Quantum Annealed Criticality
Premala Chandra, Piers Coleman, Mucio A. Continentino, Gilbert G. Lonzarich
QQuantum Annealed Criticality
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: May 31, 2018)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[1] mechanism tothe quantum case. We show that if the T = 0 system lies above its upper critical dimension, the lineof first-order transitions can end in a quantum annealed critical point where zero-point fluctuationsrestore the underlying criticality of the order parameter. The interplay of first-order phase transitions withquantum fluctuations is an active area [2–9] in the studyof exotic quantum states near zero-temperature phasetransitions [10–14]. In many metallic quantum ferro-magnets, coupling of the magnetization to low energyparticle-hole excitations transforms a high temperaturecontinuous phase transition into a low temperature dis-continuous one, and the resulting classical tricriticalpoints have been observed in many systems [2–9]. Exper-imentally there also exist insulating materials that haveclassical first-order transitions that display quantum crit-icality [15–19], and here we provide a theoretical basis forthis observed behavior.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[1] (LP), involves the interaction of strain with a fluctu-ating critical order parameter. LP found that a divergingspecific heat in the clamped system of fixed dimensionsleads to a first-order transition in the unclamped systemat constant pressure. Specifically, the Larkin-Pikin crite-rion [1, 20] for a first order phase transition is κ < ∆ 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 ofthe specific heat capacity in the clamped system, T c isthe transition temperature and dT c d ln V is its strain deriva-tive. The effective bulk modulus κ is defined as κ − = K − − ( K + µ ) − where K and µ are the bare bulkand the shear moduli in the absence of coupling to theorder parameter fields; more physically κ ∼ K c L c T where c L and c T are the longitudinal and the transverse soundvelocities [21]. We note that shear strain plays a crucialrole in this approach that requires µ >
0. Short-rangefluctuations in the atomic displacements renormalize thequartic coupling of the critical modes, but it is the cou-pling of the uniform ( q = 0) strain to the energy density,the modulus squared of the critical order parameter, thatresults in a macroscopic instability of the critical point leading to a discontinuous transition. FIG. 1. Proposed Temperature-Field-Pressure Phase Di-agram with a sheet of first-order transitions bounded bysecond-order phase lines linking the three critical points,two classical and one quantum; Inset: Temperature-Pressure“slice” indicating a line of classical phase transitions ending ina “quantum annealed critical point” with the standard tem-perature fan where the underlying order parameter criticalityis restored by zero-point fluctuations.
Here we rewrite the Larkin-Pikin criterion in termsof correlation functions so that it can be generalizedto the quantum case. We show that if the T = 0quantum system lies above its upper critical dimension,the corrections to the renormalized bulk modulus arenon-universal; the line of classical first-order transitionscan end in a “quantum annealed critical point” wherezero-point fluctuations restore the underlying criticalityof the order parameter. We end with a discussion ofthe temperature-field-pressure phase diagram and spe-cific measurements to probe it (cf. Fig. 1).Low-temperature measurements on ferroelectric insu-lators provide a key motivation for our study [15–19].At finite temperatures and ambient pressure these ma-terials often display first-order transitions due to strongelectromechanical coupling [22]; yet in many cases [15– a r X i v : . [ c ond - m a t . s t r- e l ] M a y
19] their dielectric susceptibilities suggest the presenceof pressure-induced quantum criticality associated withzero-temperature continuous transitions [15–19]. It isthus natural to explore whether a quantum generalizationof the Larkin-Pikin approach [1], involving the couplingof critical order parameter fluctuations to long wave-length elastic degrees of freedom, can be developed todescribe this phenomenon.In the simplest case of a scalar order parameter ψ andisotropic elasticity, the Larkin-Pikin (LP) mechanism [23]refers to a system where the order parameter ψ ( (cid:126)x ) iscoupled to the volumetric strain with interaction energy H I = λ (cid:90) d x e ll ( (cid:126)x ) ψ ( (cid:126)x ) (2)where e ab ( (cid:126)x ) = (cid:16) ∂u a ∂x b + ∂u b ∂x a (cid:17) is the strain tensor, u a ( (cid:126)x )is the atomic displacement, e ll ( x ) = Tr[ e ( (cid:126)x )] is the vol-umetric strain and λ is a coupling constant associatedwith the strain-dependence of T c , λ = (cid:0) dT c d ln V (cid:1) . Thoughthe elastic degrees of freedom are assumed to be Gaus-sian, and thus can be formally integrated out exactly,this must be done with some care. This is because thestrain field separates into a uniform ( (cid:126)q = 0) term definedby boundary conditions and a finite-momentum ( (cid:126)q (cid:54) = 0)contribution determined by fluctuating atomic displace-ments 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 , (3)where { a, b } ∈ [1 ,
3] and u a ( q ) is the Fourier transformof u a ( x ). Here we employ periodic boundary conditionsto a finite size system with volume V = L and discretemomenta (cid:126)q = πL ( l, m, n ), where l, m, n are integers.The uniform strain vanishes when the crystal is exter-nally clamped. The main effect of integrating out thefinite wavevector fluctuations in the strain is to inducea finite correction to the short-range interactions of thecritical fluctuations that can be absorbed into the quar-tic ψ terms in the action. By contrast, fluctuations inthe uniform component of the strain induce an infinite-range attractive interaction between the critical modes(see Supplementary Materials), and it is this componentof the interactions that is responsible for driving first or-der behavior. The problem is then reduced to the inter-action of critical order parameter modes, mediated by thefluctuations of a uniform strain field φ with bulk modulus κ (for details see Supplementary Materials). Conceptu-ally, the Larkin-Pikin approach amounts to a study ofcritical phenomena in a clamped system, followed by astability analysis of the critical point once the clampingis removed.Recently it was proposed to adapt the Larkin-Pikin ap-proach to pressure( P )-tuned quantum magnets where itis often found that dT c d P → ∞ as T c →
0; the authors argued that the associated quantum phase transitionsshould then be first-order [24–26]. However such a di-verging coupling of the critical order parameter fluctua-tions and the lattice should lead to structural instabilitiesnear the quantum phase transition that have not been ob-served [9, 27]. Furthermore dynamics must be includedwhen treating thermodynamic quantities at zero temper-ature [28, 29].We recast the Larkin-Pikin criterion in the languageof correlation functions, generalizing the LP approachto the quantum case summing over all possible space-time configurations. The strain field again separates intotwo contributions as in equation (3), one associated withstatic uniform boundary conditions and the other deter-mined by short wavelength displacements fluctuating atall frequencies 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 ) (4)where q α ≡ ( (cid:126)q, iν n ) with α ∈ [1 , u b ( q ) ≡ u b ( (cid:126)q, iν n )and ν n = 2 πnT is a Matsubara frequency ( k B = 1). Adetailed analysis indicates that when these space-timeelastic degrees of freedom are integrated out, they leadto the coupling of the quantum critical order parame-ter modes to a classical strain field φ , uniform in bothspace and time, with the same effective bulk modulus κ as in the finite-temperature case (see SupplementaryMaterial). The resulting effective action takes the form S eff [ ψ, φ ] = (cid:90) β dτ (cid:90) d x (cid:20) L [ ψ ] + λφ ψ ( (cid:126)x, τ ) + 12 κφ (cid:21) , (5)where ( (cid:126)x, τ ) are the Euclidean space-time co-ordinatesand L [ ψ ] is the Lagrangian of the order parameter ψ ( (cid:126)x, τ )that undergoes a continuous transition in the clampedsystem; in the simplest case L [ ψ ] is a ψ field theory L [ ψ ] = 12 ( ∂ µ ψ ) + a ψ + b ψ . (6)The partition function of the unclamped system is then Z [ φ ] = e − βF [ φ ] = (cid:90) D [ ψ ] e − S eff [ ψ,φ ] , (7)where the trace is over the internal variable ψ , and Z [ φ ]to be evaluated at the stationary point F (cid:48) [ φ ] = 0. Therenormalized bulk modulus, ˜ κ = κ − ∆ κ , is˜ κ = 1 V ∂ F∂φ = κ − λ (cid:90) d xdτ (cid:104) δψ ( (cid:126)x, τ ) δψ (0) (cid:105) , (8)where δψ ( (cid:126)x, τ ) = ψ ( (cid:126)x, τ ) − (cid:104) ψ (cid:105) . In the classical prob-lem there is no time-dependence, and (cid:82) β dτ → β ≡ /T FIG. 2. 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. so at the transition˜ κ = 1 V ∂ F∂φ = κ − λ T c (cid:90) d x (cid:104) δψ ( (cid:126)x ) δψ (0) (cid:105) = κ − ∆ κ. (9)∆ κ in (9) is proportional to energy fluctuations, and canbe re-expressed as λ T c ∆ C V ; we thus recover the LP cri-terion (1) ( κ < ∆ κ or ˜ κ <
0) for a first-order transition.We note that the renormalized quartic mode-mode cou-pling coefficient associated with (23) changes sign con-comitantly with the renormalized bulk modulus; the for-mer has contributions from both the strain coupling andfrom higher order parameter fluctuations [30].The renormalized bulk modulus ˜ κ can also be obtaineddiagrammatically (cf. Figure 2). In the low-energy ef-fective action, the quartic term now has a contributionfrom the coupling of the order parameter fluctuations tothe effective uniform strain. We then can use a Dysonequation for the strain propagator to determine ˜ κ . Morespecifically we can write (cid:18) κ (cid:19) = (cid:18) κ (cid:19) + (cid:18) κ (cid:19) λ (cid:104) ψ ( q ) ψ ( − q ) (cid:105) (cid:12)(cid:12) q =0 (cid:18) κ (cid:19) (10)that results in ˜ κ = κ − ∆ κ = κ − λ χ ψ (11)where χ ψ = χ ψ ( (cid:126)q, iν n ) (cid:12)(cid:12) (cid:126)q,iν n =0 is the static susceptibil-ity for ψ , where χ ψ ( (cid:126)q, iν n ) = (cid:90) β dτ (cid:90) d d x (cid:104) δψ ( (cid:126)x, τ ) δψ (0) (cid:105) e iν n τ − i(cid:126)q · (cid:126)x , (12)is the Fourier transform of the fluctuations in ψ and d = 3. The sign of ˜ κ in (11) is determined by the infraredbehavior of ∆ κ ; if it diverges, as it does classically (for ascalar order parameter and isotropic elasticity), then thiscorrection is universal and the transition is first order. Another possibility is revealed in the zero-temperaturelong-wavelength Gaussian approximation of (11). If wemake the Gaussian approximation (cid:104) δψ ( x ) δψ (0) (cid:105) ≈ ( (cid:104) δψ ( x ) δψ (0) (cid:105) ) , thenlim T → ∆ κ ∝ (cid:90) dq dν q d − [ χ ψ ( (cid:126)q, iν )] (13)where χ ψ ( (cid:126)q, iν ), the order parameter susceptibility, is theFourier transform of the correlator (cid:104) ψ ( x ) ψ (0) (cid:105) . Sincedimensionally [ χ ] = (cid:104) q (cid:105) and [ ν ] = [ q z ], we find thatin the approach to the quantum phase transitionlim T → [∆ κ ] = [ q d + z ][ q ] (14)so that the quantum corrections to κ are non-singular for d + z >
4. The presence of quantum zero-point fluctua-tions increases the effective dimensionality of the phasespace for order parameter fluctuations. If the effective di-mensionality of the quantum system lies above its uppercritical dimensionality, this will have the effect of liber-ating the quantum critical point from the inevitable in-frared slavery experienced by its finite-temperature clas-sical counterpart. In particular the correction to therenormalized bulk modulus is then non-universal, allow-ing for quantum annealed criticality where zero-pointfluctuations toughen the system against the macroscopicinstability present classically, restoring its underlyingcontinuous phase transition.We have therefore identified a theoretical scenariowhere there is a quantum continuous transition eventhough all transitions at finite temperature are first-order. Application of a field conjugate and paral-lel/antiparallel to the order parameter in such a systemleads to a line of first-order transitions ending in twoclassical critical points. Therefore by continuity there isa surface of first-order phase transitions in the phase di-agram (cf. Figure 1) connecting the three critical points,one quantum and two classical, bounded by second-orderphase lines. This phase diagram then presents an alter-native scenario of the interplay of discontinuous transi-tions and fluctuations to that studied in metallic magnetswhere applied field is needed to observe quantum criti-cality in addition to the tuning parameter [9].The specific heat exponent α plays a key role in the uni-versality of the classical Larkin-Pikin criterion (1) sincethe coupling of the order parameter to the lattice is astrain-energy density. For the scalar ( n = 1) case consid-ered here, α >
0, so that ∆ κ is singular and the finite-temperature transition is always first-order; for d + z > n ≥ α is negative so the correction to the renormal-ized bulk modulus will be nonuniversal even at finite tem-peratures [20, 31, 32]. In this case, there can be a classi-cal tricritical point at finite pressures with a second-ordertransition that continues to zero temperature; this situ-ation should be robust to everpresent disorder followingthe Harris criterion [33]. By contrast everpresent elasticanisotropy is known to destabilize criticality in the clas-sical isotropic elastic scalar ( n = 1) lattice and to drive itfirst-order into an inhomogeneous state [20, 31, 32]; herequantum annealed criticality may still be possible due tothe increase of effective dimensionality. The coupling ofdomain dynamics to anistropic strain has been studiedclassically for ferroelectrics [34], and implications for thequantum case are a topic for future work.Because of its underlying non-universal nature, thepossibility of pressure-tuned quantum annealed critical-ity must be determined in specific setttings. Ferro-electrics have a dynamical exponent z = 1, so such three-dimensional materials are in their marginal dimension;logarithmic corrections to the bulk modulus are certainlypresent but they are not expected to be singular. In-deed such contributions to the dielectric susceptibility, χ ,in the approach to ferroelectric quantum critical pointshave not been observed to date [18]; furthermore herethe temperature-dependence of χ is described well by aself-consistent Gaussian approach appropriate above itsupper critical dimension [18, 19]. Therefore there maybe a very weak first-order quantum phase transition butexperimentally it appears to be indistinguishable froma continuous one. We note that near quantum critical-ity the main effect of long-range dipolar interactions, notincluded in this treatment, is to produce a gap in thelogitudinal fluctuations, but the transverse fluctuationsremain critical [35–37]; the excellent agreement betweentheory and experiment at ferroelectric quantum critical-ity confirms that this is the case [18, 19].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 ther-mal expansion and the specific heat, is known to changesigns across the quantum phase transition [38, 39]; fur-thermore it is predicted to diverge at a 3D ferroelectricquantum critical point as Γ ∝ T so this would be a goodindicator of underlying quantum criticality [19]. Boththe bulk modulus and the longitudinal sound velocityshould display jumps near quantum annealed criticality,though specifics are material-dependent since the fluctu-tation contributions to both are non-universal.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 criterion [1] in the language of correlationand response functions; from this standpoint it is clearthat the correction to the renormalized bulk modulus, singular at finite temperature, is non-universal at T = 0for d + z > U Ru Si where quantum critical end-points have been suggested [45]. Finally the possibilityof quantum annealed criticality in compressible materi-als, magnetic and ferroelectric, provides new settings forthe exploration of exotic quantum phases where a broadtemperature range can be probed with easily accessiblepressures due to the lattice-sensitivity of these systems.We have benefitted from discussions with colleaguesincluding A.V. Balatsky, L.B. Ioffe, D. Khmelnitskiiand P.B. Littlewood. PC and PC gratefully acknowl-edge the Centro Brasileiro de Pesquisas Fisicas (CBPF),Trinity College (Cambridge) and the Cavendish Labo-ratory 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 NSFgrants DMR-1309929 (P. Coleman) and DMR-1334428(P. Chandra). MAC acknowledges the Brazilian agenciesCNPq and FAPERJ for partial financial support. GGLacknowledges support from grant no. EP/K012984/1 ofthe ERPSRC and the CNPq/Science without BordersProgram. PC, PC and GGL thank the Aspen Centerfor Physics and NSF grant PHYS-1066293 for hospital-ity where this work was further developed and discussed.PC and PC thank S. Nakatsuji and the Institute for SolidState Physics (U. Tokyo) for hospitality where this workwas completed. [1] A.I. Larkin and S.A. Pikin, “Phase Transitions of theFirst Order but Nearly of the Second,” Sov. Phys. JETP , 891 (1969).[2] D. Belitz, T. R. Kirkpatrick, and Thomas Vojta, “FirstOrder Transitions and Multicritical Points in Weak Itin-erant Ferromagnets,” Phys. Rev. Lett. , 4707–4710(1999).[3] S.A. Grigera, R.S. Perry, A.J. Schofield, M. Chiao, S.R.Julian, G.G. Lonzarich, S.I. Kieda, Y. Meano, A.J. Mil-lis, and A.P. Mackenzie, “Magnetic Field-Tuned Quan-tum Criticality in the Metallic Ruthenate Sr Ru O ,”Science , 329–332 (2001).[4] Andrey V. 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The key idea of the Larkin-Pikin approach is that we integrate out the Gaussian strain degrees of freedom from theaction to derive an effective action for the order parameter field so that Z = (cid:90) D [ u ] (cid:90) D [ ψ ] e − S [ ψ,u ] −→ Z = (cid:90) D [ φ ] (cid:90) D [ ψ ] e − S eff [ ψ,φ ] . (15)The key element in this procedure is a separation of the strain field into uniform and fluctuating components. Whenwe integrate out the uniform component of the strain, it induces an infinite-range attractive interaction between theorder parameter modes mediated by a classical field φ that is uniform in both space and time. The main effect ofthe integration of the fluctuating strain component is to renormalize the short-range interactions between the orderparameter modes; however completion of the Gaussian integral also leads to an infinite range repulsive order parameterinteraction. The overall infinite range interaction is attractive, but this subtlety needs to be checked carefully in boththe classical and quantum cases, as is performed explicitly in this Supplementary Material; here we summarize itsmain results. The relevant quantum generalization of the effective action in (15) is S eff [ ψ, φ ] = (cid:90) d x (cid:20) κ φ + P K + L [ ψ, b ∗ ] + λ (cid:18) φ + PK (cid:19) ψ [ x ] (cid:21) (16)where L [ ψ, b ∗ ] = 12 ( ∂ µ ψ ) + a ψ + b ∗ ψ is the ψ Lagrangian, with a renormalized short-range interaction b ∗ = b − λ K + µ (17)and an effective bulk modulus 1 κ = 1 K − K + µ . (18)In the classical case (cid:90) d x −→ T (cid:90) d x (19)and so we recover the classical effective action S eff [ ψ, φ ] = 1 T (cid:90) d x (cid:20) κ φ + P K + L [ ψ, b ∗ ] + λ ( φ + PK ) ψ [ (cid:126)x ] (cid:21) (20)with definitions as above. We note that in the main text we have replaced the renormalized b ∗ by b and we have set P = 0 for presentational simplicity. Preliminaries
The partition function can be written as an integral over the order parameter and strain fields Z = (cid:90) D [ ψ ] (cid:90) D [ u ] e − S [ ψ,u ] (21)where ψ is the order parameter field and u ( x ) is the local displacement of the lattice that determines the strain fieldsaccording to the relation e ab ( x ) = 12 (cid:18) ∂u a ∂x b + ∂u b ∂x a (cid:19) . (22)Here the action is is determined by an integral over the Lagrangian L , S = (cid:82) d xL . In the quantum case, (cid:82) d x ≡ (cid:82) dτ (cid:82) d x is a space-time integral over configurations that are periodic in imaginary time τ ∈ [0 , β ], where the β inverse temperature ( k B = 1). In the classical case the time-dependence disappears and the integral over τ is replacedby 1 /T so that S = T (cid:82) d xL .The action divides up into three parts S = S A + S I + S B = (cid:90) d x ( L A [ u ] + L I [ ψ, e ] + L B [ ψ ]) , (23)where the contributions to the Lagrangian are: (i) a Gaussian term describing the elastic degrees of freedom in anisotropic system L A [ u ] = 12 (cid:20) ρ ˙ u l + (cid:18) K − µ (cid:19) e ll + 2 µe ab (cid:21) − σ ab e ab (24)where σ ab is the external stress and we have assumed a summation convention in which repeated indices are summedover, so that for instance, e ll ≡ (cid:80) l =1 , e ll and e ab ≡ e ab e ab = (cid:80) a,b =1 , e ab ; (ii) an interaction term L I [ ψ, e ] = λe ll ψ (25)describing the coupling between the volumetric strain e ll = Tr[ e ] and the “energy density” ψ of the order parameter ψ ; (iii) the Lagrangian L B [ ψ ] of the order parameter that, in the simplest case, is a ψ field theory L B [ ψ, b ] = 12 ( ∂ µ ψ ) + a ψ + b ψ , (26)where we have explicitly noted its dependence on the interaction strength b . At a finite temperature critical point,all time-derivative terms are dropped from these expressions.Since the integral over the strain fields is Gaussian, the latter can be integrated out of the partition function leadingto an effective action of the ψ fields S eff [ ψ ] = S B [ ψ ] + ∆ S [ ψ ] where e − ∆ S [ ψ ] = (cid:90) D [ u ] e − ( S A + S I ) . (27)If we write the elastic action in the schematic, discretized form S A + S I = 12 (cid:88) i,j u i M ij u j + λ (cid:88) j u j ψ j (28)then the effective action becomes simply∆ S = 12 ln det[ M ] − λ (cid:88) i,j ψ i M − i,j ψ j (29)where the second term is recognized as an induced attractive interaction between the order-parameter fields. Thesubtlety in this procedure derives from the division of the strain field into two parts: a uniform contribution determinedby boundary conditions and a fluctuating component in the bulk. For the classical case 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 (30)where the u b ( (cid:126)q ) are the Fourier transform of the atomic displacements, while in the quantum problem e ab ( (cid:126)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 a ( q )) e i ( (cid:126)q · (cid:126)x − ν n τ ) , (31)where ν n = 2 πnT is the bosonic Matsubara frequency. Note that the exclusion of all terms where (cid:126)q = 0 fromthe summation also excludes the special point where both iν n and (cid:126)q are zero. As we now demonstrate, the overallattractive interaction ( ∝ − ψ i M − ij ψ j ) contains both short-range and infinite range components. The Gaussian Strain integral: Classical Case
Our task is to calculate the Gaussian integral, e − ∆ S [ ψ ] = (cid:90) D [ e ab , u q ] e − ( S A + S I ) (32)where the classical action S A + 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 ) + P ) e ll ( (cid:126)x ) (cid:21) , (33)where we have denoted σ ab = − P δ ab in terms of the pressure P . We begin by splitting the strain field into the q = 0and finite q components, 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 . (30)This separation enables us to use periodic boundary conditions, putting the system onto a spatial torus with discretemomenta (cid:126)q = πL ( l, m, n ). After this transformation, the action divides up into two terms, S = S [ e ab , ψ ] + S [ u, ψ ]. Weshall 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 ] . (34)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 + P ) e ll = 12 e ab M abcd e cd + v ab e ab , (35)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 = 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) , (36) v ab = VT ( λψ q =0 + P ) δ ab . (37)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) (38)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. (39)When we integrate over the uniform part of the strain field,12 e ab M abcd e cd + v ab e ab → S [ ψ ] = − v ab M − abcd v cd (40)0Now the two terms P Labcd and P Tabcd in M (36) are independent projection operators ( P Γ abef P Γ efcd = P Γ abcd , Γ ∈ L, T ),projecting the longitudinal and transverse components of the strain. The inverse of M is then given by 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) , (41)so the Gaussian integral over the uniform part of the strain field gives S [ ψ ] = − v ab M − abcd v cd = − V T K ( λψ q =0 + P ) . (42)Now the matrix entering the fluctuating part of the action S [ u, ψ ], 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) (43)where ˆ q a = q a /q are the direction cosines of (cid:126)q . The 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) , (44)so the Gaussian integral over fluctuating part of the strain field leads to1 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 (45)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 . (46)The first term is a local attraction while the second term, involving only the (cid:126)q = 0 Fourier component, correspondsto to a repulsive long range interaction.When we combine the results of the two integrals (42) and (46) we obtain∆ S [ ψ ] = − V T λ κ ( ψ q =0 ) − T λ K + µ (cid:90) d xψ ( x ) − V T K (2 λψ q =0 P + P ) (47)where 1 κ = 1 K − K + µ (48)is the effective Bulk modulus.The final step in the procedure, is to carry out a Hubbard Stratonovich transformation, factorizing the long-rangeattraction in terms of a stochastic uniform field φ , − V T λ κ ( ψ q =0 ) → T (cid:90) d x (cid:104) κ φ + λφψ ( x ) (cid:105) . (49)Combining (47) and (49) we obtain the following expression for∆ S [ ψ, φ ] = 1 T (cid:90) d x (cid:20) κ φ + P K + λ (cid:18) φ + PK (cid:19) ψ ( x ) − λ K + µ ) ψ ( x ) (cid:21) . (50)1Finally, adding this term to the original order parameter action S B [ ψ ] = T (cid:82) d xL B [ ψ, b ], our final partition functioncan be written Z = (cid:90) dφ (cid:90) D [ ψ ] e − S eff [ ψ,φ ] (51)where S eff [ ψ, φ ] = S B [ ψ ] + ∆ S [ ψ, φ ] is given by S eff [ ψ, φ ] = 1 T (cid:90) d x (cid:20) κ φ + P K + L [ ψ, b ∗ ] + λ ( φ + PK ) ψ [ x ] (cid:21) (52)where L [ ψ, b ∗ ] = 12 ( ∂ µ ψ ) + a ψ + b ∗ ψ . is the ψ Lagrangian, with a renormalized short-range interaction b ∗ = b − λ K + µ . (53)Note that in the main text we have dropped the “ ∗ ” on b for presentational simplicity; there b refers to this renormalizedinteraction (53).Thus the main effects of integrating out the strain field are a renormalization of the short-range interaction of theorder parameter field and the development of an infinite-range interaction mediated by an effective strain field φ . Ifwe differentiate the action with respect to the pressure, we obtain the volumetric strain δSδP ( (cid:126)x ) = e ll ( (cid:126)x ) = 1 K (cid:0) P + λψ ( (cid:126)x ) (cid:1) , (54)which, as a result of integrating out the strain fluctuations, now contains a contribution from the order parameter.Again in the main text we set P = 0 for presentational simplicity. The Gaussian Strain integral: Quantum Case
In the quantum case, the action in the Gaussian strain integral e − ∆ S [Ψ] = (cid:90) D [ e ab , u q ] e − ( S A + S I ) (55)now involves an integral over space time, with S = (cid:82) d xL ≡ (cid:82) β dτ (cid:82) d xL . We now restore the kinetic energy termsin Lagrangian (24) and (26), so that now the quantum action takes the form S A + S I = (cid:90) dτ d x (cid:20) ρ u l + (cid:18) K − µ (cid:19) e ll ( x ) + 12 2 µe ab ( x ) + ( λψ ( x ) + P ) e ll ( x ) (cid:21) . (56)Again our task is to cast this into matrix form S A + 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 . (57)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 before, we must separate out the static, (cid:126)q = 0 component of the strain tensor, 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 ) , (58)Note that there is no time-dependence to the uniform part of the strain, since the boundary conditions are static.However the fluctuating component excludes (cid:126)q = 0, but includes all Matsubara frequencies; with these caveats, thequantum integration of the strain fields closely follows that of the classical case.2Again the action divides up into two terms, S = S [ e ab , ψ ] + S [ u, ψ ], corresponding to the distinct unifom and finite (cid:126)q contributions to the strain. We shall again 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 ] . (59)The uniform part of the action S [ e ab , ψ ] = (cid:90) dτ (cid:20) (cid:18) K − µ (cid:19) e ll + 12 2 µe ab (cid:21) + VT ( λψ q =0 + P ) e ll = 12 e ab M abcd e cd + v ab e ab , (60)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 β ( λψ q =0 + P ) δ ab , (61)is unchanged, but now ψ q = 1 V β (cid:90) d xψ ( x ) e − i ( (cid:126)q · (cid:126)x − ν n τ ) (62)is the space-time Fourier transform of the order parameter intensity. The non-uniform part is now 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) , (63)where 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. (64)When we integrate over the uniform part of the strain field, we obtain12 e ab M abcd e cd + v ab e ab → S [ ψ ] = − v ab M − abcd v cd = − V β K ( λψ q =0 + P ) , (65)or S [ ψ ] = − K (cid:90) d x ( λψ q =0 + P ) . (66)For presentational simplicity, we will now set P = 0 since the role of pressure here follows that in the classicaltreatment already described.The matrix entering the fluctuating part of the action can be projected into the longitudinal and transverse com-ponents 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) , (67)where ˆ q a = q a /q is the unit vector. The 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) , (68)3where c L = K + µρ , c T = 2 µρ (69)are the longitudinal and transverse sound velocities. The two terms appearing in M − are recognized as the propa-gators for 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 )= − 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) . (70)Now in this last term, (cid:18) q ρν n + ( K + µ ) q (cid:19) (71)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:40) ν n (cid:54) = 0 K + µ ν n = 0 . (72)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 . (73)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) . (74)If we now combine S and S , we obtain S + S = V βλ κ ( ψ q =0 ) − V βλ (cid:88) q ψ − q ψ q (cid:18) q ρν n + ( K + µ ) q (cid:19) . (75)where 1 κ = 1 K − K + µ (76)is the effective Bulk modulus, as in the classical case.Next we carry out a Hubbard-Stratonovich transformation, rewriting the the long-range attraction in terms of astochastic static and uniform scalar field φ as follows − V β λ κ ( ψ q =0 ) → (cid:90) d x (cid:104) κ φ + λφψ ( x ) (cid:105) . (77)The remaining interaction term can be divided up into two parts as follows (cid:88) q ψ − q ψ q (cid:18) q ρν n + ( K + µ ) q (cid:19) = 1 K + µ (cid:88) q ψ − q ψ q (cid:20) − (cid:18) ν n /c L q + ν n /c L (cid:19)(cid:21) . (78)4The first term inside the brackets is independent of momentum and frequency, leading to a finite local attraction termthat will act to renormalize the b term in the Lagrangian L ψ,b as in the classical case. The second term is a non-localand retarded interaction. Due to Lorentz invariance, simple power-counting shows that this term has the same scalingdimensionality as a local repulsive term, and thus it will not modify the critical behavior of the second-order phasetransition.If we transform back into into space-time co-ordinates, then we obtain S + S = (cid:90) d x (cid:20) κ φ + λφψ ( x ) − λ K + µ ) ψ ( x ) (cid:21) + S NL = S eff [ ψ, φ ] + S NL (79)where S NL = λ V β ( K + µ ) (cid:90) d xd x (cid:48) ∂ τ ( ψ )( x ) V ( x − x (cid:48) ) ∂ τ ( ψ )( x (cid:48) ) (80)and V ( x ) = (cid:90) d q (2 π ) (cid:18) c − L | (cid:126)q | + ν n /c L (cid:19) e i ( (cid:126)q · (cid:126)x − iν n τ ) = 12 πc L | (cid:126)x | + c L τ ) (81)is the non-local interaction mediated by the acoustic phonons. Then the final quantum partition function resultingfrom integrating out the strain fields in the quantum case can be written ( P (cid:54) = 0) Z = (cid:90) dφ (cid:90) D [ ψ ] e − S eff [ ψ,φ ] − S NL (82)where S eff [ ψ, φ ] = (cid:90) d x (cid:20) κ φ + P K + L [ ψ, b ∗ ] + λ (cid:18) φ + PK (cid:19) ψ [ x ] (cid:21) (83)and L [ ψ, b ∗ ] = 12 ( ∂ µ ψ ) + a ψ + b ∗ ψ . is the ψ Lagrangian, with a renormalized short-range interaction b ∗ = b − λ K + µ (84)and an effective bulk modulus 1 κ = 1 K − K + µ . (85)Thus, as in the classical case, the main effect of integrating out the strain field, is a renormalization of the short-rangeinteraction of the order parameter field, and the development of an infinite range interaction, mediated by an effectivestrain field φ . The introduction of a nonlocal contribution with the same scaling dimensions as the ψ term will notaffect the properties of the fixed point, and thus it will not change the universality class of the fixed point, as in theclassical Larkin-Pikin case. However we emphasize that in its quantum generalization the effective dimension of thetheory is d eff = d + z . Again we note that in the main text we have replaced the coefficient of the renormalizedinteraction b ∗ in (84) by bb