition is continuous orweakly first order in the DMRG simulation. Still, we canextract useful information of the ground state by simulat-ing the model with increasing values of L and χ. For thediscussionon bond dimension scaling, see Appendix. D.In the DMRG simulation, we measured (i) energy, (ii)plaquette order parameter, and (iii) correlation lengthspectra.

A. Detection of the pVBS Phase

Although the matrix product state description of thestate is exact in the infinite bond dimension limit, fora finite bond dimension, the cylindrical geometry of the

FIG. 2: Visualization of bond strengths 〈Si · Sj〉 within theiDMRG unit cell. x-direction is along the cylinder, and y-direction is around the circumference. The width of the bondrepresents the value of S1 ·S2. The line color is black (red) ifS1 ·S2 is negative (positive). One can notice that the singletis formed at a plaquette.

iDMRG simulation provides some bias to the preferredentanglement structure for the ground state. As a result,in the pVBS phase, one of two symmetry broken phaseswould be automatically chosen and the order parameterwould not vanish in the iDMRG simulation. To char-acterize the pVBS phase, we define the plaquette orderparameter as follows (See Eq. (5) and Eq. (6) in Sec. IIIfor a more rigors definition),

Im〈M〉 ∼∑i

(−)x〈Si · Si+x〉 − (−)y〈Si · Si+y〉. (2)

In the pVBS phase, the dimer strengths 〈Si · Sj〉 oneach bond 〈ij〉 can be visualized in Fig. 2, which clearlydemonstrates the pattern of the pVBS ordering aroundempty (square) plaquettes. Indeed, we measured thatIm〈M〉 6= 0 and Re〈M〉 = 0 in the paramagnetic regimeas expected for the pVBS phase. (See Fig. 4)

Although the plaquette order parameter Im〈M〉 is auseful indicator, it is not precise because (i) the systemsize does not reach the thermodynamic limit and (ii) thegeometry of the iDMRG simulation provides some biastoward a specific entanglement structure for the groundstate. Albeit small, it has a non-vanishing value in theNeel phase, see Fig. 4. Thus, we use a discontinuity inthe second derivative of the energy ∂2E/∂J2

1 to locatethe pVBS-Neel transition point. Up to L = 10, the firstorder derivative of energy is continuous across the phasetransition, implying that the transition is either a weaklyfirst order or second order transition. On the other hand,from the energy plot in Fig. 3, the dVBS-pVBS transitionpoint can be easily extracted because the dVBS state isan exact ground state of Eq. 1 with energy per site Esite =−0.375J2. As we can see, the first order derivative of theenergy is discontinuous here, signaling a clear first-orderphase transition between the two spin singlet dVBS andpVBS phases. For L < 6, we do not observe the pVBSphase. Transition points for different system sizes aresummarized in Tab. I.

4

L = 6 L = 8 L = 10 L = 12

(J1/J2)c1 0.682 0.677 0.675 0.675

(J1/J2)c2 0.693 0.728 0.762 0.77

TABLE I: (J1/J2)c1 ((J1/J2)c2) is the transition point be-tween the dVBS and pVBS (pVBS and Neel) phases. Transi-tion points are extracted from the peak of the energy deriva-tive at χ = 4000. At L = 12, the DMRG simulation do notconverge for different initial states at χ = 4000, resulting indifferent transition points. Thus, the critical point is deter-mined as the midpoint between two transition points obtainedfrom the VBS-like and Neel-like initial states.

FIG. 3: At L = 10, χ = 4000. (a) Energy (E) per site. Thehorizontal dotted line represents the exact ground state en-ergy of dimer VBS state, Esite = −0.375. (b) ∂E/∂J1 persite near the transition between plaquette VBS and Neel or-der. The continuous first order derivative is a characteristicof the continuous phase transition. Black dashed lines denotephase boundaries among dimer VBS, plaquette VBS, and Neelorder.

B. Correlation Length Spectra, MonopoleFluctuations, and Emergent O(4) Symmetry

Here, we present a signature of the DQCP in the cor-relation length spectrum data obtained from the iDMRGsimulation. A continuous phase transition is character-ized by the divergence of a correlation length, and anequal-time correlation function exhibits the power-lawdecaying behavior 〈O(r)O(0)〉 ∝ r−2∆O where ∆O isa scaling dimension of an operator O. In particular, ifthere exists an emergent symmetry, operators unified un-der the emergent symmetry should share the same scal-ing dimension ∆O [30]. For example, at the conven-tional DQCP with the emergent SO(5) symmetry, theNeel n = (nx, ny, nz) and VBS v = (vx, vy) order pa-rameter should have the same power-law behavior:

〈n(r) · n(0)〉 ∼ 〈v(r) · v(0)〉 ∼ 1/r1+η, (3)

where η = 2∆O − 1 is the anomalous exponent definedrelative to the engineering exponent in (2+1)D which is1. However, in the iDMRG simulation, the finite cir-cumference L of the cylinder and finite bond dimensionχ introduce a cutoff length scale [31], which prevents usfrom observing the power-law behavior in the correlation

FIG. 4: The inverse of the largest correlation length for spinsinglet and triplet operators as a function of J1/J2 at L = 10and χ = 4000. Here, the dashed line represents the transitionpoint extracted from the second order energy derivative. Thesmall plot at the top right shows the plaquette order parame-ter (ImM) across the transition. Deep in the VBS phase, thecorrelation length of a spin-triplet operator is larger than thatof a spin-singlet operator as expected by a mean-field theory.As we approach the critical point, we can observe that the cor-relation length of the spin-singlet sector becomes larger thanthat of the spin-triplet sector. This behavior agrees with whatis expected from the scenario in Fig. 8.

function. Therefore, at long distances along the cylinder,the DMRG correlation function of an operator O will al-ways decay exponentially with certain finite correlationlength ξO in the simulation. Nevertheless, instead of try-ing to compare the scaling dimension ∆O, we can deter-mine the emergent symmetry by comparing the correla-tion lengths ξO between Neel (spin-1) and VBS (spin-0)excitations. In Zauner et al. [32], it has been found thatthe correlation length spectrum is inversely proportionalto the energy of the excitations that mediate this cor-relation behavior. Thus, the correlation length spectragive access to the individual dynamics of different typesof excitations.

Using the DMRG transfer matrix technique, one canreadily obtain the correlation lengths ξ along the cylin-der. Moreover, since our DMRG simulation has an ex-plicit U(1)z symmetry, extracted correlation lengths arelabeled by Sz quantum numbers. Since the microscopicmodel has full SO(3) symmetry, there will be an ex-act degeneracy among correlation lengths with differentquantum numbers. For example, a three-fold degener-acy among Sz = 0,±1 would imply that this correlationlength corresponds to the excitation carrying the quan-tum number S = 1 of the SO(3) symmetry. In this way,we can identify the SO(3) spin quantum number of eachoperator appeared in the correlation length spectrum.

In Fig. 4, we plot the inverse of the largest correla-tion lengths for spin-singlet and triplet operators. Thespin-singlet operator corresponds to the pVBS order pa-rameter (or the monopole operator M) at low-energy.

5

The monopole operator is gapped in both the Neel andpVBS phases and only become gapless at the criticalpoint. Therefore, the divergence of the spin-singlet cor-relation length can signal the onset of DQCP in the ther-modynamic limit. Indeed, we identified that the peakof the singlet correlation length exactly coincides withthe critical point extracted from the singularity of theenergy derivative (Fig. 3). Furthermore, the correlationlength ξS=0 at the critical point increases as the bond di-mension χ increases. Although here we present only thecorrelation length spectrum at L = 10 and χ = 4000, wealso performed numerical simulations for different systemsizes and obtained the result that the critical point sum-marized in Tab. I coincides with the peak of the spin sin-glet correlation length. Therefore, we can infer that thetransition is induced by the proliferation of monopoles,consistent with DQCP physics.

Moreover, to contrast our simulation result in theShastry-Sutherland lattice with the conventional O(3)Wilson-Fisher transition between an explicitly dimer-ized VBS and Neel order phases [33], we performed theiDMRG simulation for a 2D J1-J ′1 model with the antifer-romagnetic Heisenberg coupling J1 on nearest neighborbonds together with the coupling J ′1 on a fixed set ofdimer covering bonds (See Appendix. B), and hence aunique VBS pattern is pinned by J ′1. Indeed, for thismodel, one can observe that the correlation length ofthe spin-triplet sector is always larger than the correla-tion length of the spin-singlet sector across the phasetransition. This agrees with the picture discussed inRef. [34, 35], where the transition is triggered by the con-densation of spin-triplet excitations (triplon) from theVBS phase. Within the mean-field theory framework,it was shown that the energy of the spin-triplet excita-tion Etriplon is smaller than the energy of the spin-singletexcitation Esinglon throughout the whole transition. Un-der the further assumption that the singlon and triplonhave the similar characteristic velocity v [32], one ex-pect the aforementioned ordering of correlation lengthsξ ∼ (v/E). Thus, the inversion of the magnitude of cor-relation lengths for spin singlet and triplet operators atthe transition signifies the unconventional feature of theDQCP in the Shastry-Sutherland lattice. For a detailedanalysis regarding DQCP physics, see Sec. IV.

Finally, we remark that the spin-singlet correlationlength ξS=0 and the spin-tiplet correlation length ξS=1

approaches each other at the (finite size) critical point aswe increase the system size. At L = 6 and χ = 4000, theratio ξS=1/ξS=0

∣∣crit

= 0.33, but at L = 10 and χ = 4000,

ξS=1/ξS=0

∣∣crit

= 0.95. From this trend, we expect tohave ξS=1/ξS=0 = 1 in the thermodynamic limit, whichindicates that the spin-singlet and spin-triplet excitationswill become degenerate at the critical point, forming thefour-component vector representation of a larger O(4)symmetry group. Put differently, we can observe thatthe crossing point of ξS=0 = ξS=1 in Fig. 4 approaches tothe critical point as we increase the system size, consis-tent with the emergent O(4) symmetry relating Neel and

VBS order parameters.

III. SYMMETRY ANALYSIS

The field theory of DQCP, the so called “non-compact”CP1 theory, has the following form [5, 36]

LCP1 = |(∂ − ia)z|2 + κ(∇× a)2

+ . . . , (4)

where a two-component complex spinon z = (z1, z2)T

is coupled to U(1) gauge field a. On top of this crit-ical theory, one can have additional terms dependingon symmetry of the system. The Shastry-Sutherlandlattice has a p4g space group symmetry, as shown inFig. 1(a). The lattice respects two glide reflection Gx, Gyand two diagonal reflection σxy, σxy symmetries as illus-trated in Fig. 1(a). The glide reflections and the (spinful)time-reversal T symmetries can combine into compositesymmetries T Gx, T Gy, dubbed as the time-reversal glidesymmetries. Note that glide-reflection symmetry is alsobroken in the Neel phase, while the time-reversal glide isnot. Therefore, relative to the Neel phase, it is properto think about the pVBS phases as to break the time-reversal glide symmetries.

To define the symmetry transformations more con-veniently, we consider an ideal version of the Shastry-Sutherland lattice on a regular square lattice withoutdistortion, as shown in Fig. 5. This does not changethe symmetry group but allows us to label every siteby the Cartesian coordinate (x, y) conveniently (wherex, y ∈ Z). The length of the nearest Cu-Cu bond is setto 1, such that the unit cell is of the size 2 × 2 (andhence the lattice constant is 2 here). With this, we candefine the glide reflections Gx : (x, y) → (x + 1,−y)and Gy : (x, y) → (−x, y + 1), the diagonal reflectionsσxy : (x, y)→ (y, x) and σxy : (x, y)→ (−y + 1,−x+ 1),as well as the translations Tx : (x, y) → (x + 2, y) andTy : (x, y)→ (x, y + 2). Together, they generate the p4gspace group. The p4g space group also contains a 90

rotation symmetry C4 : (x, y) → (−y + 2, x − 1) withrespect to the center of the plaquette without a diagonalbond.

On each site i = (x, y), we define the spin operatorSi = (Sxi , S

yi , S

zi ), whose symmetry transformations are

listed in Tab. II (assuming the spin rotation is not lockedto the spatial rotation in lack of the spin-orbit coupling).The time-reversal symmetry T is also included, whichcan flip all spin components. The Neel order parametern = (nx, ny, nz) and VBS order parameters v = (vx, vy)are defined as

n ∼ (−)x+ySi,

vx ∼ (−)x( 14 − Si · Si+x),

vy ∼ (−)y( 14 − Si · Si+y),

(5)

where each prefactor translates into the momentum(π, π), (π, 0) and (0, π) respectively in k-space, as marked

6

σ xy

σxy

Gx

GyJ1

J2C4

x"

y"

-1

0

1

2-1 0 1 2

1

FIG. 5: The ideal Shastry-Sutherland lattice, deformed fromFig. 1(a). Each site i can be labeled by a Cartesian coordinate(x, y). The x and y coordinates are calibrated on the top andright respectively. The displacement vectors x = (1, 0) andy = (0, 1) are defined to connect nearest neighbor sites. J1

and J2 couplings are assigned to the nearest neighbor (in blue)and dimer (in red) bonds.

(kx, ky) n M† Si fi

Gx (kx,−ky) −n −M† SGx(i) (−)yfGx(i)

Gy (−kx, ky) −n −M† SGy(i) fGy(i)

σxy (ky, kx) n M Sσxy(i) (−)xyfσxy(i)

σxy (−ky,−kx) n M Sσxy(i) (−)x(y+1)fσxy(i)

T (−kx,−ky) −n M −Si Kiσ2fi

T Gx (−kx, ky) n −M −SGx(i) (−)yKiσ2fGx(i)

T Gy (kx,−ky) n −M −SGy(i) Kiσ2fGy(i)

TABLE II: Symmetry transformation of momentum (kx, ky),Neel order n, monopole operator M†, spin operator Si andfermionic spinon fi (in the sense of PSG).

out in Fig. 1(b), and the displacement vectors are definedto be x = (1, 0) and y = (0, 1). The VBS order parame-ters vx, vy can be combined to form the following operatorgiven a certain gauge choice (Appendix.A)

M† =1√2

((vx + vy) + i(vx − vy)

), (6)

known as the monopole operator in the literature [5, 37],which corresponds to a hegehog-monopole of the Neel or-der parameter n in the spacetime. The monopole eventchanges the skyrmion number of the n field configurationby +1, and it is equivalent to inserting 2π-flux of U(1)gauge field a in the CP1 theory Eq. (4). The symmetryproperties of n and M† follows from those of the spinoperator Si and is summarized in Tab. II. For a detailedderivation, see Appendix A. Apart from these discretesymmetries, there is also an SO(3) spin rotation symme-try, under which n transforms as a SO(3) vector.

The expectation value of the monopole operator 〈M〉defined in Eq. (6) serves as a unified order parameter forvarious types of VBS orders. Depending on the phase

λ2 > 0: square pVBS λ2 < 0: diamond pVBS

FIG. 6: Two types of plaquette valence bond solid (pVBS)phases. Thick purple links encircle the plaquette on whichthe spin singlet is formed. A diamond plaquette contains thediagonal bond, while a square plaquette does not. Each typeof the pVBS order induces a corresponding lattice distortion.The undistorted lattice is shown as the background for con-trast. Depending on the sign of λ2, either diamond plaquetteor square plaquette pVBS is favored.

angle of 〈M〉, the columnar VBS (cVBS) is described by〈M〉 ∼ ±e±iπ/4 and the plaquette VBS (pVBS) is de-scribed by 〈M〉 ∼ ±1 (diamond plaquette) or 〈M〉 ∼ ±i(square plaquette) as illustrated in Fig. 6. On the otherhand, 〈M〉 6= 0 could also be interpreted as the con-densation of monopoles. So the DQCP, as a transitionfrom the Neel phase into the VBS phase, can be thoughtas driven by the VBS ordering or equivalently by themonopole condensation (starting from the Neel phase),which can be tuned by a monopole chemical potential rin the Lagrangian as rM†M. The transition happensas r changes sign. The condensation of monopole estab-lishes the VBS order on the one hand and simultaneouslydestroys the Neel order on the other hand, due to a non-trivial topological term among the Neel and VBS orderparameters, which was analyzed in details in Ref. 5, 6.This scenario provides a plausible description of a directcontinuous transition between the Neel and VBS phases.

However, apart from the apparent tuning parameterterm rM†M, we must also include other symmetry-allowed (multi-)monopole terms in the Lagrangian, whichcould crucially influence the properties of the DQCP. Onthe Shastry-Sutherland lattice, to the leading order, theytake the form of

LM = rM†M+ λ2 ReM2 + · · · . (7)

Here we adopt the short-hand notations ReO = (O +O†)/2 and ImO = (O−O†)/(2i) for generic operator O.Given the symmetry properties in Tab. II, one can seethat the single monopole term, no matter ReM or ImM,is forbidden by the glide reflection symmetry Gx or Gy.Furthermore, the imaginary part of the double-monopoleterm ImM2 is forbidden by the diagonal reflection sym-metry σxy or σxy. Note that these symmetries exist inthe critical theory as the spontaneous symmetry break-ing has not yet occurred and the microscopic model hasthe symmetries.

7

The higher-order monopole terms (M4,M6, ...) are ex-pected to be less relevant and are therefore not includedin Eq. (7) explicitly. Therefore the double-monopole termλ2 ReM2 is the most relevant monopole perturbationallowed on the Shastry-Sutherland lattice. Dependingon its sign, the system will favor a square plaquette(or diamond plaquette) VBS order in the VBS phase,describe by the order parameter ImM (or ReM), ifλ2 > 0 (or λ2 < 0), as demonstrated in Fig. 6. Thesquare and diamond pVBS orders have distinct symme-try properties. Under the reflection symmetries σxy andσxy, ImM 7→ − ImM while ReM stays invariant (seeTab. II), so the square pVBS spontaneously breaks thereflection symmetries while the diamond pVBS does not.Additionally, square-plaquette-centered C4 rotation sym-metry is spontaneously broken in the diamond pVBS,while it is not in the square pVBS. Therefore the twodifferent pVBS orders will lead to different lattice dis-tortions that are symmetry-wise distinguishable in theexperiments in the X-ray/neutron diffraction or NMR[1, 38, 39].

Previous studies [20, 40] as well as our iDMRG datashow that the pVBS phase has a square plaquette order.Thus λ2 > 0 should be relevant to our discussion of thepVBS-Neel transition in the Shastry-Sutherland modelEq. (1). λ2 > 0 can be also argued based on the mi-croscopic Hamiltonian in Eq. (1). In the analysis of thepVBS phases, there are two types of singlet plaquetteconfigurations: s-wave and d-wave types [14] representedby the following singlet pairing configurations in a pla-quette:

s-wave: + d-wave:

(8)

For simplicity, consider a single plaquette with four spin-1/2s on the corners. For both square and diamondplaquettes, there is a AFM coupling J1 along plaque-tte sides; for the diamond plaquette, there is an addi-tional AFM J2 coupling across one diagonal. Then, thes-wave and d-wave singlet configuration has the energy−2J1 + 1/4J2 and −3/4J2 respectively for the diamondplaquette. As the pVBS phase exists at a parameterregime J1/J2 ∼ 0.7, an estimation of the singlet con-figuration energy as listed in Tab. III indicates that thes-wave pairing in the square plaquette has the lowestenergy. In the iDMRG simulation, the wavefunction in-deed exhibits the s-wave pairing symmetry in the pVBSphase. A recent exact-diagonalization study [41] on thesmall cluster of spins (Ns = 40) also reported that thephase next to the Neel order phase hosts a spin-spin cor-relation which contradicts to the d-wave singlet in a di-amond plaquette. Therefore it is natural to have λ2 > 0in the phenomenological field theory.

At the first glance, it seems that the double-monopoleterm λ2 in Eq. (7) is relevant and may destroy the DQCP.However, it is realized in Ref. 42 that r and λ2 actuallyrecombine into a new tuning parameter r and a new rel-

TABLE III: Singlet configuration energy around differentplaquette with different pairing symmetry, estimated fromJ1/J2 ∼ 0.7 for the pVBS phase.

s-wave d-wave

square −2J1 ∼ −1.4J2 0

diamond −2J1 + 0.25J2 ∼ −1.15J2 −0.75J2

r

λ2

AFMpVBS

(square)

(diamond)

SO(5)

O(4)

O(4)

⟨Imℳ⟩ ≠ 0

⟨Reℳ⟩ ≠ 0

⟨ℳ⟩ = 0

FIG. 7: Phase diagram of the appearance of O(4) DQCP onperturbing the SO(5) theory. Arrows indicate the RG flowdirection.

evant perturbation λ2. In the case of λ2 > 0, the La-grangian LM in Eq. (7) can be written as

LM = r(ImM)2 + λ2(ReM)2 + · · · , (9)

with r = r − λ2 and λ2 = r + λ2. The parameter r stilldrives a transition at r = 0 (or equivalently r = λ2),as shown in Fig.7 with a modified emergent symme-try. At the transition point, the relevant perturbationλ2 = 2λ2 > 0 simply gaps out the diamond plaquettepVBS fluctuation ReM from the low-energy sector, leav-ing the square plaquette pVBS fluctuation ImM quan-tum critical. It is further argued that the pVBS fluctu-ation ImM will become degenerate with the Neel fluc-tuation n at the critical point[42], because the pertur-bations n4, n2(ImM)2, (ImM)4, · · · that can break thesymmetry that rotates Neel and pVBS are all rank-fouroperators, which are expected to be irrelevant at the crit-ical point. Therefore the Neel and VBS order parameterscan combine into a O(4) vector (n, ImM), manifestingan emergent O(4) symmetry. The remaining topologicalO(4) Θ-term still ensures that the development of thepVBS order ImM will simultaneously destroy the Neelorder n, establishing a direct pVBS-Neel transition withemergent O(4) symmetry.

IV. DANGEROUSLY IRRELEVANT SCALINGAND ITS ABSENCE

In this section, we discuss a peculiarity of the DQCPin the Shastry-Sutherland lattice compared to the other

8

Global Symmetry LM Emergent Symmetry

Square (p4m) λ4 ReM4 SO(3)× Z4 → SO(5)

Easy-plane Square λ4 ReM4 U(1)× Z4 → O(4)

Rectangular (pmm) λ′2 ImM2 SO(3)× Z2 → O(4)

Shastry-Surtherland (p4g) λ2 ReM2 SO(3)× Z2 → O(4)

TABLE IV: Symmetry-allowed most-relevant monopoleterms (apart from rM†M) and the corresponding DQCPemergent symmetries on different lattices

DQCP scenarios that have been extensively discussed.[5,

6, 27, 36, 42–45] In the presence of the λ2 term in Eq. (9),the U(1) symmetry of the monopole operator (whichacts as M → eiθM) is explicitly broken down to Z2

(at the lattice level). This Z2 symmetry can be iden-tified as the glide reflection symmetry (Gx or Gy) on theShastry-Sutherland lattice, which can be further brokenspontaneously in the pVBS phase by its order param-eter 〈ImM〉. At the DQCP, this Z2 symmetry will berestored and combined with the SO(3) spin rotation sym-metry to form the larger emergent O(4) symmetry, de-noted as SO(3)×Z2 → O(4). Although the Z2 symmetryis restored at the DQCP, it is never further enlarged tothe U(1) symmetry of monopole conservation, because

the explicit symmetry breaking term λ2 is relevant. Thepresence of the relevant coupling λ2 leads to an importantdifference between the DQCP on the Shastry-Sutherlandlattice with the more conventional DQCP on the squarelattice.

To expose the differences and connections, let us brieflymention the other two lattices: the square lattice and therectangular lattice. Due to the different lattice symme-tries, the allowed leading monopole terms will be differ-ent, as summarized in Tab. IV. They will lead to differ-ent VBS orders and different properties of the DQCP.For example, on a rectangular lattice, the other double-monopole term λ′2 ImM2 is allowed but λ2 ReM2 is for-bidden, which favors the horizontal/vertical cVBS or-der depending on λ′2 > 0 (or λ′2 < 0). The DQCP onthe rectangular lattice has a similar emergent O(4) sym-metry, which was carefully analyzed in Ref. 42. How-ever, on a square lattice, the four-fold rotational sym-metry forbids all the double-monopole terms, leavingthe quadruple-monopole term λ4 ReM4 most relevant,which favors cVBS (or pVBS) if λ4 > 0 (or λ4 < 0).In the absence of the double-monopole term, the DQCPon the square lattice has an even larger emergent sym-metry SO(3) × Z4 → SO(5). In the easy-plane model,the lattice symmetry would be that of the square lattice,but the spin-rotation symmetry is reduced from SO(3)to U(1). Here, the symmetry enhancement would beU(1)× Z4 → O(4) [25, 44].

Although the DQCP on the square lattice with theeasy-plane deformation has the same O(4) emergent sym-metry as the DQCP on the rectangular or Shastry-Sutherland lattice, there is a crucial difference between

FIG. 8: (a) Renormalization group (RG) flow of the quadru-pled monopole term λ4M4 in the square lattice DQCP sce-nario, which is dangerously irrelevant. For a small deviationof the tuning parameter J1/J2 towards the VBS phase, λ4

initially decreases under RG flow until the RG scale reachesthe spin-spin correlation length. Only after that, λ4 beginsto increase to reach the VBS fixed point. This has noticeableconsequences for observables in a finite size numerical simula-tion. (b) Schematic plot for the inverse correlation length ξ−1

of spin singlet and triplet operators as a function of tuningparameter for the square lattice DQCP scenario with eitherO(4) or SO(5) emergent symmetries. Note that ξ−1 is re-lated to the energy (mass gap) of the associated excitation[32]. Here, the skyrmion corresponds to the Higgsed ‘photon’excitation in the CP1 theory. In the VBS phase, this photonexcitation manifests as a spin singlet VBS order parameterfluctuation, i.e. the VBS domain wall thickness. Similarly,the magnon becomes a ‘triplon’ in the VBS phase. The plotassumes the simulation of the DQCP at the finite circumfer-ence L and bond dimension χ, which prevents ξ to diverge atthe DQCP.

them. On the square lattice, the U(1) → Z4 symmetrybreaking term λ4 ReM4 is dangerously irrelevant, whichenhances Z4 to U(1) at the DQCP. As we move awayfrom the DQCP toward the VBS phase, the system willexhibit two different length scales: the spin correlationlength ξspin and the VBS domain wall width ξVBS [5, 47].The system is critical at the length scale below ξspin,meaning that the spin correlation decays in a power-law.Beyond this length scale, however, the system is still notfully in the VBS phase. Because the ReM4 operator isirrelevant at the critical point, its coupling coefficient λ4

has decreased under the renormalization group (RG) flowinitially, as in Fig. 8(a). However, once RG flow goes be-yond the length scale of ξspin, λ4 starts to increase withthe RG flow until it becomes strong enough to break theU(1) symmetry down to Z4 at the length scale ξVBS [48].

A careful analysis [5] showed that ξVBS ∼ ξ(∆−1)/2spin , where

∆ > 3 (irrelevant) is the scaling dimension of ReM4 [86].Thus, ξVBS grows more rapidly than ξspin as we approachthe critical point.

However, on the Shastry-Sutherland lattice, there is nosymmetry enhancement from Z2 to U(1) at the DQCP

because U(1)→ Z2 symmetry breaking term λ2(ReM)2

is relevant. As a result, there is no dangerously irrele-vant RG flow around the DQCP. As soon as we moveaway from the critical point, Z2 anisotropy is there to

9

1.8 2.0 2.2 2.4 2.6 2.8J1/J2

0.3

0.6

0.9

1.2

1/ξ

VBS

Square Lattice, L = 10

spin-0

spin-1

spin-2

0.70 0.75 0.80 0.85 0.90 0.95J1/J2

0.3

0.6

0.9

1.2

1/ξ

VBS

Shastry-Sutherland Lattice, L = 10

spin-0

spin-1

spin-2

FIG. 9: The correlation length spectrum for (Left) J1-J2

model in the spin-1/2 square lattice (χ = 2000) and (Right)the Shastry-Sutherland model (χ = 4000). The correlationlength spectrum coincides with the excitation level crossingspectrum in Fig.2 of [46] (After switching J2/J1 to J1/J2).For the value of J1/J2 smaller than the range shown in thefigure, we would get the collinear striped AFM order (dVBS)for the square (Shastry-Sutherland) lattice. The level-crossingbehaviors for both models are similar in the Neel ordered side.

break the emergent O(4) symmetry down to the micro-scopic SO(3)× Z2 symmetry. Thus, there does not exista separation of length scales nor scenario expected inFig. 8. The singlet gap (the gap of pVBS fluctuation)should open up immediately as we tune r away from thecritical point in the thermodynamic limit.

What is the consequence of all these observations? Inthe study of finite size systems, the RG flow should stopat a certain point beyond the system size. This meansthat the dangerously irrelevant scaling can significantlyaffect the correlation behavior or excitation spectrum insmall system sizes, as illustrated in Fig. 8(b). Due tothe dangerously irrelevant scaling, near the DQCP to-wards the VBS side, the correlation length of the VBSorder parameter fluctuation (S = 0) should be largerthan the correlation length of the Neel order parameterfluctuation (S = 1). However, in the thermodynamiclimit, such a behavior is not guaranteed as the eventualfate under the RG flow is often difficult to understand.

For example, in the mean-field treatment of the VBSphase, it has been argued that the spin-triplet excitation(triplon) has a lower energy than the singlet excitation(singlon) [35]. On the other hand, in the finite size sys-tems, the dangerously irrelevant scaling enforces the re-gion with ξS=0 > ξS=1 in Fig. 8(b) to appear regardlessof the eventual RG behavior in the VBS phase. More-over, since we consider a quasi two-dimensional systemin the iDMRG simulation, the finite circumference sizecan also affect the behavior in Fig. 8. In principle, theMermin-Wagner theorem prevents the spontaneous sym-metry breaking of a continuous symmetry in dimensionsD ≥ 2 [49]. In other words, in a quasi two-dimensionalsystem, the strong fluctuation of the continuous orderparameter (e.g. SO(3) spin-rotation) would gap out thesystem and reduce the correlation length for the associ-ated Goldstone bosons (e.g. magnons). As the disorder-ing effect would be stronger for the physical SO(3) spin-rotation symmetry than for the emergent U(1) symmetryenhanced from Z4, we would again expect the parameterregion of ξS=0 > ξS=1 to appear near the DQCP.

By contrast, in the Shastry-Sutherland model, the rel-evant Z2 perturbation can always gap out the monopolefluctuation away from the critical point. To confirm this,we performed the iDMRG simulation on the spin-1/2 J1-J2 model at the same circumference size L = 10 and com-pared the correlation length spectra between two models.In Fig. 9(a), we observe that ξS=0, the correlation lengthof a S = 0 local excitation, is always larger than ξS=1

in the entire VBS phase on the square lattice. However,In Fig. 9(b), ξS=0 is larger than ξS=1 only at the transi-tion point and immediately becomes smaller than ξS=1 aswe tune the system away from the critical point towardsthe VBS phase. This is one of the non-trivial predictionfrom the presence of relevant anisotropy operator at afinite-system size simulation. In the Neel ordered phase,these two models exhibit very similar correlation lengthspectra. For more details, see Appendix. B.

Our numerical result for the square J1-J2 model alignswith the recent work [46] on the finite-DMRG simulationwhich calculated first several excited states with differ-ent spin quantum numbers. If we replace the excitationenergy in Ref. [46] with ξ−1, we obtain the same crossingbehavior. This can be justified by the fact that when a lo-cal excitation has a mass gap m, the correlation functionmediated by the local excitation decays as ∼ e−mr, thusm ∝ ξ−1. Therefore, our theoretical scenario explainswhy a local [87] spin-singlet excitation in the Ref. [46]has lower energy than a spin-triplet excitation aroundthe critical point. It also elucidates the reason why theprevious DMRG results of the J1-J2 model on the squarelattice [14, 50–54] were unable to identify the nature ofthe VBS phase without applying a pinning field. Becauseall previous numerics were also performed in the similarsystem size, they were in the regime where the VBS or-der parameter fluctuations were severe, disallowing oneto confirm whether it is a plaquette or columnar VBS.Therefore, we conclude that the absence or presence of

10

an irrelevant operator is essential to understand physicalobservables in any finite size system.

In summary, we note that previously the two promis-ing scenarios to realize DQCP with spin 1/2 was eitherthe SO(3) symmetric or easy plane deformations, bothwith four-fold degenerate VBS orders. Here, however,we have the two-fold degenerate VBS order. Neverthe-less, as discussed in Ref. [42], the two-fold monopole withSO(3) symmetry is equivalent to the easy plane deforma-tion, if an enlarged SO(5) symmetry is assumed in theabsence of these perturbations. Furthermore, we remarkthat the scaling behavior of dangerously irrelevant opera-tor enables us to understand multiple observations madein previous numerical simulations which is absent in theShastry-Sutherland model here.

V. SPECTRAL SIGNATURES OF DQCP

A hallmark of the DQCP is the emergence of decon-fined spinons at the critical point, which entails distinctfeatures in both the magnon and phonon excitation spec-tra that can be probed in INS or RIXS experiments. Tobetter appreciate the predicted spectral features at low-temperature around the pVBS-Neel transition, we needto first understand the background elastic scattering sig-nal of SrCu2(BO3)2 in its high-temperature paramag-netic phase without any symmetry breaking. We willfocus on the scattering of neutrons or photons off of thecopper sites. As shown in Fig. 1(a), there are four coppersites in each unit cell, coordinated at rA = (1+δ, 1+δ)/2,rB = (1 − δ, 3 + δ)/2, rC = (3 + δ, 1 − δ)/2, rD =(3−δ, 3−δ)/2 respectively, with the distortion parametergiven by δ = 0.544 according to Ref. 16. In the high-temperature paramagnetic phase, an elastic scatteringexperiment will reveal lattice diffraction peaks at a set ofmomenta Q = π(H,K) (H,K ∈ Z, note that the latticeconstant is 2 in our convention) with the amplitude givenby S(Q) =

∫d2rρ(r)eiQ·r '

∑a=A,B,C,D e

iQ·ra , where

ρ(r) can represent either the electron density from Cuorbitals (which scatters X-ray photons) or the density ofCu nuclear (which scatters neutrons). The correspondingintensity |S(Q)|2 is plotted in Fig. 1(b). Notably, thereare extinction points in the diffraction pattern, protectedby the glide reflection symmetry. Note that the systemat the DQCP has the glide-reflection symmetries Gx andGy although both the Neel and pVBS phases do not. [88]The glide reflections Gx and Gy act as lattice translationsby a half lattice constant followed by the reflections aboutthe translation directions, as illustrated in Fig. 1(a). Thefact that the density distribution ρ(r) at equilibrium re-spects all lattice symmetries (including Gx and Gy) im-plies that ρ(x, y) = ρ(x + 1,−y) = ρ(−x, y + 1). Theyimpose the following constraints on the scattering ampli-tude

S(Qx, Qy) = eiQxS(Qx,−Qy) = eiQyS(−Qx, Qy), (10)

which implies the extinction of diffraction peaks at Q ∈π(2Z+1, 0) or π(0, 2Z+1), as marked out in Fig. 1(b). Ingeneral, ρ(r) can describe the spatial pattern of any scat-terers that interact with the probing particle (e.g. X-rayor neutrons). Eq. (10) holds under the assumption thatthe scatterer field ρ(r) is even (symmetric) under glidereflection symmetry. This constraint can be generalizedto other type of scatterers including magnetic fluctua-tions at finite frequency. For example, the destructionof the scattering amplitude at these extinction pointscould extend to finite-frequency inelastic scattering, aslong as no scatterer at that energy scale breaks the glidereflection symmetry. However, as we lower the temper-ature and approach the pVBS-Neel transition, certainglide-reflection-breaking excitations (meaning that thescatterer is odd under the glide reflection) may emergeat low energy as part of the quantum critical fluctua-tion. Indeed, we will show that the emergent SO(4) con-served current fluctuation and the pVBS order fluctua-tion are examples of glide-reflection-breaking scatterers,which will become critical at the DQCP and “light up”the extinction points. They will provide unique signa-tures of the DQCP that are also relatively easy to resolvein experiments, as there are no background scattering sig-nals at the extinction points.

π

π

π

π

π

π

π

π

π

i

j

kl

i

j

kl

(a) (b)

FIG. 10: (a) The π-flux model of fermionic spinons. Thespinon hopping on the thick red bond gets a minus sign, suchthat each plaquette has a π-flux. Additional spinon inter-action terms are applied to blue shaded diamonds. (b) Thearrangement of the site indices [ijkl] around each shaded di-amond.

The proposed spectral signatures at the pVBS-Neeltransition is most convenient to analyze using a fermionicspinon theory for the DQCP, which has been shown tobe equivalent (dual) to the conventional CP1 theory[55–58]. In the fermionic spinon theory, the spin operator

Si = 12f†i σfi is fractionalized into fermionic spinons

fi = (fi↑, fi↓)ᵀ, which are then placed in the π-flux

state[59–62] described by the following mean-field Hamil-tonian

HMF =−∑〈ij〉

tij(f†i fj + h.c.)

+ g∑[ijkl]

(f†i fj + h.c.)(f †kfk − f†l fl) + · · · ,

(11)

where 〈ij〉 runs over all the nearest neighbor bonds and[ijkl] runs over all the shaded diamond plaquette de-

11

picted in Fig. 10(a). The spinon hopping amplitudetij takes tij = −t on the bonds highlighted in red inFig. 10(a) and tij = t on the rest of the bonds, such thatthe spinon sees π-flux threading through each plaquette.The phenomenological parameter t ∼ J1 sets the energyscale of the spinon, which is expected to be of the sameorder as the spin interaction strength J1. The π-fluxstate model Eq. (11) was originally proposed as an exam-ple of algebraic spin liquids[59–62] (including the DQCPas a special case). It is recently confirmed via quantumMonte Carlo (QMC) simulations[56–58] that the π-fluxstate actually provides a pretty good description of thespin excitation spectrum at the DQCP.

The π-flux state mean-field ansatz determines a pro-jective symmetry group (PSG)[63] that describes how thespinon should transform under the space group symme-try, as concluded in the last column of Tab. II. It is foundthat the spinon hopping along the dimer bonds are for-bidden by the σxy, σxy symmetry, which is not brokenat the DQCP. Therefore the effect of J2 can only en-ter the Hamiltonian as a four-fermion interaction to theleading order, given by the g term in Eq. (11). The gterm describes the spinon interaction around each dia-mond plaquette [ijkl] where the site indices i, j, k, l arearranged according to Fig. 10(b). It turns out that the in-teraction g is directly related to the λ2 ReM2 term giventhat the monopole operator M† ∼ (vx + vy) + i(vx − xy)can be written in terms of VBS order parameters vx, vy,which are further related to fermionic spinons via vx ∼(−)xf†i+xfi + h.c. and vy ∼ (−)x+yf†i+yfi + h.c., such

that the interaction can be written as gvxvy ∼ gReM2.Therefore g > 0 would correspond to λ2 > 0, which fa-vors the square plaquette VBS order ImM.

Let us ignore the interaction g for a moment. Bydiagonalizing the spinon hopping Hamiltonian, we findthe spinon dispersion εk = ±2t

√cos2 kx + cos2 ky, which

gives rise to 4 Dirac fermions (or equivalently 8 Majoranafermions) at momentum (π/2, π/2). Note that the distor-tion parameter δ does not affect the spinon band struc-ture. Naively, the spinon mean-field theory has an emer-gent SO(8) symmetry rotating among the 8 low-energyMajorana fermions. However, a SU(2) gauge structuremust be introduced with the fractionalization, which re-duces the emergent symmetry to SO(5). The interac-tion term g plays an important role to further break theSO(5) symmetry explicitly to O(4), matching the emer-gent symmetry observed in the DMRG simulation.

Based on the fermionic spinon mean-field ground state,the spin excitation spectrum S(ω, q) has been calculatedin Ref. 56–58 and is reproduced in Fig. 11(a) for illustra-tion, where the high symmetry points Γ, X,M are definedin Fig. 1(b). The lower edge of the spinon continuum isgiven by ωmin(q) = mink |εk+q − εk|, which reads

ωmin(q) = 2t√

sin2 qx + sin2 qy, (12)

as shown in Fig. 11(a). This provides us a way to esti-mate the mean-field parameter t from the experimentally

measured spin excitation spectrum, by fitting this loweredge. This is a rather robust result as its shape is unaf-fected by the distortion parameter δ.

Remarkably, a gapless continuum will appear on top ofthe extinction point X (as well as Y by σxy symmetry)as seen in Fig. 11(a). As previously analyzed in Eq. (10),the spin excitation at the X point must be odd underthe glide reflection symmetry Gx. This symmetry con-straint enforces that the gapless continuum should cor-respond to the emergent SO(4) conserved current fluctu-ation Jy = n∂y ImM− ImM∂yn which involves boththe Neel n and pVBS ImM order parameters and hasbeen thoroughly studied in Ref. 57, 60, 64. Since n, ∂yand ImM are all odd under the glide reflection Gx (thatpreserves the extinction point X), the current operatorJy is also odd under Gx. Therefore Eq. (10) does nothold anymore, such that the fluctuation of Jy can ap-pear at the extinction point X as a spinon continuum inthe magnon channel (as Jy carries spin-1). On the otherhand, fluctuations like n ImM is not allowed to appearat the X point in the spin excitation spectrum becausethe bound state n ImM is even under glide reflection (cf.Eq. (10)). The Jy continuum will only become gapless ifboth the Neel and pVBS fluctuations are gapless, whichonly happens at the DQCP. Protected by the emergentO(4), the conserved current operator must have a scalingdimension exactly pinned at 2, which indicates that themagnon spectral weight at the extinction point X mustincrease with the frequency linearly (at below the energyscale of J1)

Smagnon(ω, q = X) ∝ ω. (13)

as shown in Fig. 11(b). We propose this as a hallmarkfeature of the spin fluctuation at the pVBS-Neel tran-sition in SrCu2(BO3)2. Confirmation of this linear fre-quency growth of the spectral weight will provide directevidence for the emergent O(4) symmetry at the DQCP.

Apart from the features in the spin excitation(magnon) spectrum, the DQCP also introduces new fea-tures to the phonon spectrum, due to the pVBS-phononcoupling. The pVBS order has a linear coupling to thelattice displacement as its representation under the lat-tice symmetry group matches with that of strain fields

LVBS-phonon ∼ vxux + vyuy, (14)

where ux (or uy) is the lattice displacement in the x (ory) direction with a momentum (π, 0) (or (0, π)). It is cru-cial that ux and uy here are not acoustic phonon modesaround momentum (0, 0) in a continuum theory, other-wise they can only enter the field theory in the formof derivatives ∂iuj . The coupling is allowed by latticesymmetry and is evident from the pVBS induced latticedistortion as demonstrated in Fig. 6(a). This leads toa hybridization between phonon and pVBS fluctuations.As the pVBS fluctuation becomes critical (gapless) atthe DQCP, the quantum critical fluctuation will also ap-pear in the phonon spectrum due to the hybridization

12

10-2

10-1

100

101

102

0 1 2 3ω / t

00.10.20.30.40.5

S magno

n(ω,q

=X)

(b)

~ ω

10-3

10-2

10-1

100

101

0 0.1 0.2 0.3ω

0

0.1

0.2

0.3

S pho

non(ω,q

=X)

(d)

~ω-2+η

1

FIG. 11: (a) Spin excitation spectrum (dynamical structuralfactor) Smagnon(ω, q) at the DQCP. Darker color indicateshigher intensity. The dashed line trace out the lower-edgeof the continuum which is described by Eq. (12). (b) Thefrequency dependence of the spectral intensity at the extinc-tion point X. At the low-frequency limit, the spectral inten-sity grows with frequency linearly, which manifests the con-served current associated to the emergent O(4) symmetry. (c)Schematic illustration of the phonon spectrum Sphonon(ω, q).The bare phonon dispersion is inferred from Ref. [65]. TheVBS-phonon coupling leads to a continuum in the phononspectrum. (d) The frequency dependence of the phonon spec-tral intensity at the extinction point X. The intensity fallsoff in a power-law with frequency, whose exponent should bethe same as that of the spin fluctuation at M point.

effect, as illustrated in Fig. 11(c) (see Appendix C for de-tailed analysis). As the pVBS order carries the momen-tum (π, 0) and (0, π), the phonon continuum will alsoget softened at these momenta, which happen to be theextinction points X and Y of the lattice diffraction pat-tern. Note that the pVBS fluctuation is odd under glidereflection, therefore new phonon continuum is allowed toemerge at the extinction points with the spectral weightdiverging at low-frequency following a power-law,

Sphonon(ω, q = X) ∝ ω−2+η. (15)

The anomalous dimension η should match that of thepVBS order parameter at the DQCP, which, by the emer-gent O(4) symmetry, is also the same η of the Neel orderparameter. Based on the QMC simulations in Ref. 25, 66,η has been estimated to be η = 0.13 ∼ 0.3. Observa-tion of such critical phonon fluctuations at the extinc-tion points with a power-law divergent spectral weightas shown in Fig. 11(d) will be another direct evidence ofDQCP.

In conclusion, extinction points are protected by theglide reflection symmetry, but both the conserved cur-rent and the pVBS order parameter breaks the glide re-

flection. Their critical fluctuations are therefore allowedto appear at the extinction point. This is rather a niceproperty that there will be no background signals formlattice diffraction, which makes these spectral signaturesof DQCP more easier to resolve in experiments. Ouranalysis indicates that the conserved current fluctuation(which carries spin-1) should appear in the magnon spec-trum and the pVBS fluctuation (which carries spin-0)should appear in the phonon spectrum. Observation ofthese critical fluctuations in the scattering experimentwill be strong evidence for the potential realization ofDQCP in SrCu2(BO3)2.

VI. EFFECTS OF INTERLAYER COUPLING

So far, we have discussed the DQCP physics assumingthat the system is two-dimensional. However, the ma-terial realization of Eq. (1), SrCu2(BO3)2, has a three-dimensional structure which is a stack of the Shastry-Sutherland lattices with a relative shift given by the lat-tice vector (1,1) between neighboring layers. Each layeris separated by the layer of oxygen, but due to the super-exchange term mediated by the oxygen, there is a smallinterlayer anti-ferromagnetic interaction J3 ∼ 0.1J2 [67]between the spin-1/2s located at the crossed dimers inFig. 12. Since the stacking structure preserves Gx,y andσxy,xy symmetries, previous monopole analysis still holdsfor each two-dimensional layer. Here, we would like tobetter understand the effect of the three-dimensional in-terlayer coupling to the DQCP physics.

To analyze, we first consider coupling layers of spinsystems at the conventional DQCP point with the emer-gent SO(5) symmetry (other than the O(4) case in theprevious discussion). Here, each layer is described by thefollowing nonlinear sigma (NLSM) model in Euclideanspacetime [61, 68]:

S(l)DQCP =

∫d3x

1

2g(∂µΦ(l))2 + 2πiΓWZW[Φ(l)] (16)

where Φ(l) = (nx,ny,nz, ImM,ReM) is the order pa-

rameter of the l-th layer and ΓWZW[Φ(l)] is a Wess-Zumino-Witten (WZW) term at level k = 1. With theinterlayer coupling, the total action is given by the fol-lowing general form [69]

S =∑l

S(l)DQCP −

∑l

∫d3x gabΦ(l)

a Φ(l+1)b , (17)

where the interlayer coupling coefficients gab can be ar-ranged into a matrix g. Given that the scaling dimen-sion of Φ is estimated to be around ∆Φ ' 0.6 in theliterature[70–79], the interlayer coupling is expected tobe relevant (as 2∆Φ < 3), locking the order parame-ters across different layers. Then, the coefficient matrixg becomes important since the sign of its determinantdet g crucially affects how the topological (WZW) term

13

J1

J1

J2

J2

J3

(a) (b)

(c) (d)

1

FIG. 12: (a) Three-dimensional interlayer AFM coupling J3

among spins located at the cross between upper and lower di-agonal bonds, mediated by the interpenetrating oxygen atom.The other interlayer couplings are negligible. Preferred stack-ing of (b) the Neel ordered phase with n(l) = n(l+1) (redand blue dots for opposite spins), (c) and the diamond pVBS

phase with ReM(l) = ReM(l+1), (d) square pVBS phase

with ImM(l) = − ImM(l+1) (thick purple bonds mark sin-glet plaquettes). For (c) and (d) cases, energetically favorablestacking can be deduced by the interlayer dimer resonance.

from each layer will be added up (cf. [69]). More pre-cisely, if the sign of det g is positive (negative), the WZWterms are added up in a uniform (staggered) manner.The reason is that we can always redefine Φ(l) in alter-nate layers to make g positive definite, but the price topay will be that the WZW term will change sign alter-natively if the original det g is negative. Note that theprocedure requires the topological term to be invariantunder the change of origin (translation symmetric), i.e.Tx,y : ΓWZW[Φ] 7→ ΓWZW[Φ]. Otherwise, the addition orsubtraction of topological terms across layers would notbe well-defined due to the arbitrariness of the choice oforigin across the layers for the locked order parameter.

For example, consider coupling the two-dimensionalsquare lattice J-Q models [80] into a 3D cubic latticewith AFM spin-spin interaction along vertical bonds.Each layer putatively realizes the DQCP physics withΦ = (n, vx, vy) describing Neel and cVBS order param-eters. Under the AFM interlayer coupling, the couplingmatrix g has the sign structure of gaa = (−,−,−,+,+).This is because the vertical AFM coupling prefers n(l) =−n(l+1) while the vertical plaquette ring exchange [89]

favors v(l)x,y = v

(l+1)x,y . In this case, det g < 0, so the

WZW terms in neighboring layers tend to cancel eachother. However the cancellation will not be exact, as theΦ field still admits (smooth) fluctuation over layers, thisresults in a residual topological term, namely a topolog-ical Θ-term. Staggering a WZW-term at level k wouldgive a Θ-term at Θ = πk. Now the problem of the cou-

pled DQCPs boils down to understand the fate of theSO(5) NLSM with Θ = π in (3+1)D. There are somehints from the fermionic parton analysis. One can con-sider fractionalizing the Φ vector to the bilinear from ofa fermionic parton field ψ following Φa ∼ ψiγ5Γaψ, suchthat the ψ fermion is in the SO(5) spinor representation(or the Sp(2) fundamental representation). The emergentgauge structure will be SU(2), which points to the SU(2)quantum chromodynamics (QCD) model in (3+1)D withSp(2) flavor symmetry,

L = ψ(γµDµ +m+ iγ5ΦaΓa)ψ. (18)

Integrating out the fermion and gauge fluctuation is ex-pected to reproduce the SO(5) NLSM with Θ = π(1 +sgnm), such that Θ = π is realized at m = 0. How-ever, the number of Dirac fermion flavors (Nf = 2) is notenough to avoid a chiral symmetry breaking in 3D. There-fore, under the interlayer coupling, it is likely that theSO(5) DQCP flows to a discontinuous transition pointinduced by the quantum fluctuation. Considering thatthe O(4) DQCP from easy-plane anisotropy or rectangu-lar deformation is descended from the SO(5) DQCP [42],breaking SO(5) down to O(4) makes the situation worse.

In a similar way, one can analyze the three-dimensionalstacking of the Shastry-Sutherland lattice. If we allowthe possibility of the diamond pVBS phases, the orderparameter is written as Φ = (n,ReM, ImM) whereReM (ImM) represents the square (diamond) pVBSorder parameter. Note that now each layer is shiftedby (1, 1) vector (see Fig. 5) relative to the layer below,and the interlayer AFM coupling is given by Fig. 12(a)instead of the direct vertical coupling. In Fig. 12(b), weshow two identical layers of Neel ordered pattern rela-tively shifted by (1, 1). If we focus on the four spinssurrounding the diagonal bond crossing, we found thatthe two spins from the upper layer and the two spinsfrom the lower layer are aligned oppositely, which is fa-vored by the AFM interlayer spin exchange J3. So theinterlayer coupling prefers to lock the Neel order param-eter in the same direction across the layer as n(l) =n(l+1). In Fig. 12(c), we show two identical diamondpVBS patterns displaced from each other. This config-uration can gain the effective interlayer ring exchangeenergy induced by the J3 coupling, which resonates thenearby dimers across the layer. Thus we conclude thatReM(l) = ReM(l+1) is more favorable. In Fig. 12(d),we show two opposite square pVBS patterns displacedfrom each other. This configuration also gains interlayerring exchange energy by resonating the dimers lying ontop of each other. But this would require the squarepVBS order parameter to be opposite between neigh-boring layers as ImM(l) = − ImM(l+1). In conclusion,the interlayer coupling prefers (n(l),ReM(l), ImM(l)) =(n(l+1),ReM(l+1),− ImM(l+1)). Here, instead of using(vx, vy), we use (ReM, ImM) to parameterize the VBSorder parameter, which is a basis change with a positivedeterminant. In this basis, the interlayer coupling matrixg takes the sign structure of gaa = (+,+,+,+,−). As a

14

result, we again have det g < 0, which implies that SO(5)DQCP would flow to the first order transition point in3D. Since our O(4) scenario is considered to be a per-turbed SO(5) DQCP (see Fig. 7), it is likely that the O(4)DQCP would also flow to the first order transition point.

If det g happens to be positive, the leading order ef-fect is that the WZW terms would add up together, asthe interlayer coupling g tends to lock the order parame-ters Φ(l) across the layers. Admittedly, the locking effectwill become weaker at longer distance (along the per-pendicular direction of layers), but we can still analyzethe problem by first grouping the neighboring layers toa renormalized layer and then considering the residualcoupling between renormalized layers. Across neighbor-ing layers, the order parameters are expected to bind to-gether, such that the renormalized model can be viewedas a NLSM with WZW term at large level k. The intu-ition from (0+1)D O(3) WZW term is that the large levellimit corresponds to the large spin limit, where the quan-tum fluctuation of the order parameter is suppressed.Coupling spin-1/2 into a spin chain ferromagnetically inSx,y,z-channels results in the ferromagnetic ground statewith giant spin and classical spin wave excitations. Inthis limit, the low-energy physics can be captured withinLandau-Ginzberg (LG) theory. Further adding differenteasy-axis anisotropies to the ferromagnetic spin chainwill drive first-order transitions between different Isingordered phases according to the LG theory. We con-jecture that in higher dimension, similar effect will ren-der each renormalized layer into a classical magnet whichshould be described within Landau-Ginzberg paradigm,such that the DQCP is not available. So in the presenceof interlayer coupling, the Neel and VBS phases will likelybe separated by a first order transition or intermediatecoexisting phases. Thus, our analysis shows that in bothdet g > 0 and det g < 0 cases, the interlayer couplingwould ultimately destabilize the DQCP, and drive it, forexample, to a first order transition. However, our analy-sis also indicates that the det g < 0 case, correspondingto the real material, will have stronger quantum fluctu-ations, potentially leading to a weaker first order tran-sition or a smaller region of coexisting order parametersthan the det g > 0 case.

One additional remark is that while the interlayer Neelorder coupling enters directly from the J3 term, the in-terlayer VBS coupling arises from the higher-order per-turbation (resonance) of the J3 term. As a result, thecritical point would be shifted to expand the Neel orderphase. This is consistent with the phase diagram studiedin [40], where the interlayer coupling drives a system intothe AFM order and shrink down the VBS region.

VII. PREDICTIONS FOR EXPERIMENT

Before discussing experimental consequences of theDQCP, we will make a few remarks on the nature of the

plaquette VBS phase. In the Sec. III, we discussed twodifferent possibilities for the pVBS phases (see Fig. 6):the square and diamond pVBS. While the square pVBSbreaks the reflection symmetries σxy and σxy, the dia-mond pVBS breaks the empty-plaquette-centered C4 ro-tation symmetry, see Fig. 5. As a result, when the sys-tem is at the pVBS phase, magnetic excitations initiallydegenerate under the p4g symmetry would split differ-ently depending on whether the plaquette is formed ata square or diamond. While our iDMRG simulation ofthe Shastry-Sutherland model points to the square pVBSphase, in the recent experiments on SrCu2(BO3)2 usingINS[1] and NMR[38, 39], the magnetic excitations in thepVBS phase seems to break the C4 rotation symmetry,indicating the diamond pVBS phase. This discrepancyimplies that the effective spin model for the real materialcould deviate from the Shastry-Sutherland model studiedhere. For example, three-dimensional interlayer couplingmay induce some effective further-neighbor couplings be-yond Shastry-Sutherland model. Therefore the type ofpVBS phase to be stabilized at low energy is model de-pendent. Nevertheless, this does not affect to the DQCPscenario and the emergent O(4) symmetry because it onlycorresponds to a different sign of λ2 in Eq. (7).

As discussed earlier, the DQCP naturally realizes aquantum spin liquid, a long sought after state of quan-tum magnets. Furthermore it realizes a particularly ex-otic variety - a critical spin liquid - with algebraicallydecaying correlations arising from the gapless emergentdegrees of freedom. Moreover, an experimental realiza-tion of the DQCP would be a crucial manifestation of themany-body Berry phase effect that intertwines differentorder parameters. A dramatic experimental consequenceof the DQCP is the emergent symmetry and resultantspectroscopic signatures expected from INS or RIXS. Inparticular, the model for SrCu2(BO3)2 studied here ex-hibits the O(4) emergent symmetry with two promisingspectroscopic signatures at X-point in the Brillouin zone,summarized as the following:

• Magnon (S = 1) Channel: This gives the informa-tion about the critical fluctuation of the emergent O(4)conserved current. As a result, the spectral intensityincreases linearly with the frequency, S ∼ ω. If theemergent symmetry did not exist, there should not ex-ist low energy spectral weight at this momentum. Thedeviation from the linear relation would give us a mea-sure of how accurate the emergent symmetry is.

• Phonon (S = 0) Channel: This gives the informationabout the pVBS order parameter fluctuation. For agiven anomalous dimension ηVBS of the pVBS orderparameter, the spectral intensity diverges with the fre-quency, S ∼ 1

ω2−ηVBS. The DQCP scenario implies that

the VBS order parameter is fractionalized, resulting ina non-zero ηVBS.

Moreover, the emergent symmetry implies that ηVBS isequal to the anomalous dimension of the Neel order pa-rameter fluctuation, ηNeel. However, the Neel order pa-

15

rameter fluctuation is located at the M -point, which hasa pronounced Bragg peak in addition. In principle, theBragg peak corresponds to spin-0 channel and it must bepossible to extract ηNeel and compare the ηNeel with ηVBS

to tell whether the emergent O(4) symmetry exits. In theearlier work [67], it is estimated from the low-T magneticsusceptibility and heat capacity data that J1 ≈ 4.7 meV,J2 ≈ 7.3 meV, and J3 ≈ 0.7 meV. Moreover, the Debyefrequency of the acoustic phonon branch has been mea-sured to be ωD ≈ 10 meV in Ref. [65]. Therefore, forexperiments to confirm the theoretical predictions, it isrequired to have the energy resolution smaller than themilli-electron volt.

According to the present numerical simulation, thephase transition between the pVBS and Neel order canbe a second-order or weakly-first order transition. Theresult does not contradict recent numerical work with asimilar phase diagram where a first-order transition be-havior was observed, because these models [22, 23] aredifferent from the microscopic Hamiltonian in Eq. (1),which is more likely to capture the couplings in thereal material. Since the nature of the phase transitionmay be tunable, it is possible that the experiments onSrCu2(BO3)2 may realize the transition that is either asecond order or a weakly first order with large correlationlength.

Would all these predictions become meaningless if thetransition is actually a weakly first order? In fact, even ifthe two-dimensional system hosts the DQCP, we arguedin the previous section that the interlayer coupling mightdrive the system into a first-order transition point inthree-dimension. Indeed, at the weakly first-order tran-sition point, the system would have a finite excitationgap, and experimental spectroscopic data at zero tem-perature would deviate from Fig. 11 due to the absenceof a gapless critical fluctuation. However, if we exam-ine the system within the length scale smaller than the(large) correlation length ξ, the system would still exhibitthe DQCP physics. In other words, if we only examinethe system above the energy scale set by the correlationlength ω > ωgap ∼ 1/ξ, we would observe the predictedspectral intensity trends in Fig. 11. It is our hope thatfuture experiments will be able to use these results toclarify the physics behind the interesting properties ofSrCu2(BO3)2.

VIII. CONCLUSIONS

We studied a two dimensional S = 1/2 model whichcaptures key features of the Shastry-Sutherland mate-rial SrCu2(BO3)2. We obtained the phase diagram us-ing numerical iDMRG simulations and observed a po-tentially continuous transition between a plaquette VBSstate with two-fold degeneracy and a Neel ordered phase.The transition, studied using both numerical and fieldtheoretical techniques, is proposed to be a deconfinedquantum critical point, and we discussed its special fea-

tures including the lack of a dangerously irrelevant scal-ing and an emergent O(4) symmetry. Concrete predic-tions are made for future experiments in SrCu2(BO3)2,where a pressure tuned transition between Neel order anda putative plaquette VBS state has already been reported[1]. The predicted experimental signatures include theform of spectral intensity of spin singlet and spin tripletexcitations at extinction points, which should be accessi-ble in future resonant X-ray and neutron scattering ex-periments. These can provide a smoking gun signature ofthe deconfined criticality and emergent O(4) symmetry.Complications arising from the coupling between layersin the third dimension of the bulk material are briefly dis-cussed, although further work in this direction is needed.We hope this study will trigger future experimental inves-tigation of this quantum critical point in an interestingmaterial, and more generally provide a road map for theexperimental study of deconfined quantum criticality.

Acknowledgments

We thank Y.-C. He, C. Wang, M. Takigawa, A. Sand-vik, B. Zhao, L. Wang, Z.-Y. Meng, S.E. Han, and M.Metlitski for stimulating discussions. The DMRG nu-merics were performed using code developed in collabora-tion with Roger Mong and the TenPy collaboration. Thecomputations in this paper were run on the Odyssey clus-ter supported by the FAS Division of Science, ResearchComputing Group at Harvard University. J.Y. Lee andA. Vishwanath were supported by a Simons InvestigatorFellowship. S. Sachdev was supported by the NationalScience Foundation under Grant No. DMR-1664842.

Appendix A: Monopole Transformation

In this section, we calculate how the monopole op-erator transforms under the symmetry of the Shastry-Sutherland lattice. To do this, we think of the Shastry-Sutherland lattice as the lattice being deformed from thesquare lattice with spin-1/2 per site. Starting from the2 + 1D antiferromagnetic ordered phase (Neel) of thesquare lattice, one can derive the action in terms of localNeel order parameter n(rn) in a path integral formalism,

S =1

2g

∫dτd2r

[(∂n

∂τ

)2

+ c2(∇rn)2

]+ SB (A1)

where n ∼ εrS is a Neel order parameter, and εr =(−1)rx+ry is a factor required for an alternating spin di-rection. In addition to the continuum action of a clas-sical O(3) nonlinear σ-model, there exists a Berry phasecontribution due to the quantum nature of spin dynam-ics, which manifestly has the lattice origin. [81, 82] Letω(nr) be a solid angle swept by a local Neel vector lo-cated at r throughout the imaginary time evolution from0 to β, with respect to the reference direction n0 (See

16

FIG. 13: (a) The monopole operator inserted in the Eu-clidean spacetime. White arrows represent spin directions.The imaginary time trajectory of each spin is represented by acolored line. White arrows in the black trajectory shows howthe direction of the spin changes along the imaginary timewhen the monopole is inserted. (b) The solid angle ω(nr)swept through the imaginary time with respect to the refer-ence vector n0. (c) The dual lattice where a monopole op-erator resides on. Here, plaquette centers correspond to thespin sites. The lattice spin acts as an alternating flux pat-tern (−1)rx+ry4πS for monopoles. The hopping amplitudealong each black arrow gives a phase factor of eiπS = i in ourS = 1/2 case.

Fig. 13(b)). Then, SB = iS∑r ηrω(nr) where S is the

spin of an each site and ηr = (−1)rx+ry is an alternatingphase factor coming from the antiferromagnetic natureof the magnetic order.

For any spatial slice, one can define the Skyrmion num-

ber Q(τ) = 14π

∫n · (∂xn × ∂yn)

∣∣∣τ, which is a topolog-

ical invariant. Then, a monopole creation operator isdefined as a topological defect at the spacetime pointwhich changes Q(τ) by +1 across it. By the furtherduality mapping, this monopole in the NLσM will bemapped into the monopole of the CP1 theory. As thecenter of the monopole cannot have a finite spin direc-tion, the monopole is located at a dual lattice. When wehave a monopole event in the spacetime, it must give abranch-cut structure on the image of ω(r) because ω(r)must change by 4π around the monopole location in thespace. More intuitively, when monopole residing on thedual lattice encircles a single site, the imaginary timetrajectories of all spins except the one encircled by themonopole would oscillate just back and forth, while thetrajectory for the encircled one would entirely wind its ωby 4π upon the completion of encircling as in Fig. 13(b).Thus, one can view this problem as a monopole hoppingaround the dual 2D lattice in which each site in the orig-inal lattice gives a 4πSηr flux through the plaquettes ofthe dual lattice. (Monopole is a charged object under this

Symmetries Transformations

Tx M† 7→ −iM n 7→ −nTy M† 7→ iM n 7→ −nRsiteπ/4 M† 7→ iM† n 7→ n

Rplaqπ/4 M† 7→ −M n 7→ −nσx M† 7→ iM n 7→ n

σy M† 7→ −iM n 7→ n

T M† 7→ M n 7→ −n

TABLE V: Transformation of the monopole operator M andNeel vector n in the field theory of nonlinear sigma model inthe Neel order phase.

Gp4g Gp4m Action

Tx T 2x M† 7→ M† n 7→ n

Ty T 2y M† 7→ M† n 7→ n

σxy Rπ/2σx M† 7→ M n 7→ n

σxy TxTyRπ/2σy M† 7→ M n 7→ n

gx Txσx M† 7→ −M† n 7→ −ngy Tyσy M† 7→ −M† n 7→ −nRπ/4 Rplaq

π/4 M† 7→ −M n 7→ −nT T M† 7→ M n 7→ −n

TABLE VI: The table shows the correspondence betweenthe symmetries of the Shastry-Sutherland lattice (p4g) andsquare lattice (p4m). Bold symbols are for symmetries of theShastry-Sutherland lattice.

“flux” emanated from a spin-S.) The associated phasefactor is independent of the exact imaginary time loca-tion of the monopole event. Thus for S = 1/2 case, onecan fix the system into a certain gauge and view this asif ±π/2 Aharonov-Bohm phase factor gets accumulatedfor each hopping process for monopoles.

The monopole transformation rule is summarized inTab. V. Rsite

π/2 is a site (spin) centered rotation. Note that

it is important to fix the convention for the rotation cen-ter because translation symmetry is projective. Here,we choose a rotation to be defined with respect to the

black spin in Fig. 13(a). Rplaqπ/2 is a plaquette centered

rotation, which is defined with respect to the center offour spins in Fig. 13(a). Under the unit translations Tx,y,

time reversal T , and plaquette centered rotation Rplaqπ/2 ,

the Neel order changes its sign. It means that the fluxpattern is reversed under such transformations, thus weneed to transform a monopole into an anti-monopole tocompensate for the change. For reflections σx,y, althoughn does not flip, the definition of the Skyrmion numbertells that we need to change the sign for the number ofmonopoles. Thus, a monopole transforms into an anti-monopole again. After figuring out whether a monopoletransforms into a monopole or anti-monopole, we need to

17

multiply it by an additional phase factor αg to accountfor the Berry phase effect.

Assume the topological term SB is absent momentar-ily, which is how CP1 theory in Eq. (4) is derived. This isa usual practice because unlike the first term inside theparenthesis in Fig. A1, the second term, SB , cannot bestraightforwardly extended to the continuum field the-ory description. Under the absence of SB , the monopoleinsertion operator M† just makes a global adjustmentof the Neel order configuration to increase the Skyrmionnumber by one, without any additional phase factor.

However, we know from the existence of SB in thelattice description that the Berry phase effect is impor-tant. In order to take into account the Berry phase effect,we need to examine how the monopole transforms undereach symmetry action. Under the active transformationwhere the coordinate system remains the same, we have

g :M†r 7→ αg · g[M†]g(r), g[M†] =M† or M, (A2)

where the action of g on M† is determined by the pre-vious argument on whether the monopole transformsinto the monopole or anti-monopole upon the symmetrytransformation.

To determine αg, we need to fix a gauge first. Fixing agauge is important because a monopole is always createdin a pair with an anti-monopole. Thus, the monopoleevent always has a reference point (anti-monopole) con-nected by the branch-cut. The Berry phase factor is in-dependent of the way we draw the branch-cut becausegoing around a single spin-lattice site gives a 2π phase.Let’s fix the reference gauge such that the monopole cre-ated at (0, 0) gives a Berry phase β. Then, for a genericcoordinate r, inserting a monopole would give the Berryphase factor η(r)β where η(r) = 1, i,−1, or − i depend-ing on whether the coordinates (rx, ry) are (even, even),(odd, even), (odd, odd), (even, odd) [81]. This is shownexplicitly in Fig. 13(c) as moving the monopole along thearrow gives an additional phase factor i. Inserting ananti-monopole at r gives a phase factor η∗(r)β∗ since itwould give an exactly opposite contribution to the Berryphase term by having ω 7→ −ω. Now, by fixing β = 1,the insertion of the monopole and anti-monopole at eachdual lattice site simply gives a phase η(r) and η∗(r).

To illustrate the further procedure, let us consider twoexamples, Rsite

π/2 and Tx. Under Rsiteπ/2, a monopole remains

monopole, but its location changes as r 7→ Rsiteπ/2r. By cal-

culating the relative phase factor between the monopolecreated at r and Rsite

π/2r, we obtain its transformation rule

in the continuum description:

η(Rsiteπ/2r)β

η(r)β= i =⇒ M† 7→ iM†. (A3)

In the case of Tx, a monopole transforms into an anti-monopole under the symmetry action. At the same time,its location changes as r 7→ r + x. Following the similar

procedure, we obtain the following rule:

η∗(r + x)β∗

η(r)β= −i =⇒ M† 7→ −iM. (A4)

Following the similar analysis, we can obtain αg for allsymmetry transformations summarized in Tab. V. In thecase of time-reversal symmetry, we do not need an ex-tra phase factor because time-reversal already complex-conjugates the phase factor of a monopole operator,which matches with the phase factor given by the anti-monopole at the same site. Moreover, any monopole con-densation amplitude would be time-reversal symmetricbecause T : 〈M〉 7→

⟨M†

⟩∗= 〈M〉.

So far, we assumed β = 1. However, a different choiceof β can be made, which would affect to the transfor-mation rule for the symmetries that filps monopole toanti-monopole. In fact, we can show that this is relatedto the identification rule between monopole M† and theVBS order parameters vx and vy. For β = 1, we getthe relation Eq. (6); for example, Tx : M† 7→ −iM im-plies that the Tx-invariant monopole condensation corre-sponds to the condition that ReM + ImM = 0. How-ever, if β = β = eiπ/4, we would get Tx : M† 7→ −M,implying that the Tx-invariant condition is ReM = 0.In such a case, we can deduce that M† ∼ vx + ivy.

Once we fix the gauge choice and determine how themonopole transforms under the square lattice symmetries(p4m), how the monopole transforms under the Shastry-Sutherland lattice symmetries (p4g) can be deduced eas-ily. This is because the Shastry-Sutherland lattice sym-metries can be expressed in terms of the square latticesymmetries if we take the Shastry-Sutherland lattice inFig. 5 is deformed from the smaller square lattice. Theresult is summarized in Tab. VI.

Appendix B: Comparison with a DMRG simulationresult of the J1-J2 model in the square lattice

In this section, we present our analysis on the squarelattice with spin-1/2. Using the iDMRG simulation, westudied these models on an infinite cylinder with a cir-cumference size up to L = 10 lattice sites. Here, wefocused on the correlation length spectra. The simula-tion is explicitly U(1)z symmetric, and we can plot thecorrelation spectra for each U(1)z quantum number, Sz.Since following models are SO(3) symmetric in the mi-croscopic Hamiltonian, there must exist some degeneracybetween different Sz sectors, which can be interpreted asthe spectrum for a higher spin.

First, let us consider the case where the square latticesymmetry is broken. The following model realizes thephase transition between Neel order and dimerized phase:

H = J1

∑〈i,j〉∈blue

Si · Sj + J ′1∑

〈i,j〉∈red

Si · Sj (B1)

where red and blue bonds are shown in Fig. 14(a). Here,the dimerized phase does not break any symmetry be-

18

cause the square lattice symmetry is already broken inthe model. Therefore, the transition should be describedby the Landau-Ginzburg theory. Indeed, it is known fromthe quantum Monte Carlo simulation [33] that the sys-tem realizes O(3) Wilson-Fisher critical point at J1/J

′1 =

0.523. In the iDMRG simulation, we also observed thatthe Neel order parameter develops at J1/J

′1 ∼ 0.52. At

the transition, a single monopole event is not suppressedbecause different configurations for single monopole eventcannot exactly cancel each other due to the absence ofsymmetries. As a result, the gauge fluctuation (VBS or-der parameter fluctuation) becomes confining, and theCP1 theory is no longer valid. Instead, the critical the-ory is described by the classical NLsM with O(3) Neelvector. The correlation spectra in Fig. 14(c) shows thatthe spin-triplet correlation length is the largest across thephase transition while the spin-singlet correlation lengthis much smaller than that. This behavior is consistentwith the critical theory described by the classical NLsM.

Next, we study the J1–J2 Heisenberg model with asquare lattice symmetry. The model is defined by thefollowing Hamiltonian:

H = J1

∑〈i,j〉

Si · Sj + J2

∑〈〈i,j〉〉

Si · Sj (B2)

where Si is a spin-1/2 operator, J1 is the nearest-neighbor AFM coupling, and J2 is the next nearest-neighbor AFM coupling, see Fig. 14(b). When J2 = 0(J1 = 0), the model is known to realize Neel ordered(conventional AFM stripe) phase. For the intermediatevalue of J1/J2, the system is frustrated and known torealize the disordered phase.

In accordance with the recent IPEPS study [83], weobtained the Neel, columnar VBS (cVBS), and conven-tional stripe phases as we increase J2/J1. However, inorder to obtain the VBS order, we had to apply somebias (pinning field). Under the absence of the bias, thesystem looks totally symmetric, implying the existence ofthe symmetric superposition of symmetry broken states,namely a Cat state. The Cat state can be preferred overthe symmetry broken phase if the circumference size iscomparable to the length scale associated with the fluc-tuations, which is the size of the monopole in this case.

Before getting into the discussion of the correlationlength spectra, we want to elaborate on some simulationdetails. In our iDMRG simulation, the iDMRG unit cellconsists of two columns of lattices along the circumfer-ence, as otherwise translational symmetry broken phases(AFM order or VBS order) cannot develop. The price topay is that kx = 0 and kx = π momenta become indistin-guishable. (In our simulation, ky cannot be measured)However, at the critical point where the explicit symme-try breaking order has not developed yet, we can use asingle column to distinguish kx = 0 and kx = π mo-menta. Indeed, in the iDMRG simulation of the single-column unit cell, we observe that the largest correlationlength for S = 1 spectra carries momentum kx = π inthe single-column simulation, which is consistent with the

FIG. 14: (a) Square lattice J1-J ′1 model with a dimer couplingJ ′1. J ′1 explicitly breaks the square lattice symmetry. (b)Square lattice J1-J2 model. (c,d) Correlation length spectrafor the model in (a,b)

momentum (π, π) of the gapless magnon in Neel order.For S = 0 case, we obtain that the lowest one carrieskx = 0 while the second lowest one carries kx = π, whichruns almost parallel to the lowest one. They correspondto the Z4 VBS order parameter fluctuations (spin-singlet)at (0, π) and (π, 0), but the degeneracy is lifted due tothe iDMRG geometry which breaks the C4-rotation lat-tice symmetry.

Surprisingly, in this model, we obtained the correlationlength spectra that exactly agrees with the level crossingbehavior of the excitation spectrum in the finite DMRGalgorithm (Fig.2 in Ref. 46). Our result is consistent withRef. 32, which discussed the agreement between correla-tion length spectrum and local excitation spectrum inthe DMRG simulation. Moreover, we want to commenton the argument in Ref. 46. In this previous work, it wasargued that the small region where ξS=1 > ξS=0 > ξS=2

corresponds to the gapless spin liquid phase [46]. How-ever, this reasoning is inconsistent with the iDMRG sim-ulation result of the Shastry-Sutherland lattice becausethis region is clearly a symmetry broken phase with anon-zero Neel order parameter from the numerics. Thus,we can conjecture that such a region would shrink into acritical ‘point’ rather than remain as an extended phaseof Dirac spin liquid both in the J1-J2 square lattice modeland Shastry-Sutherland model. Indeed, if we perform asingle-column iDMRG simulation for the J1-J2 model,the simulation does not converge well for the J1/J2 > 2.0,which means that the hypothesized gapless spin liquidphase is, in fact, more like the AFM phase where thedouble-column iDMRG unit cell is required.

19

Finally, we remark on the evidence that supports ourdiscussion in Sec. IV. In the previous finite DMRG workson this model [50], the plaquette VBS appeared in-stead of the columnar VBS. In fact, it was found inRef. [83] that these two states have almost the sameenergy (∆E/E < 0.1%). This again implies that thedangerously irrelevant operator M4, which is responsi-ble for the VBS ordering, has not flowed large enough tocondense monopole to a certain direction. This can besupported by Fig. 9(a), where the correlation length forthe spin-singlet operator is smaller than the correlationlength of the spin-triplet operator throughout the wholeintermediate regime between the Neel and conventionalAFM stripe order.

Appendix C: VBS-Phonon Coupling and PhononSpectrum

The square pVBS order breaks the glide reflection sym-metries Gx, Gy and the diagonal reflection symmetriesσxy, σxy. Due to the lattice symmetry breaking, thepVBS order should induce lattice distortion as shown inFig. 6. Therefore the pVBS fluctuation must couple tothe lattice vibration mode, i.e. the phonon mode. Herewe would like to determine the specific form of the pVBS-phonon coupling.

We will focus on the copper lattice in the following dis-cussion. Although the lattice also contains other atomsand the phonon spectrum can be complicated, we chooseto work on the symmetry level to demonstrate the uni-versal consequences of pVBS fluctuation on the phononspectrum without diving into the details. For this pur-pose, we first specifies the coordinate of copper sites inthe each unit cell. As shown in Fig. 15(a), there are fourcopper sites in each unit cell. At equilibrium, they locateat

rA = (1 + δ, 1 + δ)/2,

rB = (1− δ, 3 + δ)/2,

rC = (3 + δ, 1− δ)/2,rD = (3− δ, 3− δ)/2,

(C1)

where 0 ≤ δ < 1 parameterize the deformation of theShastry-Sutherland lattice from the square lattice. Ac-cording to Ref. 16, the lattice constant is 8.995A, theshortest Cu-Cu bond is 2.905Aand the second shortestCu-Cu bond is 5.132A. This implies that, in unit of thelattice constant, we have

AD =1 + δ

2≈ 2.905A

8.995A,

AB =

√1 + δ2

2≈ 5.132A

8.995A.

(C2)

The optimal solution is δ ≈ 0.544. Using this deforma-tion parameter, we can write down equilibrium positionsof copper atoms through out the lattice.

k1

k2

A C

DB

X

X

Y Y

(a)

10-3

10-2

10-1

100

101

1

FIG. 15: (a) A,B,C,D label four copper sites in each unitcell (marked out in dashed lines). k1, k2 are the stiffnesses ofthe two types of bonds (nearest neighbor and dimer). X,Ylabel the two types of plaquettes. (b) Schematic illustrationof the phonon spectrum. A continuum will emergent at Xpoint due to the VBS-phonon coupling.

Our strategy to figure out the pVBS-phonon couplingis to first investigate the pattern of lattice distortion in-duced by the pVBS order. Because the pVBS order doesnot enlarge the four-copper unit cell, its induced latticedistortion will also be identical among unit cells. There-fore the distortion can be described by four displacementvectors uA,uB ,uC ,uD, translating each sublattice sep-arately as

ri → r′i = ri + ui (i = A,B,C,D). (C3)

The energy cost associated with the distortion can bemodeled by summing up the bond energies

Ebond[ui] =k1

2

∑ij∈n.n.

((r′i − r′j)2 − (ri − rj)2

)2+k2

2

∑ij∈dimer

((r′i − r′j)2 − (ri − rj)2

)2,

(C4)

where ui dependence is implicit in r′i = ri +ui. The en-ergy will increase whenever a bond is stretched or com-pressed. The shape of the potential in Eq. (C4) capturesthis physics when the distortion ui is small. The twostiffness coefficients k1 and k2 are expected to be differ-ent in general. Of course, in the realistic material, Sr,B, O atoms will all be involved and the bond energymodel will be more complicated. However the toy modelEq. (C4) respects all the symmetry property and providesthe stiffness to the copper lattice, which can be used toanalyze the pVBS induced lattice distortion. Finally wenote that the energy model Eq. (C4) written with respectto a single unit cell with periodic boundary condition (i.e.on a torus geometry), so for those bond across the unitcell, their bond length must be correctly treated by con-sidering the periodic boundary condition.

Upon introducing the pVBS order, we will add an ad-ditional term to the energy model,

E[ui] = Ebond[ui] + EVBS[ui],

EVBS[ui] = ImM∑p

(−)p∑i∈p

(r′i −Rp)2, (C5)

20

where p denotes the square plaquettes and i ∈ p denotesthe four corner sites around the plaquette p. Rp coordi-nates the plaquette center,

Rp =

(1, 0) p ∈ X,(0, 1) p ∈ Y.

(C6)

The staggering factor (−)p is +1 for X-type plaquette(yellow) and −1 for Y -type plaquette (green) as shown inFig. 15(a). ImM = vy − vx denotes the square pVBS or-der parameter. The physical meaning of EVBS is that thepVBS order will contract one type of the square plaque-tte and expand the other type, exerting forces on copperatoms that points towards or away from the plaquettecenter.

Given the full energy model in Eq. (C5), we can ex-pand E[ui] to the quadratic order of ui. The linear termwill be proportional the pVBS order parameter ImM, asImM is the force that distort the lattice. The quadraticterm determines how the lattice responses to the distor-tion force in the linear response regime. We found thatindependent of the choice of δ and k2, the response isalways given by

uA =ImM4k1

(−1, 1),

uB =ImM4k1

(−1,−1),

uC =ImM4k1

(1, 1),

uD =ImM4k1

(1,−1).

(C7)

Under the Fourier transformation to the momentumspace

u(q) =∑i

uieiq·ri , (C8)

the solution in Eq. (C7) corresponds to

ux(π, 0) ∝ − ImM, uy(0, π) ∝ ImM. (C9)

This calculation indicates that the square pVBS orderwill lead to a lattice distortion that corresponds to a thesimultaneous condensation of phonon modes ux at mo-mentum (π, 0) and uy at momentum (0, π). Given thatImM = vy − vx, we conclude that there must be a lin-ear coupling between lattice displacement and the VBSorder parameter in the form of

LVBS-phonon = κ(vxux + vyuy), (C10)

in order to produce the linear response in Eq. (C9). Thecoupling in Eq. (C10) can be further justified by symme-try arguments. Tab. VII shows the momentum quantumnumber and the symmetry transformation of the VBS or-der parameter v and finite-momentum phonon mode u.One can see v and u have identical symmetry propertiesand hence a linear coupling as in Eq. (C10) is allowed.

vx vy ux uy

q (π, 0) (0, π) (π, 0) (0, π)

Gx −vx −vy −ux −uyGy −vx −vy −ux −uyσxy vy vx uy ux

σxy vy vx uy ux

TABLE VII: Momentum and symmetry transformations ofthe VBS order parameter v and finite-momentum phononmode u.

Given the VBS-phonon coupling, we can investigatethe effect of low-energy VBS fluctuation on the phononspectrum near the DQCP. We first write down the fieldtheory action describing both degrees of freedom,

S[u,v] =1

2

∑q

(ω2 − Ω2q)u(−q) · u(q)

−1

2

∑q

G−1VBS(q)v(−q) · v(q)

+∑q

κqv(−q) · u(q),

(C11)

where q = (ω, q) represents the energy-momentum vec-tor. Ωq describes the phonon dispersion relation. GVBS

is the correlation function of the VBS critical fluctuation,whose low-energy behavior is given by

GVBS(ω, q) =1(

(q −Q)2 − ω2)1−η/2 , (C12)

where η is the anomalous exponent of the O(4) vectorat the O(4) DQCP. Based on the previous numericalmeasurements[25, 66], η is estimated to be η = 0.13 ∼0.3. Q = (π, 0) or (0, π) denotes the momentum pointwhere the VBS fluctuation gets soften. The VBS-phononcoupling κq is expected to be momentum dependent,because the VBS order parameter only couples to thehigh-energy phonon around X and Y points but notthe acoustic phonon around Γ point. By the acousticphonon around Γ point, we mean the low energy partof the phonon, i.e. the segment of the acoustic brancharound the gapless point, which usually appears in thefield theory description of phonons. Given these setup,we can integrate out the VBS fluctuation and obtain thedressed propagator of phonon,

D(ω, q) =1

Ω2q − ω2 − κ2

qGVBS(ω, q). (C13)

Then the phonon spectral function can be obtained from

S(ω, q) = 2 ImD(ω + i0+, q). (C14)

The phonon dispersion Ωq is unknown to us, as we didnot have the full model of the lattice vibration. For

21

demonstration purpose, we can use the following toymodel

Ω2q = sin2(qx/2) + sin2(qy/2), (C15)

which captures the gapless acoustic phonon at the Γ pointand gapped phonons at X and Y points (see Fig. 1(b)).We also take the anomalous exponent η = 0.13 and useκq = κ0Ωq with κ0 = 0.05 so that

limq→(0,π),(π,0)

κq = κ0, limq→(0,0)

κq = 0. (C16)

With these, we show the phonon spectrum calculatedfrom Eq. (C14) in Fig. 15(b). The prominent feature isa V-shape continuum at the X point (as well as theY point) in the Brillouin zone. This continuum in thephonon spectrum represents the critical fluctuation of theVBS order parameter at the DQCP. Although the spec-tral weight is expected to be weak, since the X point isan extinction point, it is still feasible to collect spectralsignals of this continuum. In particular, the frequencydependence of the spectral weight at the X point is pre-dicted to follow

S(ω, q = X) ∝ ω−2+η, (C17)

which can be checked experimentally. It will be meaning-ful to compare the measured η with large-scale quantumMonte Carlo simulation result.

Appendix D: Detailed Numerical Data

In this appendix, we discuss the evolution of theiDMRG simulations results as we increase the bond di-mension χ for system sizes L = 8 and L = 10. Since theaccuracy of the iDMRG simulation is determined by thebond dimension, a reliable analysis requires one to exam-ine the results as a function of the bond dimension. Here,the iDMRG simulation results of the Shastry-Sutherlandlattice model in Eq.1) at L = 10 for a range of bonddimensions are presented in Fig. 16. Although the thetruncation error εtrun is very large (> 10−4) at the lowbond dimensions, as we increase the bond dimensisonupto χ = 4000, εtrun goes below 10−5 and the iDMRGresults becomes sensible.

Fig. 16(b) shows the first derivative of energy, whosechange of the slope correspsonds to the transition be-tween the pVBS and Neel ordererd phases. As the bonddimension increases, the transition point shifts leftward,implying that the parameter regime for the pVBS phaseshrinks. The behavior aligns with the intuition that thegapped pVBS phase would be favored over the the gaplessNeel orderd phase for a low entanglement MPS state. Ac-cordingly, the peak of the spin-singlet correlation lengthwhich coincides with the phase transition point also shiftsleftward, presented in Fig. 16(d). At the same time, thepeak of the spin-singlet correlation length becomes largerand more pronounced as the bond dimension increases,

FIG. 16: The iDMRG simulation results with ∆J1 = 0.002at L = 10, shown for a range of MPS bond dimension χ.Bond dimension scalings of (a) energy per site E, (b) energyderivative per site ∂E/∂J1

a, (c) truncation error p, (d) cor-relation length of spin-singlet operator ξS=0, and (e) pVBSorder parameters. Note that the correlation length is plottedinstead of the inverse. In (f), we plot pVBS order parametersas functions of truncation erros for a range of the tuning pa-rameter J1. Blue dotted line is a linear fitting for the threedata points at χ = 2000, 3000, 4000.

aData for χ = 500, 1000 are not shown here as these data-pointsbehave irregularly, and cover the other data points.

signaling the continuous or weakly first order phase tran-sition.

Although the peak location of the spin-singlet correla-tion length changes with the bond dimension, a furtherindication of the phases boundary can be obtained fromthe order parameter plotted versus the truncation error[84, 85]. In principle, the ground state is fully symmetricand local order parameter cannot be non-zero. However,in the iDMRG simulation, the numerical process favorsa minimally entangled state, giving rise to a non-zero lo-cal order parameter in a spontaneous symmetry breakingphase at finite bond dimensions. This can even happenwhen the system is outside but close to the spontaneoussymmetry breaking phase, meaning that one needs to

22

FIG. 17: The iDMRG simulation results with ∆J1 = 0.002(0.001 for χ = 4000) at L = 8, shown for a range of MPS bonddimension χ. Different observables are labeled as in Fig. 16.Note that the pVBS order parameter vanishes at J1 = 0.726,which is smaller than the value for L = 10.

plot the order parameter as a function of the truncationerror in order to see whether a non-zero order parameteris truly physical. In Fig. 16(f), we plotted the pVBS orderparameter defined in Eq. 5 as a function of the trunca-tion error, and extrapolated them. We observe that theextrapolated order parameter disappears at J1 = 0.76,which agrees well with the peak location J1 = 0.762of the spin-singlet correlation length at χ = 4000 withL = 10. As mentioned earlier, this extrapolation methodwould be benefited a lot if a wider range of bond di-mensions is available. However, at χ = 4000, each datapoint already takes about 50 hours of simulations timesfor 12 multi-threads with 80GB RAM, and both the timeand RAM scale roughly as χ2. Therefore, a significantlyhigher bond dimension is currently inaccessible at our

capacity.Finally, we remark the scaling of the result for different

system sizes. In Fig. 17, we plotted the iDMRG simula-tion results at L = 8 for a range of MPS bond dimensionas in Fig. 16. Note that the truncation error is an orderof magnitude smaller than the results at L = 10. More-over, the value of J1 where the pVBS order parameterdisappears is much smaller for the smaller system sizeL. However, this does not imply that the Neel orderedphase develops for J1 > 0.726, as we can see from thecorrelation length plot in Fig. 18. In Fig. 18(a), while thepeak of the spin-singlet correlation length implies thatthe peak corresponds to the phase transition point for thepVBS phase, the spin-triplet correlation length remainsalmost constant, which implies that the Neel order doesnot develop, since the Neel order would give rise to theincrease of the spin-triplet correlation length originatedfrom the gapless magnon excitation. On the other hand,at L = 10, we observe that the peak of the spin-singletcorrelation length is immediately followed by the rise ofthe spin-triplet correlation length which signals the on-set of the Neel ordered phase. L = 6 behavior is similarto that of L = 8, and the signature of the Neel orderedphase, such as the staggering magnetization or the rise ofthe spin-triplet correlation lengh, is not observed near thepVBS critical point. This is related to the discussion inthe main text. For a quasi one-dimensional system with afinite-size circumference size, the spontaneous symmetrybreaking of the continuous group would be suppresseddue to the disordering effect. As a result, the disappear-ance of the pVBS ordering is not immediately followedby the Neel ordering for a finite circumference system.

FIG. 18: Comparison between the L = 8 and L = 10 resultsat χ = 4000 for the spin-singlet and triplet correlation lengths.For L = 8, the peak of the ξS=0 is located at J1 = 0.727, whilefor L = 10 the peak is located at J1 = 0.762. Unlike the resultat L = 10, the spin-triplet correlation length at L = 8 doesnot grow much after the peak.

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arXiv e-prints arXiv:1808.00463 (2018), 1808.00463.[86] Beyond the length scale ξspin, the system behaves like

XY model with order parameterM. From the inspectionof the XY model, the domain wall length scale is given

by ξVBS ∼ λ−1/24 , where λ4 is a renormalized coefficient

for ReM4 at that length scale. From the generic scalingargument, we have ξVBS ∼ Lf(λ4L

3−∆), where L can beany length scale within the scaling regime. At the length

25

scale L = ξspin where the system behaves like XY model,

we already know that ξVBS ∼ λ−1/24 . Thus, we can deduce

f(x) ∼ x−1/2, and derive that ξVBS ∼ ξ(∆−1)/2spin

[87] In the VBS phase, there are four degenerate groundstates, whose degeneracy is lifted by the finite systemsize. These nearly degenerate excited states should bedistinguished from the other excitations that can trulybe ‘local’.

[88] The Neel ordered phase has the time-reversal glideT Gx,y, while the pVBS phase has the time-reversal T .At the DQCP, we have both symmetries, thus the com-position T T Gx,y = Gx,y should be there.

[89] The AFM spin-spin interaction along vertical bonds in-duces the effective vertical plaquette ring exchange forthe low energy subspace.