Hong Yao, Shou-Cheng Zhang and Steven A. Kivelson- Algebraic spin liquid in an exactly solvable spin model

8/3/2019 Hong Yao, Shou-Cheng Zhang and Steven A. Kivelson- Algebraic spin liquid in an exactly solvable spin model

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Work supported in part by US Department of Energy contract DE-AC02-76SF00515.

Algebraic spin liquid in an exactly solvable spin model

Hong Yao, Shou-Cheng Zhang, and Steven A. KivelsonDepartment of Physics, Stanford University, Stanford, California 94305

(Dated: November 8, 2009)

We have proposed an exactly solvable quantum spin-3/2 model on a square lattice. Its groundstate is a quantum spin liquid with a half integer spin per unit cell. The fermionic excitations aregapless with a linear dispersion, while the topological vison excitations are gapped. Moreover,

the massless Dirac fermions are stable. Thus, this model is, to the best of our knowledge, the firstexactly solvable model of half-integer spins whose ground state is an algebraic spin liquid.

The term spin liquid is widely used to give sharpmeaning to the more general intuitive notion of a Mottinsulator; a spin liquid is an insulating state that cannotbe adiabatically connected to a band insulator, i.e. toan insulating Slater determinant state. In a system thatpreserves time reversal symmetry, any insulating statewith an odd number of electrons (or a half-integer spin)per unit cell is a spin liquid. Interesting proposals [1, 2]have been made concerning the relevance of such states to

the theory of high temperature superconductivity in thecuprates and other materials. Indeed, various spin-liquidphases have been proposed, which are distinguished bythe character of any gapless spinons and the exchangestatistics of the topological vison excitations.

Since they are new and exotic quantum phases ofmatter, it is desirable to construct solvable models withshort range interactions with stable spin-liquid ground-state phases. A breakthrough occurred when Moessnerand Sondhi [3] demonstrated the existence of a gappedspin liquid ground state in the quantum dimer model [4],analogous to the short range version of the RVB state[5, 6]. An exactly solvable spin-1/2 model in a gapped Z2spin liquid phase was later constructed by Wen [7]. How-ever, much of the recent interest, spurred in part by thepossible observation of such a state in -(ET)2Cu2(CN)3[8, 9, 10] and Zn(Cu)3(OH)6Cl2 [11], has focussed onspin-liquids with gapless spin excitations, so-called al-gebraic spin liquids.

The exactly solvable Kitaev model on the honeycomblattice [12] can exhibit gapless excitations. However, be-cause the honeycomb lattice has two sites per unit cell,this model has an integer spin, hence an even number ofelectrons per unit cell. In the present paper we constructan exactly solvable model, in much the same spirit as the

Kitaev model, whose ground state is a spin liquid withan oddnumber electrons per unit cell and stable gaplessfermionic excitations. To the best of our knowledge, thisis the first exactly solvable model with this sort of spinliquid ground state -algebraic spin liquid.

The Kitaev model has a spin-1/2 on each site of atrivalent lattice, where the coordination number is dic-tated by the existence of three Pauli matrices. In order tostudy a model on a square lattice, we instead consider amodel with a spin-3/2 on each lattice site. The resulting

larger Hilbert space, with 4 spin polarizations per site,permits us to express the model in terms of the 4 4 an-ticommuting Gamma matrices, a (a = 1, , 5) whichform Clifford algebra, {a, b} = 2ab. Specifically, the5 Gamma matrices can be represented [13] by symmetricbilinear combinations of the components of a spin 3/2operator, S, as

1 =1

3{Sy, Sz}, 2 = 1

3{Sz, Sx}, 3 = 1

3{Sx, Sy},

4 =1

3[(Sx)2 (Sy)2], 5 = (Sz)2 5

4. (1)

Model Hamiltonian: We define our model on a squarelattice, with a spin-3/2 on each site, and corresponding matrices defined as in Eq. (1). In terms of these,

H =i

Jx

1i

2i+x + Jy

3i

4i+y

(2)

+i

Jx

15i

25i+x + J

y

35i

45i+y

J5

i

5i ,

where ab [a, b]/(2i) and i labels the lattice siteat ri = (xi, yi). We call this model the Gamma ma-trix model (GMM). Suppose that the square lattice hasN = LxLy sites, where Lx and Ly are the lattices linearsizes and are assumed, for simplicity, to be even in thispaper. Moreover, we consider periodic boundary condi-tions. Obviously, the GMM can be written explicitly as aspin-3/2 model. The GMM model respects translationalsymmetry and time reversal symmetry (TRS). It does nothave global spin SU(2) or even U(1) rotational symmetry,but is invariant under 180 rotations about the z axis inspin space, i.e. it has global Ising symmetry. The lack ofany continuous symmetry implies that the fermionic ex-citations (discussed below) do not have well-defined spin

quantum number, so their relation to spinons is un-clear. A feature of the model which makes it solvable isan infinite set of conserved fluxes: [Wi,H] = 0 for anyi and [Wi, Wj ] = 0, where Wi 13i 23i+x14i+y24i+x+y isthe plaquette operator at site i.

Note that, in contrast to the spin-1/2 Kitaev model onthe honeycomb lattice, the GMM has an odd number (3)of electrons per unit cell. The present model in the limitJx = Jy = J

x = J

y and J5 = 0 is similar to a model pro-

posed by Wen in Ref. [14], where, however, the Gamma

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matrices were constructed from two spin-1/2 operatorson each site, and therefore behave differently under timereversal than in the present realization. Moreover, thepresent model generically does not possess the specialsymmetries of Wens model that are responsible for someof the behaviors of that model.

Fermionic Representation: Spin-3/2 operators can beexpressed as bilinear forms involving three flavors of

fermion operators, Sz = aa + 2bb 3/2 and S+ =3fa +

3af+ 2ab, subject to the constraint that the

physical states are only those with odd fermion parity,

(1)N = 1, where N = ff+aa+bb. Rather than us-ing this representation in terms of three Dirac fermions,we will directly represent the Gamma matrices in termof a related set of 6 Majorana fermions:

i = ici di,

5i = ic

i d

i, = 1, 2, 3, 4,

5i = idid

i (3)

where ci , di, and di are Majorana fermions on site i.

Six Majorana fermions form an eight dimensional Hilbertspace, which is an enlarged one from the physical Hilbert

space of an spin-3/2. In terms of spin-3/2 operators,1i

2i

3i

4i

5i = 1 for all i. Consequently, all allowed

physical states | in term of Majorana fermions mustsatisfy the following constraint, for all i,

Di| ic1i c2i c3i c4ididi | = |. (4)

In terms of Majorana fermions, the Hamiltonian in theenlarged Hilbert space can be written as

H =i

Jxui,xididi+x + Jyui,yididi+y

+Jxui,xididi+x + J

yui,yid

idi+y

J5idid

i, (5)

where ui,x ic1i c2i+x and ui,y ic3i c4i+y. It is obviousthat ui, are conserved quantities with eigenvalues ui, =1, = x, y. The enlarged Hilbert space can be dividedinto sectors {u}. In each sector, the Hamiltonian Eq. (5)describes free Ma jorana fermions:

H({u}) =i

Jxui,xididi+x + Jyui,yididi+y

+Jxui,xididi+x + J

yui,yid

idi+y J5ididi

, (6)

where ui, are emergent Z2 gauge fields. The Z2 gauge

transformations are given by di idi, di idi, andui, iui,i+, where i = 1. In the enlargedHilbert space, the eigenstates | = |c |d,d canbe written as a direct product of a part that involvesthe c fermions and a part that involves the d and d

fermions, respectively. Here |c is defined by ui,|c =ui,|c and |d,d are eigenstates of Eq. (6) with thecorresponding ui,s.

The spectrum ofH({u}) depends only on gauge invari-ant quantities - the flux on local plaquettes, exp(ii)

ui,xui+x,yui,yui+y,x, and two global fluxes exp(ix) i(yi=1)

ui,x and exp(iy)

i(xi=1)ui,y, where i and

x,y = 0, . [Note that the previously defined Wi = exp(ii).] It is obvious that

i i = 0 (mod 2), so

there are N 1 independent local fluxes. Including thetwo global fluxes, the number of independent fluxes isN + 1. Since there are 2N Z2 gauge fields, the num-ber of different gauge field choices corresponding to each

flux sector {} is 2N1. In other words, in the enlargedHilbert space, each state is 2N1-fold degenerate. Notethat in the thermodynamic limit, the energy is indepen-dent of the two global fluxes, which gives rise to a fourfoldtopological degeneracy of the physical ground states.

Projection operators: Most of the states in the en-larged Hilbert space are not physical states. To obtaina physical eigenstate, we must find a linear combinationof the degenerate eigenstates which is simultaneously aneigenstate of every Di with eigenvalue 1. This is realizedthrough the pro jection operator P:

| = P| i (1 + Di)/2|. (7)Clearly, Eq. (7) implies Di| = | for any i. Explicitly,P is given by

P =

1 +i

Di +i1


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3

sublattice, only states with Nf = odd survive projec-tion. Conserving the parity of fermion number reflectsthe fact that physical fermionic excitations are createdby non-local (string) operators [14, 23].

-flux state and gapped visons: In each flux sector{}, the lowest energy of the Hamiltonian is denotedby E0({}). The ground state energy of the model isachieved by minimizing E0(

{

}) with respect to

{

}.

Formally, by integrating out the fermions, an effectiveaction for the Z2 gauge fields can be derived. However,in general, it is nontrivial to obtain an explicit form ofthe effective action.

For J5 = 0, fortunately, there is a theorem due to Lieb[15] which implies that the energy minimizing flux sectorof a half-filled band of electrons hopping on a planar,bipartite lattice is per square plaquette. We definethis uniform -flux, the ground state sector, as beingvortex-free. Z2 vortex excitations, visons, are definedto be plaquettes with i = 0. Due to the constraint

ii = 0 (mod 2), only an even number of visons

are allowed. It is special for this model that visons donot have dynamics even though they interact with oneanother. (Numerical calculations of the two vison energyreveals that the interaction between visons is repulsive.)The minimum energy cost of creating two visons in theground state is always finite and is defined as the vison

gap energy v (|JxJy|+

|JxJy|). Since the visons

are gapped, the uniform -flux is still the ground statesector as long as J5 is much smaller than v . In the restof paper, we restrict our attention to cases (e.g. smallJ5) in which the ground state lie in the -flux sector.

Spin correlation: The state with uniform -fluxes pre-serves the translational symmetry and TRS of the model.

To prove the ground state of the model is a spin liquid,we need to show that there is no magnetic order andthat the spin correlations are short-ranged. Spin-3/2 op-erators can be expressed as bilinear forms of Majoranafermions. For instance, Szi = ic

3i c

4i +

12

ic1i c2i . The action

of Szi on the ground state | creates visons in the foursurrounding plaquettes around site i [16]. The effect ofSxi and S

yi on the ground state is similar. Because visons

are non-dynamical in the present model, |Si Sj | isidentically zero unless sites i and j are nearest neighbors,i.e. the spin correlations are unphysically short-ranged.Presumably, if additional small, local terms are added to

the Hamiltonian, perturbative corrections to this corre-lator would lead to a finite correlation length.

Gapless fermions: To obtain the excitation spectrumin the -flux (ground state) sector, we fix the gaugeby choosing ui,x = 1, ui,y = (1)i. The correspondingHamiltonian is given by

H0 =

i

txf

i fi+x + ty(1)ifi fi+y J5fi fi

x(1)ifi fi+x yfi fi+y + H.c.

, (10)

which describes a p-wave superconductor of spinlessfermions [17]. Here t J + J and J J.(Note that in Ref. [14], the pairing terms are absentdue to projective symmetries.) In terms of the Blochstates, fk =

i exp(ik ri)fi/

N, Eq. (10) in momen-

tum space is given by

H0 = k

kHkk, k = (f

k, f

k+Q, fk, fkQ), (11)

where Q = (, ) and the summation is over only halfof the Brillouin zone since k is equivalent to k + Q. Ingeneral, the analytical form of the eigenvalues of the 44matrix Hk are complicated. Due to time reversal sym-metry, however, it is straightforward to derive the quasi-particle excitation spectrum as follows:

E,k = 2

J25 + 2g+,k 2

g2,k + J

25gk, (12)

where g,k = (J2x Jx2)cos2 kx + (J2y Jy2)sin2 ky and

gk = (Jx

+ Jx

)2 cos2 kx

+ (Jy

+ Jy

)2 sin2 ky

.

When J5 = 0, E+,k = 4(J2x cos

2 kx + J2y sin

2 ky)1/2 and

E,k = 4(Jx2

cos2 kx + Jy2

sin2 ky)1/2 are the energies ofthe d and d Majorana fermions respectively since theydecouple from each other. Both spectra are gapless atnodes K = (/2, 0), around which the spectrum islinear in momentum; the excitations are massless Diracfermions. However, the massless fermions are unstable inthe sense that additional small, local (further neighborhopping and pairing) terms can gap the nodes [23].

When 0 < J5 v, (so the ground state lies in the-flux sector,) it is clear that E+,k is always gapped. Theconditions for E,k to have gapless excitations are:

JxJx cos

2 kx + JyJy sin

2 ky = J25/4, (13)

(JxJy JyJx)cos kx sin ky = 0. (14)

For simplicity, we consider the case in which Jx, Jy J5 > 0 so that J5 v is satisfied for arbitrary Jx andJy. From the two conditions we obtain the phase diagramshown in Fig. 1 as a function of Jx and J

y: (i) When

Jx > J25/(4Jx), J

y > J

25/(4Jy), and J

x/J

y = Jx/Jy, the

fermion spectrum has eight Dirac nodes at (/2x, 0)and (/2,y) with = arcsin(J5/

4JJ); (ii)

When Jx > J25/(4Jx) and J

y < J

25 /(4Jx), there are four

Dirac nodes at (

/2

x, 0); (iii) When Jx < J

25 /(4Jx)

and Jy > J25/(4Jx), there are also four Dirac nodesbut at (/2,y); (iv) When Jx < J25/(4Jx) andJy < J

25 /(4Jy), all fermionic excitations are gapped.

For 0 < J5 v, the Dirac nodes, if they exist, aretopologically stable in the sense of Wigner-Von Neumanntheorem. Any weak translational and time reversal in-variant perturbation only shifts the positions of the nodes[23]. Consequently, the present phase with Dirac nodesis characteristic of a stable quantum phase of matter, i.e.an algebraic spin liquid [18, 19, 20].


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4

xJ'

Eight nodes

Eightnodes

Fournodes

Four nodesGapped

Pesudo-FS

yJ'

FIG. 1: The quantum phase diagram of the Gamma matrixmodel as a function of Jx and J

y in the case J5 Jx, Jy,where the ground state lies in the uniform -flux sector. TheDirac nodes in the phase diagram are topologically stable. Atthe critical (red) line Jx/J

y = Jx/Jy, fermions form a Fermisurface (FS). The other two critical (blue) lines are defined asJx = J

2

5/(4Jx) or J

y = J2

5/(4Jy).

It is worth noting that along the critical line Jx/Jy =

Jx/Jy and Jx > J

25/(4Jx), corresponding to the red line

in Fig. 1, the discrete nodes broaden into a line of nodes.In short, in this special case the present model realizesa spinon Fermi surface. Since spin-3/2 operators can bewritten as a bosonic bilinear in term of two Schwingerbosons, S = bs

ssbs/2, s =, with the constraint

bb + bb = 3, the present spin model can be written as

a bosonic model, albeit one with four-body interactions.It follows as a corollary that by tuning a single coupling

constant to a critical value, a purely bosonic model canexhibit an emergent Fermi surface.Upon approach to the critical line Jx = J

25/(4Jx)

from the eight node phase, both of the two Dirac conesat (/2 + x, 0) approach (, 0) leading, at critical-ity, to a single node with the unusual dispersion Ek

Aq4x + Bq2y) for small |q| = |k (, 0)|, where A and

B are constants depending on J, J, and J5. Anothertwo nodes at (/2 x, 0) approach (0, 0) leading tothe same unusual dispersion. Similar physics is obtainedat the other critical line Jy = J

25/(4Jy).

Large-J5 limit: The ground-states for 1/J5 = 0 formthe low energy manifold Szi =

3/2 for each i. Defin-

ing an effective spin-1/2 , such that Szi = 3/2 cor-responds to zi = 1, and employing degenerate per-turbation theory in J and J, the effective Hamilto-nian can be shown be Heff = Jeff

i

xi

yi+x

xi+x+y

yi+y,

Jeff = (Jx Jx)2(Jy Jy)2/(16J35 ), which is exactlyWens plaquette model [7]. In terms of fluxes, Heff =

i Jeff exp[ii]. Consequently, the ground state is stillin the -flux sector. We expect that the gapped phase

in small J5 limit can be adiabatically connected to thelarge J5 gapped phase.

Discussion: It is straightforward to generalize theGMM to other lattices in 2D and also to higher dimen-sions. For the 2D triangular lattice, a quantum spin-7/2(or spin-12 -

12 -

12 [14]) model can be defined via the seven

Gamma matrices. Due to the non-bipartiteness of thelattice, the model spontaneously breaks TRS. Thus, it is

expected that a chiral spin liquid [22] could be realizedin this model on a triangular lattice [23].

Acknowledgments: We would like to thank EduardoFradkin, Zhengcheng Gu, Taylor Hughes, Alexei Ki-taev, Robert B. Laughlin, Joseph Maciejko, Joel Moore,Chetan Nayak, Xiaoliang Qi, and Shivaji Sondhi forhelpful discussions. We are grateful to Xiao-Gang Wenfor pointing out a different interpretation of the presentGamma matrix model in Ref. [14]. S.A.K. and H.Y. aresupported in part by DOE grant DEFG03-01ER45925.S.C.Z. is supported by NSF DMR-0342832 and DOEgrant DE-AC03-76SF00515. H.Y. thanks the support by

a Stanford Graduate Fellowship.

[1] P.W. Anderson, Science 235, 1196 (1987).[2] For a review, see P. A. Lee, N. Nagaosa, X.-G. Wen, Rev.

Mod. Phys. 78, 17 (2006).[3] R. Moessner and S. Sondhi, Phys. Rev. Lett. 86, 1881

(2001).[4] D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61,

2376 (1988).[5] P. W. Anderson, Mater. Res. Bull. 8, 153 (1973).[6] S. A. Kivelson, D. S. Rokhsar, and J. P. Sethna, Phys.

Rev. B 35, 8865 (1987).

[7] X. G. Wen, Phys. Rev. Lett. 90, 016803 (2003).[8] Y. Shimizu et al, Phys. Rev. Lett. 91, 107001 (2003).[9] S. Yamashita et al, Nat. Phys. 4, 459 (2008).

[10] Y. Qi, C. Xu, and S. Sachdev, Arxiv: 0809.0694.[11] J. S. Helton et al, Phys. Rev. Lett. 98, 107204 (2007).[12] A. Kitaev, Annals of Physics 321, 2 (2006).[13] S. Murakami, N. Nagaosa, and S. C. Zhang, Phys. Rev.

B 69, 235206 (2004).[14] X.-G. Wen, Phys. Rev. D, 68, 065003 (2003).[15] E. H. Lieb, Phys. Rev. Lett. 73, 2158 (1994).[16] G. Baskaran, S. Mandal, and R. Shankar, Phys. Rev.

Lett. 98, 247201 (2007).[17] H.-D.Chen and J.-P.Hu, Phys. Rev. B 76, 193101 (2007).[18] W. Rantner and X.-G. Wen, Phys. Rev. Lett. 86, 3871

(2001).[19] M. Hermele et al, Phys. Rev. B 70, 214437 (2004).[20] M. Hermele, T. Senthil, and M. P. A. Fisher, Phys. Rev.

B 72, 104404 (2005).[21] Y. Shimizu et al, Phys. Rev. B 73, 140407(R) (2006).[22] H. Yao and S. A. Kivelson, Phys. Rev. Lett. 99, 247203

(2007)[23] H. Yao et al, unpublished.[24] The total number of-fluxes must be even, so the choice

of which sublattice is immaterial.

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Hong Yao, Shou-Cheng Zhang and Steven A. Kivelson- Algebraic spin liquid in an exactly solvable spin model