Realizing quantum spin liquid phases in spin-orbit driven correlated materials

3.8 MB·

by  Andrei Catuneanu

Realizing quantum spin liquid phases in spin-orbit driven correlated materials
Andrei Catuneanu,1 Youhei Yamaji,2,3 Gideon Wachtel,1 Hae-Young Kee,1,4 and Yong Baek Kim1,4
1Department of Physics and Center for Quantum Materials,
University of Toronto, 60 St. George St., Toronto, Ontario, M5S 1A7, Canada
2Department of Applied Physics and Quantum-Phase Electronics Center (QPEC),
The University of Tokyo, Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan
3JST, PRESTO, Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan
4Canadian Institute for Advanced Research, Toronto, Ontario, M5G 1Z8, Canada
Thespinliquidphaseisoneoftheprominentstronglyinteractingtopologicalphasesofmatterwhose
unambiguous confirmation is yet to be reached despite intensive experimental efforts on numerous
candidate materials. The challenge is derived from the difficulty of formulating realistic theoretical
models for these materials and interpreting the corresponding experimental data. Here we study a
7 theoretical model with bond-dependent interactions, directly motivated by recent experiments on
1
two-dimensional correlated materials with strong spin-orbit coupling. We show numerical evidence
0
fortheexistenceofanextendedfamilyofquantumspinliquids,whicharepossiblyconnectedtothe
2
Kitaevspinliquidstate. Theseresultsareusedtoprovideanexplanationofthescatteringcontinuum

n seeninneutronscatteringonα-RuCl3. Implicationsoftheseresultstothree-dimensionalmaterials
a such as hyperhoneycomb iridate, β-Li IrO , are also discussed.
2 3
J

6
2 Introduction — The role of strong interaction between two-dimensional honeycomb lattice:
electrons in the emergence of topological phases of mat-
(cid:88)
] ter, where both theoretical and experimental understand- H = Hγ, (1)
l
e ing is far from complete compared to weakly interacting γ∈x,y,z
-
r systems, has recently been a topic of intensive research.
t The archetypal example of a topological phase with strong where
s
. electron interaction is the quantum spin liquid1, in which (cid:88)
at the elementary excitations are charge-neutral fractional- Hz = [KzSizSjz+Γz(SixSjy+SiySjx)] (2)
m ized particles. While a lot of progress has been made on (cid:104)ij(cid:105)∈z−bond
- the theoretical understanding of the quantum spin liquid
d and Hx,y are defined similarly with corresponding K
phase, its direct experimental confirmation has remained x,y
n and Γ . Each Hγ represents the n.n spin interactions
elusive despite various studies on a number of candidate x,y
o along one of the three bond directions, γ = x,y,z. The
c materials2–6. Significant progress, however, has recently
model is parameterized by K = −(1+2a)cosφ, K =
[ been made due to the availabilty of a new class of corre- z x,y
−(1−a)cosφ, Γ = sinφ, with a characterizing bond
lated materials, where strong spin-orbit coupling leads to x,y,z
1 anisotropy. When φ = 0,π (i.e. Γ = 0), this model re-
various bond-dependent spin interactions7–9. These mate- γ
v ducestotheexactlysolvableKitaevmodelwithaquantum
7 rials are Mott insulators with 4d and 5d transition metal
spinliquidgroundstate. Moreover,arecentanalysisinthe
3 elements, which include iridates and ruthenates with two-
φ=π/2 limit (i.e. K =0) revealed a macroscopically de-
8 dimensional honeycomb lattice10,11 and three-dimensional γ
generate ground state in the classical model25.
7 variants12,13.
0 The above model is directly motivated by experiments
. onα-RuCl (RuCl )andearlierab initiocomputations. In
1 3 3
Magnetic frustration in these new systems arises from RuCl , Ru3+ ionscarryaspin-orbitentangledpseudospin-
0 3
bond-dependent interactions7,14–18 rather than relying on
7 1/2 degree of freedom and sit on a two-dimensional hon-
1 the geometric frustration of the underlying lattice struc- eycomb lattice. Ab initio computations suggest that the
: ture used in earlier approaches. These materials are of dominant spin exchange interactions are given by K < 0
v γ
great interest because they may intrinsically generate the
and Γ > 0 with comparable magnitude as well as a non-
i γ
X Kitaev interaction which, in the absence of other interac- negligible 3rd n.n. antiferromagnetic Heisenberg interac-
r tions, would lead to a material realization of an exactly tion J > 026–29. In addition, it was found that both K
3 γ
a solvable model for the quantum spin liquid phase19. This
and Γ are slightly anisotropic and that J may promote
γ 3
raises the question for the stability of the Kitaev spin liq- the zig-zag magnetic order observed experimentally27,28.
uid against other perturbations always present in a real
On the other hand, a recent inelastic neutron scattering
material. In some known models, the Kitaev spin liquid
experimentobservedfiniteenergyscatteringcontinuarem-
phase is stable only for sufficiently small magnitudes of
iniscent of the excitation spectra in quantum spin liquid
other interactions14,20–24, making its experimental realiza-
phases, both above and below the magnetic ordering tran-
tion a challenging endeavor.
sition temperature, potentially indicating proximity to a
quantum spin liquid phase30. While this interpretation is
In this work, we analyze a theoretical model that may natural,itisnotobviouswhatkindofquantumspinliquid
host an extended family of quantum spin liquid phases may be nearby given that the relevant microscopic model
and make connections to recent experiments on a num- is far from the ideal Kitaev limit.
ber of 4d and 5d transition metal oxide materials. We Here we take the Hamiltonian in Eq. (1) as the mini-
consider the following nearest-neighbor (n.n.) model on a mal model for the putative quantum spin liquid phase and
2
HaL HbL
-0.20 0.002 Φ(cid:144)Π=0 Φ(cid:144)Π=0.1 Φ(cid:144)Π=0.2
N -0.25 -¶2E
(cid:144)E0-0.30 0.0 N¶Φ20 Y M Y M Y M
-0.35 1 X G K X G K X G K
2.0 -0.002 q G¢ G¢ G¢
S
Y e
M v 5
1.5 X G K ati 0.
q G¢ Rel Y Y Y
S 1.0 M M M
X G K X G K X G K
0
G¢ G¢ G¢
0.5
0 0.25 0.5 0.75 1
Φ(cid:144)Π=0.5 Φ(cid:144)Π=0.75 Φ(cid:144)Π=1
Φ(cid:144)Π
FIG.1. (Coloronline)(a)Top: E /N (yellow)and−1 ∂2E0 (purple)foranisotropyparametera=0.1. Bottom: S foranisotropy
0 N ∂φ2 q
parametera=0.1atΓ-(black),M-(blue),Y-(cyan),K-(red)andΓ(cid:48)-(green)inthereciprocallattice(inset). (b)Representation
of S , averaged over domains in a real material, when a=0.1 for various φ in the phase diagram.
q
treatJ asaperturbation. Westudythismodelusingthree in −1 ∂2E0 encompassing a large region of phase space
3 N ∂φ2
different numerical methods which combined together give separating the Γ-limit and the exactly solvable antiferro-
the following results: magnetic Kitaev limit at φ/π = 1. These peaks coincide
1) When Kγ has the ferromagnetic sign (0 ≤ φ ≤ π/2), with kinks in E0/N (solid yellow) shown in Fig. 1a. Two
there is no phase transition between the Kitaev spin liquid smaller peaks can also be seen near φ/π = 0.75, however
limit at φ/π =0 and the Γγ-only limit at φ/π =0.5. This these are not present in the a = 0 limit, while the larger
result is clearly seen in the ground state energy and static peaks near φ/π = 0.5 and 1 appear consistently for differ-
structure factor (SSF) computed by exact diagonalization ent a. The small peaks can thus be considered spurious
(ED) on a 24-site honeycomb cluster with a = 0.1. An and a consequence of the finite cluster size, with only one
independentcalculationofthegroundstateenergyanden- phaseseparatingtheΓandantiferromagneticKitaevlimits.
tanglemententropyontheinfinitetreelatticewitha=0.1 Similar finite size effects were also found for φ/π ∈ [0,0.5]
byinfinitetime-evolutionblockdecimation(iTEBD)shows when a = 0, as discussed in the Supplementary Materials,
no sign of a phase transition and supports the ED results. which makes it difficult to draw a conclusion on the phase
2)Aninterveningmagneticallyorderedphaseseparatesthe diagram for the isotropic model (a=0).
spin liquid phase near the pure Γ limit and the antiferro-
γ Magnetic order and perturbations — The ground state
magnetic Kitaev spin liquid at φ/π =1.
wavefunction of Eq. (1) computed by ED is used to eval-
3) The specific heat C(T) and entropy S(T) at finite tem-
uate real-space spin-spin correlation functions (cid:104)S · S (cid:105),
peratures computed by the method of thermal pure quan- i j
where i and j are site indices on the honeycomb lat-
tum states31–35 (see Supplementary Materials) suggest a
tice. By Fourier transform, we obtain the SSF given by
sthmeopotuhrecΓrossliomveitr.from the ferromagnetic Kitaev limit to Sq = N1 (cid:80)i,jei(ri−rj)·q(cid:104)Si·Sj(cid:105)whereqisavectorinthere-
γ ciprocal lattice. The SSF at various points in the Brillouin
4) Zig-zag spin correlations become dominant upon per-
zone (BZ) is plotted over the phase space in the bottom
turbing the quantum spin liquid phase in 0 < φ < π/2 by
panel of Fig. 1a.
J , suggestingsignificantenhancementofthezig-zagorder
3
for large J . The discontinuities in the SSF can be directly matched
3
Extendedspinliquidstateinglobalphasediagram—The withthedivergencesin−N1 ∂∂2φE20. VisualizationsoftheSSF
groundstateenergypersiteE /N ofEq. (1)wascomputed over the BZ for representative φ in the phase diagram are
0
for φ/π ∈ [0,1], and for anisotropy parameters a = 0 and presented in Fig. 1b. The SSF in Fig. 1b is obtained
0.1 by ED on a 24-site cluster using periodic boundary by computing the average of (cid:104)Si·Sj(cid:105) over all n.n. bonds,
conditions (see Supplementary Materials). Divergences in 2ndn.n.,etc. andperfomingtheresultingone-dimensional
−1 ∂2E0 were used to identify possible phase transitions. Fouriertransform. Thiscalculationreflectsthepresenceof
N ∂φ2 different domains in the crystal, in which either of x,y,z
Sharp peaks of varying intensity can be identified in
bondinteractionscanbestrongerandthus,overthewhole
−1 ∂2E0 throughout the phase space as shown in Fig. 1a crystal, these domains result in an isotropic S despite
N ∂φ2 q
fora=0.1(solidpurple). Remarkably,whenφ/π ∈[0,0.5] the inherent bond anistropy in Eq. (1). The SSF varies
with slight anisotropy, the second derivative of the energy smoothly when a = 0.1 for φ ∈ [0,π/2] and the spin cor-
presentsnosharpfeaturessuggestingthatthe ground state relations at the Γ- and M-points are comparable in inten-
of the Γ-limit (φ/π = 0.5) is smoothly connected to the sity when Γ (cid:39) K , leading to a “star”-shaped structure
γ
ferromagnetic Kitaev spin liquid (φ/π =0). In the antifer- in the SSF as seen in Fig. 1b (e.g, φ/π = 0.2). The ex-
romagneticregionofphasespace,therearetwolargepeaks tended phase separating φ/π = 0.5 and 1 is characterized
3
10
1.8 Φ(cid:144)Π=0.2 0.0
J3=0 J3=0.02 1
0.1
q11..46 0.5RelativeSq C 5 000...2575
S 1.2
0 ΦΠ
1.0
(cid:144)
0
0.8
1 0.1 1 10
-0.05 0 0.05 0.8 0.0
J 0.1
3 2 0.6
n 0.2
l (cid:144)
S 0.4 0.5
FIG. 2. (Color online) SSF at the Γ- (black) and M-points
0.75
(blue) in the BZ for small J3. Solid and dashed curves corre- 0.2
ΦΠ
spond to φ/π = 0.5 and 0.2 respectively. The inset shows the
dramaticchangeintherelativeintensityattheΓ-andM-points 0
0.1 1 (cid:144) 10
for small J >0.
3 T
FIG. 3. (Color online) Results of the method of thermal pure
by dominating spin correlations at the K- and Γ(cid:48)-points
quantum states on a 24-site cluster with anisotropy parame-
in the reciprocal lattice (φ/π = 0.75 in Fig. 1b). Con- ter a = 0.1. Plotted in solid curves are the temperature de-
tained within this phase is the exactly solvable point with pendence of heat capacity C(T) (top) and entropy S(T) (bot-
hidden SU(2) symmetry at φ/π = 0.75 which features K- tom) for various φ in the phase diagram. The shaded regions
point correlations36 consistent with the results presented represent the estimated errors on the results. The tempera-
(cid:112)
here. Furthermore, the energy spectrum (see Supplemen- ture T is expressed in units where (Kz/(1+2a))2+Γ2z =
(cid:112)
tary Materials) in this extended phase is qualitatively dif- (Kx,y/(1−a))2+Γ2x,y =1.
ferent from the rest of the phase space which, together
with the peaks in −1 ∂2E0 and changes in S , indicates
N ∂φ2 q
that a real transition into this magnetically ordered state seenwhenφ(cid:38)0.7onaccountoftheabruptchangeinC(T),
hasoccured. Thustheextendedspinliquidphaseforferro- resembling that of the heat capacity in trivially ordered
magneticKisseparatedfromtheantiferromagneticKitaev phases41. This finding is consistent with our ED results
spin liquid at φ/π =1 by a magnetically ordered phase. and the 120-order at φ/π = 0.75 seen in Refs. 14 and 36.
These results can be connected to real materials, partic- ThedependenceofS(T)onφisplottedinthebottompanel
ularlyRuCl inwhichazig-zagmagneticorderinghasbeen of Fig. 3 with a clear 1-plateau observed when φ/π = 0,
3 2
observed37–39. Previous studies have shown that in addi- consistent with the expected Kitaev spin liquid behaviour.
tion to the n.n. ferromagnetic Kitaev and antiferromag- Inaddition, aplateauofabout1/5thetotalentropyisob-
netic Γ interactions, a 3rd n.n. antiferromagnetic Heisen- servedwhenφ/π =0.5. Anotherplateauisobservedinthe
berg interaction J is non-vanishing and plays a role in magnetically ordered phase around φ/π = 0.75; however,
3
determining the magnetic ordering in RuCl 26,27. Fig. 2 thisfeaturecanbeattributedtofinite-sizeeffectsasfollows.
3
shows that the effect of perturbing Eq. (1) by J is to en- The (N +1)-fold ground state degeneracy at φ/π = 0.75
3
hance (suppress) the M-point (Γ-point) spin correlations, due to the hidden SU(2) symmetry36 is only slightly bro-
consistent with a zig-zag magnetically ordered state ob- ken away from this point, inducing a plateau in S(T) with
servedinexperiments. Thisresultindicatesthatbytuning height given by ln(N + 1)/Nln2 (cid:39) 0.1935 ∼ 1/5 when
J in the real material, an alternate path to achieve a spin N = 24. By contrast, the height of the plateau around
3
liquid phase may be realized. φ/π =0.5 is independent of N42.
Specific heat and thermal entropy — Previous study on The physical origin of the two peak structure in C(T)
the finite temperature properties of the Kitaev model has and the plateau in S(T) can be traced to the energy scales
shown that Kitaev spin liquids feature two peaks in the ofthethermalfluctuationsoftheunderlyingquasiparticles
heat capacity C(T) and a 1-plateau in the entropy S(T), in the spin liquid40. In the Kitaev spin liquid at zero tem-
2
which is attributed to the thermal fractionalization of spin perature, the low-lying quasiparticle excitations are char-
degrees of freedom40. Here we go beyond the Kitaev limit acterizedbyitinerantMajoranafermionswhichdispersein
andinvestigatetheheatcapacityandentropyatfinitetem- a background of zero flux19,43. It has been shown that as
peratureinthepresenceofΓ,whichisexpectedtocompete temperature increases, the flux degrees of freedom begin
withKγ inRuCl3,usingthemethodofthermalpurequan- to fluctuate and lead to the low temperature peak seen in
tum states (see Supplementary Materials). C(T). These fluctuating fluxes utlimately increase the en-
ThedependenceofC(T)onφwhena=0.1isplottedin tropy of the system, resulting in the plateau seen in S(T).
the top panel of Fig. 3. The expected two peak structure Furthermore, the high temperature peak in C(T) is at-
inC(T)isobservedwhenφ/π =0, andisseentobemain- tributed to the development of short range spin correla-
tained continuously as φ/π approaches 0.5 so that the Γ- tions and above this temperature scale, the system enters
limitshowsaqualitativelysimilarbehaviourinC(T)tothe a classical paramagnetic state40. Our results show that
Kitaev spin liquid. Evidence for a phase transition can be thetwo-peakstructureinC(T),andcorrespondingentropy
4
reduced - perhaps due to a finite contributions from the
0 1
a Majorana fermions45. An increase is expected in the en-
N -0.2 el
/ Né tanglement entropy as one approaches a phase transition,
E0 -0.4 0.5 M usuallyrelatedtotheclosingofanexcitationgap, however
-0.6 no such peaks are seen for 0 < φ < π/2. Similarly, there
0 are no sharp features in the ground state energy E as a
0
7 8
1 strongbonds function of φ, which indicates that this phase is smoothly
4 3 6 5
n2 0.8 b 1 2 connected to the Kitaev spin liquid at φ=0.
/lE 0.6 weakbonds 6 75 84 3 There is an apparent first order transition around φ =
S 3 8 0.6π intoaN´eelstatewithspinsorderedinthe[111]direc-
0.4
tion, accompanied by a dramatic lowering of S on both
E
0.2
strongandweakbondsintothisregion. ThisN´eelstatebe-
0 comes a simple product state when φ/π =0.75, as seen by
0 0.25 0.5 0.75 1
the vanishing of S . Due to a hidden SU(2) symmetry, a
φ/π E
120orderstate14,36 isexpectedontheisotropichoneycomb
FIG. 4. (Color online) iTEBD results for small anisotropy lattice at this point; however, this state is not compatible
a = 0.1, and bond dimension χ = 10. (a) N´eel order param- withtheeightsiteunitcelltreelattice,andaN´eelproduct
eter, M and ground state energy per site, E /N. In the stateis selected instead. A final transition into a param-
N´eel 0
orderedregion,themomentsalignalongthe[111]direction. (b) agnetic state is seen before the antiferromagnetic Kitaev
Entanglement entropy SE associated with splitting the system limit, so that there are three phases observed on the in-
along the different bonds in the eight-site unit cell. In most of finite tree, in direct similarity to the ED results on the
the phase diagram all the strong z bonds are identical, and so
honeycomb lattice above.
are all the weak, x and y bonds. In the transition region near
Discussion – The highlight of our numerical results is
φ=0.6π, the symmetry between like bonds is broken, perhaps
indicatingafirstordertransition. Atφ/π=0.75,theentangle- that there exists an extended family of spin liquid phases,
ment entropy vanishes since the system is in a product state. which may be smoothly connected to the ferrmagnetic Ki-
Inset: schematic of Cayley tree with z=3 connectivity. taev spin liquid and the introduction of J3 would greatly
favor a transition from such a quantum spin liquid phase
into the zig-zag ordered phase. As briefly mentioned ear-
plateau,isqualitativelymaintainedwheninterpolatingbe- lier,thisispreciselytheregimewhereourmodelisrelevant
tweentheKitaevspinliquidphaseandtheΓ-limit,further to physics of RuCl3. In a recent inelastic neutron scatter-
suggesting that no phase transition has taken place. ing experiment on RuCl3, it is found that the continuum
of finite energy excitations exists both below and above
Similaritiesontheinfinitetree—WefurtherstudiedEq.
the magnetic transition temperature despite that the low
(1) on an infinite Cayley tree with z = 3 connectivity, us-
ingtheinfinitetime-evolvingblockdecimationalgorithm44 temperaturegroundstateisthezig-zaglong-rangeordered
state30. The inelastic neutron scattering data for the con-
(iTEBD; see Supplementary Materials). Classically, the
tinuum show the star-shape intensity that extends from
ground state in the Γ-limit on the infinite tree is macro-
thezonecentertowardstheMpointsoftheBrillouinzone.
scopically degenerate because a different state with the
This would represent the remnant short-range spin corre-
same energy can be constructed by flipping the sign of one
lations which exist both below and above the transition
spin component on an infinite string of neighboring spins.
temperature.
For example, in the inset of Fig. 4, we can flip the signs of
{...Sy,Sx,Sz,Sy,Sx,Sz...},andobtainanewstatewith Recall that the static structure factor in our ED study
7 3 1 2 6 8
thesameenergy. Forafinitetree,thenumberofpathsone shows enhanced (decreased) short-range spin correlations
canconstructscaleswiththesizeoftheboundary,whichis at the M point (zone center) of the Brillouin zone as one
extensive. TheΓ-limitonthetwo-dimensional(2D)honey- moves from the ferromagnetic Kitaev limit to the pure Γγ
comb and three-dimensional (3D) hyper-honeycomb25 lat- limit. When the strength of the ferromagnetic Kitaev in-
tices also feature similar classical degeneracy. The similar- teraction and the Γγ interaction become comparable, both
ity at the classical level of the Γ-limit on the infinite tree of the short-range spin correlations at the M and the zone
to the 2D and 3D lattices prompts us to study the quan- center would show significant intensity, which leads to the
tum model on the infinite tree for further insight. Figure star-shape structure in momentum space. This behavior
4 shows results of the eight-site iTEBD calculation with maybefavorablycomparedtothefinite-energyshort-range
bond dimension χ = 10, and anisotropy a = 0.1. In this spin correlations seen in RuCl3. Given that the ab initio
calculation, we have also introduced an anisotropy to Γ computations suggest comparable magnitudes of the fer-
γ
such that Γx = Γy = (1−a)sinφ and Γz = (1+2a)sinφ romagnetic Kitaev and Γγ interactions in RuCl326, it is
in order to apply the iTEBD method (see Supplementary conceivable that RuCl3 may be very close to the quantum
Materials). No transition is found when φ/π ∈[0,0.5] and spin liquid phase found in our model. As shown in our
the obtained state is a highly entangled paramagnet, with work, the introduction of small J3 would greatly favor the
SE ∼0.8forstrong(z)bonds, whileforweak(x,y)bonds, zig-zag magnetically ordered phase as observed in RuCl3.
S ∼0.4. Deep in the gapped phase of the Kitaev model, Anotherimplicationofourworkmaybefoundinarecent
E
with large anisotropy a, one finds S ∼ log2 ∼ 0.693 for highpressureexperimentonβ-Li IrO ,wheretheIr4+ ions
E 2 3
the strong bonds and much smaller values of S for the carrying the pseudospin-1/2 sit on the three-dimensional
E
weak bonds. Both, however, increase as the anisotropy is hyperhoneycomb lattice13. The local connectivity of this
5
latticeisthesameasthehoneycomblatticewhilethesmall- phasesidentifiedinournumericalworkwouldbeextremely
est loop here is a ten-sided loop. The spin interactions in valuable for future applications on real materials. This
this system is also described by the Kitaev, Γ , and an- would be an excellent topic for future study.
γ
tiferromagnetic Heisenberg interactions. At ambient pres- Acknowledgements
sure, the material is in a complex spiral magnetically or- This work was supported by the NSERC of Canada
dered phase. Upon applying the hydrostatic pressure, the and the Center for Quantum Materials at the University
magnetic order disappears at 2.5GPa13 and the NMR ex- of Toronto. Y. Y. was supported by JSPS KAKENHI
periment suggests that the system may be in a quantum (Grant Nos. 15K17702 and 16H06345) and was supported
spin liquid phase46. On the other hand, a recent ab initio by PRESTO, JST. Y. Y. was also supported in part by
computation shows that the spin interactions under high MEXT as a social and scientific priority issue (Creation of
pressure favors the Γ interaction and the antiferromag- new functional devices and high-performance materials to
γ
netic interaction becomes negligibly small47. Hence again support next-generation industries) to be tackled by using
the Kitaev-Γ model becomes relevant in β-Li IrO under post-K computer. Computations were mainly performed
2 3
high pressure. While our ED study is done on the two- on the GPC supercomputer at the SciNet HPC Consor-
dimensional honeycomb lattice, the iTEBD result on the tium. SciNet is funded by: the Canada Foundation for In-
infinite tree lattice would strongly suggest that the spin novationundertheauspicesofComputeCanada; theGov-
liquid phenomenology found in the honeycomb lattice may ernmentofOntario;OntarioResearchFund-ResearchEx-
also be applicable to the hyper-honeycomb system. That cellence; and the University of Toronto. We thank helpful
is,thequantumspinliquidobtainedbyapplyinghighpres- discussions with Frank Pollmann, Matthias Gohlke, Shun-
sure may be smoothly connected to the Kitaev spin liquid suke Furukawa, and Subhro Bhattacharjee. We particu-
phase. larly thank Natalia Perkins and Ioannis Rousochatzakis
Finally,moreanalyticalunderstandingoftheconnection forinformingusoftheirunpublishedEDresultsonrelated
betweenthepureKitaevlimitandthequantumspinliquid models.
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