Quantum triangular ice in the easy-axis ferromagnetic phase
QQuantum triangular ice in the easy-axis ferromagnetic phase
S. A. Owerre
1, 2, ∗ African Institute for Mathematical Sciences, 6 Melrose Road, Muizenberg, Cape Town 7945, South Africa. † Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada.
We use spin wave theory to investigate the ground state properties of the Z -invariant quantumXXZ model on the triangular lattice in the ferromagnetic phase. The Hamiltonian comprises nearestand next-nearest-neighbour Ising couplings, external magnetic fields, and a Z -invariant ferromag-netic coupling. We show that quantum fluctuations are suppressed in this system, hence linear spinwave theory gives reasonable estimates of the ground state thermodynamic properties. Our resultsshow that, at half-filling (zero magnetic fields), the spontaneous breaking of Z symmetry leads to aferromagnetic phase whose energy spectrum is gapped at all excitations with a maxon dispersion at k = 0 . This is in sharp contrast to rotational invariant systems with a vanishing phonon dispersion.We show that the k = 0 mode enhances the estimated values of the thermodynamic quantities. Weobtain the trends of the particle density and the condensate fraction. The density of states and thedynamical structure factors exhibit fascinating peaks at unusual wave vectors, which should be ofinterest. PACS numbers: 75.10.Jm, 75.30.Ds, 05.30.Jp, 75.40.Gb
Introduction –. Quantum spin ice (QSI) on three-dimensional pyrochlore lattice has become the subject ofintense research in recent years.
In this system, thespin ordering at the vertices of the pyrochlore lattice,that is two-in, two-out, is reminiscent of hydrogen atomsin water ice. Competing interactions between frustratedspins give rise to fascinating physics with rich quantumphases. Huang, Chan and Hermele recently proposedan alternative simplified Hamiltonian that captures thephysics of QSI on three-dimensional pyrochlore lattice.In the presence of a [111] crystallographic field, the spinconfigurations exhibit an ordering which can be mappedonto a two-dimensional kagome lattice. The Hamilto-nian, however, retains the same form as the Huang,Chan and Hermele model. Recently, Carrasquilla et al., have uncovered the phase diagram of this model on thekagome lattice by Quantum Monte Carlo (QMC) simula-tions. The Hamiltonian studied by Carrasquilla et al., is termed Quantum Kagome Ice and it has the form H = − J ±± (cid:88) (cid:104) lm (cid:105) (cid:0) S + l S + m + S − l S − m (cid:1) + J z (cid:88) (cid:104) lm (cid:105) S zl S zm − h z (cid:88) l S zl , (1)where S ± l = S xl ± iS yl are the raising and the loweringspin operators respectively. In accordance with previousterminology, the term Quantum Triangular Ice refers tothe study of Eq. (1) on the triangular lattice. Here, J z > is the frustrated nearest-neighbour (nn) Ising coupling,and J ±± > is the unfrustrated Z -invariant easy-axisferromagnetic coupling. The sign of J ±± is immaterialby virtue of the unitary transformation S ± lm → ± iS ± lm .The crucial difference between the U(1)-invariant XXZmodel and Eq. (1) is that the ferromagnetic interac-tion in the former is easy-plane and has a unique signfor non-bipartite lattices, whereas in the latter, the fer-romagnetic interaction is easy-axis and the sign is im- material, which results in a Z -invariant Hamiltonian inthe x - y plane. Thus, Eq. (1) cannot be mapped ontoa U(1)-invariant XXZ model (hard-core boson) on non-bipartite lattices, whereas for bipartite lattices, Eq. (1)is related to the U(1)-invariant counterpart by a simplespin flip. QMC simulations of Carrasquilla et. al on thekagome lattice uncovered three distinct phases, which in-clude the unconventional disordered magnetized lobes forsmall ± h z and J ±± /J z < . , and it is believed to bea candidate for Quantum Spin Liquid (QSL) state, en-abling the search for two-dimensional QSL within a classof pyrochlore QSI materials. The remaining two phasesare the S x and the S z ferromagnetic ordered phases forsmall and large h z respectively.We have provided an analytical explanation of theQMC results using spin wave theory on the kagomelattice. Remarkably, we observed that the average spin-deviation operator (cid:104) n l (cid:105) = (cid:104) b † l b l (cid:105) is very small in QuantumKagome Ice , i.e., quantum fluctuations are suppressed.Hence, we were able to capture a broad trend of the quan-tum phases uncovered by QMC simulations. We havealso obtained the complete phase diagram of Eq. (1) onthe triangular lattice (
Quantum Triangular Ice ), usingthe semiclassical large- S expansion. In the case of tri-angular lattice, there is an additional phase at h z = 0 and J z → ∞ . This phase is selected by quantum fluctu-ations via order-by-disorder mechanism and it is called a ferrosolid state — a state similar to a supersolid state inthe U(1)-invariant XXZ model, but exhibits brokentranslational and Z symmetries. It also appears adja-cent to the unconventional disordered magnetized lobesfor small ± h z . Again, spin wave theory provides an ac-curate picture of the quantum phases. The XY limit ofthis model on both lattices has also been analyzed usingspin wave theory and QMC. In this paper, we explore the ground state propertiesof the easy-axis ferromagnetic phase of this model on thetriangular lattice. A related U(1)-invariant XXZ model a r X i v : . [ c ond - m a t . s t r- e l ] N ov in the superfluid phase has been studied previously on thetriangular lattice, using series expansion methods andspin wave theory. The present model has not been stud-ied in the ferromagnetic phase. The study of this modelhas some physical relevance since many magnetic mate-rials are ferromagnets. This includes the unconventionalCeRh B , which has been studied by many authors. The ferromagnetic XY coupling also plays a prominentrole in the experimental realization of hard-core bosons inultracold atoms on quantum optical lattices.
Hence,it is expedient to understand the ferromagnetic natureof the Z -invariant model, as it might be applicable tothese systems. We will consider a more general Hamilto-nian given by H = − J ±± (cid:88) (cid:104) lm (cid:105) (cid:0) S + l S + m + S − l S − m (cid:1) + (cid:88) lm J l,m S zl S zm − h z (cid:88) l S zl − h x (cid:88) l S xl , (2)where J l,m = J z on the nn sites and J l,m = J (cid:48) z on thenext-nearest-neighbour (nnn) sites. The external mag-netic fields are introduced to enable the calculation ofparallel and longitudinal magnetizations. The Z sym-metry of Eq. (2) at h x = 0 is synonymous with the factthat the total S z is not conserved.In this model, spin wave theory is exact at the Heisen-berg point J (cid:48) z = h x,z = 0 and J z = J ±± = J . The exactground state in this limit is a ferromagnet along the x -direction. The excitation spectrum, however, exhibits nosoft (Goldstone) modes at k = 0 and the average spin-deviation is not divergent at finite temperature. Hence,the discrete Z symmetry is spontaneously broken evenat finite temperature, thus Mermin-Wagner theorem isnot applicable. This unusual feature is obviously absentin rotational invariant systems. Away from the Heisen-berg point, the average spin-deviation (cid:104) n l (cid:105) does not ex-ceed . for all parameter regimes considered at half-filling h x,z = 0 . Thus, spin wave theory gives a very gooddescription of the system. We calculate the excitationspectrum (cid:15) ( k ) , the density of states, and the static struc-ture factors S zz ( k ) and S ± ( k ) . We also calculate theparticle density and the non-divergent condensate frac-tion at k = 0 . In contrast to the U(1)-invariant model,with a phonon dispersion near k = 0 , the spectrum ofthe Z -invariant model exhibits a maxon dispersion near k = 0 with a gap of ∆ ∝ (cid:112) J ±± ( J z + J (cid:48) z + J ±± ) at half-filling ( h x,z = 0 ). We see that the gap does not vanish for J ±± (cid:54) = 0 . For fixed J (cid:48) z , J ±± and varying J z , we observewell-defined roton minima at some points inside the Bril-louin zone. The spectrum also exhibits some profoundflat mode at some paths of the Brillouin zone and the gapat the Brillouin zone corners K = ( ± π/ , vanishes at J (cid:48) z = J ±± − J z / . Thus, for J (cid:48) z = 0 the transition be-tween the easy-axis ferromagnet and the ferrosolid occursat J z = 2 J ±± , which recovers the result obtained in theU(1)-invariant XXZ model. However, spin wave theorygives a better description of the present model. The static structure factors exhibit some distinctive features withsharp peaks at the minima of the energy spectra and nodiscontinuity in the entire Brillouin zone. Likewise, thedensity of states exhibit interesting features with peaksat various energies.
Linear spin wave theory –. We now consider theanisotropic Hamiltonian in Eq. (2) in the easy-axis ferro-magnetic phase. At the Heisenberg point J (cid:48) z = h x,z = 0 and J z = J ±± = J , the resulting Hamiltonian can bewritten as H = J (cid:88) (cid:104) lm (cid:105) − S xl S xm + 12 ( S + l S − m + S − l S + m ) . (3)We have chosen x -quantization axis, hence S ± l = S zl ± iS yl . For bipartite lattices, Eq. (3) can be transformed toSU(2) Heisenberg ferromagnet by a π -rotation about the x -axis on one sublattice, i.e. , S xm → S xm , S ± m → − S ± m . Fornon-bipartite lattices, such rotation cannot be performed,thus Eq. (3) differs from the SU(2)-invariant Heisenbergferromagnet, and exhibits Z × U(1) symmetry with U (1) symmetry in the z - y plane and Z symmetry in the x -axis. The easy-axis ferromagnetic ordered state resultsfrom the spontaneous breaking of Z symmetry alongthe x -axis, (cid:104) S x (cid:105) (cid:54) = 0 and (cid:104) S zy (cid:105) = 0 . Thus, for spin- / ,the state | ψ F M (cid:105) = (cid:81) l | S xl = ↑(cid:105) is an exact eigenstate ofEq. (3) with E MF = − JN S , and (cid:104) S x (cid:105) = S . As we willshow, the resulting excitation above this ground statehas a gapped quadratic dispersing mode near k = 0 ,signifying no Goldstone mode due to the absence of acomplete continuous symmetry. This is in sharp contrastto the superfluid phase with broken U (1) symmetry inthe x - y plane (cid:104) S xy (cid:105) (cid:54) = 0 and a gapless linear excitationat k = 0 .For the anisotropic model, the mean field energy in theferromagnetic phase is given by E MF ( θ ) = 3 N S ( λ z + λ z − h/ , (4)where N is the total number of sites and λ z = J z cos θ − J ±± sin θ ; λ z = J (cid:48) z cos θ ; (5) h = h z cos θ + h x sin θ. (6)The minimization of the mean field energy with respectto θ yields h z − J cos ϑ = h x cot ϑ, (7)where ϑ is the angle that minimizes Eq. (4) and J =( J z + J (cid:48) z + J ±± ) . We can eliminate h x from Eq. (4) usingEq. (7), the classical energy becomes E MF ( ϑ ) = 3 N S [ J sin ϑ + J z + J (cid:48) z − h z sec ϑ/ . (8)It should be noted that ϑ is a function of h x . At h x =0 , the solution of Eq. (7) gives cos ϑ = h z /h c , where h c = 6 J is the critical field above which the spins arefully polarized along the z -direction. The correspondingclassical energy is given by E MF ( ϑ ) = − N S [ J cos ϑ + J ±± ] . (9)For h x (cid:54) = 0 , the mean field energy is obtained perturba-tively for small h x , E MF ( ϑ ) = E MF ( h x = 0) + h x ∂ E MF ( ϑ ) ∂h x (cid:12)(cid:12)(cid:12)(cid:12) h x =0 + · · · , (10)where E MF ( h x = 0) = E MF ( ϑ = ϑ ) . We perform spinwave theory in the usual way, by rotating the coordinateabout the y -axis in order to align the spins along theselected direction of the magnetization. S xl = S (cid:48) xl cos θ + S (cid:48) zl sin θ,S yl = S (cid:48) yl , (11) S zl = − S (cid:48) xl sin θ + S (cid:48) zl cos θ. Next, we express the rotated coordinates in terms of thelinearized Holstein Primakoff (HP) transform. S (cid:48) zl = S − b † l b l ,S (cid:48) yl = i (cid:114) S (cid:16) b † l − b l (cid:17) , (12) S (cid:48) xl = (cid:114) S (cid:16) b † l + b l (cid:17) . The truncation of the HP transformation at linear or-der is guaranteed provided the average spin-deviationoperator (cid:104) n l (cid:105) = (cid:104) b † l b l (cid:105) is small. Indeed, (cid:104) n l (cid:105) is smallin the present model, hence linear spin wave theory issuitable for the description of this system. Taking themagnetic fields, h x,z , to be of order S and keeping onlythe quadratic terms, the resulting bosonic Hamiltoniancan be diagonalized by the Bogoliubov transformation, b k = u k α k − v k α †− k , (13)where u k − v k = 1 , one finds that the resulting Hamilto-nian is diagonalized by u k = 12 (cid:18) A k ( ϑ ) ω k ( ϑ ) + 1 (cid:19) ; v k = 12 (cid:18) A k ( ϑ ) ω k ( ϑ ) − (cid:19) , (14)with ω k ( ϑ ) = (cid:112) A k ( ϑ ) − B k ( ϑ ) . The diagonal Hamilto-nian yields H = S (cid:88) k ω k ( ϑ ) (cid:16) α † k α k + α †− k α − k (cid:17) . (15)The excitation of the quasiparticles is given by (cid:15) k ( ϑ ) = 2 Sω k ( ϑ ) = 2 S (cid:113) A k ( ϑ ) − B k ( ϑ ) , (16)while the spin wave ground state energy is given by E SW ( ϑ ) = E MF ( ϑ ) + S (cid:88) k [ ω k ( ϑ ) − E lo ( ϑ )] , (17) where A k ( ϑ ) = E lo ( ϑ ) + 3 J ±± γ k + B k ( ϑ ); (18) B k ( ϑ ) = 32 (¯ g k ( ϑ ) + g k ( ϑ )) ; (19) E lo ( ϑ ) = − J z + J (cid:48) z ) + h z sec ϑ/ , (20) ¯ g k ( ϑ ) = ( J z + J ±± ) sin ϑγ k − J ±± γ k , (21) g k ( ϑ ) = J (cid:48) z sin ϑ ¯ γ k . (22)We have eliminated h x using Eq. (7). The structure fac-tors are given by γ k = 13 (cid:32) cos k x + 2 cos k x √ k y (cid:33) , (23) ¯ γ k = 13 (cid:32) cos √ k y + 2 cos 3 k x √ k y (cid:33) . (24) Excitation spectra –. We now investigate the natureof the energy spectra of Eq. (2). We will be interestedin half-filling or zero magnetic fields and spin- / . Weadopt the Brillouin zone paths of Refs. [11,12] as shown inFig. (1). Our main focus is the appearance of a minimuminside the Brillouin zone (roton minimum) and the possi-bility of any soft modes. The vanishing of the spectrumat the corners of the Brillouin zone represents a phasetransition to a new spin configuration. The simplest caseis the Heisenberg point h x,z = J (cid:48) z = 0; J z = J ±± = J .In this limit spin wave theory is exact and the excita-tion spectrum of Eq. (3) is ω k = A k = 3 J (1 + γ k ) ,where B k = 0 . We see that the exact ground state isthe fully polarized easy-axis ferromagnet along the x -axis, and the corresponding excitation exhibits no zeromodes since − / ≤ γ k ≤ ; see Fig. (2). Near k = 0 ,the energy behaves as ω k ≈ a − b k , with a = 6 J and b = 3 J/ . Hence, the average spin deviation operatornear this mode at low-temperature is given by ∆ S x ( T ) = (cid:90) d k e ω k /k B T − ∼ T ln(1 − e − a ) b . (25)This unusual feature means that the discrete Z symme-try is spontaneously broken even at finite temperature,thus Mermin-Wagner theorem does not apply. Anotherimportant feature of Eq. (25) is that we can suppressquantum fluctuations by making the gap “ a ” as large aspossible.As depicted in Figs. (2) and (3), the energy spectraof the Z -invariant XXZ model have a similar behaviourto the U(1)-invariant XXZ model at the corners ofthe Brillouin zone and along RM except for the case ofXY model. At the corners of the Brillouin zone, thespectrum vanishes when J (cid:48) z = J ±± − J z / . Hence at J (cid:48) z = 0 , the transition from the easy-axis ferromagnetto a ferrosolid occurs at J ±± − J z / , which hap-pens to be the same as the U(1)-invariant model. As -1.5 0 1.5 k x / π -1.501.5 k y / π R Γ NQ KMP
FIG. 1: Color online. The Brillouin zone of the triangularlattice and the corresponding paths that will be adopted inthis paper. ǫ ( k ) R M Γ N Q K M PR M Γ N Q K M PR M Γ N Q K M PR M Γ N Q K M P
FIG. 2: Color online. The plots of the energy dispersion at h x,z = 0 ( ρ = 0 . and J ±± = 1; J (cid:48) z = 0 . XY model: J z = 0 (dashed). Heisenberg model: J z = 1 (solid). The blue curvesdenote the present model and the red curves denote the U(1)model. we will show in the subsequent section, spin wave the-ory gives a better description of the present model. Incontrast to the U(1)-invariant model, the spectra of the Z -invariant model is invariably gapped in the entire Bril-louin zone and the spectra display a maxon dispersion atthe Γ point ( k = 0 ) as opposed to the usual phonondispersion in rotational invariant systems. The gappednature of the Z -invariant model is as a consequence ofthe Z -symmetry of the Hamiltonian and it plays a verycrucial role in the quantum phase diagram of the Quan-tum Kagome Ice uncovered by QMC.
The contribu-tion from the k = 0 mode is the most important fea-ture of the Z -invariant model. The gap at the Γ pointbehaves as ∆ ∝ (cid:112) J ±± ( J z + J (cid:48) z + J ±± ) , which vanishesonly for J ±± = 0 . In the subsequent section, we willshow that the k = 0 mode enhances the estimated val-ues of the thermodynamic quantities. These trends are ǫ ( k ) R M Γ N Q K M PR M Γ N Q K M PR M Γ N Q K M PR M Γ N Q K M PR M Γ N Q K M PR M Γ N Q K M PR M Γ N Q K M P
FIG. 3: Color online. The plots of the energy dispersion at h x,z = 0 ( ρ = 0 . , J ±± = J z = 1 and several values of J (cid:48) z = 0 . (dashed), J (cid:48) z = − . (solid), and J (cid:48) z = 1 (symbol).The colors have the same meaning as in Fig. (2). slightly modified away from half-filling. Thermodynamic quantities –. The effects of thegapped excitations in the preceding section are man-ifested explicitly in the ground state thermodynamicquantities. We compute the magnetizations per site givenby (cid:104) S z (cid:105) = − SN ∂ E SW ( ϑ ) ∂h z , (26) (cid:104) S x (cid:105) = − SN ∂ E SW ( ϑ ) ∂h x (cid:12)(cid:12)(cid:12)(cid:12) ϑ = ϑ . (27)Using Eq. (17) we obtain (cid:104) S z (cid:105) = S cos ϑ + cos ϑ N (cid:88) k Θ k (cid:115) A k ( ϑ ) − B k ( ϑ ) A k ( ϑ ) + B k ( ϑ ) . (28)with Θ k = ( J z + J ±± ) γ k + J (cid:48) z ¯ γ k . The total density ofparticles is given by ρ = S + (cid:104) S z (cid:105) . To linear order in h x we find (cid:104) S x (cid:105) = S sin ϑ − cos ϑ sin ϑ N (cid:88) k Θ k (cid:115) A k ( ϑ ) − B k ( ϑ ) A k ( ϑ ) + B k ( ϑ ) (29) −
12 sin ϑ N (cid:88) k (cid:20) A k ( ϑ ) ω k ( ϑ ) − (cid:21) . Similar to the U(1)-invariant model, the condensate frac-tion at k = 0 is related to the S x -order parameter by ρ = lim S → / (cid:104) S x (cid:105) . To linear order in spin wave theory ρ is given by ρ = ρ c − cos ϑ N (cid:88) k Θ k (cid:115) A k ( ϑ ) − B k ( ϑ ) A k ( ϑ ) + B k ( ϑ ) (30) −
12 1 N (cid:88) k (cid:20) A k ( ϑ ) ω k ( ϑ ) − (cid:21) . ρ h n l i J ′ z = 0 J ′ z = 0 . J ′ z = 1 FIG. 4: Color online. The spin-deviation operator against theparticle density at h x = 0 , J ±± = J z = 1 . -10 -8 -6 -4 -2 0 h z ρ FIG. 5: Color online. The particle density ρ vs. h z at J z = J ±± = 1 , J (cid:48) z = 0 , and S = 1 / . where ρ c = sin ϑ is the classical condensate fraction.An important feature of the Z -invariant model is thatall the thermodynamic quantities are finite at all pointsin the Brillouin zone. This enhances the estimated val-ues of the thermodynamic quantities. At the XY point, J z = J (cid:48) z = h x,z = 0 , the estimated value of the orderparameter is (cid:104) S x (cid:105) = S − . , which should be com-pared to the O(2)-invariant model (cid:104) S x (cid:105) = S − . . We see that quantum fluctuations are suppressed inthe Z -invariant model. A detail analysis of the XYmodel has been given elsewhere for the triangular andthe kagome lattices. As we can see from Fig. (4), thespin-deviation operator (cid:104) n l (cid:105) is extremely small close tohalf-filling and also very small away from half-filling. Inother words, linear spin theory is very suitable for de-scribing the ground state properties of the Z -invariantXXZ model on non-bipartite lattices. We have shownthe trend of the particle density in Fig. (5) as a functionof h z . In Fig. (6), we plot the condensate fraction at k = 0 as a function of ρ . At ρ = 0 . , the estimated val-ues of ρ at the XY point are ρ = 0 . linear spin wave ρ h n l i J z = 0; J ′ z = 0 J z = 1; J ′ z = 0 J z = J ′ z = 1 FIG. 6: Color online. The condensate fraction, ρ , againstthe particle density ρ at J ±± = 1 and S = 1 / . theory of the Z -invariant model, ρ = 0 . series ex-pansion of the O(2)-invariant model, and ρ = 0 . second-order spin wave theory on the square lattice. Atthe Heisenberg point, ρ = ρ c = 0 . as expected. Structure factors –. Due to the gapped nature ofthe Z -invariant model, all the thermodynamic quantitiesbehave nicely without any divergent contributions. Thedensity of states for this model is depicted in Fig. (7) forseveral values of the anisotropies. There are some strikingfeatures in the density of states of this model. We observeseveral spikes (van Hove singularities) depending on theanisotropies, which stem from the k = 0 mode and theflat mode along RM . In the case of XY model, the majorcontribution to the spike comes from the k = 0 mode andthe discontinuity is a result of the lowest energy states atthe roton minima at the corners of the Brillouin zone.Let us turn to the calculation of the dynamical struc-ture factors, which is given by S βγ ( k , ω ) = 12 π (cid:90) ∞−∞ dt e iωt (cid:104) S β k ( t ) S γ − k (0) (cid:105) , (31)where S β k = √ N (cid:80) l e − i k l S βl is the Fourier transform ofthe operators, and β, γ = ( x, y, z ) label the componentsof the spins. At zero temperature, the structure factorsare obtained from the Green’s function: S βγ ( k , ω ) = − π Im G βγ ( k , ω ) , (32)with G βγ ( k , ω ) = − i (cid:104)T S β k ( t ) S γ − k (0) (cid:105) being the time-ordered retarded Green’s function. At half-filling or zeromagnetic fields, θ = π/ , Eq. (11) gives S xl → S (cid:48) zl and S zl → − S (cid:48) xl . The quantization axis in this case is alongthe x -axis and the off-diagonal terms are S ± l = S zl ± iS yl .Using linear spin wave theory to order S we find thetwo static structure factors at half-filling: S zz ( k ) = S ( u k − v k ) / and S ± ( k ) = 2 Sv k . The static structurefactors S zz ( k ) and S ± ( k ) are shown in Figs. (8) and (9)respectively. In contrast to the U (1) -invariant model, the ω ρ ( ω ) J z = 0; J ′ z = 0 J z = 0; J ′ z = 1 J z = 0 . J ′ z = 1 J z = 1; J ′ z = 1 FIG. 7: Color online. The density of states D ( ω ) vs. ω athalf-filling, h x,z = 0 ( ρ = 0 . ; J ±± = J (cid:48) z = 1 . S zz ( k ) R M Γ N Q K M PJ z = 0; J ′ z = 0 J z = 1; J ′ z = 0 J z = 1 . J ′ z = 0 J z = 1 . J ′ z = 1 FIG. 8: Color online. The static dynamical structure factor, S zz ( k ) , along the Brillouin zone paths in Fig. (1) at h x,z =0 ( ρ = 0 . ; J ±± = 1 and S = 1 / . static structure factors show no discontinuity in the entireBrillouin zone. Our calculation shows that S zz ( k ) devel-ops sharp step-like peaks at the minima of the energyspectra and it is completely flat at the Heisenberg point J ±± = J z = 1 , J (cid:48) z = 0 . The off-diagonal term S ± ( k ) exhibits a similar behaviour but become completely zeroexcept at the peaks. At K = Q = ( ± π/ , we find S ( Q ) = S (cid:115) J ±± J ±± + J (cid:48) z ) − J z . (33)Figure (10) shows the plot of S ( Q ) as a function of J z . Conclusion –. We have explored the ground statethermodynamic properties of the easy-axis ferromagneticphase of the quantum triangular ice model. We observedfascinating features which are different from the U(1)-invariant XXZ model. In particular, divergent and dis-continuous quantities in the U(1)-invariant XXZ modelare finite in the Z -invariant XXZ model, hence all thepoints in the Brillouin zone contribute to the thermo- S ± ( k ) R M Γ N Q K M PJ z = 0; J ′ z = 0 J z = 1; J ′ z = 0 J z = 1 . J ′ z = 0 J z = 1 . J ′ z = 1 FIG. 9: Color online. The static dynamical structure fac-tor, S ± ( k ) , along the Brillouin zone paths in Fig. (1). Theparameter are the same as in Fig. (8). J z S zz ( Q ) J ′ z = 0 J ′ z = 0 . J ′ z = 1 FIG. 10: Color online. The static structure factor, S zz ( Q ) asa function of J z at J ±± = 1 . dynamic quantities. Interestingly, we found that linearspin wave theory provides an accurate picture of this sys-tem with the spin-deviation operator (cid:104) n l (cid:105) < . forall parameter regimes considered. In contrast to rota-tional invariant systems, at the Heisenberg point, theferromagnetic state of the Z -invariant model exhibitsno Goldstone mode at k = 0 . The excitation spectrum isgapped in the entire Brillouin zone, and the average spin-deviation is finite at low temperature near the gappedstates. Hence, the Z discrete symmetry is spontaneouslybroken even at finite temperature. As a result of the dis-tinctive features of this model, the particle density andthe condensate fraction at half-filling give reasonable es-timates at the level of our spin wave theory. Also, thedynamical structure factors and the density of states ex-hibit interesting peaks at unusual momenta. The mostdistinctive feature of the Z -invariant model comes fromthe k = 0 mode. This mode plays a very prominent rolein the unconventional phases obtained from the fully frus-trated Z -invariant model. The features uncovered inthis paper might be useful for experimental purposes ingapped physical systems the could be modeled with the Z -invariant model. It is also interesting to investigatethe nature of this model in quantum optical lattices. An investigation of magnon decay would be of interest inthe Z -invariant model, this requires one to go beyondthe linear spin wave approximation; however, this is notvery important at the level of our investigation since lin- ear spin wave theory provides reasonable estimates of thethermodynamic quantities. Acknowledgments –. The author would like to thankAfrican Institute for Mathematical Sciences (AIMS),where this work was conducted. Research at Perime-ter Institute is supported by the Government of Canadathrough Industry Canada and by the Province of Ontariothrough the Ministry of Research and Innovation. ∗ Electronic address: [email protected] † Electronic address: [email protected] Gingras, M. J. P. and McClarty, P. 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