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01 Nov 2011
Quantum Phase Transition of the Sub-Ohmic Rotor Model
Manal Al-Ali
Thomas Vojta Missouri University of Science and Technology, [email protected]
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Recommended Citation M. Al-Ali and T. Vojta, "Quantum Phase Transition of the Sub-Ohmic Rotor Model," Physical review B: Condensed matter and materials physics, vol. 84, no. 19, pp. 195136-1-195136-6, American Physical Society (APS), Nov 2011. The definitive version is available at https://doi.org/10.1103/PhysRevB.84.195136
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Quantum phase transition of the sub-Ohmic rotor model
Manal Al-Ali and Thomas Vojta Department of Physics, Missouri University of Science and Technology, Rolla, Missouri 65409, USA (Received 18 October 2011; published 28 November 2011) We investigate the behavior of an N-component quantum rotor coupled to a bosonic dissipative bath having a sub-Ohmic spectral density J (ω) ∝ ωs with s<1. With increasing dissipation strength, this system undergoes a quantum phase transition from a delocalized phase to a localized phase. We determine the exact critical behavior of this transition in the large-N limit. For 1 >s>1/2, we find nontrivial critical behavior corresponding to an interacting renormalization group fixed point, while we find mean-field behavior for s<1/2. The results agree with those of the corresponding long-range interacting classical model. The quantum-to-classical mapping is therefore valid for the sub-Ohmic rotor model.
DOI: 10.1103/PhysRevB.84.195136 PACS number(s): 05.10.Cc, 05.30.Jp
I. INTRODUCTION model, the classical counterpart is a one-dimensional Ising model with long-range interactions that decay as 1/r1+s for Quantum phase transitions are abrupt changes in the ground large distances r. In recent years, the applicability of the state properties of a quantum many-particle system that occur quantum-to-classical mapping to the sub-Ohmic spin-boson when a nonthermal control parameter is varied.1 In analogy model has been controversially discussed after numerical to thermal phase transitions, they can be classified as either renormalization group results14 suggested that its critical first-order or continuous transitions. Continuous quantum behavior for s<1/2 deviates from that of the corresponding phase transitions, also called quantum-critical points, are Ising model. While there is now strong evidence15–18 that characterized by large-scale temporal and spatial fluctuations this conclusion is incorrect and that the quantum-to-classical that lead to unconventional behavior in systems ranging from mapping is actually valid, the issue appears to be still not strongly correlated electron materials to ultracold quantum fully settled.19 Moreover, possible failures of the quantum-to- gases (for reviews see, e.g., Refs. 2–7). classical mapping have also been reported for other impurity Impurity quantum phase transitions8 are an interesting models with both Ising20–22 and higher23,24 symmetries, and class of quantum phase transitions at which only the degrees the precise conditions under which it is supposed to hold are of freedom of a finite-size (zero-dimensional) subsystem not resolved. become critical at the transition point. The rest of the system In the present paper, we therefore investigate the large-N (the “bath”) does not undergo a transition. Impurity quantum limit of the sub-Ohmic quantum rotor model. Analogous phase transitions can occur, e.g., in systems composed of to the spin-boson model, this system undergoes a quantum a single quantum spin coupled to an infinite fermionic or phase transition with increasing dissipation strength from a bosonic bath. Fermionic examples include the anisotropic delocalized phase to a localized phase.25,26 We exactly solve Kondo model9 and the pseudogap Kondo model.10 the critical properties of this transition. Our analysis yields The prototypical system involving a bosonic bath is the nontrivial critical behavior corresponding to an interacting dissipative two-state system,11,12 also called the spin-boson renormalization group fixed point for 1 >s>1/2, while we model, which describes a two-level system coupled to a single find mean-field behavior for s<1/2. All critical exponents dissipative bath of harmonic oscillators. Its ground-state phase agree with those of the corresponding long-range interacting diagram depends on the behavior of the bath spectral density classical model,27 implying that the quantum-to-classical ∝ s J (ω) for small frequencies ω. Power-law spectra J (ω) ω mapping is valid. are of particular interest. In the super-Ohmic case (s>1), Our paper is organized as follows. We define the sub-Ohmic the system is in the delocalized (disordered) phase for any rotor model in Sec. II. In Sec. III, we derive its partition dissipation strength. In contrast, for sub-Ohmic dissipation function, and we solve the self-consistent large-N constraint (0 1098-0121/2011/84(19)/195136(6)195136-1 ©2011 American Physical Society MANAL AL-ALI AND THOMAS VOJTA PHYSICAL REVIEW B 84, 195136 (2011) average S2=N, because fluctuations of the magnitude of where the dot marks the derivative with respect to imaginary S are suppressed by the central limit theorem. The large-N time τ, and β = 1/T is the inverse temperature. quantum rotor is thus equivalent to the quantum spherical The bath action is quadratic in the qj ; we can thus exactly model of Ref. 28, which is given by the Hamiltonian integrate out the bath modes. After a Fourier transformation from imaginary time τ to Matsubara frequency ωn, this yields = 1 2 + 1 2 2 − + 2 − −A = 0 −A 0 HS 2 P 2 ω0S hS μ(S 1). (1) D[q˜i (ωn)] exp( B ) ZB exp( B ), where ZB is the partition function of the unperturbed bath and Here, S and P represent the position and momentum of one 2 2 rotor component, μ is a Lagrange multiplier enforcing the λj ω 2= A = T n S˜(ω )S˜(−ω ). (8) constraint S 1, and h is an external symmetry-breaking B 2 2 + 2 n n 29 2mj ωj ωn ωj field. ωn j We now couple (every component of) the rotor to a bath The sum over j can be turned into an integral over the spectral 30 of harmonic oscillators. In the conventional linear-coupling density J (ω). Carrying out this integral gives form, the Hamiltonian describing the bath and its coupling to 1 S reads A = 1−s | |s ˜ ˜ − B T αωc ωn S(ωn)S( ωn), (9) 2 2 2 p m λ ωn H = j + j ω2q2 + λ q S + j S2 , (2) B j j j j 2 = 2mj 2 2mj ω with the dimensionless coupling constant α j j A A 2πα¯ cosec(πs/2). Combining S and B yields the with qj , pj , and mj being the position, momentum, and mass effective action of the sub-Ohmic rotor model as of the jth oscillator. The ωj are the oscillator frequencies T 1−s s and λ the coupling strengths between the oscillators and S. Aeff =−βμ + + αω |ωn| S˜(ωn)S˜(−ωn) j 2 c The last term in the bracket is the usual counter term which ωn insures that the dissipation is invariant under translations in − T h˜(ω )S˜(−ω ), (10) S.11 The coupling between the rotor and the bath is completely n n ω characterized by the spectral density n where = ω2 + 2μ is the renormalized distance from quan- 2 0 π λj 2 A J (ω) = δ(ω − ω ), (3) tum criticality. The ωn term in S is subleading in the limit 2 m ω j ω → 0. It is thus irrelevant for the critical behavior at the j j j n quantum critical point and has been dropped. The theory then which we assume to be of power-law form needs a cutoff for the Matsubara frequencies which we chose to be ω . Because the effective action is Gaussian, the partition = 1−s s c J (ω) 2παω¯ c ω , (0 <ω<ωc). (4) = 0 ˜ −A function Z ZB D[S(ωn)] exp( eff) is easily evaluated. We find Here,α ¯ is the dimensionless dissipation strength and ωc is a cutoff frequency. We will be interested mostly in the case of 2π 1/2 Z = Z0 exp(βμ) sub-Ohmic dissipation, 0 We now derive a representation of the partition function B. Solving the spherical constraint in terms of an imaginary-time functional integral. Because 2= the sub-Ohmic rotor model H = H + H is equivalent to The spherical (large N) constraint S 1 can be easily S B =− a system of coupled harmonic oscillators (with an additional derived from the free energy F T ln Z by means of the = self-consistency condition), this can be done following Feyn- relation 0 ∂F/∂μ. In the case of a time-independent external ˜ = man’s path integral approach31 with position and momentum field h with Fourier components h(ωn) δn,0h/T , this yields eigenstates as basis states. After integrating out the momentum 1 h2 variables, we arrive at the partition function T + = 1. (12) 1−s s 2 + αωc |ωn| ωn −A −A = S B Z D[S(τ)]D[qj (τ)]e . (5) We now solve this equation, which gives the renormalized distance from criticality as a function of the external The Euclidian action is given by parameters α, T , and h, in various limiting cases. β 1 A = ˙2 + 2 2 − + 2 − S dτ S ω0S hS μ(S 1) (6) 1. T = 0 and h = 0 0 2 At zero temperature, the sum over the Matsubara frequen- β λ2S2 mj 2 2 2 j cies turns into an integral, and the constraint equation reads AB = dτ q˙ + ω q + Sλj qj + , 2 j j j 2m ω2 0 j j j 1 ωc 1 dω = 1. (13) 1−s s (7) π 0 + αωc ω 195136-2 QUANTUM PHASE TRANSITION OF THE SUB-OHMIC ... PHYSICAL REVIEW B 84, 195136 (2011) For sub-Ohmic dissipation s<1, a solution 0tothis Here, 0 is the zero-temperature value given in Eq. (14), equation only exists for dissipation strengths α below a and A =−(1/s) cosec(π/s) as above. For s<1/2, we expand critical value αc because the integral converges at the lower (16)inα − αc and find bound even for = 0. The value of αc defines the location α = T (α>α,s < 1/2), (18a) of the quantum critical point. Performing the integral for α − α c = 0, we find α = 1/[π(1 − s)]. As we are interested in c c = −1/2 1/2 1/2 = the critical behavior, we now solve the constraint equa- B αcωc T (α αc,s < 1/2), (18b) α tion for dissipation strengths close to the critical one α = + T α<α,s < / , 0 − ( c 1 2) (18c) αc. We need to distinguish two cases: 1 >s>1/2 and αc α s<1/2. with given in Eq. (15) and B = 1/[π(1 − 2s)] as above. In the first case, the calculation can be performed by 0 subtracting the constraint equations at α and at α from each c 3. T = 0 and h = 0 other. After moving the cutoff ωc to ∞, the resulting integral can be easily evaluated giving At zero temperature, but in the presence of an external field, the spherical constraint reads s/(s−1) s/(1−s) = αωcA (αc − α) (s>1/2), (14) 1 ωc 1 h2 dω + = 1. (19) where A =−(1/s) cosec(π/s). In the case s<1/2, Eq. (13) 1−s s 2 π 0 + αωc ω can be evaluated by a straight Taylor expansion in αc − α, resulting in Proceeding in analogy to the last subsection, we determine the distance from criticality in the limit h → 0 and |α − αc| −1 = αcωcB (αc − α)(s<1/2), (15) small but fixed. In the case 1 >s>1/2, we obtain = − 1/2 with B 1/[π(1 2s)]. For s<1/2, the functional depen- = α − h (α>αc,s > 1/2), (20a) dence of on αc α thus becomes linear, independent of s.As α − αc we will see later, this causes the transition to be of mean-field + = −s 1−s 2s 1/(s 1) = type. A αcωc h (α αc,s > 1/2), (20b) For dissipation strengths above the critical value α ,the 2 c = + α s h spherical constraint can only be solved by not transforming 0 (α<αc,s > 1/2), (20c) αc − α 1 − s 0 the sum over the Matsubara frequencies in Eq. (12) into the = frequency integral in Eq. (13). Instead, the ω = 0 Fourier where 0 is the zero-field value given in Eq. (14) and A n − component has to be treated separately.32 Alternatively, one (1/s) cosec(π/s) as above. For s<1/2, the corresponding can explicitly introduce a nonzero average for one of the results read N order parameter components (see, e.g., Ref. 1). Both α 1/2 approaches are equivalent; we will follow the first route in = h (α>αc,s < 1/2), (21a) α − α the next subsection. c = −1 2 2 1/3 = B αc ωch (α αc,s < 1/2), (21b) > = 2. T 0 and h 0 α h2 = 0 + (α<αc,s < 1/2), (21c) At small but nonzero temperatures, an approximate solution αc − α 0 of the spherical constraint (12) can be obtained by keeping the with 0 given in Eq. (15) and B = 1/[π(1 − 2s)] as above. ωn = 0 term in the frequency sum discrete while representing all other modes in terms of an ω integral. This gives IV. OBSERVABLES AT THE QUANTUM PHASE T 1 ωc 1 + dω = 1. (16) TRANSITION π + αω1−s ωs 0 c A. Magnetization We now solve this equation on the disordered side of the After having solved the spherical constraint, we now turn transition (α<αc), at the critical dissipation strength αc, and to the behavior of observables at the quantum critical point. on the ordered side of the transition (α>αc). We again need The magnetization M =S follows from Eq. (11)viaM = to distinguish the cases 1 >s>1/2 and s<1/2. −∂F/∂h = T∂(ln Z)/∂h. This simply gives In the first case, we subtract the quantum critical (T = 0,h = 0,α = αc) constraint from Eq. (16). After evaluating M = h/ . (22) the emerging integral, the following results are obtained in the To find the zero-temperature spontaneous magnetization in the limit T → 0 and |α − αc| small but fixed, ordered phase, we need to evaluate Eq. (22)forT = 0, α>α, α c = T α>α,s > / , and h → 0. Using Eqs. (20a) and (21a), we find − ( c 1 2) (17a) α αc M = (α − α )/α (23) = −s 1−s s = c A αcωc T (α αc,s > 1/2), (17b) for the entire range 1 >s>0. The order parameter exponent α s β thus takes the value 1/2 in the entire s range. For T>0, = 0 + T (α<αc,s > 1/2). (17c) αc − α 1 − s does not vanish even in the limit h → 0. The spontaneous 195136-3 MANAL AL-ALI AND THOMAS VOJTA PHYSICAL REVIEW B 84, 195136 (2011) magnetization is therefore identical to zero for any nonzero the action (8)]. As this representation is analytic in ωn,the temperature, independent of the dissipation strength α. Wick rotation can be performed easily. We then carry out the The critical magnetization-field curve of the quantum phase integration over the spectral density after the Wick rotation. transition can be determined by analyzing Eq. (22)forT = 0, The resulting dynamical susceptibility reads α = α , and nonzero h. In the case of 1 >s>1/2, inserting c 1 Eq. (20b) into Eq. (22) yields χ(ω) = . 1−s s + + αωc |ω| [cos(πs/2) − i sin(πs/2)sgn(ω)] = s −1 −(1−s) 1−s 1/(1 s) M A αc ωc h (s>1/2), (24) (31) = + − which implies a critical exponent δ (1 s)/(1 s). For s< = = = 1/2, we instead get the relation At quantum criticality (α αc, T 0, h 0), the real and imaginary parts of the dynamic susceptibility simplify to 1/3 M = Bα−2ω−1h (s<1/2). (25) c c cos(πs/2) sin(πs/2)sgn(ω) Reχ(ω) = , Imχ(ω) = (32) The critical exponent δ thus takes the mean-field value of 3. 1−s s 1−s s αcωc |ω| αcωc |ω| in the entire range 1 >s>0. Comparing this with the B. Susceptibility temperature dependencies (28b) and (30), we note that the The Matsubara susceptibility can be calculated by taking results for s<1/2 violate ω/T scaling, while those for the second derivative of ln Z in Eq. (11) with respect to the 1 >s>1/2 are compatible with it. Fourier components of the field, yielding 1 C. Correlation time χ(iωn) = . (26) + 1−s | |s αωc ωn To find the inverse correlation time (characteristic energy) = = = −1 + We first discuss the static susceptibility χst χ(0) 1/ in ξt , we parametrize the inverse susceptibility as 1−s | |s = +| |s the case of 1 >s>1/2. To find the zero-temperature, zero- αωc ωn (1 ωn/ ). This implies the relation field susceptibility in the disordered (delocalized) phase α< = −1 s−1 1/s αc,weuseEq.(14)for , which results in α ωc . (33) = −1 −1 s/(1−s) − −s/(1−s) χst α ωc A (αc α) (s>1/2). (27) The dependence of the inverse correlation time on the tuning parameter α at zero temperature and field in the case of 1 > The susceptibility exponent thus takes the value γ = s>1/2 is obtained by inserting Eq. (14) into Eq. (33). In the s/(1 − s). disordered phase, α<αc, this gives For dissipation strengths α αc, the susceptibility diverges in the limit T → 0. The temperature dependencies follow from −1/(1−s) 1/(1−s) = ωcA (αc − α) (s>1/2). (34) substituting Eqs. (17a) and (17b)intoχst = 1/ . This yields α − α The correlation-time critical exponent therefore reads νz = χ = c T −1 (α>α,s > 1/2), (28a) − st α c 1/(1 s). Note that this exponent is sometimes called just = s−1 −1 s −s = ν rather than νz in the literature on impurity transitions. We χst ωc αc A T (α αc,s > 1/2). (28b) follow the general convention for quantum phase transitions In the ordered (localized) phase, we thus find Curie behavior where ν describes the divergence of the correlation length 2 with an effective moment of M = (α − αc)/α in agreement while νz that of the correlation time. By substituting Eq. (17b) with Eq. (23). into Eq. (33), we can also determine the dependence of on −1 The static susceptibility in the case s<1/2 is obtained temperature at α = αc and h = 0. We find = A T .The analogously. Using Eq. (15), the zero-temperature, zero-field characteristic energy thus scales with T , as expected from susceptibility reads naive scaling. In the case of s<1/2, the zero-temperature, zero-field = −1 −1 − −1 χst α ωc B(αc α) (s<1/2), (29) correlation time in the disordered phase behaves as [using implying that the susceptibility exponent takes the mean-field Eq. (15)] = value γ 1. From Eq. (18b), we obtain the temperature −1/s 1/s = ωcB (αc − α) (s<1/2), (35) dependence of χst at the critical damping strength, = χ = ω−1α−1/2B1/2T −1/2 (α = α ,s < 1/2). (30) resulting in the mean-field value νz 1/s for the correlation st c c c time critical exponent. The dependence of on temperature In the ordered phase, the behavior for s<1/2 is identical to at α = αc and h = 0 follows from Eq. (18b); it reads −1/(2s) (2s−1)/(2s) 1/(2s) that for s>1/2giveninEq.(28a). = B ωc T . The characteristic energy thus We now turn to the dynamic susceptibility. To compute the scales differently than the temperature, in disagreement with retarded susceptibility χ(ω), we need to analytically continue naive scaling. the Matsubara susceptibility by performing a Wick rotation to real frequencies, iωn → ω + i0. A direct transformation of Eq. (26) is hampered by the nonanalytic frequency dependence D. Scaling form of the equation of state s |ωn| . We therefore go back to a representation of the dynamic A scaling form of the equation of state for 1 >s>1/2 term in the susceptibility in terms of discrete bath modes [see can be determined by subtracting the quantum critical (T = 0, 195136-4 QUANTUM PHASE TRANSITION OF THE SUB-OHMIC ... PHYSICAL REVIEW B 84, 195136 (2011) h = 0, α = αc) spherical constraint from the general constraint criticality given in Eq. (14), and D is an s-dependent constant. (12). After performing the resulting integral, we find Upon approaching criticality α → αc, the prefactor of the T s power-law diverges, suggesting a weaker temperature α − α h2 T c + + = −1+1/s −1/s 1−1/s A α ωc . (36) dependence at criticality. The specific heat can be calculated α 2 1−s s from CS = T (∂SS /∂T ), it thus behaves as Dsαω T / 0. = c We substitute h/M [from Eq. (22)], and after some We now turn to the critical dissipation strength α = αc,For lengthy but straightforward algebra, this equation can be 1 >s>1/2, we find a temperature-independent but nonuni- written in the scaling form versal (s-dependent) entropy in the limit of low temperatures. + − − For s<1/2, the impurity entropy diverges logarithmically as X(M/r1/2,h/r(1 s)/(2 2s),T /r1/(1 s)) = 0, (37) ln(ω0/T) with T → 0. In the ordered phase α>αc, we find with X being the scaling function and r = (α − αc)/α being a logarithmically diverging entropy for all s between 0 and 1. the reduced distance from criticality. This scaling form can At first glance, these logarithmic divergencies appear to be used to reproduce the critical exponents β = 1/2, γ = violate the third law of thermodynamics. We emphasize, s/(1 − s), and δ = (1 + s)/(1 − s) found above. For s<1/2, however, that the impurity entropy represents the difference the same approach gives a scaling equation containing the between the entropy of the coupled rotor-bath system and mean-field exponents β = 1/2, γ = 1, and δ = 3. Moreover, that of the unperturbed bath. Because the bath is infinite, the an explicit dependence on the cutoff for the Matsubara entropy thus involves an infinite number of degrees of freedom frequencies remains. and does not have to remain finite. Whether the logarithmic divergence occurs only in the large-N limit or also for finite-N E. Entropy and specific heat rotors remains a question for the future. We note in passing that the entropy of classical spherical Within our path integral approach, thermal properties models27,34 also diverges in the limit T → 0 (even when are somewhat harder to calculate than magnetic properties measured per degree of freedom). In these models, the diverges because the measure of the path integral explicitly depends occurs because the classical description becomes invalid at on temperature. As the spherical model is equivalent to a set sufficiently low temperatures. It can be cured by going from of coupled harmonic oscillators, we can use the “remarkable the classical spherical model to the quantum spherical model.28 formulas” derived by Ford et al.,33 which express the free This implies that the diverging entropy in the ordered phase of energy (and internal energy) of a quantum oscillator in a heat the sub-Ohmic rotor model is caused by a different mechanism bath in terms of its susceptibility and the free energy (and than that in the classical spherical model. internal energy) of a free oscillator. For our spherical model, they read ∞ V. CONCLUSIONS =− + 1 d FS μ dωFf (ω,T )Im ln χ(ω) , (38) In summary, we have investigated the quantum critical π 0 dω ∞ behavior of a large-N quantum rotor coupled to a sub- 1 d US =−μ + dωUf (ω,T )Im ln χ(ω) . (39) Ohmic bosonic bath characterized by a power-law spectral π 0 dω density J (ω) ∼ ωs with 0 To calculate the impurity entropy SS = (US − FS )/T,we insert Eq. (40) into Eqs. (38) and (39) and perform the resulting 1 >s>1/2 s<1/2 integral. In the disordered phase α<αc, the entropy behaves β 1/21/2 as γs/(1 − s)1 = 1−s s δ (1 + s)/(1 − s)3 SS Dαωc T / 0 (41) νz 1/(1 − s)1/s in the limit T → 0 for all s in the sub-Ohmic range 1 >s>0. η 2 − s 2 − s Here, 0 is the zero-temperature renormalized distance from 195136-5 MANAL AL-ALI AND THOMAS VOJTA PHYSICAL REVIEW B 84, 195136 (2011) Moreover, the behaviors of the dynamic susceptibility The properties of our quantum rotor model must be and inverse correlation time are compatible with ω/T contrasted with the behavior of the Bose-Kondo model which scaling for 1 >s>1/2, while they violate ω/T scaling describes a continuous symmetry quantum spin coupled to for s<1/2. We conclude that the quantum phase tran- a bosonic bath. For this system, the quantum-to-classical sition of the sub-Ohmic quantum rotor model is con- mapping appears to be inapplicable.23 A related observation trolled by an interacting renormalization group fixed point has been made in a Bose-Fermi-Kondo model.24 The main in the case 1 >s>1/2. In contrast, the transition is difference between a rotor and a quantum spin is the presence controlled by a noninteracting (Gaussian) fixed point for of the Berry phase term in the action of the latter. Our results s<1/2. thus support the conjecture that this Berry phase term, which is We now turn to the question of the quantum-to-classical complex and has no classical analog, causes the inapplicability mapping. The classical counterpart of the sub-Ohmic quantum of the quantum-to-classical mapping. rotor model is a one-dimensional classical Heisenberg chain 1+s with long-range interactions that decay as 1/r with distance ACKNOWLEDGMENTS r. The spherical (large-N) version of this model was solved by Joyce;27 its critical exponents are identical to that of the sub- We acknowledge helpful discussions with M. Vojta and Ohmic quantum rotor found here. The quantum-to-classical S. Kirchner. This work has been supported in part by the NSF mapping is thus valid. under Grant No. DMR-0906566. 1S. Sachdev, Quantum phase transitions (Cambridge University 22M. Cheng, M. T. Glossop, and K. Ingersent, Phys.Rev.B80, 165113 Press, Cambridge, 1999). (2009). 2S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Shahar, Rev. Mod. 23S. Sachdev, C. Buragohain, and M. Vojta, Science 286, 2479 Phys. 69, 315 (1997). (1999). 3T. Vojta, Ann. Phys. (Berlin) 9, 403 (2000). 24L. Zhu, S. Kirchner, Q. Si, and A. Georges, Phys. Rev. Lett. 93, 4M. Vojta, Rep. Prog. Phys. 66, 2069 (2003). 267201 (2004). 5P. Coleman and A. J. Schofield, Nature (London) 433, 226 (2005). 25L. F. Cugliandolo, D. R. Grempel, G. Lozano, H. Lozza, and C. A. 6P. Gegenwart, Q. 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