Divergence of the Grüneisen ratio at symmetry-enhanced first-order quantum phase transitions
DDivergence of the Grüneisen ratio at symmetry-enhanced first-order quantum phasetransitions
Charlotte Beneke and Matthias Vojta
Institut für Theoretische Physik and Würzburg-Dresden Cluster of Excellence ct.qmat,Technische Universität Dresden, 01062 Dresden, Germany (Dated: February 4, 2021)Studies of the Grüneisen ratio, i.e., the ratio between thermal expansion and specific heat, havebecome a powerful tool in the context of quantum criticality, since it was shown theoretically thatthe Grüneisen ratio displays characteristic power-law divergencies upon approaching the transitionpoint of a continuous quantum phase transition. Here we show that the Grüneisen ratio also divergesat a symmetry-enhanced first-order quantum phase transition, albeit with mean-field exponents, asthe enhanced symmetry implies the vanishing of a mode gap which is finite away from the transition.We provide explicit results for simple pseudo-spin models, both with and without Goldstone modesin the stable phases, and discuss implications.
I. INTRODUCTION
Quantum phase transitions (QPTs) constitute an im-portant topic in condensed-matter research:
A QPT isassociated with qualitative changes of the ground state ofa many-body system, for instance its entanglement prop-erties. Moreover, the quantum critical regime of contin-uous QPTs displays phenomenology very different fromthat of stable phases and is often the source for novelphysics. A number of signatures and tools have beenidentified to diagnose QPTs, such as critical power lawsas function of absolute temperature T , with associatedexponents, and universal scaling behavior.Among the thermodynamic diagnostics for continu-ous QPTs, the Grüneisen ratio (sometimes also calledGrüneisen parameter) – defined as the ratio betweenthermal expansion α and specific heat c p , Γ = α/c p – is particularly revealing: In Ref. 4 it was theoreti-cally shown that it diverges upon approaching a pressure-driven quantum critical point in a characteristic power-law manner, and also displays sign changes near a quan-tum critical point. For magnetic-field-driven transitions,the role of Γ is taken by the magnetic Grüneisen ratio, Γ H = − ( ∂M/∂T ) H /c H , which can be determined fromthe magnetocaloric effect. Beyond standard quantumcritical points, the Grüneisen ratio has also been con-sidered for disorder-dominated quantum Griffiths phasesand found to display a much weaker logarithmic diver-gence as function of temperature. On the experimental side, measurements of theGrüneisen ratio have been frequently used to detect andcharacterize quantum phase transition. Prominent ex-amples are the heavy-fermion compounds CeNi Ge , YbRh Si , , CeCu − x Ag x , and CeIn − x Sn x where adivergence of Γ was found which could be attributed toa quantum phase transition. Interestingly, divergenciesof the magnetic Grüneisen ratio have also been found ina number of compounds where no obvious quantum crit-ical point exists, and we refer the reader to Ref. 11 fora review. At present, a consistent explanation for theseobservations is lacking. In the field of QPTs, a particularly interesting develop-ment concerns the emergence of enhanced symmetries atthe transition point. These are symmetries not present inthe underlying Hamiltonian, but emergent at long timesand distances in the critical regime. This has been dis-cussed in particular in the context of deconfined quan-tum critical points. For instance, the transition betweena Néel antiferromagnet and a columnar valence-bondsolid has been argued to display an emergent SO(5)symmetry, and a number of field-theoretic dualitieshave been invoked to rationalize enlarged symmetries. Likewise, enlarged symmetries have been detected in nu-merical simulations of Z gauge theories coupled to Diracfermions and also at the ordering transition of a classi-cal dimer model. Symmetry enhancement is also possible at first-orderQPTs. This refers to situations where the system discon-tinuously switches between two types of order, with thetransition point displaying an emergent higher symme-try, leading to a family of stable states. Such behaviorhas been recently detected in numerical simulations ofa SU(2)-symmetric spin model on the two-dimensionalShastry-Sutherland lattice where an emergent O(4) sym-metry appeared at the transition between an antiferro-magnet and a plaquette singlet state. In a related spinmodel on a square lattice, such a first-order transitionhas been found to display emergent SO(5) symmetry. Other examples of enhanced symmetries at first-ordertransitions appeared in Refs. 21 and 22. Together, thismotivates to consider the phenomenology of symmetry-enhanced first-order QPTs in more detail, not the leastto provide guidance to experiments.In this paper we argue that the Grüneisen ratio doesnot only diverge at quantum critical points, but generi-cally also at symmetry-enhanced first-order QPTs. Thereason is that the enhanced symmetry implies the exis-tence of an excitation mode which is gapless only at thetransition point but gapped away from it. We provide ex-plicit results for simple effective pseudospin models wherewe determine the full crossover behavior of the Grüneisenratio. We demonstrate that the Grüneisen ratio displaysnot only a jump accompanied by a sign change upon a r X i v : . [ c ond - m a t . s t r- e l ] F e b crossing the transition at finite temperature, but alsocharacteristic divergencies upon approaching the zero-temperature transition point. The type of divergenciesdepends on the presence or absence of Goldstone modesin the stable phases; the presence of Goldstone modesmay lead to further sign changes of the Grüneisen ratio.We also comment on further experimental implications.The remainder of the paper is organized as follows:In Section II we summarize properties of the Grüneisenratio and argue why a divergence can be expected atsymmetry-enhanced first-order QPTs. Section III intro-duces the two spin models which we use to exemplifythis divergence, with explicit results shown in Sections IVand V for an Ising-Ising and a XY-Ising transition, re-spectively. A general discussion in Sec. VI concludes thepaper. Technical details are relegated to the Appendix. II. GRÜNEISEN RATIO: GENERALCONSIDERATIONS
A QPT occurs at T = 0 as function of a non-thermalcontrol parameter such as pressure, magnetic field, orchemical substitution. In the pressure-driven case, withthe transition located at p c , one may define a dimension-less control parameter r = ( p − p c ) /p c . At a continuousQPT, the free-energy density contains a critical contri-bution which can be probed via the volume thermal ex-pansion α = 1 V ∂V∂T (cid:12)(cid:12)(cid:12)(cid:12) p = − V ∂S∂p (cid:12)(cid:12)(cid:12)(cid:12) T . (1)and the specific heat capacity c p = TN ∂S∂T (cid:12)(cid:12)(cid:12)(cid:12) p . (2)where S is the entropy, V the sample volume, and N the particle number. The Grüneisen ratio is commonlydefined as Γ = αc p = − V n T ( ∂S/∂p ) T ( ∂S/∂T ) p (3)where V n = V /N the volume per particle. For field-driven transitions, the dimensionless control parameteris r = ( H − H c ) /H c , such that one can define a quantity Γ H = (1 /T )( ∂S/∂H ) T / ( ∂S/∂T ) H , also known as mag-netic Grüneisen ratio, which takes the role of Γ .Using thermodynamic hyperscaling arguments, Ref. 4showed that the Grüneisen ratio diverges in the quantumcritical regime of a continuous QPT according to Γ cr ( T, r = 0) ∝ T − / ( νz ) (4)where ν and z are the correlation-length and dynamicexponents, respectively. Likewise, approaching the QCPat low temperatures yields the divergence Γ cr ( T = 0 , r ) ∝ | r | − . (5) More precisely, Eqs. (4) and (5) are obeyed in the regimes T (cid:29) | r | νz and T (cid:28) | r | νz , respectively. Ref. 4 also showedthat these divergencies also hold (up to possible logarith-mic corrections) for systems where hyperscaling does notapply, i.e., above the upper critical dimension.Importantly, Γ does not diverge upon approaching afinite- T phase transition, thus a divergence of Γ is usu-ally considered a unique signature of a continuous QPT.Notably, for self-dual QCPs it can be shown that theGrüneisen ratio remains finite, as the prefactor of theleading divergence vanishes; this applies for instanceto the transverse-field Ising model in one space dimen-sion.The scaling arguments put forward in Ref. 4 describethe vicinity of a QCP, but more generally are validif an excitation mode becomes soft as a function of anon-thermal control parameter. This is precisely whatalso happens at a symmetry-enhanced first-order QPT:The enhanced symmetry implies a larger number of softmodes at the transition point (as compared to away fromit), hence we expect a divergent Grüneisen ratio. Thiswill be demonstrated explicitly in the remainder of thepaper. III. EFFECTIVE SPIN MODELS
The microscopic models for which non-trivialsymmetry-enhanced first-order QPTs have been estab-lished are complicated and not amendable to simpleapproximate solutions. We will therefore analyse toymodels where the enhanced symmetry is explicit insteadof emergent; these models should be understood as effective models describing the relevant degrees offreedom near a symmetry-enhanced first-order QPT.Specifically, we will consider spin models with tunablemagnetic anisotropy. In the context of the first-orderQPTs of interest, the models’ degrees of freedom are tobe interpreted as effective (pseudo)spins, and the mod-els’ anisotropy, encoded in a tuning parameter λ , canbe tuned by hydrostatic pressure (or a similar controlparameter). Thus, the enhanced symmetry at the tran-sition point does not correspond to an enhanced explicitsymmetry of the original system. This aspect will be-come relevant when interpreting the results for thermalexpansion, and we will get back to this below. A. Models
We will study the thermodynamics of two sim-ple nearest-neighbor lattice spin models which displaysymmetry-enhanced first-order transitions. The first,which we dub XZ model, is defined as H XZ = (cid:88) (cid:104) ij (cid:105) ( J x S xi S xj + J z S zi S zj ) , (6) (a) (b) FIG. 1. Schematic phase diagrams of the (a) XZ and (b)XXZ models in space dimension d > . A symmetry-enhancedfirst-order QPT occurs upon tuning λ through λ = 1 . and the second is the XXZ model, with H XXZ = (cid:88) (cid:104) ij (cid:105) [ J x ( S xi S xj + S yi S yj ) + J z S zi S zj ] , (7)where the (cid:126)S i are spins of size s located on sites i ofa regular lattice. For simplicity, we will work on a d -dimensional hypercubic lattice and consider d = 3 unlessnoted otherwise.In both models, we use J x ≡ J > as the unit ofenergy, and parameterize the exchange anisotropy by J z = λJ x with λ > . At low temperatures, both modelsdisplay antiferromagnetic long-range order: For λ > Ising order is realized with spins along the z direction.For λ < the XZ model displays Ising order along the x direction whereas the XXZ model shows planar (XY-type) order. The XZ model is obviously symmetric (orself-dual) under the transformation λ → /λ , J → λJ which exchanges the role of x and z directions in spinspace. The key thermodynamic difference between bothmodels is the existence of a Goldstone mode in the XXZmodel for λ < , whereas the XZ model is gapped forany λ (cid:54) = 1 .The point λ = 1 displays enhanced U(1) ( SU(2) ) sym-metry in the XZ (XXZ) model, respectively, and varying λ through constitutes a symmetry-enhanced first-ordertransition. This implies that the order parameters ofthe phases realized for λ ≷ jump discontinuously if λ is varied through 1. For instance, for the XZ modelat T = 0 and in the classical limit s → ∞ , the stag-gered magnetizations along the x and z directions fol-low m x = s Θ(1 − λ ) and m z = s Θ( λ − , respectively,where Θ is the heavy-side function. Importantly, there isno simple phase coexistence or hysteresis associated withthe transition at λ = 1 . Instead, the system displays or-der which spontaneously breaks a symmetry higher thanin the adjacent λ ≷ phases. Specifically, at λ = 1 the XZ (XXZ) model features one (two) Goldstone modes,respectively.The finite-temperature phase diagrams are schemat-ically shown in Fig. 1. The QPT continues as a ver-tical line of first-order transitions; in general such aline can be curved, but must display infinite slope as T → because the low- T entropies on both sides ofthe transition are equal due to symmetry enhancement, lim T → lim λ → − S = lim T → lim λ → + S .We assume that the anisotropy parameter λ can betuned by pressure: A change of external pressure p changes the sample volume V , and the resulting changesof bond lengths lead to a change in λ , in other words,a volume change leads to a change of the effectiveanisotropy. Hence, ∂S/∂p = ( ∂S/∂λ )( ∂λ/∂p ) , and thefactor ( ∂λ/∂p ) is a system-specific constant. We recallthat the models are effective models, hence the highersymmetry at λ = 1 should not be confused with a higherexplicit symmetry of the original system. B. Calculation of thermodynamics In d = 3 space dimensions, the ordering temperature T N is finite at (and near) the transition point λ = 1 ,see Fig. 1. We work at temperatures below T N , such thatantiferromagnetic order is well-established, and computethe thermodynamic quantities using standard linear spin-wave theory. Using the Holstein-Primakoff representa-tion of the spins, the bilinear piece of the Hamiltoniancan be diagonalized using Fourier- and Bogoliubov trans-formations to yield a system of non-interacting magnonmodes, H SW = (cid:88) (cid:126)k,i ω (cid:126)k,i α † (cid:126)k,i α (cid:126)k,i + const (8)where the momentum summation (cid:80) (cid:126)k runs over the an-tiferromagnetic Brillouin zone of the ordered state, and i is a mode index, for details see the Appendix. The modeenergies ω (cid:126)k,i characterize a free Bose gas, and one cancompute the specific heat from the entropy according to c v = T − N s (cid:88) (cid:126)k,i ω (cid:126)k,i ω (cid:126)k,i / (2 T ) (9)where N s is the number of lattice sites, and we have setBoltzmann’s constant k B = 1 . Note that we compute thespecific heat at constant volume and neglect the differ-ence between c v and c p (which is small in solids). Simi-larly, the thermal expansion follows from α = T − N s (cid:88) (cid:126)k,i ω (cid:126)k,i ∂ω (cid:126)k,i /∂λ ω (cid:126)k,i / (2 T ) . (10)As explained above, we have replaced the pressure deriva-tive ∂/∂p with a λ derivative ∂/∂λ , assuming that ∂λ/∂p = const . Given that, due to the gap closing, atleast one mode energy varies in a non-analytic fashionwith λ upon crossing the QPT, we can expect that both c v and α are non-analytic as function of λ at λ = 1 .In three space dimensions and in the absence of frus-tration, the linear spin-wave approximation provides re-liable results at T = 0 even for small spin size s including s = 1 / . This remains true at finite temperature as longas thermal occupations remain small, i.e., for T (cid:28) T N ,with the mean-field estimate for the Néel temperature T N being dJs . In particular, the qualitative low-energybehavior of the modes near the QPT is dictated by Gold-stone’s theorem and will thus not change upon includinginteractions beyond linear spin-wave theory. We notethat, in the leading order of the spin-wave expansion,the excitation energies scale as ( Js ) which thus sets thenatural unit for temperature. In the following we restrictour attention to the regime T / ( Js ) < . IV. TRANSITION BETWEEN TWO GAPPEDPHASES
We start by analyzing the XZ model (6) which displaysa symmetry-enhanced transition between two gappedIsing phases. The qualitative behavior of specific heat,thermal expansion, and Grüneisen ratio can be derivedanalytically in the regime T (cid:28) Js . Here we summarizethe results, with details given in the Appendix.At the transition point, λ = 1 , the system is an anti-ferromagnet which spontaneously breaks U(1) symmetry,with a single gapless magnon mode with linear dispersion.In the Ising phases realized for λ (cid:54) = 1 the mode gap scalesas ∆ ∝ | λ − | / .We first focus on the high-temperature regime, T (cid:29) ∆ .This includes λ = 1 where a scaling analysis of Eq. (9)yields c v ∝ T d which also holds for λ (cid:54) = 1 as long as T (cid:29) ∆ . The behavior of the thermal expansion is de-termined by the mode evolution with λ . For λ (cid:54) = 1 wehave ∂ ∆ /∂λ ∝ ±| λ − | − / , with the two signs applyingto λ ≷ , respectively. A scaling analysis of the relevantintegral (10) in the limit T (cid:29) ∆ then yields α ∝ ± T d − .As a result, we obtain Γ ∝ ± /T . Remarkably, thispower-law divergence agrees with the scaling result at aquantum critical point Γ ∝ /T /νz if we assume mean-field exponents ν = 1 / and z = 1 . This underlines thecommon origin of the Grüneisen divergence, namely amode gap closing as ∆ ∝ | r | νz where r = λ − in thepresent case.In the low-temperature regime T (cid:28) ∆ both c v and α are exponentially small, and their ratio to leading order isgiven by Γ = (1 / ∆) ∂ ∆ /∂λ which results in a divergence Γ ∝ / ( λ − . Again, this agrees with the scaling resultat a quantum critical point. The self-duality of the model implies Γ( λ, T ) = − Γ(1 /λ, λT ) . This is consistent with Γ ≷ for λ ≷ ,i.e., Γ jumps from negative to positive values upon cross-ing the transition at any T . This is different from the T / ( J s ) -0.01-0.1-1-10-100 FIG. 2. Grüneisen ratio Γ calculated for the XZ model (6) asfunction of tuning parameter λ and temperature T . Γ changessign at the symmetry-enhanced first-order transition at λ = 1 and diverges as T → both for λ → + and λ → − . Notethat the λ axis is logarithmic, emphasizing the self-duality ofthe model w.r.t. λ ↔ /λ . T/(Js)
T/(Js) T = 1 + =1.001=1.01=1.5(a) T=0.006 JsT=0.06 JsT=0.6 Js(b)
FIG. 3. Grüneisen ratio Γ calculated for the XZ model (6) asin Fig. 2, here plotted (a) as function of T for various λ > and (b) as function of λ for various T . The insets show thedata in a log-log fashion to illustrate the power laws. behavior in the quantum critical regime of a continuousQPT where Γ varies analytically at finite T which impliesthat Γ displays a characteristic zero crossing within thequantum critical regime. The analytical considerations are well borne out byour numerical calculations. Numerical results for theGrüneisen ratio Γ , obtained from a lattice evaluation ofEqs. (9) and (10), are displayed in Figs. 2 and 3. Theyshow the full crossover behavior in the vicinity of the T / ( J s ) -0.01-0.1-1-10-100 FIG. 4. Grüneisen ratio Γ calculated for the XXZ model (7)as function of tuning parameter λ and temperature T . While Γ diverges as T → for λ → + , the behavior for λ → − ismore complicated due to the presence of a Goldstone modefor λ < , for details see text. first-order QPT.We finally comment on the behavior at λ = 1 . Here,both thermal expansion and Grüneisen ratio change sign,i.e., are not well defined. This is a result of the assump-tion that a volume change controls the tuning parame-ter λ , which implies that a system placed at λ = 1 willchange its λ value if heated or cooled at fixed p . In otherwords, changing T at fixed p , starting at λ = 1 , drives thesystem in one of the stable phases. Again, we recall thatthe enhanced U(1) symmetry of our model at λ = 1 is anemergent symmetry of the original system, being realizedonly on a particular line in p – T parameter space. V. TRANSITION BETWEEN GAPLESS ANDGAPPED PHASES
We now turn to the XXZ model whose main differencew.r.t. the XZ model is the Goldstone mode existing for λ < due to the spontaneously broken U(1) symmetry.As above, we use simple analytical considerations to de-termine the asymptotic thermodynamic behavior in thevarious regimes.We start with the Ising ordered phase: For λ ≥ thebehavior of the XZ and XXZ are very similar, i.e., gaplesslinearly dispersing modes at λ = 1 and a gap scaling as ∆ ∝ ( λ − / for λ > , but the number of low-energymodes is doubled in the XXZ model. Hence, for λ > werecover the results Γ ∝ /T for T (cid:29) ∆ and Γ ∝ / ( λ − for T (cid:28) ∆ .The situation is different for λ < . Here, we have oneGoldstone mode, corresponding to spin fluctuations inthe XY plane, and one out-of-plane mode which developsa gap ∆ ∝ (1 − λ ) / . Within linear spin-wave theory, themode contributions are strictly additive both for specificheat and thermal expansion, c = c ∆ + c gs and α = α ∆ + α gs . The regime T (cid:29) ∆ has contributions from bothmodes. In this limit we find α gs ∝ T d and α ∆ ∝ − T d − ,hence | α gs | (cid:28) | α ∆ | at low T , because the gapped mode c v T=0.18 JsT=0.30 JsT=0.36 JsT=0.48 Js(a) (b)
FIG. 5. (a) Specific heat c v and (b) thermal expansion α calculated for the XXZ model (7) as function of λ for different T . c v is a continuous function of λ while α jumps and changessign at λ = 1 as a result of the gap closing. A further signchange occurs for λ < , for details see text. displays a stronger λ dependence. Moreover, both c ∆ and c gs scale as T d . This results in a divergence Γ ∝ − /T ,as in Section IV.The thermodynamics of the low- T regime, T (cid:28) ∆ , re-quires a more careful discussion. If the contributions ofthe gapped out-of-plane mode are negligible comparedto that of the Goldstone mode, we have to considerthe latter only. As noted above, its contributions toboth specific heat and thermal expansion scale as T d ,with prefactors both being non-singular as λ → − , seeAppendix for details. Therefore, Γ approaches a finitevalue in this case which, moreover, is positive, sincethe Goldstone-mode velocity increases with λ . How-ever, the condition T (cid:28) ∆ does not automatically im-ply that the out-of-plane mode can be neglected, be-cause in this regime we have α ∆ ∝ − T d/ − e − ∆ /T while α gs ∝ T d . As a result, the Grüneisen ratio changessign for λ < along a line in the T - λ phase diagramwhich ends at the T = 0 transition point. The loca-tion of this line, given by α gs + α ∆ = 0 , can be es-timated as ∆ = (2 + d/ T | ln T / ( Js ) | up to additivecorrections, equivalently − λ ∝ [ T / ( Js )] ln T / ( Js ) .Below this line, the Goldstone-mode contribution to α dominates and the Grüneisen ratio is positive and fi-nite, such that the behavior upon approaching the zero-temperature transition point is characterized by non- T/(Js)
T/(Js) T = 1 + =1.001=1.01(a) T=0.006 JsT=0.06 JsT=0.6 Js(b)
FIG. 6. Grüneisen ratio Γ calculated for the XXZ model (7)as in Fig. 4, here plotted (a) as function of T for various λ > and (b) as function of λ > for various T . The insets showthe data in a log-log fashion to illustrate the power laws. Inthis gapped Ising phase, the behavior is very similar to thatof the XZ model in Fig. 3. commuting limits, lim λ → − lim T → Γ( T, λ ) (cid:54) = lim T → lim λ → − Γ( T, λ ) . (11)The full numerical result for Γ is displayed in Fig. 4,with c v and α shown individually in Fig. 5. As antic-ipated, α and Γ change sign twice. Further details of Γ are in Figs. 6, 7. Its non-trivial behavior for λ < ,shown in Fig. 7, becomes clear if one approaches theQPT along different trajectories. For square-root trajec-tories, i.e., fixed κ = T / ∆ , the Grüneisen ratio diverges, Γ( κ ∆( λ ) , λ ) → − T − at sufficiently low T , as analyti-cally shown in the Appendix. In contrast, along straighttrajectories with fixed κ (cid:48) = T / [ Js (1 − λ )] the Grüneisenratio approaches a constant value, Γ( κ (cid:48) (1 − λ ) Js, λ ) → const . , as this trajectory is located below the line with Γ = 0 sufficiently close to the QPT.
VI. CONCLUSIONS
In this paper we have shown that the Grüneisen ratio Γ , i.e., the ratio between thermal expansion and specificheat, generically diverges upon approaching a symmetry-enhanced first-order QPT, provided that it can be drivenby pressure. Such a divergence, previously discussed andanalyzed for continuous QPTs, occurs here because the T/(Js)
T/(Js) T = 1=0.99=0.95=0.9(a) T=0.006 JsT=0.06 JsT=0.18 JsT=0.30 JsT=0.48 JsT=0.6 Js 0.001 0.01 0.1 - (b1) (b2) FIG. 7. Grüneisen ratio Γ calculated for the XXZ model (7)as in Fig. 6, but now for the XY phase at λ < . Γ is shown(a) as function of T for various λ < and (b1,b2) as functionof λ for various T . enhanced symmetry is accompanied by a vanishing modegap. Remarkably, the power laws characterizing the di-vergence of Γ have the same form as found at continu-ous QPTs, but with exponents locked to their mean-fieldvalues. Our explicit results also demonstrate an inter-esting interplay of the soft mode(s) arising from symme-try enhancement with the Goldstone modes of the stablephases, leading to additional sign changes of Γ .Our results may be applicable to a number ofcorrelated-electron materials where unconventional first-order-like transitions have been detected. One casein point is Ce Pd Si where a field-driven switch-ing between two magnetic phases, accompanied by amode softening, has been observed in neutron scatteringexperiments. In this context we note that symmetry-enhanced first-order transitions in metals will induce de-viations from Fermi-liquid behavior. The details of thiswill be investigated in future work.
ACKNOWLEDGMENTS
We thank P. M. Consoli, M. Garst, and L. Janssenfor useful discussions as well as for collaborations onrelated subjects. Financial support from the DeutscheForschungsgemeinschaft through SFB 1143 (project-id247310070) and the Würzburg-Dresden Cluster of Excel-lence ct.qmat – Complexity and Topology in QuantumMatter (EXC 2147, project-id 390858490) is gratefullyacknowledged.
Appendix A: Details of spin-wave expansion
To determine the thermodynamics of the spin mod-els (6) and (7), we employ standard linear spin-wavetheory. In all phases, we expand about a two-sublatticeNéel state and introduce two types of Holstein-Primakoffbosons a and b for the A and B sublattices, respectively.As a result, Fourier transformations are performed withmomenta (cid:126)k from the magnetic Brillouin zone, with N s / momentum points, where N s is the total number of lat-tice sites.
1. XZ Model
For the λ ≥ Ising phase of XZ model (6), the spin-wave Hamiltonian reads H SW = H + H , with H =2 dλJN s s the classical ground-state energy and H = 2 dJs (cid:88) (cid:126)k (cid:104) λa † (cid:126)k a (cid:126)k + λb † (cid:126)k b (cid:126)k + γ k a † (cid:126)k b (cid:126)k + a (cid:126)k b − (cid:126)k + h.c. ) (cid:105) (A1)the bilinear fluctuation piece, with the form factor γ k = 1 d d (cid:88) j =1 cos k j (A2)where we have set the lattice constant to unity. Note that γ k ≥ in the magnetic Brillouin zone. H is diagonalizedby a standard × Bogoliubov transformation, yieldingtwo sets of eigenmodes α (cid:126)k,i with mode energies ω (cid:126)k, = 2 dJs (cid:112) λ − λγ k ,ω (cid:126)k, = 2 dJs (cid:112) λ + λγ k . (A3)At the U(1) -symmetric point, λ = 1 , ω (cid:126)k, represents theGoldstone mode of the system, with linear dispersionaround (cid:126)k = 0 . In contrast, ω (cid:126)k, is always gapped.We proceed with an analytical calculation of theGrüneisen parameter at low temperatures and near thetransition point. In this regime, contributions of themode ω (cid:126)k, are exponentially suppressed and can be ne-glected. For small k and small ( λ − , ω (cid:126)k, can be ex-panded as ω (cid:126)k, ≈ (cid:113) ∆ + c | (cid:126)k | (A4)with the gap ∆ ≡ ∆ given by ∆ ( λ ) = 2 dJs (cid:112) λ ( λ − (A5) and the velocity c ( λ ) = Js √ dλ. (A6)To evaluate the thermal expansion, we need the λ deriva-tives of gap and velocity, ∂ ∆ ( λ ) ∂λ = 2 Jsd λ − (cid:112) λ ( λ − ⇒ ∆ ∂ ∆ ( λ ) ∂λ = (2 Jsd ) (cid:18) λ − (cid:19) . (A7)and ∂c ( λ ) ∂λ = c √ λ ⇒ c ( λ ) ∂c ( λ ) ∂λ = 12 λ (A8)with c = c (1) .The expressions (9) and (10) can now be evaluated inthe continuum limit, making use of the spherical sym-metry of the dispersion (A4). We start with the regime T (cid:29) ∆ , reached, e.g., by taking the limit λ → + at finite T . Using the substitution x = βc ( λ ) k where β = 1 /T we obtain lim λ → + c v = C (1) (cid:90) ∞ dx x d +1 ( x/ (A9)and lim λ → + α = C (1) (cid:90) ∞ dx β (2 dJs ) x d − + x d +1 ( x/ (A10)with the prefactor C ( λ ) = Ω d T d c ( λ ) d (A11)where Ω d is defined as the d -dimensional solid angle ofthe hypersphere. The leading low- T behavior is thus c v ∝ T d and α ∝ T d − , with the dominant term in α arisingfrom the λ dependence of the gap. The integrals can beevaluated, resulting in lim λ → + Γ = 2 d ˜Γ( d ) ζ ( d − d +2) ζ ( d +1) (cid:16) TJs (cid:17) − (A12)where ˜Γ( x ) and ζ ( x ) are the Gamma and Riemann zetafunctions, respectively. The expression (A12) matchesthe numerical result in Fig. 3(a) for d = 3 where the pref-actor evaluates to . . Corrections to the leading powerlaw (A12) take the form of a standard high-temperatureexpansion, Γ ∝ [ T / ( Js )] − [1 + O (∆ /T )] .In the opposite limit T (cid:28) ∆ , both specific heat andthermal expansion are exponentially suppressed. For λ > , we have lim T → c v ( λ ) = C ( λ ) β ∆ e − β ∆ × (cid:90) ∞ dx x d − e − x / (2 β ∆ ) (A13)and lim T → α ( λ ) = C ( λ ) β ∆ ∂ ∆ ∂λ e − β ∆ × (cid:90) ∞ dx x d − e − x / (2 β ∆ ) . (A14)The leading low- T behavior of their ratio is thus foundas lim T → Γ( λ ) = ∂ ∆ ( λ ) ∂λ ∆ ( λ ) = 2 λ − λ ( λ − → λ → + / λ − (A15)in agreement with the numerical result in Fig. 3(b).
2. XXZ Model
In the XXZ model (7) the low-temperature phase for λ > ( λ < ) breaks a Z ( U(1) ) symmetry, respectively.Therefore two different calculations are required. a. Ising Phase
For λ ≥ the expansion is performed about a Néelstate polarized along ˆ z . The bilinear piece of the spin-wave Hamiltonian now reads H = 2 dJs (cid:88) (cid:126)k (cid:104) λa † (cid:126)k a (cid:126)k + λb † (cid:126)k b (cid:126)k + γ k ( a (cid:126)k b − (cid:126)k + h.c. ) (cid:105) (A16)with γ k defined in Eq. (A2), and the summation runsover the magnetic Brillouin zone as before. A × Bo-goliubov transformation yields two degenerate magnonmodes with dispersion ω (cid:126)k, , , = 2 dJs (cid:113) λ − γ (cid:126)k . (A17)At the SU(2) -symmetric point at λ = 1 , these modes aregapless at (cid:126)k = 0 . For small k and small ( λ − , the modeenergy can be expanded as in Eq. (A4), but with a gap ∆ ≡ ∆ given by ∆ ( λ ) = 2 Jsd (cid:112) λ − (A18)and a velocity c ( λ ) = 2 Js √ d . (A19)As above, we evaluate the expressions (9) and (10) inthe continuum limit. For T (cid:29) ∆ we find lim λ → + Γ = 4 d ˜Γ( d ) ζ ( d − d +2) ζ ( d +1) (cid:16) TJs (cid:17) − (A20)which is identical to the corresponding result in the XZmodel up to a factor of two arising from the different ∆ dependence of the gap, Eq. (A5) vs. Eq. (A18). For T (cid:28) ∆ we find lim T → Γ( λ ) = λλ − → λ → + / λ − (A21)which agrees with the XZ-model result. b. XY Phase For λ ≤ the expansion is performed about a Néelstate in the XY plane. The linear spin-wave Hamiltoniantakes the form H = 2 dJs (cid:88) (cid:126)k (cid:104) a † (cid:126)k a (cid:126)k + b † (cid:126)k b (cid:126)k + γ k (1 − λ )( a † (cid:126)k b (cid:126)k + h.c. )+ γ k (1 + λ )( a (cid:126)k b − (cid:126)k + h.c. ) (cid:105) . (A22)A × Bogoliubov transformation now yields two modeswith dispersion ω (cid:126)k, = 2 dJs (cid:113) γ (cid:126)k (cid:0) − λ (cid:0) γ (cid:126)k (cid:1)(cid:1) ,ω (cid:126)k, = 2 dJs (cid:113) − γ (cid:126)k (cid:0) − λ (cid:0) − γ (cid:126)k (cid:1)(cid:1) , (A23)which are non-degenerate except for λ = 1 where ω (cid:126)k, = ω (cid:126)k, . For λ < the second mode represents the (in-plane)Goldstone mode of the XY phase whereas the first (out-of-plane) mode develops a gap, ∆ ≡ ∆ . For small k andsmall (1 − λ ) a Taylor expansion of the mode energiesyields ω ≈ (cid:113) ∆ + c | (cid:126)k | ,ω ≈ c ( λ ) | (cid:126)k | (A24)with ∆ ( λ ) = 2 dJs (cid:112) − λ ) ,c ( λ ) = Js (cid:112) d (3 λ − ,c ( λ ) = Js (cid:112) d (1 + λ ) . (A25)Both specific heat and thermal expansion acquire con-tributions from both modes, such that the Grüneisen ra-tio becomes Γ = α gs + α ∆ c gs + c ∆ . (A26)In the regime T (cid:29) ∆ we recover the power laws c gs , c ∆ ∝ T d , moreover we have α gs ∝ T d and α ∆ ∝ − T d − . Eval-uating the integrals yields lim λ → − Γ = − d ˜Γ( d ) ζ ( d − d +2) ζ ( d +1) (cid:16) TJs (cid:17) − (A27)with the sign being reversed compared to Eq. (A20). Theopposite regime T (cid:28) ∆ contains a region where thecontributions to c v and α from the gapped out-of-planemode are negligible compared to that from the Gold-stone mode. The latter scale as T d and are non-singularas λ → − because c ( λ ) is non-singular. This results ina positive and non-divergent Grüneisen ratio lim T → Γ( λ ) = 12(1 + λ ) → λ → − / (A28)in the region where the gapped mode can be neglected –this does not apply to the entire regime T (cid:28) ∆ as willbecome clear shortly.To understand the crossover from the non-divergentbehavior (A28) to the divergent behavior (A27), weevaluate the gapped-mode contributions in the regime T (cid:28) ∆ . By expanding the argument of the sinh in Eqs.(9) and (10) we find c ∆ = C ( λ )( β ∆) d/ e − β ∆ , (A29) α ∆ = − C ( λ )(2 d ) ( βJs ) ( β ∆) d/ e − β ∆ , (A30)with the prefactor C ( λ ) = 2 d/ − ˜Γ( d/ d T d c ( λ ) d . (A31) Hence, we can write c ∆ = T d f ( β ∆) and α ∆ = − T d − g ( β ∆) where both functions f ( x ) , g ( x ) are expo-nentially suppressed with x . Recalling that α gs ∝ T d weconclude that at fixed β ∆ the gapped mode will dom-inate the thermal expansion at sufficiently low T , suchthat Γ ∝ − /T upon approaching the QPT along tra-jectories of ∆ /T = const . even if T (cid:28) ∆ . The line sepa-rating non-divergent from divergent behavior of Γ uponapproaching the QPT has the form ∆ /T ∝ | ln T / ( Js ) | ,obtained from solving α gs = α ∆ , as noted in the maintext. Along this line the ratio ∆ /T grows upon approach-ing the transition point. S. Sachdev,
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