Quantum Annealed Criticality: A Scaling Description
Premala Chandra, Piers Coleman, Mucio A. Continentino, Gilbert G. Lonzarich
QQuantum Annealed Criticality: A Scaling Description
Premala Chandra, Piers Coleman,
1, 2
Mucio A. Continentino, and Gilbert G. Lonzarich Center for Materials Theory, Rutgers University, Piscataway, New Jersey, 08854, USA Department of Physics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK Centro Brasileiro de Pesquisas Fiscicas, 22290-180, Rio de Janeiro, RJ, Brazil Cavendish Laboratory, Cambridge University, Cambridge CB3 0HE, United Kingdom (Dated: December 4, 2020)Experimentally there exist many materials with first-order phase transitions at finite temperaturethat display quantum criticality. Classically, a strain-energy density coupling is known to drive first-order transitions in compressible systems, and here we generalize this Larkin-Pikin mechanism tothe quantum case. We show that if the T = 0 system lies above its upper critical dimension,the line of first-order transitions ends in a “quantum annealed critical point” where zero-pointfluctuations restore the underlying criticality of the order parameter. The generalized Larkin-Pikinphase diagram is presented and experimental consequences are discussed. I. INTRODUCTION
The interplay of first-order phase transitions withquantum fluctuations is an active area in the studyof exotic quantum states near zero-temperature phasetransitions.
In many metallic quantum ferromag-nets, coupling of the local magnetization to the low en-ergy particle-hole excitations transforms a high temper-ature continuous phase transition into a low tempera-ture discontinuous one, and the resulting classical tri-critical points have been observed in many systems.
Experimentally there also exist insulating materials thathave classical first-order transitions that display quantumcriticality, and here we provide a theoretical basisfor this behavior. In a nutshell, we study a system withstrain-energy density coupling that has a line of first-order transitions at finite temperatures. We show that asthe temperature is lowered, quantum fluctuations reducethe amplitudes of their thermal counterparts, weakeningthe first-order transition and “annealing” the system’selastic response, ultimately resulting in a T = 0 “quan-tum annealed” critical point. The generalized temper-ature ( T )-tuning parameter ( g )-field ( h ) phase diagramemerging from our study is presented in Figure 1 wherethe field ( h ) is conjugate to the order parameter.At a first-order transition the quartic mode-mode cou-pling of the effective action becomes negative. One mech-anism for this phenomenon, studied by Larkin and Pikin (LP), involves the interaction of strain with the fluctu-ating energy density of a critical order parameter. LPfound that a diverging specific heat in the “clamped”(fixed volume) system leads to a first-order transition inthe unclamped system at constant pressure. The Larkin-Pikin criterion for a first order phase transition is κ < (cid:101) ∆ C V T c (cid:18) dT c d ln V (cid:19) (1)where V is the volume, ∆ C V is the singular part of thespecific heat capacity in the clamped critical system, T c isthe transition temperature and dT c d ln V is its volume strainderivative. The effective bulk modulus κ is defined by FIG. 1. Temperature ( T )-Field ( h )-Tuning Parameter ( g )Phase Diagram with a sheet of first-order transitions boundedby critical end-points (CEP) terminating at a zero tempera-ture quantum critical point (QCP); here g tunes the quantumfluctuations and h is the field conjugate to the order param-eter. Inset: Temperature-Tuning Paramter “slice” indicatinga line of classical phase transitions ending in a “quantum an-nealed critical point” where the underlying order parametercriticality is restored by zero-point fluctuations. κ − = K − − ( K + µ ) − where K and µ are the barebulk and the shear moduli in the absence of couplingbetween the order parameter and strain; more physi-cally κ ∼ K c L c T where c L and c T are the longitudinaland the transverse sound velocities. An experimentalsetting for this behavior is provided by
BaT iO with aclassical ferroelectric phase transition that is continuouswhen clamped and, due to electromechanical coupling,becomes first-order when unclamped. Low-temperature measurements on ferroelectric insu- a r X i v : . [ c ond - m a t . s t r- e l ] D ec lators provide a key motivation for our study. Atfinite temperatures and ambient pressure these mate-rials often display first-order transitions due to strongelectromechanical coupling; yet in many cases their dielectric susceptibilities suggest the presence ofpressure-induced quantum criticality associated withzero-temperature continuous transitions. It is thusnatural to explore whether a generalization of the Larkin-Pikin mechanism with strain-energy density coupling,can be developed to describe this phenomenon.Here we generalize the Larkin-Pikin approach to in-clude quantum zero-point fluctuations of the energy den-sity, showing that it is the divergence of the energyfluctuations, both quantum and classical, that governthe LP mechanism. Quantum fluctuations introduce anadditional time dimension into the partition function,which now sums over all space-time configurations. At a finite temperature T , the temporal extent of thequantum fluctuations is bounded by the Planck time τ P = (cid:126) k B T with a corresponding quantum correlationlength ξ Q ∼ ( τ P ) /z where z is the the dynamical ex-ponent. Therefore for temperatures where ξ Q is greaterthan the lattice spacing, the thermal correlation volumecontains a quantum mechanical core on length- and time-scales determined by ξ Q and τ P . Due to their additionaltime dimension, quantum fluctuations are typically lesssingular than are their classical counterparts. As thetemperature is lowered, the correlation volume of thezero-point fluctuations grows, reducing the amplitudes ofthe singular thermal fluctuations in the clamped system.The induced Larkin-Pikin first order transition thus be-comes progressively weaker with decreasing temperature,leading to a continuous “quantum annealed” transitionat T = 0.More specifically, Larkin and Pikin considered thecoupling L I = λe ll ( (cid:126)x ) ψ ( (cid:126)x ) (2)between the volumetric strain field e ll and the squaredamplitude ψ of the critical order parameter. In a criticalsystem, the singular fluctuations of the energy densityare directly proportional to ψ ; thus (2) corresponds to astrain-energy coupling. Naively (2) is expected to inducea short-range attractive order parameter interaction. LPshowed that (2) also leads to an anomalous long-rangeinteraction between order parameter fluctuations. S −→ S − λ T κ (cid:20) V (cid:90) d x (cid:90) d x (cid:48) ψ ( (cid:126)x ) ψ ( (cid:126)x (cid:48) ) (cid:21) (3)with 1 κ = (cid:18) K − K + µ (cid:19) (4)where µ is the shear modulus. This long-range inter-action is finite if µ >
0, i.e if the medium is a solid.LP showed that this induced long-range interaction in (3) generates positive feedback to the tuning parameter,leading to a multi-valued free energy surface and a re-sulting first order phase transition.Here we expand the LP approach to include both quan-tum and classical fluctuations, summing over all possiblespacetime configurations in the action, to obtain a gen-eralized LP criterion κ < (cid:101) (cid:18) dg c d ln V (cid:19) χ ψ (5)where χ ψ = (cid:90) β dτ (cid:90) d x (cid:104) δψ ( (cid:126)x ) δψ (0) (cid:105) (6)is the space-time average of the quantum and thermal“energy” fluctuations, β = k B T and g is the tuning pa-rameter for the phase transition, with the convention that g c ( T = 0) = 0. At zero temperature, this expressionextends the original LP criterion (1) to quantum phasetransitions. At finite temperatures, the critical temper-ature and the critical coupling constant are related by g c ( T c ) = uT / ˜Ψ c , where ˜Ψ = ˜ νz is the shift exponentgoverning the finite temperature transition with ˜ ν and zthe quantum correlation length and the dynamical criti-cal exponents respectively; therefore d ln g c = d ln T c and the LP criterion becomes κ < (cid:101) (cid:18) dT c d ln V (cid:19) C V /T c (cid:122) (cid:125)(cid:124) (cid:123)(cid:18) g T c (cid:19) χ ψ , (7)where we have identified ∆ C v /T c = ( g/ T c ) χ ψ withthe specific heat capacity. In this way, we see that thegeneralized Larkin Pikin equation encompasses the orig-inal criterion (1) in addition to being applicable at lowtemperatures.Recently an adaptation of the Larkin-Pikin approachwas proposed for pressure( P )-tuned quantum magnetswhere it is often found that dT c dP → ∞ as T c → P − P c = uT / ˜Ψ c , sothat dT c /dP ∝ T − / ˜Ψ c diverges as T c → < However such a diverg-ing coupling of the critical order parameter fluctua-tions and the lattice should lead to structural instabil-ities near the quantum phase transition that have notbeen observed.
Using Maxwell relations, we can write dT c dP = ∆ V ∆ S (cid:12)(cid:12) T = T c . Since ∆ S → T c →
0, proponentsof the previous argument assume that ∆ V is finite in thesame limit, indicating latent work at the quantum phasetransition. However the ratio ∆ V ∆ S can still diverge at acontinous quantum transition if the numerator and thedenominator have different temperature-dependences as T c → + . In fact, from our generalization of the Larkin-Pikin approach, we show that∆ V ∝ − T η , ∆ S ∝ − T η ( T − ) , (8)where η = α − ˜ αα ˜Ψ with α and ˜ α the classical and quantumcritical exponents respectively, governing the divergenceof energy fluctuations. Generically α > ˜ α , since ther-mal fluctuations are more singular than quantum fluctu-ations, so that η >
0. This means thatlim T c → + ∆ V → < dT c dP = ∆ V ∆ S (cid:12)(cid:12)(cid:12)(cid:12) T = T c ∝ − T − c (10)diverges as T c goes to zero.The structure of the paper is as follows. In SectionII we present the original Larkin-Pikin approach, firstconstructing the classical LP action. Next, following LP,we parameterize the positive feedback contribution tothe internal tuning parameter of the elastically-coupledfree energy using the uncoupled (clamped) free energyand scaling functions associated with classical critical-ity. The non-monotonic relation between the internaltuning parameter and the physical temperature is coin-cident with a multivalued (unclamped) free energy, in-dicating the presence of a first-order transition in theelastically-coupled system. We also derive the classicalLarkin-Pikin criterion (1) as a macroscopic instability ofthe original (uncoupled) critical point with respect to thestrain-energy density coupling. This approach can berewritten in terms of correlation functions, giving insightinto the T c → is presentedand used in the generalized Larkin-Pikin equations tostudy the system’s behavior in the approach to T c → T c → T c →
0, confirming that the quantum transitionis continuous. Next a field conjugate to the order param-eter is applied in Section V, and the critical endpointsare determined. Field behavior in the approach to thequantum critical point is also studied, and these resultsare summarized in the Larkin-Pikin phase diagram. Ex-perimental consequences are presented in Section VI andwe end (VII) with a summarizing discussion and openquestionsfor future work. Derivations of the classical andquantum Larkin-Pikin actions and of various crossoverscaling expressions are presented in five Appendices forinterested readers.
II. THE CLASSICAL LARKIN-PIKINAPPROACH
The Larkin-Pikin (LP) mechanism refers to a com-pressible system where the order parameter, ψ ( (cid:126)x ), iscoupled to the volumetric strain in the simplest case ofa scalar ψ and isotropic elasticity. The action S [ ψ, u ]for this compressible system then divides up into threecontributions S [ ψ, u ] = S L [ ψ ] + S E [ u ] + S I [ ψ, e ]= 1 T (cid:90) d x ( L L [ ψ ] + L E [ u ] + L I [ ψ, e ]) . (11)Here, in order to present the original classical LP problemin a way that is amenable to its quantum generalizationconsidered later, we have used the notation L , denot-ing the Lagrangian density which is also the Hamiltoniandensity in the classical case.The Lagrangian density L L [ ψ ] describes the physics ofthe order parameter in the clamped system that, in thesimplest case, is a ψ field theory L L [ ψ ] = 12 ( ∂ µ ψ ) + a ψ + b ψ , (12)where a = c ( T − T c ) is the tuning parameter, and b > L E [ u ] = 12 (cid:20)(cid:18) K − µ (cid:19) e ll + 2 µe ab (cid:21) − σ ab e ab (13)describes the elastic degrees of freedom, where σ ab is theexternal stress, e ab ( (cid:126)x ) = (cid:16) ∂u a ∂x b + ∂u b ∂x a (cid:17) is the strain ten-sor, u a ( (cid:126)x ) is the local atomic displacement and e ll ( (cid:126)x ) =Tr[ e ( (cid:126)x )] is the volumetric strain. Finally L I [ ψ, e ] = λe ll ψ (14)describes the interaction between the volumetric strain e ll and the squared amplitude ψ , the “energy density”,of the order parameter, where λ is a coupling constant as-sociated with the strain-dependence of T c . If we combine L L + L I = 12 ( ∂ µ ψ ) + c T − T c [ e ll ]) ψ + b ψ , (15)where T c [ e ll ] = T c − (2 λ/c ) e ll (16)is the strain dependent T c , so that (2 λ/c ) = − (cid:0) dT c d ln V (cid:1) .For notational simplicity and convenience, we shall set c = 1 in the following development.The key idea of the Larkin-Pikin approach is that weintegrate out the Gaussian strain degrees of freedom fromthe action so that the partition function takes the form Z = (cid:90) D [ ψ ] (cid:90) D [ u ] e − S [ ψ,u ] −→ Z = (cid:90) D [ ψ ] e − S [ ψ ] . (17)where the effective action S is a function of the orderparameter ψ .Though the elastic degrees of freedom are Gaussian,and can be exactly integrated out, this procedure must bedone with some care because of the special role of bound-ary normal modes. In a solid of volume L , the normalmodes can be separated into two components accord-ing to their wavelength λ : sound waves have wavelength λ (cid:28) L whereas boundary waves have wavelength com-parable with the size of the system, λ ∼ L . The LarkinPikin effect is a kind of “elastic anomaly” , whereby theintegration over boundary modes generates a nonlocal in-teraction between the order parameter in the bulk action.This elastic anomaly destroys the locality of the originaltheory, yet paradoxically, as a bulk term in the action itis independent of the detailed boundary conditions.Larkin and Pikin chose periodic boundary conditionsas the most convenient way to integrate out the boundarymodes. In a system with periodic boundary conditions,the strain field separates into a uniform ( (cid:126)q = 0) “bound-ary term” and a finite-momentum ( (cid:126)q (cid:54) = 0) contributiondetermined by fluctuating atomic displacements e ab ( (cid:126)x ) = e ab + 1 V (cid:88) (cid:126)q (cid:54) =0 i q a u b ( (cid:126)q ) + q b u a ( (cid:126)q )] e i(cid:126)q · (cid:126)x (18)where u a ( (cid:126)q ) is the Fourier transform of u a ( (cid:126)x ) with dis-crete momenta (cid:126)q = πL ( l, m, n ) with { l, m, n } integers, { a, b } ∈ { , , } and volume V = L . Physically wecan understand this separation in (18) by noting thatthe strain only couples to the longitudinal modes; how-ever at q = 0 there is no distinction between transverseand longitudinal modes so this case must be treated sep-arately from the finite q situation. Formally, the solidforms a 3-torus, and the integral of the strain e ab aroundthe torus defines the number of line defects enclosed by the torus, a kind of flux, that is (cid:73) e ab ( (cid:126)x ) dx b = e ab (cid:73) dx b = b a (19)where b b is the Burger’s vector of the enclosed defects.Thus on a torus, the boundary modes of the strain havea topological character.In order to integrate out the Gaussian strain degreesof freedom from (11) to derive an effective action for theorder parameter field in (17), we write the effective action S [ ψ ] = S L [ ψ ] + ∆ S [ ψ ] (20)where S L [ ψ ] = T (cid:82) d x L L [ ψ ] from (12) and e − ∆ S [ ψ ] = (cid:90) D [ e, u ] e − ( S E [ u ]+ S I [ ψ,e ]) . (21)If we write the elastic action in a schematic, discretizedform S E [ u ] + S I [ ψ, e ] = 12 (cid:88) i,j u i M ij u j + λ (cid:88) j u j ψ j (22)then the effective action becomes simply∆ S [ ψ ] = 12 ln det[ M ] − λ (cid:88) i,j ψ i M − i,j ψ j (23)where the second term is recognizable as an induced at-tractive interaction between the order parameter fields.Because of subtleties associated with the separation (18)of the strain into uniform and finite (cid:126)q components, in-tegration of the elastic degrees of freedom in (21) leadsto an overall attractive interaction ( ∝ − ψ i M − ij ψ j ) withboth short-range and infinite range components.Integrating over the elastic degrees of freedom (18) in (21) we obtain the Larkin-Pikin action S [ ψ ] = S L [ ψ, t, b ∗ ] − λ T (cid:18) K − K + µ (cid:19) (cid:20) V (cid:90) d x (cid:90) d x (cid:48) ψ ( (cid:126)x ) ψ ( (cid:126)x (cid:48) ) (cid:21) , (24)where S L ( ψ ) = 1 T (cid:90) d x (cid:20)
12 ( ∂ µ ψ ) + t ψ + b ∗ ψ (cid:21) (25)with a renormalized local interaction b ∗ = b − λ K + µ . (26)where we have made the replacement a → t where t = ( T − T c ) and c = aT − T c = 1.The essence of the Larkin-Pikin effect is the appearanceof a distance-independent interaction between the energydensities of the order parameter field that appears in (24): it is this term that drives a non-perturbative first ordertransition at arbitrarily small λ . Since the Larkin-Pikinargument is valid for arbitrarily small coupling λ , the per-turbative O ( λ ) renormalization of the short-range inter-action in (26) becomes negligibly small in this limit andcan be safely neglected. The prefactor of the long rangeattractive interaction (24)1 κ = (cid:18) K − K + µ (cid:19) (27)has two competing terms. The first is attractive ( ∝ K ),resulting simply from integrating out the q = 0 part ofthe strain (18), governed by the bulk modulus K . Thesecond results from the finite q components of the strain(18), but in a rather subtle fashion. The finite q elas-tic fluctuations, arising from longitudinal sound modes,are governed by the elastic modulus K + µ and leadto the perturbative renormalization of b in (26). How-ever, the finite q modes explicitly exclude a contributionfrom q = 0. This “bosonic hole” in the longitudinal in-teractions gives rise to a residual long-range repulsion,resulting in the second term ( ∝ − K + µ ). Remarkably,the overall prefactor of the long-range interaction term( κ ) is only non-zero for finite shear modulus ( µ (cid:54) = 0),indicating that the Larkin-Pikin effect only occurs for solids and is absent for liquids. We also note that for theclamped system only the second contribution to (27) re-mains, leading to a repulsive interaction; this is consistentwith the continuous transition of the clamped system.The distance-independent interaction in (24) can bewritten in terms of the volume average of the energy den-sity Ψ ≡ (cid:20) V (cid:90) d x ψ ( (cid:126)x ) (cid:21) (28)so that S [ ψ ] = S L − λ V T κ (Ψ ) . (29)Since Ψ is an intensive variable, its fluctuations aboutits thermal average (cid:104) Ψ (cid:105) δ Ψ = Ψ − (cid:104) Ψ (cid:105) (30)become vanishingly small in the thermodynamic limit, (cid:104) ( δ Ψ ) (cid:105) ∼ O ( V ). Thus (cid:0) Ψ (cid:1) = (cid:0) (cid:104) Ψ (cid:105) + δ Ψ ) (cid:1) = 2Ψ (cid:104) Ψ (cid:105) − (cid:104) Ψ (cid:105) + O (1 /V ) , (31)so that we can reexpress (24) as a set of self-consistent equations S [ ψ ] = 1 T (cid:90) d x (cid:20) L L ( ψ, t ) − λ κ (cid:104) Ψ (cid:105) ψ ( (cid:126)x ) (cid:21) + λ V κ (cid:104) Ψ (cid:105) (cid:104) Ψ (cid:105) = (cid:82) dψ Ψ e − S L [ ψ ] (cid:82) dψ e − S L [ ψ ] . (32)Equations (32) may be succintly formulated by introduc-ing an auxiliary “strain” variable φ = − λ (cid:104) Ψ (cid:105) κ (33)Then we may write e − ˜ F ( φ ) T = (cid:90) D ψ e − S [ ψ,φ ] (34)where S [ ψ, φ ] = 1 T (cid:90) d x (cid:104) L A ( ψ, t ) + λφψ + κ φ (cid:105) (35)that can be reexpressed as S [ ψ, φ ] = 1 T (cid:90) d x [ L L ( ψ, t + 2 λφ )] + κV T φ (36)where we see that the auxiliary variable φ shifts the“mass” (e.g. tuning parameter) of the order parameter by a → x = a + 2 λφ . Self-consistency is then imposed asstationarity of the free energy with respect to φ , ∂ ˜ F [ φ ] ∂φ = 0 = ⇒ (cid:2) λ (cid:104) Ψ (cid:105) + κφ (cid:3) V = 0 . (37)In the original Larkin Pikin derivation, the action(35) was obtained by performing a Hubbard-Stratonovichtransformation of the long-range interaction (29) − λ V T (Ψ ) → κV T φ + λ (Ψ ) φ, (38)followed by a saddle-point evaluation of the integral over φ . Larkin-Pikin observed that main effect of the elasticityin the unclamped system is make a parameterized shiftof the physical reduced temperature t to a parameterizedvariable X t → X = t + λφ. (39)Although the phase transition of the unclamped systemis continuous for parameterized parameter X , Larkin-Pikin observed (see Section II.A) that the original (phys-ical) tuning parameter t [ X ] becomes a non-monotonicfunction of X , leading to a first-order phase transition atfinite temperatures.Subsequent authors pursued alternative approaches tothe Larkin-Pikin criterion. If, rather than integrat-ing out the elasticity variable φ , one integrates out theorder parameter fluctuations, this results in a reduction∆ κ in the bulk modulus that is proportional to energydensity fluctuations. When the specific heat diverges,the bulk modulus is negative, there is a macroscopic in-stability and the system undergoes a first-order transi-tion. In the next two sections, we summarize each ofthese approaches to the classical Larkin-Pikin problem.
A. Review of the Original Larkin-Pikin Argument
The free energy of the clamped system is defined as e − F ( t ) T = (cid:90) D [ ψ ] e − S L [ ψ,t ] (40)where S L is defined in (11) and we have explicitly in-cluded its dependence on the tuning parameter t . Inwriting (40), we have glossed over issues of renormaliza-tion. In particular, self-energy corrections to the orderparameter propagators will shift the critical value of T c in t = T − T c from its bare value T c to a renormalizedtransition temperature T ∗ c . All of these renormalizationeffects can be absorbed into redefinitions of the appropri-ate variables, in particular from now on we will redefine t = T − T ∗ c and for convenience we will drop the asterix sothat T c refers to the renormalized critical temperature.From (34) and (35) we can write the free energy forour unclamped system as˜ F [ φ, t ] = F [ X ] + κV φ (41)where X = t + 2 λφ. (42)indicates the shifting the of the tuning parameter due tothe presence of energy fluctuations.Now 1 V ∂ F ∂X = (cid:104) Ψ (cid:105) φ = − λ (cid:104) Ψ (cid:105) κ = − λV κ (cid:18) ∂ F ∂X (cid:19) ≡ − λV κ F (cid:48) [ X ] (44)where we have defined F (cid:48) [ X ] ≡ (cid:0) ∂ F ∂X (cid:1) for simplicity.Therefore ˜ F = F [ X ] + 2 λ V κ ( F (cid:48) [ X ]) (45) and X = t − λ V κ F (cid:48) [ X ] . (46)Let us define ˜ f ≡ λV κ ˜ F , f ≡ λV κ F . (47)Then the two equations descibing the unclamped systemare ˜ f = f [ X ] + λ ( f (cid:48) [ X ]) (48)and t = X + 2 λf (cid:48) [ X ] (49)which have to solved self-consistently, where ˜ f and f arethe (renormalized) free energies of the unclamped andclamped systems respectively.To examine the consequences of these equations, werecall that that we can identify a ∝ T − T c T c ≡ t with thereduced temperature t . In the clamped system, we as-sume a second-order transition so we can write f ∝ −| t | − α (50)where t is the reduced temperature and α > a ∝ t ,this leads to f ∝ −| X | − α and f (cid:48) [ X ] ∝ − (2 − α ) | X | − α sgn( X ) (51)and combining these results with (49) we obtain t = X + 2 λf (cid:48) [ X ]= X − λ (2 − α ) | X | − α sgn( X ) . (52)where we see that there is a non-monotonic relationshipbetween the physical temperature ( t ) and the parameter-ized variable ( X ), as shown in Figure 2, that leads to aninevitable first-order transition.In order to see more specifically how (52) translatesinto a discontinuous transition let us consider, followingthe example of Larkin-Pikin, the specific case of α =1 /
2. Then for t large, f ∝ | t | . For t = 0 there aretwo solutions of (52): X = 0 and X = 4 λ with f =0 and f = − λ respectively. A plot of ˜ f vs. t isshown in Figure 3, indicating the presence of a first-ordertransition.The Larkin-Pikin criterion (1) emerges from t = X + 2 λf (cid:48) [ X ] = X + 2 λ κ (cid:104) Ψ (cid:105) X (53)where again (cid:104) Ψ (cid:105) X = (cid:82) D [Ψ]Ψ e − S ( X,ψ ) (cid:82) D [Ψ] e − S ( X,ψ ) (54) FIG. 2. Schematic of the non-monotonic relationship betweenthe reduced temperature ( t ) and the parameterized variable( X ) shifted by energy fluctuations for the unclamped Larkin-Pikin problem. is the energy density computed with the shifted reducedtemperature X . This expression describes the relation-ship between the physical temperature t = ( T − T c ) /T c and the parameterized variable X . Now the derivative dt/dX is given by dtdX = 1 − λ Vκ (cid:104) ( δ Ψ ) (cid:105) (55)We can identify the fluctuations in the right-hand side ofthis equation with the specific heat capacity C V = (cid:104) ( δ Ψ ) (cid:105) T c (56)so that dtdX = 1 − λ T c κ C V ( X ) (57)Thus if the specific heat capacity diverges at the classicalcritical point of the clamped system C V ( X ) → ∞ , dt/dX will change sign as X →
0. Then t [ X ] becomes a non-monotonic function of the internal temperature X thatinevitably leads to a first order phase transition. dtdX = 0in (57) is the LP criterion (1). B. The Larkin-Pikin Criterion as a MacroscopicInstability
An alternative approach to the Larkin-Pikin crite-rion is to probe the macroscopic stability of the original
FIG. 3. Schematic of the free energy of the unclamped com-pressible system ( ˜ f ) vs reduced temperature ( t ) for α = ;the first-order transition, due to the non-monotonicity of t vs. X , is marked here. critical point with respect to the strain-energy densitycoupling. From (34) and (35) we know that the parti-tion function of the unclamped system can be written asan integral over the order parameter fluctuations Z [ φ ] = e − ˜ F [ φ ] /T = (cid:90) D [ ψ ] e − S [ ψ,φ ] . (58)The renormalized bulk modulus,˜ κ ≡ κ − ∆ κ (59)at the transition is then obtained by taking the secondderivative of (34)˜ κ = 1 V ∂ ˜ F∂φ = κ − λ T c (cid:90) d x (cid:104) δψ ( (cid:126)x ) δψ (0) (cid:105) (60)where δψ ( (cid:126)x ) = ψ ( (cid:126)x ) − (cid:104) ψ (cid:105) . We recall the tuning pa-rameter a = c ( T − T c ) where we have set c = 1, so thatthe singular component of the specific heat coefficient isalso proportional to the energy fluctuations∆ C V T c = − ∂ F∂T (cid:12)(cid:12)(cid:12)(cid:12) Sing = 14 T c (cid:90) d x (cid:104) δψ ( (cid:126)x ) δψ (0) (cid:105) (61)which allows us to relate the shift in the bulk modulusto the singular part of the specific heat∆ κ = (2 λ ) ∆ C V T c = (cid:18) dT c d ln V (cid:19) ∆ C V T c , (62)where we have used (78) to identify 2 λ = − dT c /d ln V .The condition for a macroscopic instability, and hencea first-order transition, is when the renormalized bulkmodulus is negative κ − ∆ κ < ⇒ κ < ∆ C V T c (cid:18) dT c d ln V (cid:19) (63)and we see that we have recovered the Larkin-Pikin cri-terion (1).The renormalization of the bulk modulus (59) that re-sults can also be obtained diagrammatically (Figure 4).In this approach the bare order parameter interaction b now acquires a contribution from the coupling to thestrain (Figure 4a). In the Feynman diagrams 1 /κ is thebare “propagator” for the auxilliary strain variable φ .We can use a Dyson equation for this strain propagator(Figure 4b) to determine ˜ κ . Specifically we write (cid:18) κ (cid:19) = (cid:18) κ (cid:19) + (cid:18) κ (cid:19) λ (cid:104) ψ ( (cid:126)q ) ψ ( − (cid:126)q ) (cid:105) (cid:12)(cid:12) (cid:126)q =0 (cid:18) κ (cid:19) (64)that results in ˜ κ = κ − ∆ κ = κ − λ χ ψ (65)where χ ψ is the static susceptibility for ψ , χ ψ = 1 T c (cid:90) d x (cid:104) δψ ( (cid:126)x ) δψ (0) (cid:105) , (66)recovering (60). ∆ κ is thus a self-energy correction tothe strain propagator. FIG. 4. Diagrammatic approach to the generalized Larkin-Pikin criterion a) Bare interaction is a sum of a local and anonlocal contribution mediated by fluctuations in the strain;b) Feynman diagram showing renormalization of the strainpropagator by coupling to energy fluctuations.
This discussion enables us to obtain a heuristic under-standing of how the how Larkin Pikin approach can begeneralized to include quantum fluctuations of the or-der parameter which now occur in both both space and(imaginary) time. The prefactor 1 /T c in (66) is now re-placed by an integral over time so that χ ψ −→ (cid:90) β dτ (cid:90) d d x (cid:104) δψ ( (cid:126)x, τ ) δψ (0) (cid:105) , (67)where we have also generalized the expression to d spa-tial dimensions. This quantity is represented by the same Feynman diagrams, where momentum variables now be-come four-momenta q = ( (cid:126)q, ν n ). If we make the Gaussianapproximation (cid:104) δψ ( (cid:126)x ) δψ (0) (cid:105) ≈ ( (cid:104) δψ ( (cid:126)x ) δψ (0) (cid:105) ) , thenthe zero-temperature limit of χ ψ islim T → χ ψ ≈ (cid:90) dτ d d x ( (cid:104) δψ ( (cid:126)x ) δψ (0) (cid:105) ) = (cid:90) dν π d d q (2 π ) d ( χ ψ ( (cid:126)q, ν )) (68)where in the second line we have Fourier transformed intomomentum space, and χ ψ ( (cid:126)q, ν ) = (cid:104) δψ ( − q ) δψ ( q ) (cid:105) , theorder parameter susceptibility, is the space-time Fouriertransform of the correlator (cid:104) ψ ( (cid:126)x ) ψ (0) (cid:105) . It follows thatlim T → ∆ κ ∝ (cid:90) dq dν q d − [ χ ψ ( (cid:126)q, iν )] . (69)To examine how this quantity behaves in the approachto the quantum critical point of the clamped system, wecan use dimensional power-counting. Since [ χ ] = (cid:104) q (cid:105) and [ ν ] = [ q z ],lim T → [∆ κ ] = [ q d + z ][ q ] ∼ ξ − ( d + z ) Q where we have replaced [ q − ] = [ ξ Q ], the quantumcorrelation length. As the quantum critical point of theclamped system is approached, ξ Q → ∞ , so that thequantum corrections to κ are non-singular for d + z > III. GENERALIZATION OF THELARKIN-PIKIN APPROACH TO INCLUDEQUANTUM FLUCTUATIONS
Motivated by these heuristic arguments, we now gen-eralize the Larkin-Pikin approach to include both quan-tum and thermal fluctuations. In order to probe howthe fluctuation-driven first order transition predicted byLarkin and Pikin evolves as the temperature is loweredto absolute zero, we need to understand the crossover be-tween the continuous quantum and classical phase tran-sitions of the clamped system. From a scaling perspec-tive, temperature is a relevant perturbation that drivesthe system from a quantum to a classical critical point.The action of temperature on a quantum phase tran-sition is to introduce a boundary condition in time, sothat temperature plays the role of a finite-size correctionat a quantum critical point. In contrast to their staticclassical counterparts, quantum zero point fluctuationsare intrinsically dynamical. At a finite temperature T ,the criticality of quantum fluctuations is cut off by thePlanck time τ P = (cid:126) k B T , with a corresponding quantumcorrelation length ξ Q ∼ τ /zP where z is the dynamical ex-ponent; thus the static classical correlation volume con-tains a quantum mechanical core on length- and time-scales governed by the Planck time. At low tempera-tures, ξ Q provides the essential short-distance cut-off tothe static classical fluctuations of the order parameter.In pictorial terms, we can visualize the fluctuations asbeing “annealed” at short distances.When the temperature is raised from absolute zero,there comes a point where the finite correlation time be-comes of order the Planck time, and as the temperatureis raised further, the temporal correlation length becomes“stuck” at the Planck time. The temperature when thisoccurs determines the quantum classical crossover. Be-yond this point, correlations continue to grow but onlyin the spatial direction; the dynamical aspect of the fluc-tuations is lost and the statistical mechanics is governedpurely by a sum over spatial configurations, namely thestatistical mechanics has become classical. FIG. 5. Schematic showing the dependence of the Free energyof the clamped system in the vicinity of the quantum criti-cal point. The scaling function about the QCP determinesthe amplitude factors for the finite temperature classical crit-ical point (CCP), given by A I ( T ) for a constant temperaturesweep and A II ( g ) for a sweep at constant tuning parameter.Here the location of the quantum critical point at g c (0) islabelled as simply g c . More specifically near the quantum critical point at T = 0, the zero-point fluctuations are governed by a finitecorrelation length ξ Q ∼ ( g − g c (0)) − ˜ ν , where g is theparameter that tunes the transition and g = g c (0) is thelocation of the quantum critical point. If we combine our expressions for the quantum correlation length in theordered phase close to the line of phase transitions, wefind ( g − g c ) − ˜ ν ∼ (cid:18) (cid:126) k B T c (cid:19) z (70)which leads to T c ∼ ( g − g c ) ˜ νz ≡ ( g − g c ) ˜Ψ (71)where ˜Ψ is called the shift exponent, and we see thatΨ = ˜ νz if the effective dimension of the quantum sys-tem is below its upper critical dimension where scalingis applicable. Here we keep with convention using thisnotation, hoping that there will be no confusion with thethe spacetime volume average of the energy density.Larkin and Pikin showed that the feedback effect ofthe energy fluctuations could be reformulated in termsof the critical temperature-dependence of the free energyof the decoupled system near the phase transition, al-lowing an analysis purely in terms of the universal crit-ical behavior of the decoupled system. By generalizingthis parameteric approach to include the effect of quan-tum fluctuations, we are able to analyze the evolution ofthe Larkin-Pikin system from finite to zero temperature,showing that if the energy fluctuations are not divergentat T = 0, the finite-temperature first-order phase tran-sition progressively weakens as temperature is reduced,becoming continuous at zero temperature. A. The Generalized Larkin Pikin Action
The quantum mechanical action now picks up an ad-ditional integral over time S = (cid:90) d x L ≡ (cid:90) β dτ (cid:90) d x L . (72)where β = T and we recover the classical result for large T . The spacetime generalizations of equations (12-14) S [ ψ, u ] = S L [ ψ ] + S E [ u ] + S I [ ψ, e ]= (cid:90) dτ d x ( L A [ ψ ] + L E [ u ] + L I [ ψ, e ]) (73)now contain kinetic energy terms, so that L L [ ψ, b ] = 12 ( ∂ µ ψ ) + a ψ + b ψ (74)where ( ∂ µ ψ ) ≡ ( ˙ ψ ) + ( ∇ ψ ) and we now identify a = c ( g − g c ), where g c is the bare value of the critical cou-pling constant. The elastic degrees of freedom are nowdescribed by L E [ u ] = 12 (cid:20) ρ ˙ u l + (cid:18) K − µ (cid:19) e ll + 2 µe ab (cid:21) − σ ab e ab , (75)0and strain-energy density interaction L I [ ψ, e ] = λe ll ψ (76)is unchanged. If we combine L L + L I = 12 ( ∂ µ ψ ) + c g − g c [ e ll ]) ψ + b ψ , (77)where g c [ e ll ] = g c − (2 λ/c ) e ll (78) is the strain dependent g c , so that (2 λ/c ) = − (cid:16) dg c d ln V (cid:17) .For notational simplicity and convenience, we shall set c = 1 in the following development.Following the argument of Larkin-Pikin in the classicalcase, we choose periodic boundary conditions as the mostconvenient way to integrate out the elastic degrees offreedom for the analogous quantum problem. It is thennatural to generalize the classical expression for the strainfield (18) to the quantum case summing over all space-time configurations as e ab ( (cid:126)x, τ ) = e ab ( τ ) + 1 βV (cid:88) iν n (cid:88) (cid:126)q (cid:54) =0 i q a u b ( q ) + q b u a ( q )] e i ( (cid:126)q · (cid:126)x − ν n τ ) (79)where q α ≡ ( (cid:126)q, iν n ), u b ( q ) ≡ u b ( (cid:126)q, iν n ) and ν n = 2 πnT is the Matsubara bosonic frequency. A priori, the uniformstrain tensor e ab ( τ ) involves configurations that are time-dependent. However, we recall that the integral of the strainfield around the toroidal solid (cid:73) e ab ( x, τ ) dx b = e ab ( τ ) (cid:73) dx b = b a ( τ ) (80)measures b a ( τ ), the Burger’s vector of the defects enclosed by the torus. If we restrict ourselves to smooth Gaussiandeformations of the solid, then the Burger’s vector is a topological invariant, like the conserved winding number ofsuperconductor. Changes in the Burger’s vector are akin to flux creep in a superconductor, and they involve thepassage of dislocations across the entire solid. Spacetime configurations with such moving defects will be associatedwith large actions, making their contributions to the path integral exponentially small in the thermodynamic limit.Therefore the strain field for the quantum Larkin-Pikin problem can be written e ab ( (cid:126)x, τ ) = e ab + 1 βV (cid:88) iν n (cid:88) (cid:126)q (cid:54) =0 i q a u b ( q ) + q b u a ( q )] e i ( (cid:126)q · (cid:126)x − ν n τ ) . (81)As in the classical case, our next step is to integrate out the Gaussian elastic degrees of freedom from the action Z = (cid:90) D [ ψ ] (cid:90) D [ u ] e − S [ ψ,u ] −→ Z = (cid:90) D [ ψ ] e − S [ ψ ] . (82)where the actions now involve integrals over space-time. We write the effective action S [ ψ ] = S L [ ψ ] + ∆ S [ ψ ] (83)where S L [ ψ ] = (cid:82) d x L L [ ψ ] ((74)) and e − ∆ S [ ψ ] = (cid:90) D [ e, u ] e − ( S e [ u ]+ S I [ ψ,e ]) . (84)Again our task is to cast this action into matrix form S E + S I = 12 (cid:88) q u i M ij u j + λ (cid:88) j u j ψ j → λ (cid:88) i,j ψ i M − i,j ψ j . (85)and to determine the nature of the induced order parameter interaction; now the summations run over the discretewavevector and Matsubara frequencies q ≡ ( iν n , (cid:126)q ), where ν n = πβ n , (cid:126)q = πL ( j, l, k ). Because of the form of the straintensor (81), the action in (84) separates in two terms, corresponding to the q = 0 and the finite ( (cid:126)q, iν n ) contributions.Integration over the elastic degrees of freedom in (81) now results in order-parameter interactions local and nonlocalboth in space and time.Integrating over the elastic degrees of freedom (81) in (84) (see Appendix B for details), we obtain the effectiveaction S [ ψ ] = S ∗ L [ ψ ] − λ (cid:18) K − K + µ (cid:19) (cid:20) βV (cid:90) d x (cid:90) d x (cid:48) ψ ( (cid:126)x ) ψ ( (cid:126)x (cid:48) ) (cid:21) . (86)1Here S ∗ L ( ψ ) = (cid:90) d x (cid:20)
12 ( ∂ µ ψ ) + ˜ g ψ + b ∗ ψ + 12 ψ ( (cid:126)x ) V dyn ( x − x (cid:48) ) ψ ( (cid:126)x (cid:48) ) (cid:21) (87)where ˜ g = ( g − g c ) and b ∗ = b − λ K + µ (88)is identical to that in the classical case (26). The Fourier transform of V dyn ( x − x (cid:48) ) is V dyn ( q ) = λ K + µ (cid:18) ν n /c L (cid:126)q + ν n /c L (cid:19) , (89)a dynamical order-parameter interaction where q ≡ ( ν n , (cid:126)q ) is the wavevector in space-time. The effective action inthe quantum Larkin-Pikin problem is thus the sum of a d + z - dimensional generalization of the classical effective LPaction and a dynamical interaction induced by quantum fluctuations. Although V dyn ( q ) in (89) has q -dependence, it isstill local from a scaling perspective since V dyn ( q ) is finite and non-singular in the limit q →
0. (89) also has the samescaling dimension as the original local repulsive interaction in (74). Though V dyn ( q ) does break Lorentz invariance, itdoes not reduce the order parameter symmetry. The universality of a Wilson-Fisher fixed point is known to be robustto such spacetime symmetry-breaking. For this reason the critical behavior of the clamped system is unaffected,and thus V dyn can be neglected in the local action.We have therefore established that the generalized Larkin-Pikin action, following the integration over the Gaussianstrain including both thermal and quantum fluctuations, is S [ ψ ] = S L [ ψ, ˜ g, b ∗ ] − λ (cid:18) K − K + µ (cid:19) (cid:20) βV (cid:90) d x (cid:90) d x (cid:48) ψ ( (cid:126)x ) ψ ( (cid:126)x (cid:48) ) (cid:21) . (90)with the local action S L [ ψ, ˜ g, b ∗ ] = (cid:90) d x L L [ ψ, ˜ g, b ∗ ] = (cid:90) d x (cid:20)
12 ( ∂ µ ψ ) + ˜ g ψ + b ∗ ψ (cid:21) (91)where b ∗ is defined in (88). We note that (90) is a d + z -dimensional generalizations of the effective classical LPaction, (24), where all spacetime configurations are summed to include both thermal and quantum fluctuations. Here z is the dynamical exponent associated with the temporal dimension, since the dispersion ω ∝ q z leads to [ ξ τ ] = [ ξ ] z where ξ τ and ξ are the correlation time and length respectively. B. Generalized Larkin Pikin Equations
The development of the approach is now a simple space-time generalization of its classical counterpart, described inequations (40-49). First, we perform a Hubbard-Stratonivich transformation of the spacetime-independent interactionin (90) − λ (cid:18) κ (cid:19) (cid:20) βV (cid:90) d x (cid:90) d x (cid:48) ψ ( (cid:126)x ) ψ ( (cid:126)x (cid:48) ) (cid:21) → (cid:90) d x (cid:104) ( λφ ) ψ ( (cid:126)x ) + κ φ (cid:105) (92)where 1 κ = 1 K − K + µ (93)is the effective bulk modulus and we have introduced the auxiliary “strain” field φ that is spacetime independent.Then we may write Z = e − ˜ S ( φ ) = (cid:90) D ψ e − S [ ψ,φ ] (94)2where ˜ S = β ˜ F and S [ ψ, φ ] = (cid:90) d x (cid:104) L L ( ψ, ˜ g ) + λφψ + κ φ (cid:105) (95)that can be reexpressed as S [ ψ, φ ] = (cid:90) d x [ L L ( ψ, ˜ g + 2 λφ )] + κV β φ (96)where we see that the auxiliary variable φ shifts the “mass” (e.g. tuning parameter) of the order parameter pp by˜ g → X = ˜ g + 2 λφ. (97)Because the second term in (96) scales as the spacetimevolume, we can solve for φ using a saddle-point evaluation ∂ ˜ F [ φ ] ∂φ = 0 = ⇒ (cid:2) λ (cid:104) Ψ (cid:105) + κφ (cid:3) V = 0 (98)where Ψ ≡ (cid:20) βV (cid:90) d x ψ ( x ) (cid:21) (99)is the spacetime volume average of the energy densityand (cid:104) Ψ (cid:105) = (cid:82) dψ Ψ e − S L [ ψ ] (cid:82) dψ e − S L [ ψ ] . (100)with S L as in (91). Equations (98), (99) amd (100) leadto φ = − λκ (cid:104) Ψ (cid:105) . (101)Equations (98-101) are identical to their classical coun-terparts (28-37), apart from the replacement of a spatialintegral by a space-time integral in (99). The follow-ing development, parameterizing the free energy of theclamped and unclamped system, precisely follows its clas-sical counterpart (41-49), but for completeness we includeit here in its entireity. The free energy of the clampedsystem is e − F (˜ g ) T = (cid:90) D [ ψ ] e − S L [ ψ, ˜ g ] (102)where S L is defined in (73) and (74) and we have explic-itly included its dependence on the tuning parameter ˜ g .As in the classical case, in writing (102) we have glossedover issues of renormalization. In particular, self-energycorrections to the order parameter propagators will shiftthe quantum critical value of g c from its bare value g c to a new value g c (0). All of these renormalization ef-fects can be absorbed into redefinitions of the appropri-ate variables, in particular from now on we will redefine˜ g = g − g c (0). From (94) and (96) we can write the free energy forour unclamped system as˜ F [ φ, ˜ g ] = F [ X ] + κV φ (103)where X = ˜ g + 2 λφ. (104)indicates the shifting the of the tuning parameter due tothe presence of energy fluctuations. Now1 V ∂ F ∂X = (cid:104) Ψ (cid:105) φ = − λ (cid:104) Ψ (cid:105) κ = − λV κ (cid:18) ∂ F ∂X (cid:19) ≡ − λV κ F (cid:48) [ X ] (106)where we have defined F (cid:48) [ X ] ≡ (cid:0) ∂ F ∂X (cid:1) for simplicity.Therefore ˜ F = F [ X ] + 2 λ V κ ( F (cid:48) [ X ]) (107)and X = ˜ g − λ V κ F (cid:48) [ X ] . (108)Let us define ˜ f ≡ λV κ ˜ F , f ≡ λV κ F . (109)Here we recall that the integrals in the action involvean integral over time (72), (cid:82) d x = (cid:82) β dτ (cid:82) d x where β = T is a boundary term, so that these free energiesare determined at fixed temeprature. Therefore the twoequations describing the unclamped system are˜ f = f [ X, T ] + λ ( f (cid:48) [ X, T ]) (110)and ˜ g = X + 2 λf (cid:48) [ X, T ] (111)3which have to be solved self-consistently.Equation (111) can be rewritten as˜ g = X + 2 λ κ (cid:104) Ψ (cid:105) X (112)which leads to d ˜ gdX = 1 − λ Vκ χ ψ (113)where χ ψ = (cid:90) β dτ (cid:90) d x (cid:104) δψ ( (cid:126)x ) δψ (0) (cid:105) (114)is the space-time average of the quantum and thermal“energy” fluctuations. Since d ˜ gdX = 0 corresponds to thedevelopment of a first-order transition, as previously dis-cussed in the classical case, analogously the generalizedLP criterion is κ < (cid:101) (cid:18) dg c d ln V (cid:19) χ ψ . (115) where we have assumed g c [ e ll ] = g c − λe ll (116)similar to (78). At zero temperature, this expression gen-eralizes the original LP criterion (1) to quantum phasetransitions. At finite temperatures, the critical temper-ature and the critical coupling constant are related by g c ( T c ) = uT / ˜Ψ c , so that d ln g c = d ln T c and the LPcriterion becomes κ < (cid:101) (cid:18) dT c d ln V (cid:19) C V /T c (cid:122) (cid:125)(cid:124) (cid:123)(cid:18) g T c (cid:19) χ ψ , (117)where we have identified ∆ C v /T c = ( g/ T c ) χ ψ with thespecific heat capacity. Thus we see that the generalizedLarkin Pikin equation encompasses the original LP crite-rion, (1) and also (57),in addition to being applicable atlow temperatures. Our next step is to identify a crossoverscaling form for the clamped free energy, f , that includesboth thermal and quantum critical fluctuations. IV. QUANTUM ANNEALING OF THE FIRST-ORDER TRANSITIONA. The Amplitude Factors
In order to generalize the Larkin-Pikin argument to T →
0, we need to introduce a crossover scaling form for theclamped free energy f in (110) and (111) that is applicable near both the classical and the quantum critical points.The approach we follow here that describes both the quantum and classical cases was adapted from an earlier studyused to describe Ising anisotropy at a Heisenberg critical point. At a finite temperature, the location of the phase transition is shifted by the thermal fluctuations, so that g c ( T ) = g c (0) − uT (118)where ˜Ψ is the shift exponent defined in (71); we note that if the effective dimension of the quantum system is ator below its upper critical dimension ˜Ψ = ˜ νz . For convenience, we will shift the definition of g to absorb the zerotemperature QCP critical coupling constant, g c (0), i.e g − g c (0) → g , so that g c ( T ) = − uT . Now temperature is afinite size correction to the quantum critical point, and the free energy is determined by a crossover function f ( g, T ) = g − ˜ α Φ (cid:32) T g (cid:33) . (119)which describes both the quantum critical point, and the finite temperature classical critical point of the clampedsystem (see Figure 5), here we will use the convention that an exponent with a tilde refers to the quantum case sothat α and ˜ α are classical and quantum exponents respectively. A key point is that at finite temperature, criticalbehavior now occurs at the shifted value of g c ( T ), and the scaling behavior is governed by the finite temperaturecritical exponents. Therefore for a fixed temperature scan (Fig. 5) for small g − g c ( T ), f ( g, T ) = ( g − g c ( T )) − α A I ( T ) . (120)where A I ( T ) is the amplitude factor for the classical critical point occuring at g = g c ( T ). Similarly if we perform asweep through the phase transition at constant coupling constant g (Fig. 5), then we can write f [ g, T ] ∼ ( T − T c [ g ]) − α A II ( g ) , (121)4where A II ( g ) is amplitude factor for the quantum transition at T c [ g ] = ( − g/u ) ˜Ψ . The scaling form (119) allows us todetermine the form of these amplitude factors (see Appendix C), given by A I ( T ) = a T ( α − ˜ α ˜Ψ ) ,A II ( g ) = a g (1 − ˜Ψ)(2 − α )+( α − ˜ α ) , (122)where a and a are constants. The resulting expressions for the singular parts of the free energy for constanttemperature and constant coupling constant sweeps (see Figure 5) are f [ g, T ] = | g − g c ( T ) | − α T α − ˜ α ˜Ψ (constant T) , | T − T c [ g ] | − α g (1 − ˜Ψ)(2 − α )+( α − ˜ α ) (constant g) , (123)where, since we are interested in the singular scaling behavior, we have dropped the constants a and a . B. Clausius-Clapeyron Relations as T c → We now examine how the discontinuities ∆ S ( T c ) and∆ V ( T c ) in entropy and volume evolve along the firstorder phase boundary as the transition T c is loweredtowards zero, and connect them with the Clausius-Clapeyron relation. In this discussion, we shall identifythe tuning parameter g with the pressure P , g ≡ P − P c .Using Maxwell’s relations we have dT c dP c ≡ dT c dg c = − ∆ V ∆ S (cid:12)(cid:12)(cid:12)(cid:12) T = T c (124)From (118), we have dT c dg c ∝ − T − c . (125)In the case of particular interest, that of three-dimensional ferroelectrics, the dynamical exponent z =1, so the effective dimension d eff = 3 + z = 4 lies atthe upper critical dimension. In this case, ˜Ψ = ˜ νz = ,we see that this dT c /dP c ∝ T − c , implying that ∆ V / ∆ S diverges as T c →
0. To understand how this hap-pens, we now independently evaluate the temperature-dependences of ∆ V and ∆ S .To carry out this calculation, we need to input thequantum-renormalized amplitude factors for the free en-ergy into the parameterized equations (110) and (111).We consider tuning through the first order phase tran-sition at constant T = T c . The corresponding tuningvariable g = g − g c ( T c ) of the clamped system in (123)is now replaced by the parametric variable X describingthe tuning parameter that has been shifted by the longrange interactions, g → X . The singular part of the freeenergy, by (123), is then f [ X ] = −| X | − α T α − ˜ α ˜Ψ c . (126)Using (110) and (111), the explicit form of the quantumLarkin-Pikin equations are then˜ f [ X ] = −| X | − α T α − ˜ α ˜Ψ c + λ (cid:20) (2 − α ) X − α T α − ˜ α ˜Ψ c (cid:21) (127) and g [ X ] = X − λ (2 − α ) | X | − α T α − ˜ α ˜Ψ c sgn( X ) . (128)The only difference between this calculation and the orig-inal Larkin Pikin calculation is the presence of the am-plitude factors T α − ˜ α ˜Ψ c . From the original Larkin Pikinanalysis, we know that since g [ X ] is a non-monotonicfunction of X , the inverse function X [ g ] is a discontin-uous function of g , given by X [ g ] = sgn( g ) X + ( | g | ). Inparticular, at g = 0, X jumps from − X c to + X c , Sincethe free energy ˜ f [ X c ] = ˜ f [ − X c ] is an even function of X ,it follows that the first order transition occurs at g = 0.Using g [ X c ] = 0, we obtain X c = [2 λ (2 − α )] T α − ˜ αα ˜Ψ c . (129)To obtain ∆ V = d ˜ f /d g = ˜ f (cid:48) [ X c ] / g (cid:48) [ X c ], we need g (cid:48) [ X ]and ˜ f (cid:48) [ X ]. First, we differentiate (128) with respect to X ,and using the expression (129) for X c , we find g (cid:48) [ X c ] = α is just a constant. Also, differentiating (127) with respectto X and substituting (129), we find that˜ f (cid:48) [ X ] = − ( α/ λ ) X c . (130)from which we obtain∆ V ( T c ) ∝ − T α − ˜ αα ˜Ψ c . (131)Similarly, to obtain ∆ S = − d ˜ f /dT c = − ˜ f (cid:48) [ X ] dX/dT c ,weneed dX/dT c . Now since g = g + uT / ˜Ψ c and g (cid:48) [ X c ] = α ,we obtain dXdT c = 1 g (cid:48) [ X c ] d g dT c = uα ˜Ψ T − c . (132)so that∆ S [ T c ] = − ˜ f (cid:48) [ X c ] dXdT c ∝ T α − ˜ αα ˜Ψ c T − c . (133)For the case ˜Ψ = ˜ νz = 1 / α = 1 /
2, ˜ α = 0, both∆ V ∼ T c and ∆ S ∼ T c vanish at absolute zero, but in5such a way that their ratio diverges as T c →
0, in agree-ment with (125). Naively, the divergence of ∆
V / ∆ S as T c → S and ∆ V simply vanish at different rates, stillsignifying an approach to a continuous quantum phasetransition. More generally, so long as the finite tempera- ture exponent α exceeds the quantum exponent ˜ α , α > ˜ α ,(131) indicates that lim T c → ∆ V → T c goesto zero, indicating that quantum fluctuations “anneal”the zero-temperature quantum phase transition to be-come continuous (see Figure 6). FIG. 6. Schematic figure showing the evolution of the first order phase transition in the approach to the quantum annealedcritical point for the case ˜Ψ = ˜ νz = 1 / α = 1 /
2, ˜ α = 0. (a) Evolution of jump in volume (b) dependence of ∆ V and ∆ S on T c and (c) T c dependence of ∆ V / ∆ S . V. THE LARKIN-PIKIN PHASE DIAGRAM
We therefore have a system with a line of classical first-order transitions that ends in a T = 0 quantum criticalpoint. Next we consider application of a field conjugateand parallel/antiparallel to the order parameter. In thisSection we present the scaling approaches to the criti-cal endpoints, classical and quantum, and the resultingtemperature-field-quantum tuning parameter ( g ) phasediagram of the generalized Larkin-Pikin problem. A. Identification of the Classical Critical Endpoints
We can work out the scaling of the critical end pointin the LP mechanism using the scaling form for the freeenergy f ∝ − t − α Φ (cid:18) ht βδ (cid:19) . (135)where h is the dimensionless external field, t = ( T − T c ) /T c is the reduced temperature and T c is the tran-sition temperature of the clamped system. We wantto know how (135) behaves in a finite field when t issmall compared with h /βδ . In this limit we know that f ∝ h δ +1 and (135) can be rewritten f ∝ − h δ +1 Λ (cid:18) th /βδ (cid:19) = − h − αβδ Λ (cid:18) th /βδ (cid:19) (136)where,using the identity 2 − α = β (1 + δ ), we have sub-stituted ( δ + 1) /δ = (2 − α ) /δβ , Comparing (135) and(136), we see that Λ = (cid:18) t βδ h (cid:19) δ Φ . (137)The scaling form of the free energy, defined by (136) and(137), results in the finite-field Larkin Pikin equations˜ f = h /δ Λ (cid:18) Xh /β/δ (cid:19) + λ (cid:18) h − αβδ Λ (cid:48) (cid:18) Xh /βδ (cid:19)(cid:19) t = X − λ (cid:20) h ( α − /βδ Λ (cid:48) (0) + Xh α/βδ Λ (cid:48)(cid:48) (0) (cid:21) . (138)where details of the derivation of (138) are presented inAppendix D.We recall that criticality of f [ X, h ] only occurs at X =0 ( t = t c ) indicating that ˜ f [ X, h ] can only be critical at X = 0. In the region of first order transitions t [ X ] is non-monotonic with two points, a maxima and minima, wherethe gradient dt/dX = 0 goes to zero. As we approach the6critical field h c , the maxima and minima merge togetherat a point of inflection, meeting at X = 0. As a resultwe deduce that the critical end point occurs when X = 0(for criticality) and at dt/dX = 0 (merger of maximum and minimum). More succinctly, the critical endpointcorresponds to the inflection point in t ( X ). When weimpose these two conditions, we can solve for h c and t c = ( T CEP − T (0) c ) /T (0) c , which from (138) implies that h c = (2 λ Λ (cid:48)(cid:48) ) δβ/α , ( dt/dX = 0) t c = − λ Λ (cid:48) h α − δβ c = − h δβ c Λ (cid:48) Λ (cid:48)(cid:48) = − (2 λ Λ (cid:48)(cid:48) ) α Λ (cid:48) Λ (cid:48)(cid:48) , ( X = 0) (139)We can be sure that these quantities are both positive, because ∆ S = − ∂f [ t, h ] /∂t = h − αβδ Λ (cid:48) is the change inthe entropy due to the field, and we expect this to be negative, so that Λ (cid:48) < t c >
0. Similarly, − ∂ f∂t ∼ ∆ CT ∼ h − α/δβ Λ (cid:48)(cid:48) . This quantity gets bigger as the field is reduced, so that Λ (cid:48)(cid:48) >
0, guaranteeing that h c > B. Field Behavior Close to the Quantum Critical Endpoint
When we look at this problem as part of the approach to a QCP, we must now include the amplitude A I ( T c ) = T ( α − ˜ α/ ˜Ψ) c . The tuning parameter t of the classical calculation now becomes g = g − g c ( T c ) = g − uT c . If we nowexpand around the finite temperature critical point at a specific T c , the singular free energy of the clamped system is f [ g , h ] = − h − αβδ Λ (cid:18) δgh /βδ (cid:19) A I ( T c ) . (140)The Larkin Pikin equations now become˜ f [ X, h ] = − h − αβδ A I ( T c )Λ (cid:18) Xh /βδ (cid:19) + λ (cid:18) h − αβδ A I ( T c )Λ (cid:48) (cid:18) Xh /βδ (cid:19)(cid:19) g [ X ] = X − λA I ( T c ) (cid:20) h ( α − /βδ Λ (cid:48) (0) + Xh α/βδ Λ (cid:48)(cid:48) (0) (cid:21) . (141)where again ˜ f [ X, h ] refers to the free energy of the unclamped system.As discussed in the last section, the critical end pointoccurs at X = 0. The critical end point is then at g = − uT / ˜Ψ c + g [0]. When we do the subsequent algebra, wesee that we get similar equations to those we obtainedat finite temperature (138) with the replacement λ → λA I ( T c ). Taking equations (139) and replacing t c → g c and λ → λA I ( T c ), we obtain h c ∝ ( λA I ( T c )) δβ/α = ( λ ) δβ/α ( T c ) δβ ( α − ˜ α ) α ˜Ψ , g c ∝ ( λA I ( T c )) α = λ α ( T c ) ( α − ˜ α ) α ˜Ψ . (142)These equations are valid in the plane of constant T c .For small λ we can transform these expressions into theplane of constant g , writing T c = ( g c (0) − g ) ˜Ψ , while thelocation of the critical end point is at a temperture T EP = T c + δT EP , where δT EP = g ( dT c /dg ) ∝ g ( g c (0) − g ) ˜Ψ − ,which then gives h c ∝ λ δβα ( g c (0) − g ) δβ ( α − ˜ α ) α δT EP ∼ λ α ( g c (0) − g ) α − ˜ αα − (1 − ˜Ψ) (143)where we have restored g c (0) For the Gaussian fixed point considered by Larkin and Pikin, with ˜ α = 0, α = 1 / β = 1 / δ = 5, ˜Ψ = 1 /
2, we have T c ∼ ( g c − g ) / h c ∼ λ / ( g c − g ) / ∼ T / c δT EP ∼ λ ( g c − g ) / (144)which yields a pointed, V -shaped “anteater’s tongue” asthe surface of first-order transitions in the LP problem.In Figure 7 we present a schematic of the evolution ofthe critical endpoints as a function of T c , and we notethat the full Larkin-Pikin phase diagram is displayed inFigure 1. VI. IMPLICATIONS FOR OBSERVABLEPROPERTIES
The specific heat exponent α of the clamped (fixedvolume) system plays a key role in the universality ofthe classical Larkin-Pikin criterion (1) since the couplingof the order parameter to the lattice is a strain-energy7 FIG. 7. Evolution of the critical end-points with T c . density. For the scalar ( n = 1) case considered here, α >
0, so that ∆ κ is singular and the finite-temperaturetransition is always first-order in the unclamped (fixedpressure) system. By contrast for d + z >
4, the systemis above its upper critical dimension and there is a con-tinuous transition at T = 0 and quantum annealed criti-cality. The amplitudes of thermodynamic quantities willdecrease with temperature in the approach to the quan-tum critical point, and we have specifically presented thisbehavior for the latent work and the entropy.A key motivation for our study has been recent low-temperature experiments on polar insulators that displayquantum criticality even though their classical transi-tions are first-order. Many ferroelectrics have scalar or-der parameters with dynamical exponent z = 1, so suchthree-dimensional materials are in their marginal dimen-sion; logarithmic corrections to the bulk modulus are cer-tainly present but they are not expected to be singular.Indeed such contributions to the dielectric susceptibil-ity, χ , in the approach to ferroelectric quantum criti-cal points have not been observed to date; furthermorehere the temperature-dependence of χ is described wellby a self-consistent Gaussian approach appropriate aboveits upper critical dimension. Therefore there may bea very weak first-order quantum phase transition butexperimentally it appears to be indistinguishable from acontinuous one. We note that near quantum criticalitythe main effect of long-range dipolar interactions, not in-cluded in this treatment, is to produce a gap in the logitu-dinal fluctuations, but the transverse fluctuations remaincritical; the excellent agreement between theory andexperiment at ferroelectric quantum criticality confirmsthat this is the case. Dielectric loss and hysteresis measurements can beused to probe the line of classical first-order transitions,and to determine the nature of the quantum phase tran-sition. The Gruneisen ratio (Γ), the ratio of the thermal expansion and the specific heat, is known to change signsacross the quantum phase transition; furthermore itis predicted to diverge at a 3D ferroelectric quantum crit-ical point as Γ ∝ T so this would be a good indicator ofunderlying quantum criticality. Both the bulk modulusand the longitudinal sound velocity should display fea-tures near quantum annealed criticality, where specificsare material-dependent. Elastic anisotropy may drivethis system into an inhomogeneous state.
The cou-pling of domain dynamics to anisotropic strain has beenstudied classically for ferroelectrics, and implicationsfor the quantum case are a topic for future work. VII. DISCUSSION AND OPEN QUESTIONS
In summary, we have developed a theoretical frame-work to describe compressible insulating systems thathave classical first-order transitions and display pressure-induced quantum criticality. We have generalized theLarkin-Pikin approach to the quantum case usingcrossover scaling forms that describe both its classicaland its quantum behavior. We show that when the sys-tem is above its upper critical dimensionality, there is nolatent work at the quantum transition indicating that itis continuous. We then include a field conjugate to theorder parameter, and derive the Larkin-Pikin phase di-agram with three critical points, two classical and onequantum. Following the original Larkin-Pikin analysis,ours has been performed for a scalar order parameterand isotropic elasticity where the phase transition is first-order for all finite temperatures; here we show that for d + z > q = 0 “bound-ary component” of the strain which drives the long-rangeinteraction, when integrated around a closed loop on atorus is a topological invariant that counts the number ofenclosed defects (19) and which is closely connected withthe concept of torsion. In a system with boundaries,we still expect the long-range interaction, but now de-rived from the boundary waves of the material. There isthus a kind of bulk-boundary correspondence in the phe-nomenon that may be topological in character. One pos-sibility here, is that the Larkin Pikin interaction, whichbreaks the Lorentz invariance of the short-range physics,is a kind of symmetry breaking anomaly. Recently the possibility of a line of discontinuous8transitions ending in a quantum critical point hasalso been studied in frustrated spin models, inmultiferroics, and in transition metal difluorides. There are also experiments on metallic systems thatsuggest quantum annealed criticality, so a quantum gen-eralization of the electronic case with possible links toprevious work on metallic magnets should be pursued; implications for doped paraelectric materials and polarmetals will also be explored. Extension of this work toquantum transitions between two distinct ordered statesseparated by first-order classical transitions may be rele-vant to the iron-based superconductors and to the enig-matic heavy fermion material U Ru Si where quantumcritical endpoints have been suggested. Finally the possibility of quantum annealed critical-ity in compressible materials, magnetic and ferroelectric,provides a new setting for the exploration of exotic quan-tum phases where a broad temperature range can beprobed with easily accessible pressures due to the lattice-sensitivity of these systems. In particular, the elimina-tion of the Larkin Pikin mechanism at T = 0 exposesa bare quantum critical point, a state of matter withquantum fluctuations on all scales, with the potential forinstabilities into novel quantum phases. We have benefitted from discussions with colleagues in-cluding A.V. Balatsky, L.B. Ioffe, D. Khmelnitskii, P.B.Littlewood and P.A. Volkov. PC and PC gratefullyacknowledge the Centro Brasileiro de Pesquisas Fisicas(CBPF), Trinity College (Cambridge) and the CavendishLaboratory where this project was initiated. Early stagesof this work were supported by CAPES and FAPERJgrants CAPES-AUXPE-EAE-705/2013 (PC and PC),FAPERJ-E-26/110.030/2103 (PC and PC), and NSFgrant DMR-1830707 (P. Coleman); this work was alsosupported by grant DE-SC0020353 (P.Chandra) fundedby the U.S. Department of Energy, Office of Science.MAC acknowledges the Brazilian agencies CNPq andFAPERJ for partial financial support. GGL acknowl-edges support from grant no. EP/K012984/1 of theERPSRC and the CNPq/Science without Borders Pro-gram. PC, PC and GGL thank the Aspen Center forPhysics and NSF grant PHYS-1066293 for hospitalitywhere this work was further developed and discussed. PCand PC thank S. Nakatsuji and the Institute for SolidState Physics (U. Tokyo) for hospitality when he earlystages of this work were underway. Appendix A: Gaussian Strain integration in the Classical case
We would like to integrate out the Gaussian elastic degrees of freedom from the action so that the partition functiontakes the form Z = (cid:90) D [ ψ ] (cid:90) D [ u ] e − S [ ψ,u ] −→ Z = (cid:90) D [ ψ ] e − S [ ψ ] . (A1)where the effective action S is a function of the order parameter ψ . We write S [ ψ ] = S L [ ψ ] + ∆ S [ ψ ] (A2)where S L [ ψ ] = 1 T (cid:90) d x (cid:20)
12 ( ∂ µ ψ ) + a ψ + b ψ (cid:21) (A3)describes the physics of the order parameter in the clamped system with tuning parameter a ∝ T − T c T C and b > e − ∆ S [ ψ ] = (cid:90) D [ e ab , u q ] e − ( S E + S I ) (A4)with S E + S I = 1 T (cid:90) d x (cid:20) (cid:18) K − µ (cid:19) e ll ( (cid:126)x ) + µe ab ( (cid:126)x ) + λψ ( (cid:126)x ) e ll ( (cid:126)x ) (cid:21) . (A5)As discussed in the main text, we separate the strain field into its q = 0 and finite q components (18), e ab ( (cid:126)x ) = e ab + 1 √ V (cid:88) (cid:126)q (cid:54) =0 i q a u b ( (cid:126)q ) + q b u a ( (cid:126)q )) e i(cid:126)q · (cid:126)x , (A6)9so that the action (A5) divides into two terms, S E + S I = S [ e ab , ψ ] + S [ u, ψ ]. We next define the integrals (cid:90) de ab e − S [ e ab ,ψ ] = e − S [ ψ ] , and (cid:90) D [ u ] e − S [ u,ψ ] = e − S [ ψ ] . (A7)that treat the q = 0 and finite q elastic contributions to (A4) respectively.The uniform part of the action is S [ e ab , ψ ] = VT (cid:20) (cid:18) K − µ (cid:19) e ll + µe ab (cid:21) + VT λψ q =0 e ll = 12 e ab M abcd e cd + v ab e ab , (A8)where ψ (cid:126)q = V (cid:82) d xψ ( (cid:126)x ) e i(cid:126)q · (cid:126)x is the Fourier transform of the fluctuations in “energy density” and M abcd = 1 T K P Labcd (cid:122) (cid:125)(cid:124) (cid:123) ( δ ab δ cd ) +2 µ P Tabcd (cid:122) (cid:125)(cid:124) (cid:123)(cid:18) δ ac δ bd − δ ab δ cd (cid:19) , (A9) v ab = VT λψ q =0 δ ab . (A10)In (A9) P Labcd and P Tabcd are independent projection operators ( P Γ abef P Γ efcd = P Γ abcd , Γ ∈ L, T ) associated wtih thelongitudinal and transverse components of the strain.When we integrate over the uniform part of the strain field (A8), S [ e ab , ψ ] = 12 e ab M abcd e cd + v ab e ab → S [ ψ ] = − v ab M − abcd v cd (A11)Because of the independent nature of the projection operators P L,Tabcd in (A9), we can write the inverse of M as M − abcd = TV (cid:20) K ( δ ab δ cd ) + 12 µ (cid:18) δ ac δ bd − δ ab δ cd (cid:19)(cid:21) , (A12)so the Gaussian integral over the uniform part of the strain field yields S [ ψ ] = − v ab M − abcd v cd = − V T λ K ( ψ q =0 ) . (A13)which can also be written as S [ ψ ] = − λ T (cid:18) K (cid:19) (cid:20) V (cid:90) d x (cid:90) d x (cid:48) ψ ( (cid:126)x ) ψ ( (cid:126)x (cid:48) ) (cid:21) . (A14)The nonuniform part of the action is S [ u, ψ ] = 1 T (cid:88) (cid:126)q (cid:54) =0 (cid:18) u ∗ a ( (cid:126)q ) M ab u b ( (cid:126)q ) + (cid:126)a ( (cid:126)q ) · (cid:126)u ( (cid:126)q ) (cid:19) (A15)where M ab = (cid:20)(cid:18) K − µ (cid:19) q a q b + µ (cid:0) q δ ab + q a q b (cid:1)(cid:21) ,(cid:126)a q = (cid:16) iλ √ V ψ − q (cid:17) (cid:126)q. (A16)0The matrix entering the fluctuating part of the action S [ u, ψ ] in (A7) can be projected into the longitudinal andtransverse components of the strain M ab ( (cid:126)q ) = q (cid:20)(cid:18) K + 43 µ (cid:19) ˆ q a ˆ q b + µ ( δ ab − ˆ q a ˆ q b ) (cid:21) (A17)where ˆ q a = q a /q are the direction cosines of (cid:126)q . Inversion of this matrix is then M − ab ( (cid:126)q ) = q − (cid:34)(cid:18) K + 43 µ (cid:19) − ˆ q a ˆ q b + µ − ( δ ab − ˆ q a ˆ q b ) (cid:35) , (A18)so the Gaussian integral over fluctuating part of the strain field leads to S [ u, ψ ] = 1 T (cid:88) (cid:126)q (cid:54) =0 u ∗ a ( (cid:126)q ) M ab ( (cid:126)q ) u b ( (cid:126)q ) + (cid:126)a ( (cid:126)q ) · (cid:126)u ( (cid:126)q ) → S [ ψ ] = − T (cid:88) (cid:126)q (cid:54) =0 a a ( − (cid:126)q ) M − ab ( (cid:126)q ) a b ( (cid:126)q )= − V T (cid:88) (cid:126)q (cid:54) =0 ψ − q ψ q λ K + µ (A19)We can rewrite this as a sum over all (cid:126)q , plus a remainder at (cid:126)q = 0: S [ ψ ] = − V T (cid:88) (cid:126)q ψ − q ψ q λ K + µ + V T ( ψ q =0 ) λ K + µ = − T λ K + µ (cid:90) d xψ ( (cid:126)x ) + V T ( ψ q =0 ) λ K + µ (A20)which can be reexpressed as S [ ψ ] = − λ T (cid:18) K + µ (cid:19) (cid:26)(cid:90) d x ψ ( (cid:126)x ) − (cid:20) V (cid:90) d x (cid:90) d x (cid:48) ψ ( (cid:126)x ) ψ ( (cid:126)x (cid:48) ) (cid:21)(cid:27) . (A21)The first term is a local attraction while the second term corresponds to a long-range repulsion.When we combine (A14) and (A21), we obtain∆ S [ ψ ] = − λ T (cid:26)(cid:18) K + µ (cid:19) (cid:90) d x ψ ( (cid:126)x ) + (cid:18) K − K + µ (cid:19) (cid:20) V (cid:90) d x (cid:90) d x (cid:48) ψ ( (cid:126)x ) ψ ( (cid:126)x (cid:48) ) (cid:21)(cid:27) . (A22)Recalling (A2), we note that we can group the first term in (A22) in the local S L [ ψ ] (A3) to obtain the results S [ ψ ] = S L [ ψ, a, b ∗ ] − λ T (cid:18) K − K + µ (cid:19) (cid:20) V (cid:90) d x (cid:90) d x (cid:48) ψ ( (cid:126)x ) ψ ( (cid:126)x (cid:48) ) (cid:21) , (A23)where S L [ ψ, a, b ∗ ] = 1 T (cid:90) d x (cid:20)
12 ( ∂ µ ψ ) + a ψ + b ∗ ψ (cid:21) (A24)with a renormalized local interaction b ∗ = b − λ K + µ . (A25)as in the main text (equations (24), (25) and (26)).1 Appendix B: Gaussian Strain Integration in the Quantum Case
We would like to integrate out the Gaussian elastic degrees of freedom so that the partition function takes the form Z = (cid:90) D [ ψ ] (cid:90) D [ u ] e − S [ ψ,u ] −→ Z = (cid:90) D [ ψ ] e − S [ ψ ] . (B1)where the integrals are over spacetime (73), S = (cid:82) d xL ≡ (cid:82) β dτ (cid:82) d xL . We write the effective action S [ ψ ] = S L [ ψ ] + ∆ S [ ψ ] (B2)where S L [ ψ ] = (cid:90) d x (cid:20)
12 ( ∂ µ ψ ) + a ψ + b ψ (cid:21) (B3)and e − ∆ S [ ψ ] = (cid:90) D [ e, u ] e − ( S E [ u ]+ S I [ ψ,e ]) (B4)with S E + S I = (cid:90) d x (cid:20) ρ u l + (cid:18) K − µ (cid:19) e ll ( (cid:126)x ) + 12 2 µe ab ( (cid:126)x ) + λψ ( (cid:126)x ) e ll ( (cid:126)x ) (cid:21) . (B5)This action can be cast into matrix form S E + S I = 12 (cid:88) q u i M ij u j + λ (cid:88) j u j ψ j → λ (cid:88) i,j ψ i M − i,j ψ j . (B6)where now the summations run over the discrete wavevector and Matsubara frequencies q ≡ ( iν n , (cid:126)q ), where ν n = πβ n , (cid:126)q = πL ( j, l, k ). As discussed in the main text, we separate out the static (cid:126)q = 0 component of the strain tensor (81),writing e ab ( x, τ ) = e ab + 1 √ V β (cid:88) iν n (cid:88) (cid:126)q (cid:54) =0 i q a u b ( q ) + q b u b ( q )) e i ( (cid:126)q · (cid:126)x − ν n τ ) . (B7)We note that there is no time-dependence in the uniform part of the strain since we restrict ourselves to smoothGaussian deformations of the solid (see discussion in main text preceeding (81)). However the fluctuating componentincludes all Matsubara frequencies; with these caveats, the quantum integration of the strain fields closely followsthat of the classical case. Given the form of the elastic tensor (B7), the action (B6) naturally divides into two terms, S E + S I = S [ e ab , ψ ] + S [ u, ψ ] (B8)corresponding to the distinct unifom and finite (cid:126)q contributions to the strain, and we define the respective integrals (cid:90) de ab e − S [ e ab ,ψ ] = e − S [ ψ ] and (cid:90) D [ u ] e − S [ u,ψ ] = e − S [ ψ ] . (B9)so that ∆ S [ ψ ] = S [ ψ ] + S [ ψ ] . (B10)The uniform part of the action S [ e ab , ψ ] = (cid:90) d x (cid:20) (cid:18) K − µ (cid:19) e ll + 12 2 µe ab (cid:21) + VT ( λψ q =0 ) e ll
2= 12 e ab M abcd e cd + v ab e ab , (B11)where M abcd = (cid:20) K ( δ ab δ cd ) + 2 µ (cid:18) δ ac δ bd − δ ab δ cd (cid:19)(cid:21) ,v ab = V βλψ q =0 δ ab , (B12)is similar to the classical case (A9), but now ψ q = 1 V β (cid:90) d x ψ ( x ) e − i ( (cid:126)q · (cid:126)x − ν n τ ) (B13)is the spacetime Fourier transform of the order parameter intensity. When we integrate over the uniform part of thestrain field, we obtain S [ e ab , ψ ] = 12 e ab M abcd e cd + v ab e ab → S [ ψ ] = − v ab M − abcd v cd (B14)which, as in the classical case, can be reexpressed as S [ ψ ] = − λ βV K ( ψ q =0 ) (B15)using (A9), (A12) and (B12) where β = T .The nonuniform part of the elastic contribution to (B4) is S [ u, ψ ] = (cid:88) iν n (cid:88) (cid:126)q (cid:54) =0 (cid:18) u ∗ a ( q ) M ab u b ( q ) + (cid:126)a ( q ) · (cid:126)u ( q ) (cid:19) , (B16)where q = ( (cid:126)q, iν n ) and we use Roman letters (e.g. a, b ) to denote spatial variables so that q a is a spatial componentof q. Here M ab = (cid:20) ρν n (cid:18) K − µ (cid:19) q a q b + µ (cid:0) q δ ab + q a q b (cid:1)(cid:21) ,(cid:126)a q = (cid:16) iλ (cid:112) V β ψ − q (cid:17) (cid:126)q. (B17)This matrix can be projected into its longitudinal and transverse components M ab = (cid:20)(cid:18) ρν n + ( K + 43 µ ) (cid:19) ˆ q a ˆ q b + (cid:0) ρν n + µ (cid:1) ( δ ab − ˆ q a ˆ q b ) (cid:21) , (B18)where ˆ q a = q a /q is the unit vector. Inversion of this matrix is then M − ab = (cid:20) ρ ( ν n + c L q ) ˆ q a ˆ q b + 1 ρ ( ν n + c T q ) ( δ ab − ˆ q a ˆ q b ) (cid:21) , (B19)where c L = K + µρ , c T = 2 µρ (B20)are the longitudinal and transverse sound velocities; the two terms appearing in M − are recognized as the propagatorsfor longitudinal and tranverse phonons.When we integrate over the fluctuating component of the strain field, only the longitudinal phonons couple to theorder parameter:12 (cid:88) iν n (cid:88) (cid:126)q (cid:54) =0 u ∗ a ( q ) M ab ( q ) u b ( q ) + (cid:126)a ( q ) · (cid:126)u ( q ) → S [ ψ ] = − (cid:88) iν n (cid:88) (cid:126)q (cid:54) =0 a a ( − q ) M − ab ( q ) a b ( q )3= − V βλ (cid:88) iν n ,(cid:126)q (cid:54) =0 ψ − q ψ q (cid:18) q ρν n + ( K + µ ) q (cid:19) . (B21)In this last term, (cid:18) q ρν n + ( K + µ ) q (cid:19) (B22)the (cid:126)q = 0 term vanishes for any finite ν n , but in the case where ν n = 0, the limiting (cid:126)q → (cid:18) q ρν n + ( K + µ ) q (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126)q → = (cid:26) ν n (cid:54) = 0 K + µ ν n = 0 . (B23)We can thus replace (cid:88) iν n ,(cid:126)q (cid:54) =0 ψ − q ψ q (cid:18) q ρν n + ( K + µ ) q (cid:19) → (cid:88) iν n ,(cid:126)q ψ − q ψ q (cid:18) q ρν n + ( K + µ ) q (cid:19) − ( ψ q =0 ) K + µ . (B24)so that S [ ψ ] = V βλ K + µ ) ( ψ q =0 ) − V βλ (cid:88) iν n ,(cid:126)q ψ − q ψ q (cid:18) q ρν n + ( K + µ ) q (cid:19) . (B25)which can be rewritten as S [ ψ ] = V βλ K + µ ) ( ψ q =0 ) − (cid:88) iν n ,(cid:126)q ψ − q ψ q (cid:18) − ν n /c L ν n /c L + q (cid:19) (B26)where c L is defined in (B20).If we now combine (B15) and (B26), recalling (B10), we obtain∆ S = − V βλ (cid:18) K − K + µ (cid:19) ( ψ q =0 ) − − (cid:18) K + µ (cid:19) (cid:88) iν n ,(cid:126)q ψ − q ψ q (cid:18) − ν n /c L ν n /c L + q (cid:19) (B27)The useful spacetime expression (cid:90) d x e i ( q − q (cid:48) ) x βV = δ qq (cid:48) (B28)allows us to rewrite (B27) in spacetime coordinates as∆ S = − λ (cid:26) βV κ (cid:90) d x d x (cid:48) ψ ( x ) ψ ( x (cid:48) ) − (cid:18) K + µ (cid:19) (cid:90) d x [ ψ ( x )] + (cid:90) d x d x (cid:48) ψ ( x ) V dyn ( x − x (cid:48) ) ψ ( x (cid:48) ) (cid:27) (B29)where V dyn ( x − x (cid:48) ) = 1 βV (cid:88) (cid:126)q,iν n e − i ( (cid:126)q · (cid:126)x − ν n τ ) ( K + µ ) ν n /c L ν n /c L + q (B30)with 1 κ = 1 K − K + µ (B31)is the effective Bulk modulus. Recalling (B2), we note that we can group the second and third terms in (B29) in S L [ ψ ] (B3) to obtain the results S [ ψ ] = S L [ ψ ] − λ (cid:18) K − K + µ (cid:19) (cid:20) βV (cid:90) d x (cid:90) d x (cid:48) ψ ( x ) ψ ( x (cid:48) ) (cid:21) , (B32)4with S L [ ψ ] = (cid:90) d x (cid:20)
12 ( ∂ µ ψ ) + a ψ + b ∗ ψ + 12 ψ ( x ) V dyn ( x − x (cid:48) ) ψ ( x (cid:48) ) (cid:21) (B33)where the Fourier transform of V dyn ( x − x (cid:48) ) is V dyn ( q ) = λ K + µ (cid:18) ν n /c L (cid:126)q + ν n /c L (cid:19) , (B34)and b ∗ = b − λ K + µ . (B35)as in the main text (equations (86), (87) and (88)). We note that (B34) is finite and non-singular in the limit q → Appendix C: Derivation of the Amplitude Factors for the Crossover Scaling
In order to generalize the Larkin-Pikin argument to T →
0, we need to introduce a crossover scaling form for theclamped free energy f in (110) and (111) that is applicable near both the classical and the quantum critical points. At a finite temperature, the location of the phase transition is shifted by the thermal fluctuations, so that g c ( T ) = g c (0) − uT . (C1)where ˜Ψ is called the shift exponent; we note that if the effective dimension of the quantum system is below its uppercritical dimension ˜Ψ = ˜ νz . For convenience, we’ll take the zero temperature QCP critical coupling constant to bezero, g c (0) = 0. Now temperature is a finite size correction to the quantum critical point, and the free energy isdetermined by a crossover function f ( g, T ) = g − ˜ α Φ (cid:32) T g (cid:33) . (C2)that describes both the quantum critical point, and the finite temperature classical critical point of the clampedsystem (see Figure 5); here we will use the convention that α and ˜ α refer to the classical and quantum exponentsrespectively. At a finite temperature the critical behavior now occurs at the shifted value of g c ( T ), governed by thefinite temperature specific heat exponent α . For a fixed T scan with small g − g c ( T ), the singular part of the freeenergy is f ( g, T ) = ( g − g c ( T )) − α A I ( T ) . (C3)where A I ( T ) is the amplitude factor for the classical critical point occuring at T c = T .The scaling form (C2) allows us to determine the form of this amplitude factor. The crucial observation is thatthe classical critical point occurs at a value T / Ψ /g = − /u , so that Φ( (cid:126)x ) must have a singularity of the form( g + uT ) − α = g − α (1 + u T / ˜Ψ g ) − α ∼ (1 + ux ) − α , so that the scaling function takes the formΦ( (cid:126)x ) = (1 + ux ) − α ˜Φ( (cid:126)x ) , (C4)To see this in detail, let us rewrite (C2) as f ( g, T ) = ( g − g c ( T )) − α g − ˜ α ( g − g c ( T )) − α Φ (cid:32) T g (cid:33) = ( g − g c ( T )) − α g α − ˜ α (1 + u T g ) − α Φ (cid:32) T g (cid:33) . (C5)5In other words, f ( g, T ) = ( g − g c ( T )) − α g α − ˜ α ˜Φ (cid:32) T g (cid:33) , (C6)where ˜Φ( (cid:126)x ) = Φ( (cid:126)x )(1 + ux ) − α . (C7)To assure a classical phase transition at finite temperature with the right exponent, the cross-over function ˜Φ must besmooth around x = − /u , in otherwords, the original cross-over function contains a hidden singularity at x = − /u and factorizes as follows: Φ( (cid:126)x ) = (1 + ux ) − α ˜Φ( (cid:126)x ) , (C8)as inferred in (C4). Thus at finite temperature, the singularity at zero temperature splits into a shifted singularitywith modified exponent 2 − α f ( g, T ) = ( g − g c ( T )) − α A [ g, T ] , (C9)where the amplitude factor is given by A [ g, T ] = g α − ˜ α ˜Φ (cid:32) T g (cid:33) . (C10)Suppose we carry out a sweep at constant temperature T c (Fig. 5) , then near the classical critical line, we mayreplace g = g c ( T c ) = uT c , so that T / Ψ c /g = − /u , inside the cross-over function and f [ g, T c ] ∼ ( g − g c ) − α A I ( T c ) (C11)where A I ( T c ) = A [ g c ( T ) , T c ] = a T ( α − ˜ α ˜Ψ ) c , (C12)and a = (cid:104) u α − ˜ α ˜Φ (cid:0) − u (cid:1)(cid:105) . Likewise, if we carry out a sweep through the phase transition at constant g c (Fig. 5), thenwe can write f [ g c , T ] ∼ ( T − T c ) − α A II ( g c ) (C13)where A II ( g c ) = (cid:18) dg c dT (cid:19) − α g α − ˜ αc ˜Φ (cid:18) − u (cid:19) = a g (1 − Ψ)(2 − α )+( α − ˜ α ) c . (C14)with a = (cid:16) u Ψ ˜Ψ (cid:17) − α ˜Φ (cid:0) − u (cid:1) . Summarizing the amplitude factors are then A I ( T c ) = a T ( α − ˜ α ˜Ψ ) c ,A II ( g c ) = a g (1 − Ψ)(2 − α )+( α − ˜ α ) c . (C15)as given in (122) in the main text.6 Appendix D: Derivation of the Larkin-Pikin Equations in Finite Field
We use our scaling form for the free energy f ∝ − t − α Φ (cid:18) ht βδ (cid:19) . (D1)where h is the dimensionless external field, t = ( T − T c ) /T c is the reduced temperature and T c is the transitiontemperature of the clamped system. We want to describe the behavior of (D1) when t is small compared with h /βδ .In this limit we know that f ∝ h δ +1 and (D1) can be rewritten f ∝ − h δ +1 Λ (cid:18) th /βδ (cid:19) = − h − αβδ Λ (cid:18) th /βδ (cid:19) (D2)where,using the identity 2 − α = β (1 + δ ), we have substituted ( δ + 1) /δ = (2 − α ) /δβ , Comparing (D1) and (D2), weobtain Λ = (cid:18) t βδ h (cid:19) δ Φ . (D3)If y = th /βδ and z = y − βδ , then we haveΛ( y ) = y β (1+ δ ) Φ( y − βδ ) = y − α Φ( y − βδ ) . (D4)We note that at large values of z = h/t βδ , small values of y = th /βδ , the free energy can be expanded perturbativelyin y around y = 0, so that f [ X, h ] = − h − αβδ (cid:20) Λ(0) + Xh /βδ Λ (cid:48) (0) + X h /βδ Λ (cid:48)(cid:48) (0) (cid:21) . (D5)where we have replaced t by the parameterized variable X of the unclamped material. This means that ∂f∂X ≡ f (cid:48) X = − (cid:20) h ( α − /βδ Λ (cid:48) (0) + Xh α/βδ Λ (cid:48)(cid:48) (0) (cid:21) . (D6)When (D5) and (D6) are input into the LP equations (48) and (49), The Larkin Pikin equations in a finite fieldbecome ˜ f = h /δ Λ (cid:18) Xh /β/δ (cid:19) + λ (cid:18) h − αβδ Λ (cid:48) (cid:18) Xh /βδ (cid:19)(cid:19) t = X − λ (cid:20) h ( α − /βδ Λ (cid:48) (0) + Xh α/βδ Λ (cid:48)(cid:48) (0) (cid:21) . (D7)which are exactly the equations (138) in the main text. A. Larkin and S. Pikin, Sov. Phys. JETP , 891 (1969). D. Belitz, T. R. Kirkpatrick, and T. Vojta, Phys. Rev.Lett. , 4707 (1999). S. Grigera, R. Perry, A. Schofield, M. Chiao, S. Ju-lian, G. Lonzarich, S. Kieda, Y. Meano, A. Millis, andA. Mackenzie, Science , 329 (2001). A. V. Chubukov, C. P´epin, and J. Rech, Phys. Rev. Lett. , 147003 (2004). D. Belitz, T. R. Kirkpatrick, and T. Vojta, Rev. Mod.Phys. , 579 (2005). D. L. Maslov, A. V. Chubukov, and R. Saha, Phys. Rev.B , 220402 (2006). J. Rech, C. P´epin, and A. V. Chubukov, Phys. Rev. B ,195126 (2006). T. R. Kirkpatrick and D. Belitz, Phys. Rev. B , 134451(2012). M. Brando, D. Belitz, F. M. Grosche, and T. R. Kirk-patrick, Rev. Mod. Phys. , 025006 (2016). P. Chandra, P. Coleman, and A. Larkin, Phys. Rev. Lett. , 88 (1990). P. Chandra and P. Coleman, inLes Houches Lecture Notes (Session LVI), edited byB. Doucot and J. Zinn-Justin (Elsevier, Amsterdam,1995) pp. 495–594. L. Balents, Nature , 199 (2010). M. Zacharias, A. Rosch, and M. Garst, Eur. Phys. J. Spe-cial Topics , 1021 (2015). M. R. Norman, Rev. Mod. Phys. , 041002 (2016). I. Paul and M. Garst, Phys. Rev. Lett. , 227601 (2017). R. M. Fernandes, P. P. Orth, and J. Schmalian,arXiv:1804.00818 (2018). T. Ishidate, S. Abe, H. Takahashi, and N. Mˆori, Phys.Rev. Lett. , 2397 (1997). T. Suski, S. Takaoka, K. Murase, and S. Porowski, SolidState Communications , 259 (1983). S. Horiuchi, K. Kobayashi, R. Kumai, N. Minami, F. Ka-gawa, and Y. Tokura, Nature Communications , 7469(2015). S. Rowley, L. Spalek, R. Smith, M. Dean, M. Itoh, J. Scott,G. Lonzarich, and S. Saxena, Nature Physics , 367(2014). P. Chandra, G. Lonzarich, S. Rowley, and J. Scott, Rep.Prog. Phys. , 112502 (2017). L. Landau and E. Lifshitz,Theory of Elasticity, 3rd Edition (Pergamon Press,1986). G. S. G. Borchhardt and A. Rost, Phys. Stat. Sol. , 143(1976). M. Lines and A. Glass,Principles and Applications of Ferroelectrics and Related Materials(Clarendon Press, Oxford, 1977). S. Sachdev,
Quantum Phase Transitions (Cambridge Uni-versity Press, Cambridge, U.K., 1999). A. J. Millis, Physical review B, Condensed matter , 7183(1993). M. A. Continentino, Quantum Scaling in Many-Body Systems(Cambridge Unversity Press, 2017). G. Gehring, Europhys. Lett. , 60004 (2008). G. Gehring and A. Ahmed, J. Appl. Phys. , 09E125(2010). V. P. Mineev, Phys.-Usp. , 121 (2017). C. Bean and D. Rodbell, Phys. Rev. , 104 (1962). D. Bergman and B. Halperin, Phys. Rev. B , 2145(1976). J. Sak, Phys. Rev. B , 3957 (1974). J. Bruno and J. Sak, Phys. Rev. B , 3302 (1980). A. Aharony, in Phase Transitions and Critical Phenemenon,edited by C. Domb and M. Green (Academic Press, 1976). W. Vieira and P. Carvalho, Eur. Phys. Lett. , 21001(2014). P. Pfeuty, D. Jasnow, and M. E. Fisher, Phys. Rev. B ,2088 (1974). N. Das, Int. J. Mod. Phys. B. , 1350028 (2013). A. Rechester, Sov. Phys. JETP , 423 (1971). D. Khmelnitskii and V. Shneerson, Sov. Phys. JETP ,164 (1973). R. Roussev and A. Millis, Physical Review B , 014105(2003). L. Zhu, M. Garst, A. Rosch, and Q. Si, Physical ReviewLetters , 066404 (2003). M. Garst and A. Rosch, Physical Review B , 205129(2005). M. de Moura, T. Lubensky, Y. Imry, and A. Aharony,Phys. Rev. B , 2176 (1976). R. T. Brierley and P. B. Littlewood, Phys. Rev. B ,184104 (2014). M. O. Katanaev, arXiv.org (2005), cond-mat/0502123v1. E. Witten, Reviews of Modern Physics , 035001 (2016). N. Kellermann, M. Schmidt, and F. M. Zimmer, Phys.Rev. E , 012134 (2019). M. Schmidt, N. Kellermna, and F. M. Zimmer, Phys. Rev.E , 032138 (2020). A. Narayan, A. Cano, A. Balatsky, and N. Spaldin, NatureMaterials , 223 (2019). P. Chandra, Nature Materials , 197 (2019). S. U. Handunkanda, E. B. Curry, V. Voronov, A. H. Said,G. G. Guzm´an-Verri, R. T. Brierley, P. B. Littlewood, andJ. N. Hancock, Phys. Rev. B , 134101 (2015). G. Schmiedeshoff, E. Mun, A. Lounsbury, S. Tracy,E. Palm, S. Hannahs, J.-H. Park, T. Murphy, S. Budko,and P. Canfield, Phys. Rev. B , 180408 (2011). Y. Tokiwa, M. Garst, P. Gegenwart, S. Bud/ko, andP. Canfield, Phys. Rev. Lett. , 116401 (2013). A. Steppke, R. Kuchler, S. Lausberg, E. Lengyel,L. Steinke, R. Borth, T. Luhmann, C. Krellner, M. Nick-las, C. Geibel, F. Steglich, and M. Brando, Science ,933 (2013). S. A. Pikin, Sov. Phys. JETP , 753 (1970). K. Quader and M. Widom, Phys. Rev. B , 144512(2014). P. Chandra, P. Coleman, and R. Flint, Nature493