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Advances in Condensed Matter Physics
Volume 2012 (2012), Article ID 619513, 15 pages
Thermodynamics of the Classical Planar Ferromagnet Close to the Zero-Temperature Critical Point: A Many-Body Approach
1Dipartimento di Scienze Fisiche, Università degli Studi di Napoli “Federico II”, Piazzale Tecchio 80, 80125 Napoli, Italy
2Dipartimento di Fisica “E.R. Caianiello”, Università degli Studi di Salerno, Via Ponte don Melillo, 84084 Fisciano, Italy
3CNISM, Unità di Ricerca di Salerno, 84084 Fisciano, Italy
Received 1 February 2012; Accepted 15 March 2012
Academic Editor: Giancarlo Consolo
Copyright © 2012 L. S. Campana et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We explore the low-temperature thermodynamic properties and crossovers of a -dimensional classical planar Heisenberg ferromagnet in a longitudinal magnetic field close to its field-induced zero-temperature critical point by employing the two-time Green’s function formalism in classical statistical mechanics. By means of a classical Callen-like method for the magnetization and the Tyablikov-like decoupling procedure, we obtain, for any , a low-temperature critical scenario which is quite similar to the one found for the quantum counterpart. Remarkably, for the discrimination between the two cases is found to be related to the different values of the shift exponent which governs the behavior of the critical line in the vicinity of the zero-temperature critical point. The observation of different values of the shift-exponent and of the related critical exponents along thermodynamic paths within the typical V-shaped region in the phase diagram may be interpreted as a signature of emerging quantum critical fluctuations.
An intriguing aspect of quantum phase transitions (QPTs)  is that quantum critical fluctuations may play a relevant role also at finite temperature. This feature leads to a drastic modification of the expected properties of many systems within a wide region around their quantum critical point (QCP) [1–6].
Remarkably, the renormalization group framework (RG) [1, 7, 8] and Moriya’s self-consistent renormalized approach [9, 10] have provided a well-defined scenario for this so-called quantum criticality giving qualitative and also quantitative agreement with a lot of experimental findings [1–6].
However, recent experiments seem to suggest that these theories fail in relevant practical situations (see [11–13] and references therein). Although several alternatives have been proposed to explain these unexpected behaviors, a completely convincing picture is still lacking. Hence, it becomes crucial to provide nonambiguous criteria to determine accurately the range of temperatures where the QCP fluctuations survive against the thermal ones. On the ground of a comparison between the exactly solvable one-dimensional quantum transverse Ising model (QTIM) [1, 14–18] and its classical version (CTIM) (not to be confused with the standard Ising model) [19, 20], it was conjectured that , at least in selected cases, at finite temperature, close to the QCP, quantum critical fluctuations may not be so relevant as commonly believed. The emerging idea was that, to single out conventional quantum criticality, it is not sufficient to observe a power-law behavior of the correlation length or susceptibility decreasing temperature towards zero in the V-shaped quantum critical region of the phase diagram ; rather, the accurate determination of the critical exponents becomes the key ingredient to decide if we are in the influence domain of the QCP or the physics is governed by thermal fluctuations. Of course, to validate the previous conjecture, more realistic many-body systems should be investigated, especially for dimensionalities where a finite-temperature critical line ends in a zero-temperature critical point.
A first step along this direction has been recently performed by exploring the low-temperature properties of the -dimensional CTIM . On the ground of a suitable Ginzburg-Landau-Wilson functional and a momentum-shell RG approach around , this system is expected to have the same properties of the QTIM above a certain temperature.
Further insights will be provided in this paper where we investigate the low-temperature properties of the classical XXZ ferromagnetic model in presence of a longitudinal magnetic field when the longitudinal exchange interaction is smaller than the transverse one. Its quantum analogue, also called planar ferromagnet (PFM), has been extensively studied, in different physical contexts, using several methods. In particular, the spin-1/2 PFM has attracted great attention since, in the pioneering papers by Matsubara and Matsuda [23, 24] on superfluidity in , a quantum lattice gas of hard-core bosons with long-range attractive interactions has been proven to be just equivalent to the spin-1/2 PFM in a longitudinal field (see also ). Remarkably, the Wilson RG [26, 27], applied to a suitable functional representation of the spin-1/2 PFM, capturing the essential low-temperature physics, and the two-time Green’s function technique , utilized to investigate the microscopic spin- model, have provided a reliable scenario of the global phase diagram and crossovers in the vicinity of the QCP.
In the present work, we will study the corresponding -dimensional classical XXZ spin model (CPFM) with particular attention to a field-induced quantum-like critical scenario for a direct comparison with the quantum counterpart. We will use the two-time Green’s function method in classical statistical mechanics , developed and tested in [30–33], on microscopic classical spin model. This allows us to perform in parallel the quantum  and the classical analysis for any , giving a transparent relation between the CPFM and the spin- QPFM, both exhibiting a zero-temperature critical point. Hence, new insights on quantum criticality, at least for a class of anisotropic magnetic systems, will be provided.
The paper is organized as follows. In Section 2 we will introduce the model and the equation of motion for the appropriate two-time Green function in the context of the classical Callen-like method  to calculate the magnetization within the Tyablikov decoupling procedure. The equations for the transverse susceptibility and the critical line will be presented in Section 3. The quantum-like scenario, with the global phase diagram and crossovers, close to the ()-critical point, will be analyzed in Section 4. In Section 5, concluding remarks will be drawn. At the end, for utility of reader, Appendix A is devoted to an outline of the two-time Green’s function framework in classical statistical mechanics and Appendix B presents a method, alternative to the one employed in , to obtain the magnetization as the solution of the Callen-like method.
2. Spin Model and Callen-Like Method
The -dimensional classical XXZ Heisenberg model in a longitudinal magnetic field is described by the Hamiltonian: Here are classical spin- vectors on an -sites hypercubic lattice with unitary spacing, satisfying the identity with . Besides, the transverse () and longitudinal () exchange interactions (with ) between the spins at sites and are assumed to be symmetric, positive, and short ranged. As well known, it is perfectly legal in the classical context to put . However, for a more transparent and direct comparison with the quantum version , through this paper we will consider arbitrary .
Many magnetic materials can be described by the Hamiltonian (1) and different cases may take place depending on the relative weight of the competing anisotropic exchange interactions. Indeed, one has a uniaxial ferromagnet (FM) if with the extreme limit (Ising model); we recover the isotropic Heisenberg model when and the PFM if whose extreme limit is the XY model in a transverse field (TXYM). In the following, we will focus on the classical PFM which exhibits a field-driven zero-temperature critical point as it happens in the quantum case.
The classical model (1) can be described in terms of the canonical variables and , where is the angle between the projection of the spin vector in the -plane and the -axis. The Poisson bracket of two generic classical dynamical variables and is then defined by It is easy to show that, with this prescription, the Poisson brackets for the spin components are given by where is the Levi-Civita tensor.
Following the Callen procedure developed for the quantum Heisenberg FM , we introduce now the retarded two-time GF [29, 33] (see Appendix A): where is the usual step function, denotes the Callen-like parameter, stands for the classical ensemble average, is the inverse temperature, and , is the Liouville operator. Here, acts as a classical time-evolution operator which transforms the dynamical variable at the initial time into at the time . The physics will be of course obtained setting at the end of the calculations.
The equation of motion (EM) for the GF (4) is given by (with ) which, in the frequency- Fourier space, becomes with and . From the basic Poisson brackets (2), a simple algebra yields where with Notice that, for the magnetization per spin , the relevant exact relation is fulfilled: On the other hand, in (6), we have also Then, (6) becomes (again without approximations) The next step consists in performing an appropriate decoupling to close (12). Here we will use the classical version of the Tyablikov decoupling (TD) which, for the quantum case, has been proven  to give near-exact results close to the QCP. This decoupling procedure consists in neglecting transverse correlations in (12) so that one can assume that providing where is the ()-component of the -wave vector Fourier transform in the first Brillouin zone () . Finally, using the Fourier transforms in the -space with , (14) reduces to an algebraic equation for with solution where This equation represents the dispersion relation, at Tyablikov-like decoupling (TD) level, of undamped oscillations for the PFM, expressed as a function of in terms of the Fourier transform of the transverse exchange interaction . The key step is to determine the function or and hence . For utility of the reader, we outline here in after the classical version of the Callen procedure used to solve this problem for isotropic quantum  and classical  Heisenberg FMs and for QPFM .
From the expression (16) for and the exact relation between and the corresponding spectral density (see Appendix A) one easily finds Then, the spectral density corresponding to can be obtained via its Fourier transform: Now, with these ingredients, we are in position to obtain the correlation function related to the original GF.
On the other hand, from the relation , we can also write Then, combining (22) and (24), where is given by (8), we obtain the following differential equation for : to be solved with the initial condition and the additional one , arising from the ensemble-average definition of .
With these conditions we have obtained in  the physical solution of (25): This key result (26) represents the classical analogue of the famous Callen formula for quantum spin- models . An alternative and very instructive method to obtain the solution (26) is presented in Appendix B.
Taking into account the exact relation (10), (26) gives the the remarkable expression: which is valid for any , , and . Here, is the well-known Langevin function and is expressed by (23) in terms of the dispersion relation . If we use the TD, is given by (17) which is in turn a function of . Hence, (27) is a self-consistent equation for and . The longitudinal magnetic field and the anisotropy enter into the problem via the function .
Of course, in the spirit of the Callen method for the calculation of , it is possible to introduce more elaborate decoupling procedures which preserve its validity. For instance, we could adopt the Callen decoupling (CD) , with its classical variant , which takes into account the transverse correlations to the leading order and implies the dispersion relation: with However, in the present case, the and the (28) provide essentially identical results close to the ()-critical point where is near the full polarized-state value .
3. Transverse Susceptibility and Critical Line
We have now all the elements to extract the physics of interest setting in the previous results and solving the set of self-consistent equations: where . Here we have conveniently introduced the reduced magnetization per spin , with . Equations (31) will give as a function of and and hence the GF (16), also at .
For our aim, the relevant quantity to be calculated is the transverse GF: This allows to determine the dynamical transverse susceptibility for CPFM: where, at the TD level, In particular, the thermodynamic transverse susceptibility is given by Using (35) it is also possible to obtain the transverse correlation length via the following relation : where . The longitudinal susceptibility will be simply given by . Of course, the stability condition requires that, in (35), the inequality must be fulfilled. The equality is physically possible for and only if , which is the regime characterizing the CPFM of interest to us.
Here in after we will focus on quantum-like criticality related to the field-driven easy-plane ordering whose key quantities are and and the related ones as functions of and . However, in some relevant cases which may have physical interest, we will calculate also the longitudinal quantities and . Other expressions can be obtained by means of known thermodynamic relations .
We start exploring the main features of the CPFM phase diagram in the -plane. These results will be used as a basis for next developments. At zero temperature with , (31) provides the solution for the reduced magnetization, characterizing a fully polarized state. This implies that the dispersion relation is given by , where is the frequency gap. Hence the transverse susceptibility, which has physical meaning for , becomes Remarkably, (37) suggests that, despite the classical nature of our anisotropic spin model, there exists, as in the quantum case [26–28], a ()-critical point at the value of the longitudinal magnetic field. Then, crossing this point, decreasing to , a field-induced second-order ()-phase transition arises from a fully polarized state with to a transverse-ordered phase. However, the latter phase is unaccessible by the present analysis due to the absence in the Hamiltonian (1) of an in-plane symmetry breaking magnetic field.
From (37), with , we have as , defining the mean field exponent . (Through the paper we will use the indices and to denote the horizontal (isothermal) and the vertical trajectories approaching a critical point in the -plane, resp.)
For arbitrary temperature we can write (so that ). Hence the transverse susceptibility can be conveniently written as with or for stability reasons. Equation (38) establishes an interesting relation between the transverse physics and the longitudinal one.
In the -plane, where (), the general equation which determines the possible critical points is or, in view of the equation for ,
Here, where at the critical points and with . More explicitly, (40) can be also written as The quantity is one of the so-called structure sums depending only on the lattice structure of the spin model. Accurate numerical values of can be found in the literature for different and lattice structures [34, 35, 37]. Previous results suggest that, while a ()-CP with exists for any , a critical line with , ending in such a point, may occur only for dimensionalities for which the integral (42) converges.
If we consider short-range interactions and an hypercubic lattice () as with , from (42) it immediately follows that for only the ()-CP exists, while for a finite-temperature critical line, ending in the ()-CP (), occurs consistently with the Mermin-Wagner theorem . The same result has been obtained in the quantum case .
For , the critical line equation (42) (or (43)) can be solved numerically with respect to or providing the representation or . Notice that, along the critical line, the reduced magnetization is simply given by when is known. The critical line in the plane for is plotted in Figure 1.
Starting from (40) or (43) we can easily derive the analytical expression of the zero-field critical temperature where irrespective of the specific structure of in (37) (as in the case of short-range interactions for which ).
From the expansion , and hence , for , (40) provides, for as at finite (see (41)), It is worth noting that for QPFM, within the TD it was found that  , which reproduces, as expected, (44) in the classical limit for .
For short-range interactions and an hypercubic lattice, estimates for can be obtained assuming as .
Now we explore the behavior of the critical line in the low-temperature regime close to the ()-CP where . With , (40) becomes Solving the equation with respect to or , we find or Along this branch of the critical line we have also From the low-temperature representations (46) and (47) for the critical line we can extract the shift exponent , which determines the shape of the phase boundary close to the ()-CP. It has to be stressed that the value of is independent of , in contrast with the result () known for the QPFM [27, 28].
4. Low-Temperature Critical Properties
In this section we study the low-temperature properties and crossovers of our CPFM within the easy-plane-disordered phase, close to the field-induced ()-CP where , (nearly polarized state), and is very small. Under these conditions, in (31) ; therefore, the equation for becomes The quantity near criticality can be suitably estimated assuming, for the oscillation spectrum , the low- expression . This provides where and is a natural wave-vector cut-off related to the first Brillouin zone and determined by or (as ) , with .
As a consequence, to the leading order in , the self-consistent equation for can be written as (except for exponentially small terms) where is the hypergeometric function.
For our purposes, it is convenient to transform (51) for the longitudinal physics into a self-consistent equation for the oscillation gap , strictly related to and hence to the transverse physics.
Of course, once has been determined as a function of , one can directly calculate and, using the relation between and , it is possible to determine and, therefore, the longitudinal physics near the polarized state in the low-temperature critical regime.
Adopting the asymptotic expansions of the hypergeometric function for , now, we can systematically explore the thermodynamics of the CPFM close to the ()-CP for different values of the dimensionality of the system. Bearing this in mind, in strict analogy with the quantum case [28, 39], in the following subsections we will show the asymptotic solutions of (52) in the classical ( or ) and quantum-like ( or ) regimes, respectively, and the estimates of the related crossovers for different values of .
Replacing the expansions (53) in the self-consistent equation (52) for , we get In the regime (), classical regime in the quantum critical scenario near the isolated ()-CP, (54) admits a solution only for , which reads Under the consistency condition . Then, for within the region , in the -plane, the transverse and longitudinal susceptibilities are given by where the reduced magnetization near polarization is immediately given as .
It is worth noting that, for fixed and , the transverse susceptibility diverges with the critical exponent , while the longitudinal one remains finite.
In the regime (), named here quantum-like regime again in analogy with the corresponding quantum scenario, different terms in (54) may enter in competition and different asymptotic behaviors are expected to occur close to the isolated ()-CP. For in the region , where , (54) provides, to the leading order, a solution which is formally identical to (55) but now the condition should hold. The same occurs for the thermodynamic quantities (56) and the related ones.
For , decreasing along a vertical trajectory which corresponds to the quantum critical one in the phase diagram of the QPFM , (54) yields, with , Then, we get which defines the critical exponent , with . For the nearly polarized state (with ), a simple algebra yields which increases towards unity decreasing according to the power-law .
Now we consider the region which is more relevant from the experimental point of view, namely, the V-shaped region , around the vertical trajectory (for both and ). Under this condition, from (54) a straightforward algebra gives This expression suggests that, within the V-shaped region, the thermodynamics is essentially identical to the one along the trajectory , except for a small correction .
Finally, for and sufficiently far from the quantum-like critical trajectory, within the region , we have This implies that which differs from the MF result , found before at , for a small power-law correction in temperature, in contrast with the exponentially small correction which occurs in the quantum counterpart . Besides, for the nearly polarized state, we obtain In summary, the previous results suggest, for , a very rich phase diagram around the isolated ()-CP, qualitatively reported in Figure 2, where different low- regimes and crossover lines are presented. It appears divided in two main regions (named in analogy with the quantum case ): , where (), and , where (). The line for signals the crossover between the regimes and ; the lines and , symmetric to the vertical trajectory , provide the signature of crossovers among three distinct subregimes , , and with different asymptotic behaviors of the thermodynamic quantities as functions of and . We stress that, within the V-shaped region , delimited by the crossover lines and , the -dependent behaviors are essentially identical to those along the trajectory except for different a small power law corrections . Besides, in the regime , decreasing at fixed , the transverse susceptibility deviates from the one at except for a small power law correction as a function of and . The latter feature differs crucially from the QPFM scenario where the correction to the ()-behavior of is an exponentially small function of and [28, 39].
In any case, below two dimensions, the global phase diagram and the crossovers of the CPFM and QPFM are quite similar. This similarity represents a very interesting ingredient for experimental studies in the sense specified in the introductory section.
For the two-dimensional CPFM, which also exhibits only a ()-CP, a low-temperature scenario similar to the one derived for takes place. But now, logarithmic corrections to the leading power-law behavior arise. This peculiarity has been also found for the analogous quantum system .
With the expansion (53), close to the ()-CP the general self-consistent equation (52) reduces to First, we consider the regime . If , no solution exists while, for , (64) provides which has to be compared with the corresponding result achieved for the QPFM, . Then, for the transverse susceptibility we have the exponentially divergent behavior as (corresponding to a critical exponent ). For the nearly polarized state we immediately find and .
Let us consider now the regime . For , (64) gives and diverges as when . Moreover, around this vertical line for , we find which signals the same leading behavior of and related thermodynamic quantities in the limit , except for small corrections in .
Finally, for and , the frequency gap behaves as providing a thermodynamics very similar to the one found for except for logarithmic corrections in with respect to the MF results.
In conclusion, for the two-dimensional CPFM close to ()-CP, we have a qualitative phase diagram which is very similar to the one shown in Figure 2, presenting three regions , , and . In this case the sector corresponding to previous is absent and the V-shaped region is delimited by the lines which, in contrast to the ones shown in Figure 2, exhibit small logarithmic corrections.
For such dimensionalities the CPFM exhibits a critical line ending in the ()-CP. To explore the low-temperature critical properties it is convenient to rewrite (52) in terms of (for the disordered phase). We will focus on dimensionalities (the case is rather trivial although completely consistent with the general theory of critical phenomena) for which, given the expansions (53), (52) assumes the form where the right-hand side must be positive for stability reasons.
In this paper we will present only explicit results related to the transverse thermodynamics which plays a direct role for our purposes. However, the relevant longitudinal quantities near the polarized state may be simply obtained from the general relations and .
We start calculating the asymptotic solutions of (69) where the critical region around the ()-CP in the -plane is approached in two ways: (i) along horizontal trajectories, as (with ) at fixed (isothermal trajectories); and (ii) along vertical trajectories, as (with ) at fixed or for .
We first consider the regime (). The right-hand side of (69) suggests that two subregimes and should be investigated, with where signals the crossover between them. For isothermal trajectories, in the subregime , one finds for the asymptotic solution which provides for transverse susceptibility the nontrivial non-MF critical exponent . This spherical-model incorrect result is typical of the Tyablikov-like decoupling also for the quantum model at finite temperature [28, 39, 40].
When , (69) yields simply which corresponds to the MF exponent . The crossover between the two previous regimes () is indicated by the Ginzburg-like line for horizontal trajectories: Notice that as ; that is, the two lines and merge at the ()-CP.
For vertical trajectories at fixed (), setting in (69), we easily obtain the asymptotic solutions. All the macroscopic quantities of interest can be now determined in the previous regimes for . In particular, for , we find the critical exponents for and for . The crossover between these two asymptotic sub-regimes for occurs crossing the conventional Ginzburg-Landau line (with ): Of course, also for vertical trajectories within the region of the -plane between the critical and Ginzburg lines, with , the TD quantitatively fails.
Let us consider now the behavior of , and hence of , along the line , decreasing , which is of most experimental interest in view of the problematics discussed in Section 1.
Since in this case , one can immediately see that, to leading order in , a self-consistent solution of (69), under the condition , is given by This result is strictly connected with the shift exponent and in drastic contrast with the corresponding relation obtained for the QPFM [27, 39], which, due to the presence of quantum fluctuations, shows that . Equation (75) predicts that as along the vertical line , providing the exponent , in contrast with the quantum result for the QPFM [28, 39].
From (69), under condition , we get
This means that, within the V-shaped region delimited by the critical line for and the symmetric one for , the spectrum gap, the transverse susceptibility, and other macroscopic quantities behave essentially as along the line , except for negligible corrections in .
Increasing and crossing the line , a crossover to the regime () takes place and one easily finds that, for , the appropriate solution of (69) sounds as Therefore, (as ) behaves essentially as at (), with small corrections in .
Summarizing, for the low- global phase diagram has the qualitative structure shown in Figure 3.
A relevant feature of the phase diagram for the CPFM is that it appears qualitatively similar to the one found for the QPFM using different approaches: RG  and two-time GF method [34, 39]. However, for sufficiently low temperatures along the vertical trajectory , the quantitative difference between the behaviors of the transverse susceptibility, of the CPFM and , may play a crucial role to distinguish classical and quantum fluctuations in realistic PFM-like systems.
In this framework, it is also worth noting that, as a subproduct of the previous analysis, an identical qualitative global phase diagram for the classical XY model () in a transverse magnetic field occurs close to its ()-CP.
The same low- V-shaped-like scenario has been recently obtained for the CTIM  by means of a Wilsonian RG approach in dimensions applied to an appropriate Ginzburg-Landau functional representation. This scenario seems to be a common feature of a variety of classical anisotropic magnetic systems which exhibit a ()-CP as in the quantum counterparts.
5. Concluding Remarks
In the present paper we have explored the low-temperature properties of the -dimensional classical planar ferromagnet (CPFM), which exhibits a field-induced zero-temperature critical point, by adopting the two-time Green’s function framework in classical statistical mechanics.
It was shown that, close to the ()-CP, the phase diagram and the critical scenario are qualitatively similar to those found for the spin- QPFM  for where a critical line exists. The quantitative differences between the two systems, taking place within the V-shaped region of the phase diagram as the -CP is approached along vertical trajectories, might allow to understand when classical and quantum fluctuations are active. These discrepancies are related to the value of the shift exponent which characterizes the way in which the critical line ends at the ()-CP as for classical () and quantum () PFM. In view of our results, we argue that the experimental investigation of the low-temperature criticality of PFM-like systems along and near the line (quantum critical trajectory for the QPFM) in the -plane and, in particular, precise measurements of the critical exponents for the correlation function or the susceptibility may provide a signature of the increase of the shift exponent from to at a certain crossover temperature from the CPFM regime to the quantum one. In particular, when the quantum fluctuations become active, a dimensionality dependence of the shift exponent and related ones should emerge, in contrast with the classical region where for any . This may be a useful guide to establish where the thermal fluctuations dominate over the quantum ones, and vice versa. Therefore, it should be of experimental interest to estimate the crossover temperature below which the quantum critical fluctuations are expected to govern the physics close to the ()-CP. Of course, this requires necessarily an appropriate study of quantum spin models within their V-shaped region.
Useful insights into this problem may be provided by a recent RG analysis  for spin-1/2 QPFM with short range interactions, and by two-time GF approaches for spin-1/2  and spin-  QPFM for short- and long-range interactions. In these papers, an unexpected regime with was found in the V-shaped region above a certain temperature which, for , spin-1/2, and short-range interactions at a TD level, reads  (using our notations) where is the value at of the function . Below , the quantum behavior takes place, as expected from the paradigmatic quantum critical scenario .
The present study for the CPFM clarified the physical meaning of . Our study corroborated the idea that, decreasing along the quantum critical trajectory, provides an estimate of the temperature which signals a crossover between the classical thermally activated regime, with (), and the quantum one, with (). This feature agrees with the RG predictions for the same quantum spin model near and below four dimensions .
It is worth mentioning that within our many-body framework one can also extract the basic quantum-like dynamics by variation of dimensionality. In particular, relevant information can be easily obtained from the scaling structure of the transverse dynamic susceptibility for small values of the arguments. Indeed, from (33)-(34) close to the ()-CP we find where defines the transverse correlation length at zero temperature for . Then, comparing (79) with the general dynamic scaling relation we immediately get and for the Fisher and dynamic critical exponents, and .
In conclusion, our results are in agreement with the statements made in  for the CTIM chain and in  for . Our analysis suggests that reliable measurements of the shift exponent (or related ones) close to the QCP of magnetic systems with PFM symmetry may provide a signature of the presence of quantum critical fluctuations. We believe also that this feature is rather general and not limited to TIM-like and PFM-like systems.
A. An Outline of the Two-Time Green’s Function Framework in Classical Statistical Mechanics
In this section, for utility of the reader, we briefly review the basic ingredients of the two-time retarded and advanced GF’s framework in classical statistical mechanics in a form strictly parallel to the quantum counterpart [40–44]. For two arbitrary dynamical variables and , they are defined as  where , , is the usual step function, denotes an equilibrium ensemble average, and is the Poisson bracket of and .
In (A.1), the dynamical variables and depend on time via the conjugate canonical coordinates , ( is the number of degrees of freedom of the classical system under study), with , is the Liouville operator, and is the Hamiltonian of the system and at the initial time . Of course, the time evolution of the generic dynamical variable is governed by the well-known Liouville equation of motion (EM): One can easily prove that the two-time GFs (A.1) depends on times , only through the difference , that is and the two-time correlation function , with , is related to the classical GFs (A.3) by the following relation : where , is the temperature, and is the Boltzmann constant (we assume ). In particular, we have also For and one can introduce the Fourier transforms: where and are called the -GF of and in the -representation and the classical spectral intensity of the time-dependent correlation function , respectively, with . Then, using (A.4) and the integral representations for the step function and the Dirac -function, can be expressed in terms of the corresponding spectral intensity as where the symbol means if and −1 if . It is interesting to compare (A.8) with the quantum corresponding expression for two operators and [40–44]: where and