-
The spinon Fermi surface U(1) spin liquid in a
spin-orbit-coupledtriangular lattice Mott insulator YbMgGaO4
Yao-Dong Li1, Yuan-Ming Lu2, and Gang Chen1,3∗1State Key
Laboratory of Surface Physics, Department of Physics,
Center for Field Theory & Particle Physics, Fudan
University, Shanghai, 200433, China2Department of Physics, The Ohio
State University, Columbus, OH, 43210, United States and
3Collaborative Innovation Center of Advanced Microstructures,
Nanjing, 210093, China(Dated: August 24, 2017)
Motivated by the recent progress on the spin-orbit-coupled
triangular lattice spin liquid candi-date YbMgGaO4, we carry out a
systematic projective symmetry group analysis and mean-fieldstudy
of candidate U(1) spin liquid ground states. Due to the
spin-orbital entanglement of the Ybmoments, the space group
symmetry operation transforms both the position and the orientation
ofthe local moments, and hence brings different features for the
projective realization of the latticesymmetries from the cases with
spin-only moments. Among the eight U(1) spin liquids that we
findwith the fermionic parton construction, only one spin liquid
state, that was proposed and analyzedin Yao Shen, et. al., Nature
540, 559-562 (2016) and labeled as U1A00 in the present work,
standsout and gives a large spinon Fermi surface and provides a
consistent explanation for the spectro-scopic results in YbMgGaO4.
Further connection of this spinon Fermi surface U(1) spin liquid
withYbMgGaO4 and the future directions are discussed. Finally, our
results may apply to other spin-orbit-coupled triangular lattice
spin liquid candidates, and more broadly, our general approach
canbe well extended to spin-orbit-coupled spin liquid candidate
materials.
I. INTRODUCTION
The interplay between strong spin-orbit coupling(SOC) and strong
electron correlation has attracted asignificant attention in recent
years1. At the mean time,the abundance of strongly correlated
materials with 5dand 4f electrons, such as iridates and rare-earth
mate-rials1,2, brings a fertile arena to explore various emer-gent
and exotic phases that arise from such an inter-play3–32. The
recently discovered quantum spin liquid(QSL) candidate YbMgGaO4
33, where the rare-earth Ybatoms form a perfect triangular
lattice, is an ideal sys-tem that involves strong spin-orbital
entanglement in thestrong Mott insulating regime of the Yb
electrons34–41.
In YbMgGaO4, the thirteen 4f electrons of the Yb3+
ions are well localized and form a spin-orbit-entangledtotal
moment J with J = 7/234,35. The eight-fold de-generacy of the J =
7/2 moment is further split by theD3d crystal electric fields. The
resulting ground stateKramers doublet of the Yb3+ ion, whose
two-fold de-generacy is protected by the time-reversal symmetry,is
well separated from the excited doublets and is re-sponsible for
the low-temperature magnetic propertiesof YbMgGaO4. No signature of
time-reversal symmetrybreaking is observed for YbMgGaO4 down to the
lowestmeasured temperature36–38. Applying the recent theoret-ical
result on spin-orbit-coupled Mott insulators42, two ofus and
collaborators have proposed YbMgGaO4 to be thefirst QSL candidate
in the spin-orbit-coupled Mott in-sulator with odd electron
fillings34–36,39. More broadly,YbMgGaO4 represents a new class of
rare-earth materi-als where the strong spin-orbit entanglement of
the localmoments meets with the geometrical frustration of
thetriangular lattice such that exotic quantum phases maybe
stabilized.
Apart from the absence of magnetic ordering, the heatcapacity
was found to be Cv ∝ T 0.7 at low tempera-tures33,34,37,43, and is
close to the well-known T 2/3 heatcapacity44–46. The latter was the
one obtained within arandom phase approximation for the
spinon-gauge cou-pling in a spinon Fermi surface U(1) QSL44–46.
Moresubstantially, the broad continuum36,37 of the
magneticexcitation with a clear dispersion for the upper
excitationedge agrees reasonably with the particle-hole continuumof
the spinon Fermi surface36. However, due to the scat-tering with
the phonon degrees of freedom, the thermaltransport measurement in
YbMgGaO4 was unable to ex-tract the intrinsic magnetic contribution
to the thermalconductivity43. Partly motivated by the spin liquid
be-haviors in YbMgGaO4 and more broadly by the familiesof
rare-earth magnets with identical structures, in thispaper, we
carry out a systematic projective symmetrygroup (PSG) analysis for
a triangular lattice Mott insu-lator with spin-orbital-entangled
local moments. Unlikethe cases for the spin-only moments in the
pioneeringwork by X.-G. Wen47, the space group symmetry opera-
FIG. 1. (a) The intralayer symmetries of the R3̄m space groupfor
YbMgGaO4
35. (b) The same lattice symmetry group witha different complete
set of elementary transformations. HereS6 ≡ C−13 I. The bold arrow
is the axis for the C2 rotation(see Appendix).
arX
iv:1
612.
0344
7v4
[co
nd-m
at.s
tr-e
l] 2
3 A
ug 2
017
-
2
tion, in particular, the rotation, transforms both the po-sition
and the orientation of the Yb local moments35,39.We find that,
among the eight U(1) QSL states, thespinon mean-field state that
was introduced in Ref. 36and labeled as the U1A00 state in our PSG
classification,contains a large spinon Fermi surface and gives a
largespinon scattering density of states that is consistent withthe
inelastic neutron scattering (INS) results.
The following part of the paper is organized as follows.In Sec.
II, we describe the space group symmetry and thethe multiplication
rules for the symmetry transformation.In Sec. III, we introduce the
fermionic spinon construc-tion and the fermionic spinon mean-field
Hamiltonian. InSec. IV, we explain the scheme for the projective
sym-metry group classification when the spin-orbit couplingis
present. In Sec. V, we explain the relationship betweenthe spinon
band structure and the projective symmetrygroup of the spinon
mean-field states. In Sec. VI, wefocus on the U1A00 state and study
the spectroscopicproperties of this state. Finally in Sec. VII, we
discussthe experimental relevance and remark on the
thermaltransport result and the competing scenarios and pro-posals.
The details of the calculation are presented inthe Appendices.
II. SPACE GROUP SYMMETRY
It was pointed out that the intralayer symmetries in-volves two
translations, T1 and T2, one two-fold rota-tion, C2, one three-fold
rotation, C3, and one spatial in-version I (see Fig. 1(a))35,39.
Here we use a differentcomplete set of elementary transformations
for the spacegroup symmetries that involve two translations, T1
andT2, one two-fold rotation, C2, and one more operation,S6 (see
the definition in Fig. 1(b)). It is ready to con-firm I = S36 , C3
= S
26 with the definition S6 ≡ C−13 I. The
multiplication rules of this symmetry group is given as
T−11 T2T1T−12 = T
−11 T
−12 T1T2 = 1, (1)
C−12 T1C2T−12 = C
−12 T2C2T
−11 = 1, (2)
S−16 T1S6T2 = S−16 T2S6T
−12 T
−11 = 1, (3)
(C2)2 = (S6)
6 = (S6C2)2 = 1. (4)
Due to the presence of time reversal inYbMgGaO4
34,36–38, we further supplement the symmetrygroup with the time
reversal T such that O−1T OT = 1and T 2 = 1, where O is a lattice
symmetry operation.
III. FERMIONIC PARTON CONSTRUCTION
To describe the U(1) QSL that we propose forYbMgGaO4, we
introduce the fermionic spinon opera-tor frα(α =↑, ↓) that carries
spin-1/2, and express theYb local moment as
Sr =1
2
∑α,β
f†rασαβfrβ , (5)
U(1) QSL WT1r WT2r W
C2r W
S6r
U1A00 I2×2 I2×2 I2×2 I2×2
U1A10 I2×2 I2×2 iσy I2×2
U1A01 I2×2 I2×2 I2×2 iσy
U1A11 I2×2 I2×2 iσy iσy
TABLE I. List of the gauge transformations for the four U1APSGs.
For the time reversal, all PSGs here have W Tr = I2×2.The last two
letters in the labels of the U(1) QSLs are extraquantum numbers in
the PSG classification48.
where σ = (σx, σy, σz) is a vector of Pauli matrices. Wefurther
impose a constraint
∑α f†rαfrα = 1 on each site
to project back to the physical Hilbert space of the spins.The
choice of fermionic spinons allows a local SU(2)gauge
freedom47.
As a direct consequence of the spin-orbital entangle-ment, the
spinon mean-field Hamiltonian for the U(1)QSL should generically
involve both spin-preserving andspin-flipping hoppings, and has the
following form
HMF = −∑(rr′)
∑αβ
[trr′,αβf
†rαfr′β + h.c.
], (6)
where trr′,αβ is the spin-dependent hopping. The choiceof the
mean-field ansatz in Eq. (6) breaks the local SU(2)gauge freedom
down to U(1). Here, to get a more com-pact form for Eq. (6), we
follow Ref. 49 and introduce theextended Nambu spinor
representation for the spinons
such that Ψr = (fr↑, f†r↓, fr↓,−f
†r↑)
T and
HMF = −1
2
∑(r,r′)
[Ψ†rurr′Ψr′ + h.c.
], (7)
where urr′ is a hopping matrix that is related to trr′,αβ .With
the extended Nambu spinor, the spin operator Srand the generatorGr
for the SU(2) gauge transformationare given by47,50–53
Sr =1
4Ψ†r(σ ⊗ I2×2)Ψr, (8)
Gr =1
4Ψ†r(I2×2 ⊗ σ)Ψr, (9)
where I2×2 is a 2× 2 identity matrix. Under the symme-try
operation O, Ψr transforms as
Ψr → UOGOO(r)ΨO(r) = GOO(r)UOΨO(r), (10)
where GOO(r) is the local gauge transformation that cor-responds
to the symmetry operation O, and we add aspin rotation UO because
the spin components are trans-formed when O involves a rotation. In
Eq. (10), thegauge transformation and the spin rotation are
commu-tative54 simply because [Sµr , G
νr] = 0. Moreover, from
Eq. (9), the gauge transformation GOr is block diagonalwith GOr
= I2×2 ⊗WOr , where WOr is a 2× 2 matrix (seeAppendix).
-
3
IV. PROJECTIVE SYMMETRY GROUPCLASSIFICATION
For the spinon mean-field Hamiltonian in Eq. (6), thelattice
symmetries are realized projectively and form theprojective
symmetry group (PSG). To respect the lat-tice symmetry
transformation O, the mean-field ansatzshould satisfy
urr′ = GO†O(r)U
†OuO(r)O(r′)UOG
OO(r′). (11)
The ansatz itself is invariant under the so-called
invariantgauge group (IGG) with urr′ = G1†r urr′G1r′ . The IGGcan
be regarded as a set of gauge transformations thatcorrespond to the
identity transformation. For an U(1)QSL, IGG = U(1).
A general group relation O1O2O3O4 = 1 for the latticesymmetry
turns into the following group relation for thePSG
UO1GO1r UO2G
O2O2O3O4(r)UO3G
O3O3O4(r)UO4G
O4O4(r)
= UO1UO2UO3UO4GO1r G
O2O2O3O4(r)G
O3O3O4(r)G
O4O4(r)(12)
∈ IGG, (13)
where we used the fact that the gauge transformationcommutes
with the spin rotation. As the series of rota-tions O1O2O3O4 either
rotate the spinons by 0 or 2π,
UO1UO2UO3UO4 = ±I4×4, (14)
where I4×4 is a 4× 4 identity matrix. Since{±I4×4} ⊂ IGG,
then
GO1r GO2O2O3O4(r)G
O3O3O4(r)G
O4O4(r) ∈ IGG. (15)
This immediately indicates that, to classify the PSGs fora
spin-orbit-coupled Mott insulator, we only need to fo-cus on the
gauge part, first find the gauge transformationwith the same
procedures as those for the conventionalMott insulators with
spin-only moments47, and then ac-count for the spin rotation.
For the mean-field ansatz in HMF, we choose the“canonical gauge”
for the IGG with
IGG = {I2×2 ⊗ eiφσz
|φ ∈ [0, 2π)}. (16)
Under the canonical gauge, the gauge transformation as-sociated
with the symmetry operation O takes the formof
GOr = I2×2 ⊗WOr≡ I2×2 ⊗
[(iσx)nOeiφO[r]σ
z], (17)
where nO = 0, 1. For translations, one can always choosea gauge
such that
WT1r = (iσx)n1 , (18)
WT2r = (iσx)n2eiφ2[x,y]σ
z
(19)
with n1, n2 = 0, 1 and φ2[0, y] = 0. The group rela-tion in Eq.
(3) further demands n1 = n2 = 0. Thus the
FIG. 2. (a,b,c) The mean-field spinon bands along the
high-symmetry momentum lines (see (d)) of the U1A00, U1A01and U1A11
states, where t1, t
′1 and t2 are hoppings in their
spinon mean-field Hamiltonians (see Appendix). The Diraccones
are highlighted in dashed circles. The dashed line refersto the
Fermi level. (d) The Brioullin zone of the triangularlattice.
group relation in Eq. (1) gives WT1r = 1,WT2r = e
ixφ1σz
,where φ1 is the flux through each unit cell of the triangu-lar
lattice and takes the value of 0 or π (see Appendix).The PSGs with
φ1 = 0 (π) are labeled by U1A (U1B).Among the sixteen algebraic
PSGs that we find, eight un-physical solutions have T 2 = 1 for the
spinons and givevanishing spinon hoppings everywhere. In Tab. I and
theAppendix, we list the remaining eight PSGs that haveT 2 = −1
consistent with the fact that fermionic spinonsare Kramers doublets
(see Appendix).
V. MEAN-FIELD STATES
Here we obtain the spinon mean-field Hamiltonianfrom Tab. I and
explain why the U1A00 state standsout as the candidate ground state
for YbMgGaO4. Westart with the U1A states. Among the four U1A
states,the U1A10 state gives a vanishing mean-field Hamilto-nian
for the spinon hoppings between the first and thesecond neighbors,
the remaining ones except the U1A00state all have symmetry
protected band touchings at thespinon Fermi level (see Fig. 2). To
illustrate the idea55,we consider the U1A01 state where the spinon
Hamilto-
nian has the form HU1A01MF =∑
k hαβ(k)f†kαfkβ in the
momentum space and h(k) is a 2× 2 matrix with
h(k) = d0(k)I2×2 +
3∑µ=1
dµ(k)σµ. (20)
For this band structure there are nondegenerate bandtouchings at
Γ, M and K points that are protected bythe PSG of the U1A01 state.
Under the operation S6,
-
4
the PSG demands that spinons to transform as
fk↑ → −e−iπ/3f†−S−16 k,↓
, (21)
fk↓ → eiπ/3f†−S−16 k,↑
. (22)
Applying S6 three times and keeping HMF invariant, werequire
h(k) = −[σyh(k)σy]T (23)
which forces d0(k) = 0. The time reversal sym-metry (T = iσy ⊗
I2×2K) further requires thatdµ(k) = −dµ(−k). Thus we have symmetry
pro-tected band touchings with h(k) = 0 at the time
reversalinvariant momenta Γ and M. The K points are invariantunder
C2 and S6 because the spinon partile-hole trans-formation is
involved for S6 (see Appendix). Using thosetwo symmetries, we
further establish the band touchingat the K points. Likewise, for
the U1A11 state, thePSG demands the band touchings at Γ and M
points.Because there are only two spinon bands for the U1Astates,
these band touchings generically occur at thespinon Fermi
level.
Due to the Dirac band touchings at the Fermi level,
thelow-energy dynamic spin structure factor, that measuresthe
spinon particle-hole continuum, is concentrated at afew discrete
momenta that correspond to the intra-Dirac-cone and the
inter-Dirac-cone scatterings36. Clearly, thisis inconsistent with
the recent INS result that observes abroad continuum covering a
rather large portion of theBrillouin zone36,37.
For the U1B states, the spinons experience a π back-ground flux
in each unit cell. The direct consequence ofthe π background flux
is that the U1B states support anenhanced periodicty of the dynamic
spin structure in theBrillouin zone47,56,57. Such an enhanced
periodicity isabsent in the INS result36,37. In particular, unlike
whatone would expect for an enhanced periodicity, the spec-tral
intensity at the Γ point is drastically different fromthe one at
the M point in the existing experiments36,37.
The above analysis leads to the conclusion that theU1A00 state
is the most promising candidate U(1) QSLfor YbMgGaO4, and this
conclusion is independent fromany microscopic model. The spinon
mean field Hamilto-nian, allowed by the U1A00 PSG, is remarkably
simpleand is given as58
HU1A00MF = −t1∑〈rr′〉,α
f†rαfrα − t2∑
〈〈rr′〉〉,α
f†rαfrα, (24)
where the spinon hopping is isotropic for the first andthe
second neighbors. This mean-field state only has asingle band that
is 1/2-filled, so it has a large spinonFermi surface. From HU1A00MF
, we construct the mean-field ground state by filling the spinon
Fermi sea,
|ΨU1A00MF 〉 =∏�k
-
5
FIG. 3. (a) S(q, ω) along the high-symmetry momentumlines from
HU1A00MF with t2 = 0.2t1. The spinon bandwidthB = 9.6t1. (b) The
RPA corrected SRPA(q, ω) along the highsymmetry momentum lines. We
have set the parameters inthe spin model to be J±/Jzz = 0.915,
J±±/Jzz = 0.35, andJz±/Jzz = 0.2. The ratio Jzz/t1 is obtained from
Refs. 34and 36 and fixed to be 1.0 for concreteness.
in the momentum space. We find in Fig. 3(b) that, thelow-energy
spectral weight at M is slightly enhanced, afeature observed in
Refs. 36 and 37. From our choiceof the parameters, it is plausible
that this peak resultsfrom the proximity to a phase with a
stripe-like magneticorder35,36,39.
VII. DISCUSSION
We have demonstrated that the spinon Fermi surfaceU(1) QSL gives
a consistent explanation of the INS re-sult in YbMgGaO4. Moreover,
the anisotropic spin in-teraction, slightly enhances the spectral
weight at the Mpoints. The U(1) gauge fluctuation in the spinon
Fermisurface U(1) QSL44,45 was suggested to be the cause forthe
sublinear temperature dependence of the heat capac-ity in
YbMgGaO4
35,36,39,46.In YbMgGaO4, the coupling between the Yb moments
is relatively weak34. It is feasible to fully polarize the
spinwith experimentally accessible magnetic fields35,37,39,61
and to study the evolution of the magnetic properties un-der the
magnetic field. Recently, two of us have predictedthe spectral
weight shift of the INS for YbMgGaO4 undera weak magnetic field41,
and the predicted spectral cross-ing at the Γ point and the
dispersion of the spinon con-tinuum have actually been confirmed in
the recent INSmeasurement62. Numerically, it is useful to perform
nu-merical calculation with fixed J± and Jzz that are closeto the
ones for YbMgGaO4, and obtain the phase dia-gram of our spin model
by varying J±± and Jz±
35,39,63.More care needs to be paid to the disordered region of
themean-field phase diagram35 where quantum fluctuationis found to
be strong35. The “2kF” oscillation in the spincorrelation would be
the strong indication of the spinonFermi surface. Noteworthily
recent DMRG works64,65
have actually provided some useful information aboutthe ground
states of the system, in particular, Ref. 65suggested the scenario
of exchange disorders. Certainamount of exchange disorder may be
created by the crys-tal electric field disorder that stems from the
Mg/Ga
mixing in the non-magnetic layers37,61, but recent polar-ized
neutron scattering measurement did not find strongexchange
disorder59. Regardless of the possibilities ofexchange disorders,
the spin quantum number fraction-alization, that is one of the key
properties of the QSLs,could survive even with weak disorders. The
approachand results in our present work are phenomenologicallybased
and are independent of the microscopic mechanismfor the possible
QSL ground state in YbMgGaO4.
Ref. 43 claimed the absence of the magnetic thermalconductivity
in YbMgGaO4 by extrapolating the low-temperature thermal
conductitivity data in the zero mag-netic field. Here, we provide
an alternative understand-ing for this thermal transport result.
The hint lies inthe field dependence of the thermal conductivity.
It wasfound that, when strong magnetic fields are applied
toYbMgGaO4, the thermal conductivity κxx/T at 0.2K isincreased
compared with the one at zero field43. If one ig-nores the disorder
effect and assumes the zero-field ther-mal conductivity is a simple
addition of the magneticcontribution and the phonon contribution
with
κxx = κspin,xx + κphonon,xx, (29)
the strong magnetic field almost polarizes the spins com-pletely
and creates a spin gap for the magnon excitation,hence suppress the
magnetic contribution. The high-field thermal conductivity would be
purely given by thephonon contribution, and we would expect a
decreasingof the thermal conductivity in the strong field
comparedto the zero field result. This is clearly inconsistent
withthe experimental result. Therefore, the zero-field ther-mal
conductivity is not a simple addition of the magneticcontribution
and the phonon contribution, i.e.,
κxx 6= κspin,xx + κphonon,xx. (30)
This also strongly suggests the presence rather than theabsence
of magnetic excitations in the thermal conductiv-ity result at zero
magnetic field. If there is no magneticexcitation in the system at
low temperatures, the low-temperature thermal conductivity at zero
field shouldjust be the phonon contribution, and we would expectthe
zero-field thermal conductivity to be the same as theone in the
strong field limit, (although the intermediatefield regime could be
different). This is again inconsis-tent with the experiments. This
means that the magneticexcitation certainly does not have a large
gap and couldjust be gapless as we propose from the spinon Fermi
sur-face state. In fact, the gapless nature of the
magneticexcitation is consistent with the power-law heat capac-ity
results in YbMgGaO4. What suppresses κxx couldarise from the mutual
scattering between the magneticexcitations and the the phonons. In
fact, similar field de-pendence of thermal conductivity κxx has
been observedin other rare-earth systems such as Tb2Ti2O7
66–68 andPr2Zr2O7
69. It was suggested there67–69 that the spin-phonon scattering
is the cause. The Yb local moment,that is a spin-orbit-entangled
object, involves the orbital
-
6
degree of freedom. The orbital degree of freedom is sen-sitive
to the ion position, and thus couples to the phononstrongly. This
is probably the microscopic origin for thestrong coupling between
the magnetic moments and thephonons in the rare-earth magnets. This
is quite dif-ferent from the organic spin liquid candidates and
theherbertsmithite kagome system where the orbital degreeof freedom
does not seem to be involved70–73.
If the ground state of YbMgGaO4 is a QSL with thespinon Fermi
surface, the field-driven transition from theQSL ground state to
the fully polarized state is neces-sarily a unconventional
transition beyond the traditionalLandau’s paradigm and has not been
studied in the pre-vious spin liquid candidates70–73. The smooth
growth ofthe magnetization with varying external fields indicatesa
continuous transition34. Since we propose YbMgGaO4to be a spinon
Fermi surface U(1) QSL and gapless, thetransition would be
associated with the openning of thespin gap at the critical field.
The continuous nature ofthe transition suggests the spin gap to
open in a contin-uous manner. Moreover, the spinon confinement
wouldbe concomitant with the spin gap that suppresses thespinon
density of states and allows the instanton eventsof the U(1) gauge
field to proliferate. Therefore, it mightbe interest to identify
the critical field and obtain thecritical properties of the
field-driven transition. Thermo-dynamic, spectroscopic, and thermal
transport measure-ments with finer field variation would be
helpful.
Finally, several families of rare-earth triangular
latticemagnets have been discovered recently35,39,74–79.
Theirproperties have not been studied carefully. Our
generalclassification results and the prediction of the
spectro-scopic properties would apply to the QSL candidates thatmay
emerge in these families of materials. It is certainlyexciting if
one finds the new QSL candidates in thesefamilies behave like
YbMgGaO4
35.
VIII. ACKNOWLEDGEMENTS
We thank one anonymous referee for the suggestion forimprovement
to this paper, and Zhu-Xi Luo for point-ing out some typos. G.C.
acknowledges the discussionwith Xuefeng Sun from USTC and Yuji
Matsuda aboutthermal transports in rare-earth magnets, and the
dis-cussion with Professor Sasha Chernyshev about the re-lated
matters. This work is supported by the Min-istry of Science and
Technology of China with the GrantNo.2016YFA0301001 (G.C.), the
Start-Up Funds of OSU(Y.M.L.) and Fudan University (G.C.), the
National Sci-ence Foundation under Grant No. NSF PHY-1125915(Y.M.L
and G.C.), the Thousand-Youth-Talent Program(G.C.) of China, and
the first-class university construc-tion program of Fudan
University.
Appendix A: The coordinate System and spacegroup symmetry
Following our convention in Fig. 1 in the main text, wechoose
the coordinate system of the triangular lattice tobe
a1 = (1, 0), (A1)
a2 = (−1
2,
√3
2). (A2)
We label the triangular lattice sites by r = xa1 +y
a2.Restricted to the triangular layer, the space group con-tains
two translations T1 along the a1 direction, T2 alongthe a2
direction, a counterclockwise three-fold rotationC3 around the
lattice site, a two-fold rotation C2 arounda1 + a2, and the
inversion I at the lattice site. Theiractions on the lattice
indices are
T1 : (x, y)→ (x+ 1, y), (A3)T2 : (x, y)→ (x, y + 1), (A4)C3 :
(x, y)→ (−y, x− y), (A5)C2 : (x, y)→ (y, x), (A6)I : (x, y)→
(−x,−y). (A7)
In the formulation introduced in the main text, weconsider an
equivalent set of generators, {T1, T2, C2, S6},where the operation
S6 is defined as S6 ≡ C−13 I and actson the lattice indices as
S6 : (x, y)→ (x− y, x). (A8)
It is evident that these two sets of generators are equiva-lent,
since we merely redefine the symmetry rather thanintroducing any
new symmetry.
The multiplication rule of this symmetry group is givenin the
main text. For the convenience of the presentationbelow, we also
list these rules here,
T−11 T2T1T−12 = T
−11 T
−12 T1T2 = 1, (A9)
C−12 T1C2T−12 = C
−12 T2C2T
−11 = 1, (A10)
S−16 T1C6T2 = S−16 T2C6T
−12 T
−11 = 1, (A11)
(C2)2 = (C6)
6 = (S6C2)2 = 1. (A12)
Including the time reversal symmetry, we further have
T−11 T T1T = T−12 T T2T = 1, (A13)
C−12 T C2T = S−16 T S6T = 1, (A14)T 2 = 1. (A15)
Appendix B: Projective symmetry groupclassification
As we describe in the main text, we consider the U(1)QSL. The
spinon mean-field Hamiltonian has the follow-ing form
HMF = −∑(rr′)
∑αβ
[trr′,αβf
†rαfr′β + h.c.
], (B1)
-
7
U(1) QSL WT1r WT2r W
C2r W
S6r
U1A00 I2×2 I2×2 I2×2 I2×2
U1A10 I2×2 I2×2 iσy I2×2
U1A01 I2×2 I2×2 I2×2 iσy
U1A11 I2×2 I2×2 iσy iσy
U1B00 I2×2 (−1)xI2×2 (−1)xyI2×2 (−1)xy−y(y−1)
2 I2×2
U1B10 I2×2 (−1)xI2×2 iσy(−1)xy (−1)xy−y(y−1)
2 I2×2
U1B01 I2×2 (−1)xI2×2 (−1)xyI2×2 iσy(−1)xy−y(y−1)
2
U1B11 I2×2 (−1)xI2×2 iσy(−1)xy iσy(−1)xy−y(y−1)
2
TABLE II. List of the gauge transformations for the sym-metry
operations of the eight U(1) PSGs, where (x, y) is thecoordinate in
the oblique coordinate system. For time rever-sal symmetry, all
PSGs have the same gauge transformationW Tr = I2×2.
where trr′,αβ is the spin-dependent hopping. Withthe extended
Nambu spinor representation49 Ψr =
(fr↑, f†r↓, fr↓,−f
†r↑)
T , HMF has a more compact form
HMF = −1
2
∑(r,r′)
[Ψ†rurr′Ψr′ + h.c.
], (B2)
where urr′ is a hopping matrix that is related to trr′,αβ ,
urr′ =
trr′,↑↑ 0 trr′,↑↓ 0
0 −t∗rr′,↓↓ 0 t∗rr′,↓↑trr′,↓↑ 0 trr′,↓↓ 0
0 t∗rr′,↑↓ 0 −t∗rr′,↑↑
. (B3)
1. Spatial symmetry
First of all, the gauge transformation and spin rotationare
commutative. So in the PSG classification, we onlyneed to focus on
the gauge part of the PSG transforma-tion. In the canonical gauge
IGG = {I2×2 ⊗ eiφσ
z |φ ∈[0, 2π)}, the gauge transformation associated with a
givensymmetry operation O takes the form
GOr = I2×2 ⊗WOr ≡ I2×2 ⊗[(iσx)nOeiφO[r]σ
z], (B4)
where nO = 0, 1. For the symmetry multiplication ruleO1O2O3O4 =
1 where Oi is an unitary transformation,the corresponding PSG
relation becomes
GO1r GO2O2O3O4(r)G
O3O3O4(r)G
O4O4(r) ∈ IGG (B5)
or equivalently,
WO1r WO2O2O3O4(r)W
O3O3O4(r)W
O4O4(r)
∈ {eiφσz
|φ ∈ [0, 2π)}. (B6)
We start with T1 and T2, where
WT1r = (iσx)nT1 , (B7)
WT2r = (iσx)nT2 eiφT2 [r]σ
z
. (B8)
Through Eq. (A10) that connects T1 and T2, one imme-diately has
nT1 = nT2 . From Eq. (A11) where the to-tal number of T1 and T2 is
odd, one immediately hasnT1 = nT2 = 0. So we have
WT1r = 1, (B9)
WT2r = eiφT2 [x,y]σ
z
. (B10)
Using Eq. (A9), we have
[WT1T1]−1[WT2T2][W
T1T1][WT2T2]
−1
= T−11 (WT1)−1WT2T2W
T1T1T−12 W
−1T2
∈ {eiφσz
|φ ∈ [0, 2π)}, (B11)
which leads to the result
φT2 [x+ 1, y]− φT2 [x, y] ≡ φ1 (B12)
with φ1 to be determined. Since it is always possibleto choose a
gauge such that φT2 [0, y] = 0, then we haveφT2 [x, y] = φ1x.
Similarly, T−11 T−12 T1T2 = 1 leads to
φT2 [x+ 1, y + 1]− φT2 [x, y + 1] = φ2. (B13)
It is ready to find φ2 = φ1.We continue to find WS6r and W
C2r . For the operation
S6 with WS6r = (iσ
x)nS6 eiφS6 [x,y]σz
, Eq. (A11) leads to
−φS6 [T1(r)] + φS6 [r] = −φ1y + φ3, (B14)−φS6 [T2(r)] + φS6 [r]
= φ4 − φ1x+ φ1y, (B15)
for nS6 = 0, and
−φS6 [T1(r)] + φS6 [r] = −φ1y + φ3 (B16)−φS6 [T2(r)] + φS6 [r] =
φ4 + φ1x+ φ1y. (B17)
for nS6 = 1. So we obtain
when nS6 = 0,
φS6 [r] = φ1xy − φ3x− φ4y −φ1y(y − 1)
2(B18)
when nS6 = 1,
φS6 [r] = φ1xy − φ3x− φ4y −φ1y(y − 1)
2. (B19)
For nS6 = 1, we further require φ1 = 0, π. S66 = 1 is
automatically satisfied with the above relations for bothnS6 = 0
and nS6 = 1.
For WC2r with WC2r = (iσ
x)nC2 eiφC2 [x,y]σz
, we need toconsider two separate cases with nc2 = 0, 1,
respectively.If nC2 = 0, Eq. (A10) leads to
−φT2 [C−12 T1(r)]− φC2 [T1(r)] + φC2 [r] = φ5, (B20)−φC2 [T2(r)]
+ φT2 [T2(r)] + φC2 [r] = φ6. (B21)
-
8
So we obtain φC2 [x, y] = −φ5x−φ6y−xyφ1 and φ1 = 0, πfor nC2 =
0. Similary, for nC2 = 1, we obtain φC2 [x, y] =−φ5x− φ6y −
xyφ1.
Using C22 = 1, we further have φ6 = −φ5 for nC2 = 0,and φ6 = φ6
for nC2 = 1. So we arrive at the result
nC2 = 0, φC2 [x, y] = −φ5(x− y)− xyφ1, (B22)nC2 = 1, φC2 [x, y]
= −φ5(x+ y)− xyφ1. (B23)
Here, to simplify the above expression, we choose a puregauge
tranformation W̃ ar = e
ixσzφ5 . Under the puregauge transformation, the gauge part of
the PSG trans-forms as
WOr → W̃ arWOr W̃a†O−1(r). (B24)
Clearly W̃ ar only modifies WT1 and WT2 by an overall
phase shift, but WC2r becomes
WC2r = (iσx)nC2 e−ixyφ1σ
z
(B25)
for both nC2 = 0, 1, except that we require φ1 = 0, π fornC2 =
0.
For the relation (S6C2)2 = 1, we need to consider the
four cases with nS6 = 0, 1 and nC2 = 0, 1.For nS6 = nC2 = 0, we
have φ1 = π, and (S6C2)
2 = 1gives φ3 + 2φ4 = 0. We then introduce a pure
gaugetransformation W̃ br ,
W̃ br = e−i(x+y)φ4σz . (B26)
After applying W̃ br , we have
φC2 = −xyφ1, (B27)
φS6 = xyφ1 − φ1y(y − 1)
2(B28)
with φ1 = 0, π.For nS6 = 0 and nC2 = 1, we obtain φ3 = 0. We
introduce a pure gauge transformation W̃ cr ,
W̃ cr = e−i(x−y)φ4σz . (B29)
After applying W̃ br , we have
φC2 = −xyφ1, (B30)
φS6 = xyφ1 − φ1y(y − 1)
2. (B31)
For nS6 = 1 and nC2 = 0, we obtain φ3 = 0. We apply
a pure gauge transformation W̃ br and obtain
φC2 = −xyφ1, (B32)
φS6 = xyφ1 − φ1y(y − 1)
2. (B33)
For nS6 = 1 and nC2 = 1, we obtain φ3 + 2φ4 = 0. We
apply a pure gauge transformation W̃ cr and obtain
φC2 = −xyφ1, (B34)
φS6 = xyφ1 − φ1y(y − 1)
2. (B35)
In summary, we have
WT1r = 1, WT2r = e
iφ1x. (B36)
and
WC2r = (iσx)nC2 e−iφ1xyσ
z
, (B37)
WS6r = (iσx)nS6 eiφ1[xy−
y(y−1)2 ]σ
z
, (B38)
where φ1 = 0, π for nC2 = 0 or nS6 = 1.
2. Time reversal symmetry
Because time reversal is an antiunitary symmetry, theproduct
O−1T −1OT becomes
(WOr )†[(W Tr )
†WOr WTO−1(r)]
∗ (B39)
for the PSGs, where W T is the gauge transformationassociated
with the time reversal. We here redefine
W Tr = W̄Tr (iσ
y), (B40)
so that
O−1T −1OT → (WOr )†(W̄ Tr )†WOr W̄ TO−1(r). (B41)
W̄ Tr has the general form W̄Tr = (iσ
x)nT eiφT [r]σz
.We start with nT = 0. The relation in Eq. (A13) leads
to
φT [x, y]− φT [x− 1, y] = −φ7, (B42)φT [x, y + 1]− φT [x, y] =
−φ8, (B43)
so we have φT [x, y] = −φ7x − φ8y. Applying this resultto Eq.
(A14), we have
−φC2 [y, x]− φT [y, x] + φC2 [y, x]+φT [x, y] = φ9,
−φS6 [x, y]− φT [x, y] + φS6 [x, y]+φT [y,−x+ y] = φ10,
(B44)
for nC2 = nS6 = 0. The above equations giveφ7 = φ8 = 0, so we
have W̄
Tr = 1. Other cases can be
obtained likewise. We find that for both nT = 0 andnT = 1, there
is φT [x, y] = 0 and φ1 = 0, π. So we have
W̄ Tr = 1, iσy, (B45)
where we have used a global and uniform rotation eiπ4 σ
z
to rotate σx to the basis of σy.Including the time reversal,
there are 16 PSG solutions.
But for W̄ Tr = 1, the mean-field ansatz is found to
vanisheverythere. This makes sense as these PSGs have T 2 = 1for
the fermionic spinons that are expected to Kramersdoublets. So only
8 of them with T 2 = −1 for the spinonssurvive. Replacing eiφ1σ
z
with ±1, we present the PSGsolutions in the table of the main
text.
-
9
Appendix C: Spinon band structures and
mean-fieldHamiltonians
As we establish in the previous section and the maintext, there
are four U1A PSGs and four U1B PSGs. Inthe main text, we have
argued that the experimental re-suls in YbMgGaO4 is against the U1B
states. So herewe focus on the U1A states. From the U1A PSGs, it
isstraight to obtain the spinon transformations. We listthe results
in Tab. III.
1. Spinon band structures
Using Tab. III, we obtain the spinon mean-field Hamil-tonian. In
particular, the U1A10 state gives vanishingspinon hoppings on the
first and second neighbors, andthe U1A01 state gives an isotropic
spinon hopping onboth first and second neighbors. The U1A10 state,
aswe described in the main text, has symmetry protectedband
touchings at the Γ, M and K points. The U1A11state has symmetry
protected band touchings at the Γand M points.
For the U1A10 state, the spinon mean-field Hamilto-nian has the
form
HU1A01MF =∑k
hαβ(k)f†kαfkβ , (C1)
where hαβ(k) is given by
h(k) = d0(k)I2×2 +
3∑µ=1
dµ(k)σµ. (C2)
In the main text, we have used (S6)3 and T to show
d0(k) = 0 and the band touchings at Γ and M. To accountfor the
band touching at the K point, we need to use S6and C2. Under
S6,
S6HS−16 =∑k
[ei2π3 h(−S−16 (k))↑↓f
†k↑fk↓ + h.c.
]= H, (C3)
where h(k)↑↓ = dx(k)− idy(k). Since K is invariant un-der
S6,
dx(K)− idy(K) = ei2π3 [dx(K)− idy(K)], (C4)
hence dx(K) = dy(K) = 0.The C2 symmetry constraints the dz term,
we have
C2HC−12 =
∑k
dz(C−12 (k))f
†k↓fk↓ − dz(C
−12 (k))f
†k↑fk↑
= H. (C5)
Since K is also invariant under C2, we obtain dz(K) =−dz(K).
Hence dz(K) = 0. We conclude that h(K) = 0and there exists a band
touching at K.
For the U1A11 state, T and S6 are implemented in thesame way as
the U1A01 state, and we arrive at the sameconclusion that there are
band touchings at the Γ and Mpoints. At the K point, however, the
band structure isgenerally gapped due to a nonzero dz.
2. Spinon mean-field Hamiltonians
The U1A00 state has the isotropic spinon hoppings onfirst and
second neighboring bonds, and the mean-fieldHamiltonain HU1A00MF
has already been given in the maintext. This states gives a large
spinon Fermi surface inthe Brioullin zone. The spinon mean-field
states of theU1A01 state and the U1A11 state are given by
HU1A01MF =∑x,y
t1
[− if†(x+1,y),↑f(x,y),↓ − if
†(x+1,y),↓f(x,y),↑ − e
− iπ6 f†(x,y+1),↑f(x,y),↓
+eiπ6 f†(x,y+1),↓f(x,y),↑ − e
iπ6 f†(x+1,y+1),↑f(x,y),↓ + e
− iπ6 f†(x+1,y+1),↓f(x,y),↑ + h.c.]
+t2
[ei2π3 f†(x+1,y−1),↑f(x,y),↓ + e
iπ3 f†(x+1,y−1),↓f(x,y),↑ + f
†(x+1,y+2),↑f(x,y),↓
−f†(x+1,y+2),↓f(x,y),↑ + eiπ3 f†(x+2,y+1),↑f(x,y),↓ + e
i2π3 f†(x+2,y+1),↓f(x,y),↑ + h.c.
], (C6)
-
10
TABLE III. The transformation for the spinons under four U1A
PSGs that are labeled by U1AnC2nS6 .
U(1) PSGs T1 T2 C2 S6
U1A00f(x,y),↑ → f(x+1,y),↑f(x,y),↓ → f(x+1,y),↓
f(x,y),↑ → f(x,y+1),↑f(x,y),↓ → f(x,y+1),↓
f(x,y),↑ → eiπ6 f(y,x),↓
f(x,y),↓ → ei5π6 f(y,x),↑
f(x,y),↑ → e−iπ3 f(x−y,x),↑
f(x,y),↓ → e+iπ3 f(x−y,x),↓
U1A10f(x,y),↑ → f(x+1,y),↑f(x,y),↓ → f(x+1,y),↓
f(x,y),↑ → f(x,y+1),↑f(x,y),↓ → f(x,y+1),↓
f(x,y),↑ → eiπ6 f†(y,x),↑
f(x,y),↓ → e−iπ6 f†(y,x),↓
f(x,y),↑ → e−iπ3 f(x−y,x),↑
f(x,y),↓ → e+iπ3 f(x−y,x),↓
U1A01f(x,y),↑ → f(x+1,y),↑f(x,y),↓ → f(x+1,y),↓
f(x,y),↑ → f(x,y+1),↑f(x,y),↓ → f(x,y+1),↓
f(x,y),↑ → eiπ6 f(y,x),↓
f(x,y),↓ → ei5π6 f(y,x),↑
f(x,y),↑ → −e−iπ3 f†(x−y,x),↓
f(x,y),↓ → e+iπ3 f†(x−y,x),↑
U1A11f(x,y),↑ → f(x+1,y),↑f(x,y),↓ → f(x+1,y),↓
f(x,y),↑ → f(x,y+1),↑f(x,y),↓ → f(x,y+1),↓
f(x,y),↑ → eiπ6 f†(y,x),↑
f(x,y),↓ → e−iπ6 f†(y,x),↓
f(x,y),↑ → −e−iπ3 f†(x−y,x),↓
f(x,y),↓ → e+iπ3 f†(x−y,x),↑
and
HU1A11MF =∑x,y
t1
[if†(x+1,y),↑f(x,y),↑ − if
†(x+1,y),↓f(x,y),↓ + if
†(x,y+1),↑f(x,y),↑
−if†(x,y+1),↓f(x,y),↓ − if†(x+1,y+1),↑f(x,y),↑ + if
†(x+1,y+1),↓f(x,y),↓ + h.c.
]+t′1
[− f†(x+1,y),↑f(x,y),↓ + f
†(x+1,y),↓f(x,y),↑ + e
iπ3 f†(x,y+1),↑f(x,y),↓
+ei2π3 f†(x,y+1),↓f(x,y),↑ + e
i2π3 f†(x+1,y+1),↑f(x,y),↓ + e
iπ3 f†(x+1,y+1),↓f(x,y),↑ + h.c.
]+t2
[eiπ6 f†(x+1,y−1),↑f(x,y),↓ + e
i5π6 f†(x+1,y−1),↓f(x,y),↑ − if
†(x+1,y+2),↑f(x,y),↓
−if†(x+1,y+2),↓f(x,y),↑ + ei5π6 f†(x−2,y−1),↑f(x,y),↓ + e
iπ6 f†(x−2,y−1),↓f(x,y),↑ + h.c.
], (C7)
where in both Hamiltonians t1, t′1 denote the first neigh-
bor hoppings and t2 denotes the second neighbor hop-ping.
The band structures for specific choices of the
hoppingparameters are plotted in the main text. Clearly, weobserve
the band touchings at the Γ, M and K points forthe U1A01 state, and
band touchings at the Γ and Mpoints for the U1A11 state.
Appendix D: The U1A00 state and thespectroscopic results
1. Free spinon mean-field theory
The spinon mean-field Hamiltonian of the U1A00 stateis
HU1A00MF = −t1∑〈rr′〉,α
f†rαfrα − t2∑
〈〈rr′〉〉,α
f†rαfrα, (D1)
from which we compute the dynamic spin structure factorfor
different choices t2/t1. The dynamic spin structure
factor is given by
S(q, ω) = 1N
∑r,r′
eiq·(r−r′)
∫dt e−iωt
〈ΨU1A00MF |S−r (t)S+r′(0)|ΨU1A00MF 〉
=∑n
δ(ω − ξnq) |〈n|S+q |ΨU1A00MF 〉|2, (D2)
where N is the total number of spins, the summationis over all
mean-field states with the spinon particle-holeexcitation, ξnq is
the energy of the n-th excited state withthe momentum q. The
results are depicted in Fig. 4(a-e)and are consistent with the
inelastic neutron scatteringresults36,37. All the results so far
are independent fromany microscopic spin interaction.
-
11
FIG. 4. (a-e) Dynamic spin structure factor for the free spinon
theory of the U1A00 state with different values of t2/t1. (f-h)The
evolution of SRPA(q, ω) as a function of J±±. In all subfigures,
the energy transfer is normalized against the
correspondingbandwidth B. The parameter α is defined as Jzz/t1.
2. Variational calculation and random phaseapproximation
Here we consider the microscopic spin Hamiltonianthat was
introduced in Refs. 34 and 35,
Hspin =∑〈rr′〉
JzzSzrS
zr′ + J±(S
+r S−r′ + S
−r S
+r′)
+J±±(γrr′S+r S
+r′ + γ
∗rr′S
−r S−r′)
− i2Jz±[(γ
∗rr′S
+r − γrr′S−r )Szr′
+Szr(γ∗rr′S
+r′ − γrr′S
−r′)], (D3)
where γrr′ = 1, ei2π/3, e−i2π/3 for rr′ along the a1,a2
and a3 bonds, respectively. Here, a3 = −a1 − a2. Itwas suggested
and demonstrated that the anisotropicJ±± and Jz± interactions
compete with the XXZ part ofthe Hamiltonian and may lead to
disordered state34,35,39.Our calculation does show the enhancement
of quantumfluctuation in certain regions of the phase
diagram35.Here we comment about the choices of the exchange
cou-plings in the main text and in the following calculation.The
Jzz and J± couplings can be determined by the
Curie-Weiss temperature measurement on a single crys-tal sample.
The complication comes from the subtractionof the Van Vleck
susceptibility. Due to the Ga3+/Mg2+
exchange disorder in the non-magnetic layers, althoughthese ions
do not directly enter the Yb exchange path,it may modify the local
crystal field environment of theYb3+ ion and thus lead to some
complication and varia-tion of the Van Vleck susceptibility. As a
result, the veryprecise determination of the Jzz and J± couplings
can bean issue. That may explain some differences of the Jzzand J±
couplings that were obtained
34–37,39. Partly forthe same reason, the results on J±± and Jz±
may alsobe affected. However, quantum spin liquid, if it exists
asthe ground state of our generic model, is expected to bea phase
that covers a finite region of the phase diagram.Therefore, the
very precise value of the couplings maynot be quite necessary from
this point of view. There-fore, we here rely on our previous
results of the quantumfluctuation for the mean-field phase diagram
that indi-cates strong fluctations in certain parameter regimes.
Wechoose the exchange parameters from these disordered
re-gions.
For this spin Hamiltonian, the mean-field variationalenergy is
given as
Evar = 〈ΨU1A00MF |Hspin|ΨU1A00MF 〉 =1
L2
∑q
〈ΨU1A00MF |Jzz(q)SzqSz−q + 2J±(q)S+q S−−q|ΨU1A00MF 〉
=1
L2
∑q
[Jzz(q)
∑n
∣∣〈n|Szq |ΨU1A00MF 〉∣∣2 + 2J±(q)∑n
∣∣〈n|S+q |ΨU1A00MF 〉∣∣2]
=1
L4
∑q
Jzz(q)4
∑n,k
∣∣∣〈n|f†k+q,↑fk,↑ − f†k+q,↓fk,↓|ΨU1A00MF 〉∣∣∣2 + 2J±(q)∑n,k
∣∣∣〈n|f†k+q,↑fk,↓|ΨU1A00MF 〉∣∣∣2 , (D4)
-
12
where we have omitted J±± and Jz± because they do notconserve
spin, therefore their contribution to Evar is zero.This is an
artifact of the free spinon theory of HU1A00MFthat only includes
isotropic spinon hoppings for the firsttwo neighbors.
Due to the isotropic spinon hoppings, HU1A00MF does
notexplicitly reflect the absence of spin-rotational symmetrythat
is brought by the J±± and Jz± interactions. To in-corporate the J±±
and Jz± interactions, as we describein the main text, we followed
the phenomenological treat-ment for the “t-J” model in the context
of cuprate super-conductors60 and consider H = HU1A00MF + H
′spin, where
H ′spin are the J±± and Jz± interactions. In the parton
construction, H ′spin is treated as the spinon interactionsand
thus introduces the spin rotational symmetry break-ing. With a
random phase approximation for the inter-action H ′spin, we obtain
the dynamic spin susceptibility
60
χRPA(q, ω) =[1− χ0(q, ω)J (q)
]−1χ0(q, ω), (D5)
where χ0 is the free-spinon susceptibility, and J (q) is thespin
exchange matrix from H ′spin,
J (q) =2(uq − vq)J±± −2
√3wqJ±± −
√3wqJz±
−2√
3wqJ±± 2(−uq + vq)J±± (uq − vq) Jz±−√
3wqJz± (uq − vq) Jz± 0
(D6)with uq = cos(q · a1), vq = 12 (cos(q · a2) + cos(q ·
a3)),and wq =
12 (cos(q · a2)− cos(q · a3)). The renormalized
SRPA(q, ω) can be read off from χRPA via SRPA(q, ω) =− 1π Im
[χRPA(q, ω)
]+−and is plotted in Fig. 3(b) in the
main text.
The very precise values of J±± and Jz± cannot be de-termined
from the existing data-rich neutron scatteringexperiment in a
strong field normal to the triangularplane. This is partly due to
the experimental resolu-tion, and is also due to the fact that the
linear spin wavespectrum for the field normal to the plane is
indepen-dent of Jz± and is not quite sensitive to J±±
35,39. InFig. 3(b) of the main text, instead, we choose (J±±,
Jz±)to fall into the disordered region of the phase diagram inRef.
35 where the quantum fluctuations are expected tobe strong35.
Appendix E: The U1B states
In this section we use PSG to determine the free
spinonmean-field Hamiltonian for the U1B states to the first
andsecond spinon hoppings. In Fig. 5, we further presenttheir
spectroscopic features for comparison. Like thenotation for U1As,
the U1B states are also labeled byU1BnC2nS6 .
1. The U1B00 state
For the π-flux states, the dynamic spin structure fac-tor has an
enhanced periodicity due to anticommutativelattice translations.
One direct consequence of the pe-riodicity is that Γ and M become
equivalent, and theV-shaped upper excitation edge in Ref. 36 cannot
be re-produced for the U1B states.
We choose the spinon basis in the momentum space
fk,I = (fA,k,↑, fB,k,↑, fA,k,↓, fB,k,↓)T , where A and B de-
note the two inequivalent sites in each unit cell due to theπ
flux.
The Hamiltonian is written in terms of the Dirac ma-trices Γa
and their anticommutators
Γab = [Γa,Γb]/(2i). (E1)
The representation is chosen to be Γ(1,2,3,4,5) = (σx ⊗1, σz ⊗
1, σy ⊗ τx, σy ⊗ τy, σy ⊗ τz). Γa and Γab is oddunder time reversal
except when a = 4 or b = 4. TheHamiltonian is thus
h(k) =
5∑a=1
da(k)Γa +
5∑a
-
13
FIG. 5. Dynamic spin structure factor for six free spinon
mean-field states other than U1A00. Note the U1A10 Hamiltonian
isidentically zero for the first and second neighbor hoppings. None
of them is consistent with the spinon Fermi surface picture.In all
subfigures, the energy transfer is normalized against the
corresponding bandwidth B.
3. The U1B10 state
d3(k) = −√
3t1 sin[(kx −
√3ky)/2
],
d4(k) =√
3t1 cos[(kx +
√3ky)/2
],
d23(k) = −t1 sin[(kx −
√3ky)/2
],
d24(k) = −t1 cos[(kx +
√3ky)/2
],
d25(k) = 2t1 sin kx. (E5)
4. The U1B11 state
d3(k) = −√
3t2 sin[(3kx +
√3ky)/2
],
d4(k) = −√
3t2 cos[(3kx −
√3ky)/2
],
d23(k) = −t2 sin[(3kx +
√3ky)/2
],
d24(k) = t2 cos[(3kx −
√3ky)/2
],
d25(k) = −2t2 sin(√
3ky),
d34(k) = 2t1 cos(kx),
d35(k) = −2t1 sin[(kx +
√3ky)/2
],
d45(k) = −2t1 cos[(kx −
√3ky)/2
]. (E6)
-
14
∗ [email protected] William Witczak-Krempa, Gang Chen,
Yong Baek Kim,
and Leon Balents, “Correlated Quantum Phenomena inthe Strong
Spin-Orbit Regime,” Annual Review of Con-densed Matter Physics 5,
57–82 (2014).
2 Jeffrey G. Rau, Eric Kin-Ho Lee, and Hae-Young Kee,“Spin-Orbit
Physics Giving Rise to Novel Phases in Cor-related Systems:
Iridates and Related Materials,” AnnualReview of Condensed Matter
Physics 7, 195–221 (2016).
3 G. Jackeli and G. Khaliullin, “Mott Insulators in theStrong
Spin-Orbit Coupling Limit: From Heisenberg to aQuantum Compass and
Kitaev Models,” Phys. Rev. Lett.102, 017205 (2009).
4 Gang Chen and Leon Balents, “Spin-orbit effects inNa4Ir3O8: A
hyper-kagome lattice antiferromagnet,”Phys. Rev. B 78, 094403
(2008).
5 Ji ř́ı Chaloupka, George Jackeli, and Giniyat
Khaliullin,“Kitaev-Heisenberg Model on a Honeycomb Lattice:
Pos-sible Exotic Phases in Iridium Oxides A2IrO3,” Phys. Rev.Lett.
105, 027204 (2010).
6 Dmytro Pesin and Leon Balents, “Mott physics and bandtopology
in materials with strong spin–orbit interaction,”Nature Physics 6,
376–381 (2010).
7 Shigeki Onoda and Yoichi Tanaka, “Quantum melting ofspin ice:
Emergent cooperative quadrupole and chirality,”Phys. Rev. Lett.
105, 047201 (2010).
8 Lucile Savary and Leon Balents, “Coulombic quantum liq-uids in
spin-1/2 pyrochlores,” Phys. Rev. Lett. 108, 037202(2012).
9 Kate A. Ross, Lucile Savary, Bruce D. Gaulin, and LeonBalents,
“Quantum Excitations in Quantum Spin Ice,”Phys. Rev. X 1, 021002
(2011).
10 Yi-Ping Huang, Gang Chen, and Michael Hermele,“Quantum Spin
Ices and Topological Phases from Dipolar-Octupolar Doublets on the
Pyrochlore Lattice,” Phys. Rev.Lett. 112, 167203 (2014).
11 Gang Chen and Leon Balents, “Spin-orbit coupling in d2
ordered double perovskites,” Phys. Rev. B 84, 094420(2011).
12 Hamid R. Molavian, Michel J. P. Gingras, and BenjaminCanals,
“Dynamically Induced Frustration as a Route toa Quantum Spin Ice
State in Tb2Ti2O7 via Virtual Crys-tal Field Excitations and
Quantum Many-Body Effects,”Phys. Rev. Lett. 98, 157204 (2007).
13 Gang Chen and Yong Baek Kim, “Anomalous enhance-ment of the
Wilson ratio in a quantum spin liquid: Thecase of Na4Ir3O8,” Phys.
Rev. B 87, 165120 (2013).
14 R. Applegate, N. R. Hayre, R. R. P. Singh, T. Lin, A. G.
R.Day, and M. J. P. Gingras, “Vindication of Yb2Ti2O7 asa Model
Exchange Quantum Spin Ice,” Phys. Rev. Lett.109, 097205 (2012).
15 K. A. Ross, J. P. C. Ruff, C. P. Adams, J. S. Gard-ner, H. A.
Dabkowska, Y. Qiu, J. R. D. Copley, andB. D. Gaulin,
“Two-Dimensional Kagome Correlations andField Induced Order in the
Ferromagnetic XY PyrochloreYb2Ti2O7,” Phys. Rev. Lett. 103, 227202
(2009).
16 Gang Chen and Michael Hermele, “Magnetic orders
andtopological phases from f -d exchange in pyrochlore iri-dates,”
Phys. Rev. B 86, 235129 (2012).
17 Lucile Savary, Xiaoqun Wang, Hae-Young Kee, Yong BaekKim, Yue
Yu, and Gang Chen, “Quantum spin ice on
the breathing pyrochlore lattice,” Phys. Rev. B 94,
075146(2016).
18 SungBin Lee, Eric Kin-Ho Lee, Arun Paramekanti, andYong Baek
Kim, “Order-by-disorder and magnetic field re-sponse in the
Heisenberg-Kitaev model on a hyperhoney-comb lattice,” Phys. Rev. B
89, 014424 (2014).
19 SungBin Lee, Shigeki Onoda, and Leon Balents, “Genericquantum
spin ice,” Phys. Rev. B 86, 104412 (2012).
20 Eric Kin-Ho Lee, Robert Schaffer, Subhro Bhattacharjee,and
Yong Baek Kim, “Heisenberg-kitaev model on the hy-perhoneycomb
lattice,” Phys. Rev. B 89, 045117 (2014).
21 Shigeki Onoda and Yoichi Tanaka, “Quantum fluctua-tions in
the effective pseudospin- 1
2model for magnetic py-
rochlore oxides,” Phys. Rev. B 83, 094411 (2011).22 Lieh-Jeng
Chang, Shigeki Onoda, Yixi Su, Ying-Jer Kao,
Ku-Ding Tsuei, Yukio Yasui, Kazuhisa Kakurai, andMartin Richard
Lees, “Higgs transition from a magneticCoulomb liquid to a
ferromagnet in Yb2Ti2O7,” NatureCommunications 3, 992 (2012).
23 J. S. Gardner, S. R. Dunsiger, B. D. Gaulin, M. J. P.
Gin-gras, J. E. Greedan, R. F. Kiefl, M. D. Lumsden, W.
A.MacFarlane, N. P. Raju, J. E. Sonier, I. Swainson, andZ. Tun,
“Cooperative Paramagnetism in the GeometricallyFrustrated
Pyrochlore Antiferromagnet Tb2Ti2O7,” Phys.Rev. Lett. 82, 1012–1015
(1999).
24 Fei-Ye Li, Yao-Dong Li, Yue Yu, Arun Paramekanti, andGang
Chen, “Kitaev materials beyond iridates: order byquantum disorder
and weyl magnons in rare-earth doubleperovskites,” Phys. Rev. B 95,
085132 (2017).
25 Gang Chen, ““Magnetic monopole” condensation of thepyrochlore
ice U(1) quantum spin liquid: Application toPr2Ir2O7 and Yb2Ti2O7,”
Phys. Rev. B 94, 205107 (2016).
26 M J P Gingras and P A McClarty, “Quantum spin ice: asearch
for gapless quantum spin liquids in pyrochlore mag-nets,” Reports
on Progress in Physics 77, 056501 (2014).
27 Yao-Dong Li and Gang Chen, “Symmetry enriched U(1)topological
orders for dipole-octupole doublets on a py-rochlore lattice,”
Phys. Rev. B 95, 041106 (2017).
28 E. Lhotel, S. Petit, S. Guitteny, O. Florea,M. Ciomaga
Hatnean, C. Colin, E. Ressouche, M. R. Lees,and G. Balakrishnan,
“Fluctuations and All-In All-OutOrdering in Dipole-Octupole
Nd2Zr2O7,” Phys. Rev. Lett.115, 197202 (2015).
29 Owen Benton, Olga Sikora, and Nic Shannon, “Seeingthe light:
Experimental signatures of emergent electromag-netism in a quantum
spin ice,” Phys. Rev. B 86, 075154(2012).
30 Anson W. C. Wong, Zhihao Hao, and Michel J. P.
Gingras,“Ground state phase diagram of generic xy pyrochloremagnets
with quantum fluctuations,” Phys. Rev. B 88,144402 (2013).
31 Gang Chen, Rodrigo Pereira, and Leon Balents, “Exoticphases
induced by strong spin-orbit coupling in ordereddouble
perovskites,” Phys. Rev. B 82, 174440 (2010).
32 S. H. Curnoe, “Structural distortion and the spin liquidstate
in Tb2Ti2O7,” Phys. Rev. B 78, 094418 (2008).
33 Yuesheng Li, Haijun Liao, Zhen Zhang, Shiyan Li, FengJin,
Langsheng Ling, Lei Zhang, Youming Zou, Li Pi,Zhaorong Yang,
Junfeng Wang, Zhonghua Wu, and Qing-ming Zhang, “Gapless quantum
spin liquid ground statein the two-dimensional spin-1/2 triangular
antiferromag-
mailto:[email protected]://dx.doi.org/
10.1146/annurev-conmatphys-020911-125138http://dx.doi.org/
10.1146/annurev-conmatphys-020911-125138http://dx.doi.org/
10.1103/PhysRevLett.102.017205http://dx.doi.org/
10.1103/PhysRevLett.102.017205http://dx.doi.org/10.1103/PhysRevB.78.094403http://dx.doi.org/
10.1103/PhysRevLett.105.027204http://dx.doi.org/
10.1103/PhysRevLett.105.027204http://dx.doi.org/10.1103/PhysRevLett.105.047201http://dx.doi.org/
10.1103/PhysRevLett.108.037202http://dx.doi.org/
10.1103/PhysRevLett.108.037202http://dx.doi.org/10.1103/PhysRevX.1.021002http://dx.doi.org/10.1103/PhysRevLett.112.167203http://dx.doi.org/10.1103/PhysRevLett.112.167203http://dx.doi.org/
10.1103/PhysRevB.84.094420http://dx.doi.org/
10.1103/PhysRevB.84.094420http://dx.doi.org/
10.1103/PhysRevLett.98.157204http://dx.doi.org/
10.1103/PhysRevB.87.165120http://dx.doi.org/10.1103/PhysRevLett.109.097205http://dx.doi.org/10.1103/PhysRevLett.109.097205http://dx.doi.org/
10.1103/PhysRevLett.103.227202http://dx.doi.org/10.1103/PhysRevB.86.235129http://dx.doi.org/10.1103/PhysRevB.94.075146http://dx.doi.org/10.1103/PhysRevB.94.075146http://dx.doi.org/10.1103/PhysRevB.89.014424http://dx.doi.org/10.1103/PhysRevB.86.104412http://dx.doi.org/
10.1103/PhysRevB.89.045117http://dx.doi.org/10.1103/PhysRevB.83.094411http://dx.doi.org/10.1038/ncomms1989http://dx.doi.org/10.1038/ncomms1989http://dx.doi.org/
10.1103/PhysRevLett.82.1012http://dx.doi.org/
10.1103/PhysRevLett.82.1012http://dx.doi.org/10.1103/PhysRevB.95.085132http://dx.doi.org/
10.1103/PhysRevB.94.205107http://stacks.iop.org/0034-4885/77/i=5/a=056501http://dx.doi.org/10.1103/PhysRevB.95.041106http://dx.doi.org/
10.1103/PhysRevLett.115.197202http://dx.doi.org/
10.1103/PhysRevLett.115.197202http://dx.doi.org/10.1103/PhysRevB.86.075154http://dx.doi.org/10.1103/PhysRevB.86.075154http://dx.doi.org/10.1103/PhysRevB.88.144402http://dx.doi.org/10.1103/PhysRevB.88.144402http://dx.doi.org/
10.1103/PhysRevB.82.174440http://dx.doi.org/
10.1103/PhysRevB.78.094418
-
15
net YbMgGaO4,” Scientific Reports 5, 16419 (2015).34 Yuesheng
Li, Gang Chen, Wei Tong, Li Pi, Juanjuan Liu,
Zhaorong Yang, Xiaoqun Wang, and Qingming Zhang,“Rare-Earth
Triangular Lattice Spin Liquid: A Single-Crystal Study of
YbMgGaO4,” Phys. Rev. Lett. 115,167203 (2015).
35 Yao-Dong Li, Xiaoqun Wang, and Gang Chen,“Anisotropic spin
model of strong spin-orbit-coupled trian-gular antiferromagnets,”
Phys. Rev. B 94, 035107 (2016).
36 Yao Shen, Yao-Dong Li, Hongliang Wo, Yuesheng Li,Shoudong
Shen, Bingying Pan, Qisi Wang, H. C. Walker,P. Steffens, M Boehm,
Yiqing Hao, D. L. Quintero-Castro,L. W. Harriger, Lijie Hao, Siqin
Meng, Qingming Zhang,Gang Chen, and Jun Zhao, “Spinon Fermi surface
in a tri-angular lattice quantum spin liquid YbMgGaO4,” Nature540,
559–562 (2016).
37 Joseph A. M. Paddison, Zhiling Dun, Georg Ehlers, Yao-hua
Liu, Matthew B. Stone, Haidong Zhou, and MartinMourigal,
“Continuous excitations of the triangular-latticequantum spin
liquid YbMgGaO4,” Nature Physics, arXivpreprint 1607.03231
(2016).
38 Yuesheng Li, Devashibhai Adroja, Pabitra K. Biswas, Pe-ter J.
Baker, Qian Zhang, Juanjuan Liu, Alexander A.Tsirlin, Philipp
Gegenwart, and Qingming Zhang, “MuonSpin Relaxation Evidence for
the U(1) Quantum Spin-Liquid Ground State in the Triangular
AntiferromagnetYbMgGaO4,” Phys. Rev. Lett. 117, 097201 (2016).
39 Yao-Dong Li, Yao Shen, Yuesheng Li, Jun Zhao, andGang Chen,
“The effect of spin-orbit coupling on theeffective-spin correlation
in YbMgGaO4,” arXiv preprint1608.06445 (2016).
40 Yao-Dong Li, Xiaoqun Wang, and Gang Chen, “Hiddenmultipolar
orders of dipole-octupole doublets on a trian-gular lattice,” Phys.
Rev. B 94, 201114 (2016).
41 Yao-Dong Li and Gang Chen, “Detecting spin fractional-ization
in a spinon fermi surface spin liquid,” Phys. Rev.B 96, 075105
(2017).
42 Haruki Watanabea, Hoi Chun Po, Ashvin Vishwanath,and Michael
Zaletel, “Filling constraints for spin-orbit cou-pled insulators in
symmorphic and nonsymmorphic crys-tals,” PNAS 112, 14551–14556
(2015).
43 Y. Xu, J. Zhang, Y. S. Li, Y. J. Yu, X. C. Hong,Q. M. Zhang,
and S. Y. Li, “Absence of Magnetic Ther-mal Conductivity in the
Quantum Spin-Liquid CandidateYbMgGaO4,” Phys. Rev. Lett. 117,
267202 (2016).
44 Olexei I. Motrunich, “Variational study of triangular
lat-tice spin-1/2 model with ring exchanges and spin liquidstate in
κ-(ET)2Cu2(CN)3,” Phys. Rev. B 72, 045105(2005).
45 Sung-Sik Lee and Patrick A. Lee, “U(1) Gauge Theory ofthe
Hubbard Model: Spin Liquid States and Possible Ap-plication to
κ-(BEDT-TTF)2Cu2(CN)3,” Phys. Rev. Lett.95, 036403 (2005).
46 Patrick A. Lee and Naoto Nagaosa, “Gauge theory of thenormal
state of high-Tc superconductors,” Phys. Rev. B46, 5621–5639
(1992).
47 Xiao-Gang Wen, “Quantum orders and symmetric spin liq-uids,”
Phys. Rev. B 65, 165113 (2002).
48 See the Supplementary information.49 Johannes Reuther,
Shu-Ping Lee, and Jason Alicea, “Clas-
sification of spin liquids on the square lattice with
strongspin-orbit coupling,” Phys. Rev. B 90, 174417 (2014).
50 Gang Chen, Andrew Essin, and Michael Hermele, “Majo-rana spin
liquids and projective realization of SU(2) spin
symmetry,” Phys. Rev. B 85, 094418 (2012).51 Michael Hermele,
“SU(2) gauge theory of the Hubbard
model and application to the honeycomb lattice,” Phys.Rev. B 76,
035125 (2007).
52 Samuel Bieri, Claire Lhuillier, and Laura Messio,
“Projec-tive symmetry group classification of chiral spin
liquids,”Phys. Rev. B 93, 094437 (2016).
53 Yi-Zhuang You, Itamar Kimchi, and Ashvin Vishwanath,“Doping a
spin-orbit Mott insulator: Topological super-conductivity from the
Kitaev-Heisenberg model and pos-sible application to
(Na2/Li2)IrO3,” Phys. Rev. B 86,085145 (2012).
54 Robert Schaffer, Subhro Bhattacharjee, and Yong BaekKim,
“Spin-orbital liquids in non-Kramers magnets on thekagome lattice,”
Phys. Rev. B 88, 174405 (2013).
55 Yuan-Ming Lu, “Symmetry protected gapless Z2 spin liq-uids,”
arXiv preprint 1606.05652 (2016).
56 Xiao-Gang Wen, “Quantum order: a quantum entangle-ment of
many particles,” Physics Letters A 300, 175 – 181(2002).
57 Andrew M. Essin and Michael Hermele, “Spectroscopicsignatures
of crystal momentum fractionalization,” Phys.Rev. B 90, 121102
(2014).
58 In the previous work36, only the nearest-neighbor
spinonhopping is included.
59 Sandor Toth, Katharina Rolfs, Rolfs, Andrew R. Wildes,and
Christian Ruegg, “Strong exchange anisotropy inYbMgGaO4 from
polarized neutron diffraction,” arXivpreprint arXiv:1705.05699
(2017).
60 Jan Brinckmann and Patrick A. Lee, “Slave Boson Ap-proach to
Neutron Scattering in YBa2Cu3O6+y Supercon-ductors,” Phys. Rev.
Lett. 82, 2915–2918 (1999).
61 Yuesheng Li, Devashibhai Adroja, Robert I. Bewley,
DavidVoneshen, Alexander A. Tsirlin, Philipp Gegenwart, andQingming
Zhang, “Crystalline Electric-Field Randomnessin the Triangular
Lattice Spin-Liquid YbMgGaO4,” Phys.Rev. Lett. 118, 107202
(2017).
62 Yao Shen, Yao-Dong Li, H. C. Walker, P. Steffens,M. Boehm,
Xiaowen Zhang, Shoudong Shen, HongliangWo, Gang Chen, and Jun Zhao,
“Fractionalized exci-tations in the partially magnetized spin
liquid candidateYbMgGaO4,” arXiv preprint arXiv:1708.06655
(2017).
63 Changle Liu, Rong Yu, and Xiaoqun Wang,
“Semiclassicalground-state phase diagram and multi-q phase of a
spin-orbit-coupled model on triangular lattice,” Phys. Rev. B94,
174424 (2016).
64 Qiang Luo, Shijie Hu, Bin Xi, Jize Zhao, and Xiao-qun Wang,
“Ground-state phase diagram of an anisotropicspin- 1
2model on the triangular lattice,” Phys. Rev. B 95,
165110 (2017).65 Z. Zhu, P.A. Maksimov, S.R. White, and A.L.
Cherny-
shev, “Disorder-induced Mimicry of a Spin Liquid inYbMgGaO4,”
arXiv preprint arXiv:1703.02971 (2017).
66 M. Hirschberger, J. W. Krizan, R. J. Cava, and N. P.Ong,
“Large thermal hall conductivity of neutral spin ex-citations in a
frustrated quantum magnet,” Science 348,106–109 (2015).
67 S. J. Li, Z. Y. Zhao, C. Fan, B. Tong, F. B. Zhang, J. Shi,J.
C. Wu, X. G. Liu, H. D. Zhou, X. Zhao, and X. F.Sun,
“Low-temperature thermal conductivity of Dy2Ti2O7and Yb2Ti2O7
single crystals,” Phys. Rev. B 92, 094408(2015).
68 Q. J. Li, Z. Y. Zhao, C. Fan, F. B. Zhang, H. D. Zhou,X.
Zhao, and X. F. Sun, “Phonon-glass-like behavior of
http://dx.doi.org/ 10.1038/srep16419http://dx.doi.org/
10.1103/PhysRevLett.115.167203http://dx.doi.org/
10.1103/PhysRevLett.115.167203http://dx.doi.org/10.1103/PhysRevB.94.035107http://dx.doi.org/10.1038/nature20614http://dx.doi.org/10.1038/nature20614https://arxiv.org/abs/1607.03231https://arxiv.org/abs/1607.03231http://dx.doi.org/
10.1103/PhysRevLett.117.097201http://arxiv.org/abs/1608.06445http://arxiv.org/abs/1608.06445http://dx.doi.org/10.1103/PhysRevB.94.201114http://dx.doi.org/10.1103/PhysRevB.96.075105http://dx.doi.org/10.1103/PhysRevB.96.075105http://dx.doi.org/10.1073/pnas.1514665112http://dx.doi.org/
10.1103/PhysRevLett.117.267202http://dx.doi.org/10.1103/PhysRevB.72.045105http://dx.doi.org/10.1103/PhysRevB.72.045105http://dx.doi.org/10.1103/PhysRevLett.95.036403http://dx.doi.org/10.1103/PhysRevLett.95.036403http://dx.doi.org/10.1103/PhysRevB.46.5621http://dx.doi.org/10.1103/PhysRevB.46.5621http://dx.doi.org/
10.1103/PhysRevB.65.165113http://dx.doi.org/10.1103/PhysRevB.90.174417http://dx.doi.org/
10.1103/PhysRevB.85.094418http://dx.doi.org/10.1103/PhysRevB.76.035125http://dx.doi.org/10.1103/PhysRevB.76.035125http://dx.doi.org/
10.1103/PhysRevB.93.094437http://dx.doi.org/
10.1103/PhysRevB.86.085145http://dx.doi.org/
10.1103/PhysRevB.86.085145http://dx.doi.org/10.1103/PhysRevB.88.174405https://arxiv.org/abs/1606.05652http://dx.doi.org/
http://dx.doi.org/10.1016/S0375-9601(02)00808-3http://dx.doi.org/
http://dx.doi.org/10.1016/S0375-9601(02)00808-3http://dx.doi.org/10.1103/PhysRevB.90.121102http://dx.doi.org/10.1103/PhysRevB.90.121102http://dx.doi.org/10.1103/PhysRevLett.82.2915http://dx.doi.org/
10.1103/PhysRevLett.118.107202http://dx.doi.org/
10.1103/PhysRevLett.118.107202http://dx.doi.org/10.1103/PhysRevB.94.174424http://dx.doi.org/10.1103/PhysRevB.94.174424http://dx.doi.org/
10.1103/PhysRevB.95.165110http://dx.doi.org/
10.1103/PhysRevB.95.165110http://dx.doi.org/10.1126/science.1257340http://dx.doi.org/10.1126/science.1257340http://dx.doi.org/10.1103/PhysRevB.92.094408http://dx.doi.org/10.1103/PhysRevB.92.094408
-
16
magnetic origin in single-crystal Tb2Ti2O7,” Phys. Rev. B87,
214408 (2013).
69 Yuji Matsuda, (2017), Talk at workshop on
topologicalmaterials, Kyoto University.
70 Tian-Heng Han, Joel S Helton, Shaoyan Chu, Daniel GNocera,
Jose A Rodriguez-Rivera, Collin Broholm, andYoung S Lee,
“Fractionalized excitations in the spin-liquidstate of a
kagome-lattice antiferromagnet,” Nature 492,406–410 (2012).
71 Y. Shimizu, K. Miyagawa, K. Kanoda, M. Maesato, andG. Saito,
“Spin liquid state in an organic mott insulatorwith a triangular
lattice,” Phys. Rev. Lett. 91, 107001(2003).
72 T. Itou, A. Oyamada, S. Maegawa, M. Tamura, andR. Kato,
“Quantum spin liquid in the spin-12 triangularantiferromagnet
EtMe3Sb[Pd(dmit)2]2,” Phys. Rev. B 77,104413 (2008).
73 Satoshi Yamashita, Yasuhiro Nakazawa, Masaharu Oguni,Yugo
Oshima, Hiroyuki Nojiri, Yasuhiro Shimizu, KazuyaMiyagawa, and
Kazushi Kanoda, “Thermodynamic prop-erties of a spin-1/2
spin-liquid state in a kappa-type organicsalt,” Nature Physics 4,
459–462 (2008).
74 Y. Q. Liu, S. J. Zhang, J. L. Lv, S. K. Su, T. Dong,
GangChen, and N. L. Wang, “Revealing a triangular lattice
ising antiferromagnet in a single-crystal CeCd3As3,”
arXivpreprint 1612.03720 (2016).
75 Shohei Higuchi, Yuki Noshima, Naoki Shirakawa, MasamiTsubota,
and Jiro Kitagawa, “Optical, transport and mag-netic properties of
new compound CeCd3P3,” MaterialsResearch Express 3, 056101
(2016).
76 Andr T. Nientiedt and Wolfgang Jeitschko, “The Seriesof Rare
Earth Zinc Phosphides RZn3P3 (R=Y, La−Nd,Sm, Gd−Er) and the
Corresponding Cadmium CompoundPrCd3P3,” Journal of Solid State
Chemistry 146, 478 –483 (1999).
77 A. Yamada, N. Hara, K. Matsubayashi, K. Munakata,C. Ganguli,
A. Ochiai, T. Matsumoto, and Y. Uwatoko,“Effect of pressure on the
electrical resistivity of CeZn3P3,”J. Phys.: Conf. Ser. 215, 012031
(2010).
78 Stanislav S. Stoyko and Arthur Mar, “Ternary
Rare-EarthArsenides REZn3As3 (RE = La−Nd, Sm) and RECd3As3(RE =
La−Pr),” Inorganic Chemistry 50, 11152–11161(2011).
79 M. B. Sanders, F. A. Cevallos, and R. J. Cava, “Mag-netism in
the KBaRE(BO3)2 (RE= Sm, Eu, Gd, Tb, Dy,Ho, Er, Tm, Yb, Lu) series:
materials with a triangularrare earth lattice,” arXiv preprint
1611.08548 (2016).
http://dx.doi.org/ 10.1103/PhysRevB.87.214408http://dx.doi.org/
10.1103/PhysRevB.87.214408http://dx.doi.org/10.1103/PhysRevLett.91.107001http://dx.doi.org/10.1103/PhysRevLett.91.107001http://dx.doi.org/10.1103/PhysRevB.77.104413http://dx.doi.org/10.1103/PhysRevB.77.104413http://dx.doi.org/
10.1038/nphys942http://stacks.iop.org/2053-1591/3/i=5/a=056101http://stacks.iop.org/2053-1591/3/i=5/a=056101http://dx.doi.org/http://dx.doi.org/10.1006/jssc.1999.8396http://dx.doi.org/http://dx.doi.org/10.1006/jssc.1999.8396http://dx.doi.org/10.1088/1742-6596/215/1/012031http://dx.doi.org/10.1021/ic201708xhttp://dx.doi.org/10.1021/ic201708xhttps://arxiv.org/abs/1611.08548
The spinon Fermi surface U(1) spin liquid in a
spin-orbit-coupled triangular lattice Mott insulator
YbMgGaO4AbstractI IntroductionII Space group symmetryIII Fermionic
parton constructionIV Projective symmetry group classificationV
Mean-field statesVI Spectroscopic propertiesVII DiscussionVIII
AcknowledgementsA The coordinate System and space group symmetryB
Projective symmetry group classification1 Spatial symmetry2 Time
reversal symmetry
C Spinon band structures and mean-field Hamiltonians1 Spinon
band structures2 Spinon mean-field Hamiltonians
D The U1A00 state and the spectroscopic results1 Free spinon
mean-field theory2 Variational calculation and random phase
approximation
E The U1B states1 The U1B00 state2 The U1B01 state3 The U1B10
state4 The U1B11 state
References
