Thermodynamics of the Classical Planar Ferromagnet Close to the Zero-Temperature Critical Point: A Many-Body Approach

Abstract

We explore the low-temperature thermodynamic properties and crossovers of a 𝑑-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 𝑑, a low-temperature critical scenario which is quite similar to the one found for the quantum counterpart. Remarkably, for 𝑑>2 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.

1. Introduction

An intriguing aspect of quantum phase transitions (QPTs) [1] 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 [21], 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 [1]; 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 (𝑑>1)-dimensional CTIM [22]. On the ground of a suitable Ginzburg-Landau-Wilson functional and a momentum-shell RG approach around 𝑑=4, 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 ⁴He, 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 [25]). 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 [28], utilized to investigate the microscopic spin-𝑆 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 𝑑-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 [29], developed and tested in [30–33], on microscopic classical spin model. This allows us to perform in parallel the quantum [28] and the classical analysis for any 𝑑, giving a transparent relation between the CPFM and the spin-𝑆 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 [34] 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 (𝑇=0)-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 [22], to obtain the magnetization as the solution of the Callen-like method.

2. Spin Model and Callen-Like Method

The 𝑑-dimensional classical XXZ Heisenberg model in a longitudinal magnetic field ℎ≥0 is described by the Hamiltonian:1ℋ=−2^𝑁_𝑖,𝑗=1𝐽^⟂_𝑖𝑗𝑆^𝑥_𝑖𝑆^𝑥_𝑗+𝑆^𝑦_𝑖𝑆^𝑦_𝑗+𝐽^‖_𝑖𝑗𝑆^𝑧_𝑖𝑆^𝑧_𝑗−ℎ^𝑁_𝑖=1𝑆^𝑧_𝑖1=−2^𝑁_𝑖,𝑗=1𝐽^⟂_𝑖𝑗𝑆⁺_𝑖𝑆⁻_𝑗+𝐽^‖_𝑖𝑗𝑆^𝑧_𝑖𝑆^𝑧_𝑗−ℎ^𝑁_𝑖=1𝑆^𝑧_𝑖. (1) Here {𝐒_𝑗≡(𝑆^𝑥_𝑗,𝑆^𝑦_𝑗,𝑆^𝑧_𝑗);𝑗=1,2,…,𝑁} are classical spin-𝑆 vectors on an 𝑁-sites hypercubic lattice with unitary spacing, satisfying the identity 𝐒²_𝑗=𝑆⁺_𝑗𝑆⁻_𝑗+(𝑆^𝑧_𝑗)²=𝑆² with 𝑆^±_𝑗=𝑆^𝑥_𝑗±𝑖𝑆^𝑦_𝑗. Besides, the transverse (𝐽^⟂_𝑖𝑗) and longitudinal (𝐽^‖_𝑖𝑗) exchange interactions (with 𝐽^⟂_𝑖𝑖=𝐽^‖_𝑖𝑖=0) between the spins at sites 𝑖 and 𝑗 are assumed to be symmetric, positive, and short ranged. As well known, it is perfectly legal in the classical context to put 𝑆=1. However, for a more transparent and direct comparison with the quantum version [28], through this paper we will consider arbitrary 𝑆.

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 𝐽^‖_𝑖𝑗>𝐽^⟂_𝑖𝑗 with the extreme limit 𝐽^⟂_𝑖𝑗=0 (Ising model); we recover the isotropic Heisenberg model when 𝐽^⟂_𝑖𝑗=𝐽^‖_𝑖𝑗 and the PFM if 𝐽^‖_𝑖𝑗<𝐽^⟂_𝑖𝑗 whose extreme limit 𝐽^‖_𝑖𝑗=0 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 2𝑁 canonical variables 𝜙≡{𝜙_𝑗} and 𝑆^𝑧≡{𝑆^𝑧_𝑗}, where 𝜙_𝑗 is the angle between the projection of the spin vector 𝐒_𝑗 in the 𝑥𝑦-plane and the 𝑥-axis. The Poisson bracket of two generic classical dynamical variables 𝐴=𝐴(𝜙,𝑆^𝑧) and 𝐵=𝐵(𝜙,𝑆^𝑧) is then defined by{𝐴,𝐵}=^𝑁_𝑗=1𝜕𝐴𝜕𝜙_𝑗𝜕𝐵𝜕𝑆^𝑧_𝑗−𝜕𝐴𝜕𝑆^𝑧_𝑗𝜕𝐵𝜕𝜙_𝑗. (2) It is easy to show that, with this prescription, the Poisson brackets for the spin components are given by𝑆^𝑧_𝑖,𝑆^±_𝑗=∓𝑖𝑆^±_𝑖𝛿_𝑖𝑗,𝑆⁺_𝑖,𝑆⁻_𝑗=−2𝑖𝑆^𝑧_𝑖𝛿_𝑖𝑗,𝑆^𝛼_𝑖,𝑆^𝛽_𝑗=𝜖_𝛾^𝛼𝛽𝑆^𝛾_𝑖𝛿_𝑖𝑗𝛼,𝛽,𝛾=𝑥,𝑦,𝑧, (3) where 𝜖_𝛾^𝛼𝛽 is the Levi-Civita tensor.

Following the Callen procedure developed for the quantum Heisenberg FM [35], we introduce now the retarded two-time GF [29, 33] (see Appendix A):𝐺_𝑖𝑗𝑡−𝑡^′=𝜃𝑡−𝑡′𝑆⁺_𝑖𝑡−𝑡^′,𝑒^{𝑎𝑆𝑧𝑗}𝑆⁻_𝑗=𝑆⁺_𝑖𝑡−𝑡^′;𝑒^{𝑎𝑆𝑧𝑗}𝑆⁻_𝑗, (4) where 𝜃(𝑥) is the usual step function, 𝑎 denotes the Callen-like parameter, ⟨⋯⟩=𝒵⁻¹∏^𝑁_𝑗=1∫₀^2𝜋𝑑𝜙_𝑗∫^𝑆_−𝑆⋯𝑑𝑆^𝑧_𝑗⋯exp{−𝛽ℋ(𝜙,𝑆^𝑧)} stands for the classical ensemble average, 𝛽=1/𝑇 is the inverse temperature, and 𝑋(𝑡)=𝑒^𝑖𝐿𝑡𝑋, 𝐿=𝑖{ℋ,…} is the Liouville operator. Here, 𝑒^𝑖𝐿𝑡 acts as a classical time-evolution operator which transforms the dynamical variable 𝑋=𝑋(0)≡𝑋(𝜙(0),𝑆^𝑧(0)) at the initial time 𝑡=0 into 𝑋(𝑡)≡𝑋(𝜙(𝑡),𝑆^𝑧(𝑡)) at the time 𝑡. The physics will be of course obtained setting 𝑎=0 at the end of the calculations.

The equation of motion (EM) for the GF (4) is given by (with 𝜏=𝑡−𝑡′)𝑑𝐺_𝑖𝑗(𝜏)(𝜏𝑆𝑑𝜏=𝛿)⁺_𝑖,𝑒^{𝑎𝑆𝑧𝑗}𝑆⁻_𝑗+𝑆⁺_𝑖(𝜏),ℋ;𝑒^{𝑎𝑆𝑧𝑗}𝑆⁻_𝑗, (5) which, in the frequency-𝜔 Fourier space, becomes𝜔𝐺_𝑖𝑗(𝜔)=𝑖𝑆⁺_𝑖,𝑒^{𝑎𝑆𝑧𝑗}𝑆⁻_𝑗+𝑖𝑆⁺_𝑖(𝜏),ℋ;𝑒^{𝑎𝑆𝑧𝑗}𝑆⁻_𝑗𝜔, (6) with 𝐺_𝑖𝑗(𝜔)=⟨⟨𝑆⁺_𝑖(𝜏);𝑒^{𝑎𝑆𝑧𝑗}𝑆⁻_𝑗⟩⟩_𝜔 and ⟨⟨𝐴(𝜏);𝐵⟩⟩_𝜔=∫^+∞_−∞𝑑𝜏𝑒^𝑖𝜔𝜏⟨⟨𝐴(𝜏);𝐵⟩⟩. From the basic Poisson brackets (2), a simple algebra yields𝑖𝑆⁺_𝑖,𝑒^{𝑎𝑆𝑧𝑗}𝑆⁻_𝑗=𝜓(𝑎)𝛿_𝑖𝑗, (7) where𝜓(𝑎)=−𝑎Ω(𝑎)+2Ω′(𝑎)+𝑎Ω^′′(𝑎), (8) withΩ(𝑎)=𝑒^{𝑎𝑆𝑧𝑖}. (9) Notice that, for the magnetization per spin 𝑚=⟨𝑆^𝑧_𝑖⟩, the relevant exact relation is fulfilled:1𝑚=2𝜓(0)=Ω^′(0). (10) On the other hand, in (6), we have also𝑆⁺_𝑖,ℋ=𝑖_ℎ𝐽^⟂_𝑖ℎ𝑆^𝑧_𝑖𝑆⁺_ℎ−𝐽^‖_𝑖ℎ𝑆⁺_𝑖𝑆^𝑧_ℎ−𝑖ℎ𝑆⁺_𝑖. (11) Then, (6) becomes (again without approximations)(𝜔−ℎ)𝐺_𝑖𝑗(𝜔)=𝜓(𝑎)𝛿_𝑖𝑗−_ℎ𝐽^⟂_𝑖ℎ𝑆^𝑧_𝑖(𝜏)𝑆⁺_ℎ(𝜏);𝑒^{𝑎𝑆𝑧𝑗}𝑆⁻_𝑗𝜔−𝐽^‖_𝑖ℎ𝑆⁺_𝑖(𝜏)𝑆^𝑧_ℎ(𝜏);𝑒^{𝑎𝑆z𝑗}𝑆⁻_𝑗𝜔. (12) 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 [28] to give near-exact results close to the QCP. This decoupling procedure consists in neglecting transverse correlations in (12) so that one can assume that𝑆^𝑧_ℎ(𝜏)𝑆⁺_𝑘(𝜏);𝑒^{𝑎𝑆𝑧𝑗}𝑆⁻_𝑗𝜔≃𝑆^𝑧_ℎ𝑆⁺_𝑘(𝜏);𝑒^{𝑎𝑆𝑧𝑗}𝑆⁻_𝑗𝜔≃𝑚𝑆⁺_𝑘(𝜏);𝑒^{𝑎𝑆𝑧𝑗}𝑆⁻_𝑗𝜔, (13) providing𝜔−ℎ−𝑚𝐽^‖(0)𝐺_𝑖𝑗(𝜔)=𝜓(𝑎)𝛿_𝑖𝑗−𝑚_ℎ𝐽^⟂_𝑖ℎ𝐺_ℎ𝑗(𝜔), (14) where ∑_ℎ𝐽^‖_𝑖ℎ=𝐽^‖(0) is the (𝐤=0)-component of the 𝐤-wave vector Fourier transform in the first Brillouin zone (1BZ) 𝐽^‖_𝑖𝑗=(1/𝑁)∑_𝐤𝑒^{𝑖𝐤⋅(𝐫𝑖−𝐫𝑗)}𝐽^‖(𝐤). Finally, using the Fourier transforms in the 𝐤-space𝐺_𝑖𝑗(𝜔)=1𝑁_𝐤𝑒^{𝑖𝐤⋅(𝐫𝑖−𝐫𝑗)}𝐺_𝐤(𝜔),𝐽^𝛼_𝑖𝑗=1𝑁_𝐤𝑒^{𝑖𝐤⋅(𝐫𝑖−𝐫𝑗)}𝐽^𝛼(𝐤),𝛼=⟂,‖, (15) with 𝛿_𝑖𝑗=(1/𝑁)∑_𝐤𝑒^{𝑖𝐤⋅(𝐫𝑖−𝐫𝑗)}, (14) reduces to an algebraic equation for 𝐺_𝐤(𝜔) with solution𝐺_𝐤(𝜔)=𝜓(𝑎)𝜔−𝜔_𝐤, (16) where𝜔_𝐤=ℎ+𝑚𝐽^‖(𝟎)−𝐽^⟂(𝐤). (17) This equation represents the dispersion relation, at Tyablikov-like decoupling (TD) level, of undamped oscillations for the PFM, expressed as a function of 𝐤 in terms of the Fourier transform 𝐽^⟂(𝐤) of the transverse exchange interaction 𝐽^⟂_𝑖𝑗. The key step is to determine the function 𝜓(𝑎) or Ω(𝑎) and hence 𝑚=(1/2)𝜓(0)=Ω′(0). 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 [35] and classical [34] Heisenberg FMs and for QPFM [28].

From the expression (16) for 𝐺_𝐤(𝜔) and the exact relation between 𝐺_𝐤(𝜔) and the corresponding spectral density Λ_𝐤(𝜔) (see Appendix A)Λ_𝐤(𝜔)=𝑖𝐺_𝐤(𝜔+𝑖𝜖)−𝐺_𝐤(𝜔−𝑖𝜖)_𝜖→0+, (18) one easily findsΛ_𝐤(𝜔)=2𝜋𝜓(𝑎)𝛿𝜔−𝜔_𝐤. (19) Then, the spectral density Λ_𝑖𝑗(𝜔) corresponding to 𝐺_𝑖𝑗(𝜔) can be obtained via its Fourier transform:Λ_𝑖𝑗(𝜔)=1𝑁_𝐤𝑒^{𝑖𝐤⋅(𝐫𝑖−𝐫𝑗)}Λ_𝐤(𝜔)(𝑎)1=2𝜋𝜓𝑁_𝐤𝑒^{𝑖𝐤⋅(𝐫𝑖−𝐫𝑗)}𝛿𝜔−𝜔_𝐤. (20) Now, with these ingredients, we are in position to obtain the correlation function ⟨𝐵𝐴⟩=⟨𝑒^{𝑎𝑆𝑧𝑗}𝑆⁻_𝑗𝑆⁺_𝑖⟩ related to the original GF𝐺_𝑖𝑗(𝜔)=⟨⟨𝐴(𝜏);𝐵⟩⟩_𝜔=⟨⟨𝑆⁺_𝑖(𝜏);𝑒^{𝑎𝑆𝑧𝑗}𝑆⁻_𝑗⟩⟩_𝜔.

From the classical spectral theorem (see Appendix A, (A.13) and (A.14)), one immediately gets𝑒^{𝑎𝑆𝑧𝑗}𝑆⁻_𝑗𝑆⁺_𝑖=𝑇^+∞_−∞𝑑𝜔Λ2𝜋^𝑖𝑗(𝜔)𝜔(𝑎)1≃𝑇𝜓𝑁_𝐤𝑒^{𝑖𝐤⋅(𝐫𝑖−𝐫𝑗)}𝜔_𝐤, (21) and hence also𝑒^{𝑎𝑆𝑧𝑖}𝑆⁻_𝑖𝑆⁺_𝑖=Φ𝜓(𝑎). (22) Here, the quantity𝑇Φ=𝑁_𝐤1𝜔_𝐤^𝑁→∞=𝑇_1BZ𝑑^𝑑𝑘(2𝜋)^𝑑1𝜔_𝐤 (23) is independent of the Callen parameter 𝑎.

On the other hand, from the relation 𝑆⁺_𝑖𝑆⁻_𝑖=𝑆⁻_𝑖𝑆⁺_𝑖=𝑆²−(𝑆^𝑧_𝑖)², we can also write𝑒^{𝑎𝑆𝑧𝑖}𝑆⁻_𝑖𝑆⁺_𝑖=𝑆²Ω(𝑎)−Ω′′(𝑎). (24) Then, combining (22) and (24), where 𝜓(𝑎) is given by (8), we obtain the following differential equation for Ω(𝑎):Ω^′′(𝑎)1+2Φ+𝑎Ω^′(𝑎)−𝑆²Ω(𝑎)=0, (25) to be solved with the initial condition Ω(0)=1 and the additional one Ω(𝑎)=∫^𝑆_−𝑆𝑑𝑆^𝑧𝑓(𝑆^𝑧)𝑒^𝑎𝑆𝑧, arising from the ensemble-average definition of 𝑒^𝑎𝑆𝑧.

With these conditions we have obtained in [34] the physical solution of (25):Ω(𝑎)=𝑆/Φ()𝑆/Φ+𝑎sinh𝑆/Φ+𝑎().sinh𝑆/Φ (26) This key result (26) represents the classical analogue of the famous Callen formula for quantum spin-𝑆 models [35]. 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:𝑆𝑚=𝑆cothΦ−Φ𝑆𝑆=𝑆𝐿Φ, (27) which is valid for any 𝑑, 𝑇, and ℎ. Here, 𝐿(𝑥)=coth𝑥−1/𝑥 is the well-known Langevin function and Φ is expressed by (23) in terms of the dispersion relation 𝜔_𝐤. If we use the TD, 𝜔_𝐤 is given by (17) which is in turn a function of 𝑚. Hence, (27) is a self-consistent equation for 𝑚 and 𝑇. The longitudinal magnetic field ℎ and the anisotropy enter into the problem via the function Φ.

Of course, in the spirit of the Callen method for the calculation of 𝑚, it is possible to introduce more elaborate decoupling procedures which preserve its validity. For instance, we could adopt the Callen decoupling (CD) [35], with its classical variant [34],𝑆^𝑧_ℎ(𝜏)𝑆⁺_𝑘(𝜏);𝑒^{𝑎𝑆𝑧𝑗}𝑆⁻_𝑗𝜔≃𝑚𝐺_𝑘𝑗(𝜔)−𝑆⁻_ℎ𝑆⁺_𝑘2𝑆²𝐺_ℎ𝑗(𝜔), (28) which takes into account the transverse correlations to the leading order and implies the dispersion relation:𝜔^(CD)_𝐤=𝜔^(TD)_𝐤+𝑚2𝑆²1𝑁_𝐤′𝐽^⟂𝐤^′−𝐽^‖𝐤−𝐤^′𝐶𝐤^′ (29) with𝐶(𝐤)=_𝑖𝑒^{−𝑖𝐤⋅(𝐫𝑖−𝐫𝑗)}𝑆⁺_𝑖𝑆⁻_𝑗. (30) However, in the present case, the TD and the CD (28) provide essentially identical results close to the (𝑇=0)-critical point where 𝑚 is near the full polarized-state value 𝑆.

Given the magnetization 𝑚, the thermodynamics of our CPFM will be derived using the general formalism of the classical two-time Green functions (see Appendix A and [33]).

3. Transverse Susceptibility and Critical Line

We have now all the elements to extract the physics of interest setting 𝑎=0 in the previous results and solving the set of self-consistent equations:𝑆𝜎=cothΦ−Φ𝑆𝑆=𝐿Φ,Φ=𝑇_1BZ𝑑^𝑑𝑘(2𝜋)^𝑑1𝜔_𝐤(𝜎),𝜔_𝐤(𝜎)=𝜔₀(𝜎)+𝜎𝑆𝐽^⟂(0)−𝐽^⟂(𝐤)≥0, (31) where 𝜔₀(𝜎)=ℎ+𝑆𝜎[𝐽^‖(0)−𝐽^⟂(0)]. Here we have conveniently introduced the reduced magnetization per spin 𝜎=𝑚/𝑆, with 0≤𝜎≤1. Equations (31) will give 𝜎 as a function of 𝑇 and ℎ and hence the GF (16), also at 𝑎=0.

For our aim, the relevant quantity to be calculated is the transverse GF:𝐺_⟂(𝐤,𝜔)=𝑆⁺_𝑖(𝜏);𝑆⁻_{𝑗𝐤,𝜔}≡𝐺_𝐤(𝜔)∣_𝑎=0. (32) This allows to determine the dynamical transverse susceptibility for CPFM:𝜒_⟂(𝐤,𝜔)=−𝐺_⟂(𝐤,𝜔)≡−𝐺_𝐤(𝜔)∣_𝑎=0, (33) where, at the TD level,𝐺_⟂(𝐤,𝜔)=2𝑆𝜎𝜔−𝜔_𝐤(𝜎). (34) In particular, the thermodynamic transverse susceptibility is given by𝜒_⟂(𝑇,ℎ)≡𝜒_⟂(𝐤=𝟎,𝜔=0)=2𝑆𝜎𝜔₀(𝜎)=2𝑆𝜎(𝑇,ℎ)ℎ+𝑆𝜎𝐽^‖(0)−𝐽^⟂(0.) (35) Using (35) it is also possible to obtain the transverse correlation length via the following relation [36]:𝜉²_⟂1=−2𝜒_⟂⁻¹(𝑑0,0)²𝜒_⟂(𝐤,0)𝑑𝐤²_𝐤=0, (36) where 𝜒_⟂(𝐤,0)=2𝑆𝜎/𝜔_𝐤. The longitudinal susceptibility will be simply given by 𝜒_‖(𝑇,ℎ)=𝑆(𝜕𝜎(𝑇,ℎ)/𝜕ℎ). Of course, the stability condition 𝜒_⟂≥0 requires that, in (35), the inequality 𝜔₀(𝜎)=ℎ+𝑆𝜎(𝑇,ℎ)[𝐽^‖(0)−𝐽^⟂(0)]≥0 must be fulfilled. The equality is physically possible for ℎ>0 and 𝜎>0 only if 𝐽^‖(0)<𝐽^⟂(0), 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 𝜔₀ and 𝜒_⟂ and the related ones as functions of 𝑇 and ℎ. However, in some relevant cases which may have physical interest, we will calculate also the longitudinal quantities 𝜎(𝑇,ℎ) and 𝜒_‖(𝑇,ℎ). Other expressions can be obtained by means of known thermodynamic relations [33].

We start exploring the main features of the CPFM phase diagram in the (ℎ,𝑇)-plane. These results will be used as a basis for next developments. At zero temperature with ℎ≠0, (31) provides the solution 𝜎=1 for the reduced magnetization, characterizing a fully polarized state. This implies that the dispersion relation is given by 𝜔_𝐤=𝜔₀+𝑆[𝐽^⟂(0)−𝐽^⟂(𝐤)], where 𝜔₀=ℎ−𝑆[𝐽^⟂(0)−𝐽^‖(0)] is the frequency gap. Hence the transverse susceptibility, which has physical meaning for ℎ≥𝑆[𝐽^⟂(0)−𝐽^‖(0)]>0, becomes𝜒_⟂=2𝑆ℎ−𝑆𝐽^⟂(0)−𝐽^‖(0.) (37) Remarkably, (37) suggests that, despite the classical nature of our anisotropic spin model, there exists, as in the quantum case [26–28], a (𝑇=0)-critical point at the value ℎ_𝑐=𝑆[𝐽^⟂(0)−𝐽^‖(0)] of the longitudinal magnetic field. Then, crossing this point, decreasing ℎ to ℎ_𝑐, a field-induced second-order (𝑇=0)-phase transition arises from a fully polarized state with 𝜎=1 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 𝜔₀=ℎ−ℎ_𝑐≥0, we have 𝜒_⟂=2𝑆(ℎ−ℎ_𝑐)⁻¹ as ℎ→ℎ⁺_𝑐, defining the mean field exponent 𝛾_ℎ=1. (Through the paper we will use the indices ℎ and 𝑇 to denote the horizontal (isothermal) and the vertical trajectories approaching a critical point in the (ℎ,𝑇)-plane, resp.)

For arbitrary temperature we can write 𝜔₀(𝜎)=ℎ−𝜎ℎ_𝑐≥0 (so that 𝜔_𝐤=(ℎ−𝜎ℎ_𝑐)+𝑆𝜎[𝐽^⟂(0)−𝐽^⟂(𝐤)]). Hence the transverse susceptibility can be conveniently written as𝜒_⟂()=()𝑇,ℎ2𝑆𝜎𝑇,ℎℎ−𝜎(𝑇,ℎ)ℎ_𝑐, (38) with ℎ≥𝜎(𝑇,ℎ)ℎ_𝑐 or 𝜎(𝑇,ℎ)≤ℎ/ℎ_𝑐for stability reasons. Equation (38) establishes an interesting relation between the transverse physics and the longitudinal one.

In the (ℎ,𝑇)-plane, where 𝜒_⟂=∞ (𝜔₀(𝜎)=0), the general equation which determines the possible critical points isℎ−𝜎(𝑇,ℎ)ℎ_𝑐=0, (39) or, in view of the equation for 𝜎,ℎℎ_𝑐𝑆=𝐿Φ_𝑐. (40)

Here,Φ_𝑐ℎ=Φ𝑇,ℎ;𝜎=ℎ_𝑐=𝑇𝐽^⟂(0)𝑆ℎ_𝑐ℎ𝐹_𝑑(−1), (41) where 𝜔_𝐤^(𝑐)=𝜔_𝐤(𝜎=ℎ/ℎ_𝑐)=𝑆(ℎ/ℎ_𝑐)𝐽^⟂(0)(1−𝛾^⟂_𝐤) at the critical points and𝐹_𝑑(−1)=_1BZ𝑑^𝑑𝑘(2𝜋)^𝑑11−𝛾^⟂_𝐤, (42) with 𝛾^⟂_𝐤=𝐽^⟂(𝐤)/𝐽^⟂(0)≤1. More explicitly, (40) can be also written asℎ+𝐹_𝑑()𝑇−1𝐽^⟂(0)𝑆²ℎ²_𝑐ℎ−ℎ_𝑐coth𝐹_𝑑()𝑇−1𝐽^⟂(0)𝑆²ℎ_𝑐ℎ⁻¹=0. (43) The quantity 𝐹_𝑑(−1) is one of the so-called structure sums 𝐹_𝑑(𝑛)=(1/𝑁)∑_𝐤(1−𝛾^⟂_𝐤)=^{𝑛𝑁→∞}∫_1BZ(𝑑^𝑑𝑘/(2𝜋)^𝑑)(1−𝛾^⟂_𝐤)^𝑛 depending only on the lattice structure of the spin model. Accurate numerical values of 𝐹_𝑑(𝑛) can be found in the literature for different 𝑑 and lattice structures [34, 35, 37]. Previous results suggest that, while a (𝑇=0)-CP with 𝜎=1 exists for any 𝑑, a critical line with 0≤𝜎≤1, 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 𝐽^𝛼(𝐤)=2𝐽^𝛼∑^𝑑_𝜈=1cos𝑘_𝜈≃𝐽^𝛼(0)−𝐽^𝛼𝑘² (𝛼=⟂,‖) as 𝑘→0 with 𝐽^𝛼(0)=2𝑑𝐽^𝛼, from (42) it immediately follows that for 𝑑≤2 only the (𝑇=0)-CP exists, while for 𝑑>2 a finite-temperature critical line, ending in the (𝑇=0)-CP (ℎ_𝑐,𝑇=0), occurs consistently with the Mermin-Wagner theorem [38]. The same result has been obtained in the quantum case [28].

For 𝑑>2, the critical line equation (42) (or (43)) can be solved numerically with respect to ℎ or 𝑇 providing the representation ℎ_𝑐(𝑇) or 𝑇_𝑐(ℎ). Notice that, along the critical line, the reduced magnetization 𝜎(𝑇,ℎ_𝑐(𝑇))=𝜎(𝑇) is simply given by 𝜎(𝑇)=ℎ_𝑐(𝑇)/ℎ_𝑐 when ℎ_𝑐(𝑇) is known. The critical line in the plane (ℎ,𝑇) for 𝑑=3 is plotted in Figure 1.

Figure 1 

Critical line of the classical planar ferromagnet (CPFM) with short-range interactions on a three-dimensional cubic lattice (present work). Here 𝑇𝑐 denotes the critical temperature at ℎ=0 and ℎ𝑐 the critical magnetic field at 𝑇=0. A comparison is shown with the corresponding curves for the classical transverse Ising model (CTIM), obtained by RG calculations, and for the spin-1/2 quantum planar ferromagnet (QPFM). The critical line for the spin-𝑆 QPFM with long-range interactions in the classical limit 𝑆→∞ has been also obtained at the Tyablikov decoupling level, and, as expected, it coincides with the one calculated for the CPFM.

Starting from (40) or (43) we can easily derive the analytical expression of the zero-field critical temperature 𝑇_𝑐(ℎ=0)≡𝑇_𝑐 where 𝜎=0 irrespective of the specific structure of 𝛾^⟂_𝐤 in (37) (as in the case of short-range interactions for which 𝛾^⟂_𝐤=𝐽^⟂(𝐤)/𝐽^⟂(0)=(2𝐽^⟂/𝐽^⟂(0))∑^𝑑_𝜈=1cos𝑘_𝜈=(1/𝑑)∑^𝑑_𝜈=1cos𝑘_𝜈).

From the expansion coth𝑥≃1/𝑥+(1/3)𝑥+𝑂(𝑥³), and hence 𝐿(𝑥)=coth𝑥−1/𝑥≃𝑥/3+𝑂(𝑥³), for 𝑥=𝑆/Φ_𝑐≪1, (40) provides, for Φ_𝑐→∞ as ℎ→0 at finite 𝑇 (see (41)),𝑇_𝑐𝐽^⟂(0)𝑆²=13𝐹_𝑑().−1 (44) It is worth noting that for QPFM, within the TD it was found that [28] 𝑇^QPFM_𝑐/𝐽^⟂(0)𝑆(𝑆+1)=1/3𝐹_𝑑(−1), which reproduces, as expected, (44) in the classical limit for 𝑆→∞.

For short-range interactions and an hypercubic lattice, estimates for 𝑑>2 can be obtained assuming 1−𝛾^⟂_𝐤≃𝑘²/2𝑑 as 𝑘→0.

Now we explore the behavior of the critical line in the low-temperature regime close to the (𝑇=0)-CP where Φ_𝑐≪1. With 𝐿(𝑆/Φ_𝑐)≃1−Φ_𝑐/𝑆+2𝑒^{−2𝑆/Φ𝑐}, (40) becomesℎℎ_𝑐≃1−𝐹_𝑑()𝑇−1𝐽^⟂(0)𝑆²ℎ_𝑐ℎ+𝑂𝑒^{−(2/𝐹𝑑(−1))(𝐽⟂(0)𝑆2/𝑇)}. (45) Solving the equation with respect to ℎ or 𝑇, we findℎ_𝑐(𝑇)≃ℎ_𝑐1−𝐹_𝑑()𝑇−1𝐽^⟂(0)𝑆²+𝑂𝑒^{−(2/𝐹𝑑(−1))(𝐽⟂(0)𝑆2/𝑇)},𝑇⟶0, (46) or𝑇_𝑐(ℎ)≃𝐽^⟂𝑆²𝐹_𝑑(−1)ℎ_𝑐ℎ_𝑐𝑒−ℎ+𝑂^{−(2(𝐹𝑑(−1))2ℎ𝑐)/(ℎ𝑐−ℎ)},ℎ⟶ℎ⁻_𝑐. (47) Along this branch of the critical line we have also𝜎(𝑇)≃1−𝐹_𝑑()𝑇−1𝐽^⟂(0)𝑆²+𝑂𝑒^{−(2/𝐹𝑑(−1))(𝐽⟂(0)𝑆2/𝑇)}. (48) From the low-temperature representations (46) and (47) for the critical line we can extract the shift exponent 𝜓=1, which determines the shape of the phase boundary close to the (𝑇=0)-CP. It has to be stressed that the value of 𝜓 is independent of 𝑑, in contrast with the result (𝜓=𝑑/2) 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 (𝑇=0)-CP where 𝜎≈1, ℎ≃ℎ_𝑐 (nearly polarized state), and 𝜔₀(𝜎)=ℎ−𝜎ℎ_𝑐 is very small. Under these conditions, in (31) Φ(𝜎)/𝑆≪1; therefore, the equation for 𝜎 becomesΦ(𝜎)𝜎≃1−𝑆+2𝑒^{−(2𝑆/Φ(𝜎))}. (49) The quantity Φ(𝜎)≡Φ(𝜔₀) near criticality can be suitably estimated assuming, for the oscillation spectrum 𝜔_𝐤, the low-𝐤 expression 𝜔_𝐤≃𝜔₀+𝑆𝜎𝐽^⟂𝑘². This provides𝑑Φ≃2𝑇𝜔₀¹₀𝑥𝑑𝑥^𝑑/2−11+𝜏/𝜔₀, (50) where 𝜏≈𝑆𝐽^⟂Λ²_1BZ=(𝑆𝐽^⟂(0)/2𝑑)Λ²_1BZ and Λ_1BZ is a natural wave-vector cut-off related to the first Brillouin zone and determined by (1/𝑁)∑_𝐤=1 or (as 𝑁→∞) Λ_1BZ=(𝑑/𝐾_𝑑)^1/𝑑, with 𝐾_𝑑=2^1−𝑑𝜋^−𝑑/2/Γ(𝑑/2).

As a consequence, to the leading order in Φ, the self-consistent equation for 𝜎 can be written as (except for exponentially small terms)1𝜎≃1−𝑆𝑇𝜔₀(𝜎)𝐹𝑑1,2;𝑑2𝜏+1;−𝜔₀(𝜎), (51) where 𝐹(𝛼,𝛽;𝛾;𝑧) 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 𝜔₀, strictly related to 𝜒_⟂  and hence to the transverse physics.

Since 𝜎=ℎ/ℎ_𝑐−𝜔₀/ℎ_𝑐, straightforward calculations provide the following expression for (51) in terms of the natural variable 𝜔₀/𝑇:𝜔₀𝑇=𝑔𝑇+ℎ_𝑐𝜏𝑆𝜏𝜔₀𝐹𝑑1,2;𝑑2𝜏+1;−𝜔₀. (52) Here 𝑔=ℎ−ℎ_𝑐 and 𝜔₀/𝑇=2𝑆𝜎/𝑇𝜒_⟂≃2𝑆/𝑇𝜒_⟂∝(𝑇𝜒_⟂)⁻¹∝(𝑇𝜉²_⟂)⁻¹. Notice that since 𝜒_⟂(𝐤,0)≃2𝑆𝜎/[𝜔₀+𝜎𝑆𝐽^⟂𝐤²], from (37), one gets 𝜉_⟂≃𝐽^⟂𝜒_⟂^1/2.

Of course, once 𝜔₀ has been determined as a function of (𝑇,ℎ), one can directly calculate 𝜒_⟂(𝑇,ℎ) and, using the relation between 𝜔₀ and 𝜎, it is possible to determine 𝜎(𝑇,ℎ) and, therefore, the longitudinal physics near the polarized state in the low-temperature critical regime.

Adopting the asymptotic expansions of the hypergeometric function 𝐹(1,𝜈;𝜈+1;−(1/𝑧)) for 𝑧≪1,𝐹11,𝜈;𝜈+1;−𝑧≈𝜋𝜈𝑧sin𝜋𝜈^𝜈1,𝜈<1,𝑧ln𝜈𝑧,𝜈=1,𝜈−1𝑧−𝜋𝜈||𝑧sin𝜋𝜈^𝜈,1<𝜈<2, (53) now, we can systematically explore the thermodynamics of the CPFM close to the (𝑇=0)-CP for different values of the dimensionality 𝑑 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 𝜔₀/𝑇≪1 (⇔𝑇𝜉²_⟂≫1 or 𝑇𝜒_⟂≫1) and quantum-like (𝜔₀/𝑇≫1⇔𝑇𝜉²_⟂≪1 or 𝑇𝜒_⟂≪1) regimes, respectively, and the estimates of the related crossovers for different values of 𝑑.

4.1. 𝑑<2

Replacing the expansions (53) in the self-consistent equation (52) for 𝜔₀, we get𝜔₀𝑇≃𝑔𝑇+𝜋𝑑/2()ℎsin𝜋𝑑/2^𝑐𝑇𝑆𝜏𝜏^(𝑑−2)/2𝜔₀𝑇^(𝑑−2)/2, (54) In the regime 𝜔₀/𝑇≪1 (𝑇𝜉²_⟂≫1), classical regime in the quantum critical scenario near the isolated (𝑇=0)-CP, (54) admits a solution only for ℎ<ℎ_𝑐, which reads𝜔₀≃𝜋𝑑/2()ℎsin𝜋𝑑/2𝑆𝜏^{𝑐𝑑/22/(2−𝑑)}𝑇ℎ_𝑐−ℎ^2/(2−𝑑). (55) Under the consistency condition 𝑇≪(ℎ_𝑐−ℎ)^2/𝑑. Then, for ℎ<ℎ_𝑐 within the region 𝑇≪(ℎ_𝑐−ℎ)^2/𝑑, in the (ℎ,𝑇)-plane, the transverse and longitudinal susceptibilities are given by𝜒_⟂≃2𝑆𝜋𝑑/2()ℎsin𝜋𝑑/2𝑆𝜏^{𝑐𝑑/2−2/(2−𝑑)}𝑇ℎ_𝑐−ℎ^{−2/(2−𝑑)},𝜒_‖≃𝑆ℎ_𝑐1−𝜋𝑑/2()ℎsin𝜋𝑑/2𝑆𝜏^{𝑐𝑑/22/(2−𝑑)}ℎ_𝑐ℎ_𝑐−ℎ^2/(2−𝑑)×𝑇ℎ_𝑐−ℎ^2/(2−𝑑), (56) where the reduced magnetization near polarization is immediately given as 𝜎(𝑇,ℎ)≃1−𝜔₀/ℎ_𝑐.

It is worth noting that, for fixed ℎ<ℎ_𝑐 and 𝑇→0, the transverse susceptibility diverges with the critical exponent 𝛾_𝑇=2/(2−𝑑), while the longitudinal one remains finite.

In the regime 𝜔₀/𝑇≫1 (𝑇𝜉²_⟂≪1), 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 (𝑇=0)-CP. For ℎ<ℎ_𝑐 in the region (ℎ_𝑐−ℎ)^2/𝑑≪𝑇≪(ℎ_𝑐−ℎ)^(4−𝑑)/2, where 1≪𝜔₀/𝑇≪𝑔/𝑇, (54) provides, to the leading order, a solution which is formally identical to (55) but now the condition 𝜔₀/𝑇≫1 should hold. The same occurs for the thermodynamic quantities (56) and the related ones.

For ℎ=ℎ_𝑐, decreasing 𝑇 along a vertical trajectory which corresponds to the quantum critical one in the phase diagram of the QPFM [28], (54) yields, with 𝜔₀(𝑇,ℎ_𝑐)≡𝜔_0𝑐(𝑇),𝜔_0𝑐(𝑇)≃𝜏𝜋𝑑/2()ℎsin𝜋𝑑/2𝑆𝜏^{𝑐2/(4−𝑑)}𝑇𝜏^2/(4−𝑑). (57) Then, we get𝜒_⟂𝑇,ℎ_𝑐𝑆≃2𝜏𝜋𝑑/2()ℎsin𝜋𝑑/2𝑆𝜏^{𝑐−2/(4−𝑑)}𝑇𝜏^{−2/(4−𝑑)}, (58) which defines the critical exponent 𝛾_𝑇=2/(4−𝑑), with 1/2<𝛾_𝑇<1. For the nearly polarized state (with 𝜎(𝑇,ℎ)≲1), a simple algebra yields𝜎𝑇,ℎ_𝑐𝜏≃1−ℎ_𝑐𝜋𝑑/2()ℎsin𝜋𝑑/2𝜏^{𝑐2/(4−𝑑)}𝑇𝜏^2/(4−𝑑), (59) which increases towards unity decreasing 𝑇 according to the power-law ∼𝑇^𝛾𝑇.

Now we consider the region which is more relevant from the experimental point of view, namely, the V-shaped region 𝑇≫|ℎ−ℎ_𝑐|^(4−𝑑)/2, around the vertical trajectory ℎ=ℎ_𝑐 (for both ℎ≲ℎ_𝑐 and ℎ≳ℎ_𝑐). Under this condition, from (54) a straightforward algebra gives𝜔₀(𝑇,ℎ)≃𝜔_0𝑐(𝑇2)1+4−𝑑ℎ−ℎ_𝑐𝜔_0𝑐(𝑇). (60) This expression suggests that, within the V-shaped region, the thermodynamics is essentially identical to the one along the trajectory ℎ=ℎ_𝑐, except for a small correction ∼|ℎ−ℎ_𝑐|.

Finally, for ℎ>ℎ_𝑐 and sufficiently far from the quantum-like critical trajectory, within the region 𝑇≪(ℎ−ℎ_𝑐)^(4−𝑑)/2, we have𝜔₀(𝑇,ℎ)≃ℎ−ℎ_𝑐1+𝜋𝑑/2()ℎsin𝜋𝑑/2𝑆𝜏^𝑐𝑑/2𝑇ℎ−ℎ_𝑐^(4−𝑑)/2. (61) This implies that𝜒_⟂(𝑇,ℎ)≃2𝑆ℎ−ℎ_𝑐⁻¹1−𝜋𝑑/2()ℎsin𝜋𝑑/2𝑆𝜏^𝑐𝑑/2𝑇ℎ−ℎ_𝑐^(4−𝑑)/2, (62) which differs from the MF result 𝜒_⟂≃2𝑆(ℎ−ℎ_𝑐)⁻¹, found before at 𝑇=0, for a small power-law correction in temperature, in contrast with the exponentially small correction which occurs in the quantum counterpart [28]. Besides, for the nearly polarized state, we obtain𝜎(𝑇,ℎ)≃1−𝜋𝑑/2()1sin𝜋𝑑/2𝑆𝜏^𝑑/2𝑇ℎ−ℎ_𝑐^(2−𝑑)/2,𝜒_‖(𝑇,ℎ)≃2−𝑑2𝜋𝑑/2()1sin𝜋𝑑/2𝜏^𝑑/2𝑇ℎ−ℎ_𝑐^(4−𝑑)/2. (63) In summary, the previous results suggest, for 𝑑<2, a very rich phase diagram around the isolated (𝑇=0)-CP, qualitatively reported in Figure 2, where different low-𝑇 regimes and crossover lines are presented. It appears divided in two main regions (named in analogy with the quantum case [28]): 𝐶, where 𝜔₀/𝑇≪1 (𝑇𝜒_⟂∝𝑇𝜉²_⟂≫1), and 𝑄≡(𝑄₁,𝑄₂,𝑄₃), where 𝜔₀/𝑇≫1 (𝑇𝜉²_⟂≪1). The line 𝑇₁≃(ℎ_𝑐−ℎ)^2/𝑑 for ℎ<ℎ_𝑐 signals the crossover between the regimes 𝐶 and 𝑄; the lines 𝑇₂(ℎ)∼(ℎ_𝑐−ℎ)^(4−𝑑)/2 and 𝑇₃(ℎ)∼(ℎ−ℎ_𝑐)^(4−𝑑)/2, symmetric to the vertical trajectory ℎ=ℎ_𝑐, provide the signature of crossovers among three distinct subregimes 𝑄₁, 𝑄₂, and 𝑄₃ with different asymptotic behaviors of the thermodynamic quantities as functions of 𝑇 and ℎ. We stress that, within the V-shaped region 𝑄₂, delimited by the crossover lines 𝑇₂(ℎ) and 𝑇₃(ℎ), the 𝑇-dependent behaviors are essentially identical to those along the trajectory ℎ=ℎ_𝑐 except for different a small power law corrections ∼|ℎ−ℎ_𝑐|. Besides, in the regime 𝑄₃, decreasing 𝑇 at fixed ℎ>ℎ_𝑐, the transverse susceptibility deviates from the one at 𝑇=0 except for a small power law correction as a function of 𝑇 and ℎ−ℎ_𝑐. The latter feature differs crucially from the QPFM scenario where the correction to the (𝑇=0)-behavior of 𝜒_⟂ is an exponentially small function of 𝑇 and ℎ−ℎ_𝑐 [28, 39].

Figure 2 

Qualitative phase diagram of CPFM with short-range interactions for 𝑑<2, close to the zero-temperature critical point. The line 𝑇1(ℎ) marks the crossover between the regimes 𝑇𝜒⟂∝𝑇𝜉2⟂≫1 (region 𝐶) and 𝑇𝜒⟂∝𝑇𝜉2⟂≪1 (region 𝑄≡(𝑄1,𝑄2,𝑄3)) increasing ℎ. The lines 𝑇2(ℎ) and 𝑇3(ℎ), symmetric with respect to the vertical line ℎ=ℎ𝑐, signal the crossovers among the subregimes 𝑄1,𝑄2,  and 𝑄3. Finally, the heavy line OP represents the zero-temperature-ordered phase.

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.

4.2. 𝑑=2

For the two-dimensional CPFM, which also exhibits only a (𝑇=0)-CP, a low-temperature scenario similar to the one derived for 𝑑<2 takes place. But now, logarithmic corrections to the leading power-law behavior arise. This peculiarity has been also found for the analogous quantum system [28].

With the expansion (53), close to the (𝑇=0)-CP the general self-consistent equation (52) reduces to𝜔₀𝑇≃𝑔𝑇+ℎ_𝑐𝜏𝑆𝜏ln𝜔₀. (64) First, we consider the regime 𝜔₀/𝑇≪1. If ℎ≥ℎ_𝑐, no solution exists while, for ℎ<ℎ_𝑐, (64) provides𝜔₀≃𝜏exp−𝑆𝜏ℎ_𝑐ℎ_𝑐−ℎ𝑇, (65) which has to be compared with the corresponding result achieved for the QPFM, 𝜔₀≃𝑇𝑒^{−(𝑆𝜏/ℎ𝑐)((ℎ𝑐−ℎ)/𝑇)} [28]. Then, for the transverse susceptibility we have the exponentially divergent behavior 𝜒_⟂≃(2𝑆/𝜏)exp[(𝑆𝜏/ℎ_𝑐)((ℎ_𝑐−ℎ)/𝑇)] as 𝑇→0 (corresponding to a critical exponent 𝛾_𝑇=∞). For the nearly polarized state we immediately find 𝜎(𝑇,ℎ)≃1−(𝜏/ℎ_𝑐)exp[−(𝑆𝜏/ℎ_𝑐)((ℎ_𝑐−ℎ)/𝑇)] and 𝜒_‖≃(𝑆/ℎ_𝑐){1−(𝑆𝜏/ℎ_𝑐)(𝜏/𝑇)𝑒^{−(𝑆𝜏/ℎ𝑐)((ℎ𝑐−ℎ)/𝑇)}}.

Let us consider now the regime 𝜔₀/𝑇≫1. For ℎ=ℎ_𝑐, (64) gives𝜔₀𝑇,ℎ_𝑐≃ℎ_𝑐𝑇𝑆𝜏ln𝑆𝜏²ℎ_𝑐𝑇, (66) and 𝜒_⟂(𝑇,ℎ_𝑐) diverges as 𝑇⁻¹ln⁻¹(1/𝑇) when 𝑇→0. Moreover, around this vertical line for 𝑇≫|ℎ−ℎ_𝑐|ln⁻¹(1/|ℎ−ℎ_𝑐|), we find𝜔₀()ℎ𝑇,ℎ≃=^𝑐𝑇𝑆𝜏ln𝑆𝜏²ℎ_𝑐𝑇1+𝑆𝜏ℎ_𝑐𝑇ℎ−ℎ_𝑐ln𝑆𝜏²/ℎ_𝑐𝑇, (67) which signals the same leading behavior of 𝜔_0𝑐(𝑇) and related thermodynamic quantities in the limit 𝑇→0, except for small corrections in ℎ−ℎ_𝑐.

Finally, for ℎ>ℎ_𝑐 and 𝑇≪(ℎ−ℎ_𝑐)ln⁻¹(1/(ℎ−ℎ_𝑐)), the frequency gap behaves as𝜔₀(𝑇,ℎ)≃ℎ−ℎ_𝑐𝑇1−ℎ𝑆𝜏ℎ−^𝑐ℎ_𝑐𝜏ln𝑆ℎ_𝑐ℎ_𝑐ℎ−ℎ_𝑐, (68) providing a thermodynamics very similar to the one found for 𝑑<2 except for logarithmic corrections in ℎ−ℎ_𝑐 with respect to the 𝑇=0 MF results.

In conclusion, for the two-dimensional CPFM close to (𝑇=0)-CP, we have a qualitative phase diagram which is very similar to the one shown in Figure 2, presenting three regions 𝐶, 𝑄₁, and 𝑄₂. In this case the sector corresponding to previous 𝑄₁ is absent and the V-shaped region is delimited by the lines 𝑇_1,2(ℎ)∼|ℎ−ℎ_𝑐|ln⁻¹(1/|ℎ−ℎ_𝑐|) which, in contrast to the ones shown in Figure 2, exhibit small logarithmic corrections.

4.3. 𝑑>2

For such dimensionalities the CPFM exhibits a critical line ending in the (𝑇=0)-CP. To explore the low-temperature critical properties it is convenient to rewrite (52) in terms of 𝑔(𝑇)=ℎ−ℎ_𝑐(𝑇)≥0 (for the disordered phase). We will focus on dimensionalities 2<𝑑<4 (the case 𝑑≥4 is rather trivial although completely consistent with the general theory of critical phenomena) for which, given the expansions (53), (52) assumes the form𝜔₀𝑇≃𝑔(𝑇)𝑇−𝜋𝑑/2|(ℎsin𝜋𝑑/2)|^𝑐𝑇𝑆𝜏𝜏^(𝑑−2)/2𝜔₀𝑇^(𝑑−2)/2, (69) 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 𝜎=ℎ/ℎ_𝑐−𝜔₀/ℎ_𝑐 and 𝜒_‖=𝑆(𝜕𝜎/𝜕ℎ).

We start calculating the asymptotic solutions of (69) where the critical region around the (𝑇=0)-CP in the (ℎ,𝑇)-plane is approached in two ways: (i) along horizontal trajectories, as ℎ→ℎ⁺_𝑐(𝑇) (with ℎ_𝑐(𝑇)≃ℎ_𝑐−(𝑑/(𝑑−2))(ℎ_𝑐/𝑆)(𝑇/𝜏)) at fixed 𝑇 (isothermal trajectories); and (ii) along vertical trajectories, as 𝑇→𝑇⁺_𝑐(ℎ) (with 𝑇_𝑐(ℎ)≃((𝑑−2)/𝑑)(𝑆𝜏/ℎ_𝑐)(ℎ_𝑐−ℎ)) at fixed ℎ≤ℎ_𝑐 or 𝑇→0 for ℎ>ℎ_𝑐.

We first consider the regime 𝜔₀/𝑇≪1 (𝑇𝜉²_⟂≫1). The right-hand side of (69) suggests that two subregimes 𝜔₀/𝑇≪𝜃 and 𝜔₀/𝑇≫𝜃 should be investigated, with𝜃=𝜋𝑑/2|(ℎsin𝜋𝑑/2)|^𝑐𝑇𝑆𝜏𝜏^(𝑑−2)/2𝜔₀𝑇^(𝑑−2)/2, (70) where 𝜔₀/𝑇≃𝜃 signals the crossover between them. For isothermal trajectories, in the subregime 𝜔₀/𝑇≪𝜃, one finds for 𝜔₀ the asymptotic solution𝜔₀(𝑇,ℎ)≃𝜏𝜋𝑑/2|(ℎsin𝜋𝑑/2)|𝑆^{𝑐−2/(𝑑−2)}𝑇𝜏^{−2/(𝑑−2)}×ℎ−ℎ_𝑐(𝑇)^(𝑑−2)/2, (71) which provides for transverse susceptibility 𝜒_⟂∼(ℎ−ℎ_𝑐(𝑇))^−𝛾ℎ the nontrivial non-MF critical exponent 𝛾_ℎ=2/(𝑑−2). This spherical-model incorrect result is typical of the Tyablikov-like decoupling also for the quantum model at finite temperature [28, 39, 40].

When 𝜔₀/𝑇≫𝜃, (69) yields simply 𝜔₀≃ℎ−ℎ_𝑐(𝑇) which corresponds to the MF exponent 𝛾_ℎ=1. The crossover between the two previous regimes (𝜔₀/𝑇≃𝜃) is indicated by the Ginzburg-like line for horizontal trajectories:ℎ_Gi(𝑇)≃ℎ_𝑐(𝑇)+ℎ_𝑐1𝑆𝜋𝑑/2|sin(𝜋𝑑/2)|^2/(4−𝑑)×ℎ_𝑐𝜏^{(𝑑−2)/(4−𝑑)}𝑇𝜏^2/(4−𝑑). (72) Notice that ℎ_Gi(𝑇)→ℎ_𝑐 as 𝑇→0; that is, the two lines ℎ_𝑐(𝑇) and ℎ_Gi(𝑇) merge at the (𝑇=0)-CP.

For vertical trajectories at fixed ℎ<ℎ_𝑐 (𝑇_𝑐(ℎ)≠0), setting 𝑔(𝑇)≃(𝑑/(𝑑−2))(ℎ_𝑐/𝑆𝜏)(𝑇−𝑇_𝑐(ℎ)) in (69), we easily obtain the asymptotic solutions.𝜔₀≃𝜏𝑑−22|sin(𝜋𝑑/2)|𝜋𝑑/2^2/(𝑑−2)𝑇−𝑇_𝑐(ℎ)𝑇_𝑐(ℎ)^2/(𝑑−2),𝜔₀𝑇𝑑≪𝜃,ℎ𝑑−2𝑆^𝑐𝑇_𝑐(ℎ)𝜏𝑇−𝑇_𝑐(ℎ)𝑇_𝑐(ℎ),𝜔₀𝑇≫𝜃. (73) All the macroscopic quantities of interest can be now determined in the previous regimes for 𝑇→𝑇⁺_𝑐(ℎ). In particular, for 𝜒_⟂, we find the critical exponents 𝛾_𝑇≡𝛾_ℎ=2/(𝑑−2) for 𝜔₀/𝑇≪𝜃 and 𝛾_𝑇=𝛾_ℎ=1 for 𝜔₀/𝑇≫𝜃. The crossover between these two asymptotic sub-regimes for ℎ<ℎ_𝑐 occurs crossing the conventional Ginzburg-Landau line (with 𝜔₀/𝑇≃𝜃):𝑇_𝐺𝑖(ℎ)≃𝑇_𝑐(ℎ)+𝑑−2𝑑𝜏𝜋𝑑/2|sin(𝜋𝑑/2)|^2/(4−𝑑)×ℎ_𝑐𝑆𝜏^{(𝑑−2)/(4−𝑑)}𝑇_𝑐(ℎ)𝜏^2/(4−𝑑). (74) Of course, also for vertical trajectories within the region of the (ℎ,𝑇)-plane between the critical and Ginzburg lines, with ℎ<ℎ_𝑐, the TD quantitatively fails.

Let us consider now the behavior of 𝜔₀, and hence of 𝜒_⟂, along the line ℎ=ℎ_𝑐, decreasing 𝑇, which is of most experimental interest in view of the problematics discussed in Section 1.

Since in this case 𝑇_𝑐(ℎ_𝑐)=0, one can immediately see that, to leading order in 𝑇, a self-consistent solution of (69), under the condition 𝜔₀/𝑇=𝑂(1), is given by𝜔_0𝑐(𝑇)≃𝑑ℎ𝑑−2𝑆^𝑐𝑇𝜏. (75) This result is strictly connected with the shift exponent 𝜓=1 and in drastic contrast with the corresponding relation obtained for the QPFM [27, 39], which, due to the presence of quantum fluctuations, shows that 𝜓=𝑑/2. Equation (75) predicts that 𝜒_⟂∼𝑇⁻¹ as 𝑇→0 along the vertical line ℎ=ℎ_𝑐, providing the exponent 𝛾_𝑇=𝜓=1, in contrast with the quantum result 𝛾_𝑇=𝜓=𝑑/2 for the QPFM [28, 39].

From (69), under condition ℎ−ℎ_𝑐≪(𝑑/(𝑑−2))(ℎ_𝑐/𝑆)(𝑇/𝜏), we get𝜔₀(𝑇,ℎ)≃𝜔_0𝑐(𝑇)1+𝑑−2𝑑𝑆𝜏ℎ_𝑐ℎ−ℎ_𝑐𝑇. (76)

This means that, within the V-shaped region delimited by the critical line for ℎ<ℎ_𝑐 and the symmetric one 𝑇^𝑋(ℎ)≃((𝑑−2)/𝑑)(𝑆𝜏/ℎ_𝑐)(ℎ−ℎ_𝑐) for ℎ>ℎ_𝑐, the spectrum gap, the transverse susceptibility, and other macroscopic quantities behave essentially as along the line ℎ=ℎ_𝑐, except for negligible corrections in ℎ−ℎ_𝑐.

Increasing ℎ−ℎ_𝑐>0 and crossing the line 𝑇^𝑋(ℎ), a crossover to the regime 𝜔₀/𝑇≫1 (𝑇𝜉²_⟂≪1) takes place and one easily finds that, for 𝑇≪𝑇^𝑋(ℎ), the appropriate solution of (69) sounds as𝜔₀(𝑇,ℎ)≃ℎ−ℎ_𝑐+𝑑ℎ𝑑−2𝑆𝜏^𝑐×𝑇1−𝜋𝑑/2|sin(𝜋𝑑/2)|𝑑−2𝑑ℎ−ℎ_𝑐𝜏^(𝑑−2)/2. (77) Therefore, 𝜒_⟂⁻¹ (as 𝜔₀) behaves essentially as at 𝑇=0 (𝜒_⟂≃(ℎ−ℎ_𝑐)⁻¹), with small corrections in 𝑇.

Summarizing, for 2<𝑑<4 the low-𝑇 global phase diagram has the qualitative structure shown in Figure 3.

Figure 3 

Schematic low-temperature phase diagram and crossovers in the (ℎ,𝑇)-plane of the CPFM with 2<𝑑<4. Here 𝐶 stands for classical critical behavior; 𝑇𝑐(ℎ), 𝑇Gi(ℎ) and 𝑇𝑋(ℎ), denote the critical, Ginzburg, and crossover lines; OP and DP indicate the ordered and disordered phases, respectively. The critical line behaves linearly with ℎ𝑐−ℎ and the characteristic quantum-like V-shaped region, with vertex in the (𝑇=0)-CP, is clearly evident.

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 [27] and two-time GF method [34, 39]. However, for sufficiently low temperatures along the vertical trajectory ℎ=ℎ_𝑐, the quantitative difference between the behaviors of the transverse susceptibility, 𝜒_⟂∼𝑇⁻¹ of the CPFM and 𝜒_⟂∼𝑇^−𝑑/2, 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 (𝐽^‖=0) in a transverse magnetic field occurs close to its (𝑇=0)-CP.

The same low-𝑇  V-shaped-like scenario has been recently obtained for the CTIM [22] by means of a Wilsonian RG approach in 𝑑=4−𝜀 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 (𝑇=0)-CP as in the quantum counterparts.

5. Concluding Remarks

In the present paper we have explored the low-temperature properties of the 𝑑-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 (𝑇=0)-CP, the phase diagram and the critical scenario are qualitatively similar to those found for the spin-𝑆 QPFM [28] for 𝑑>2 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 (𝑇=0)-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 𝜓 which characterizes the way in which the critical line ends at the (𝑇=0)-CP as 𝑇→0 for classical (𝜓=1) and quantum (𝜓=𝑑/2) 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 ℎ=ℎ_𝑐 (quantum critical trajectory for the QPFM) in the (ℎ,𝑇)-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 𝜓=1 to 𝜓=𝑑/2 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 𝜓=1 for any 𝑑>2. 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 (𝑇=0)-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 [27] for spin-1/2 QPFM with short range interactions, and by two-time GF approaches for spin-1/2 [39] and spin-𝑆 [28] QPFM for short- and long-range interactions. In these papers, an unexpected regime with 𝜒_⟂∼𝑇⁻¹ was found in the V-shaped region above a certain temperature 𝑇^∗ which, for 2<𝑑<4, spin-1/2, and short-range interactions at a TD level, reads [39] (using our notations)𝑇^∗1≃𝜏𝜋|sin(𝜋𝑑/2)|𝐹_𝑑/2(0)^{(4−𝑑)/24/(𝑑−2)2}𝜏𝑑ℎ_𝑐^2/(𝑑−2), (78) where 𝐹_𝑑/2(0) is the value at 𝑦=0 of the function 𝐹_𝑑/2(𝑦)=∫^∞₀𝑑𝑥(𝑥^(𝑑/2)−1/(𝑒^𝑥+𝑦−1)). Below 𝑇^∗, the quantum behavior 𝜒_⟂∼𝑇^−𝑑/2 takes place, as expected from the paradigmatic quantum critical scenario [1].

The present study for the CPFM clarified the physical meaning of 𝑇^∗. Our study corroborated the idea that, decreasing 𝑇 along the quantum critical trajectory, 𝑇^∗ provides an estimate of the temperature which signals a crossover between the classical thermally activated regime, with 𝜒_⟂∼𝑇⁻¹ (𝜓=1), and the quantum one, with 𝜒_⟂∼𝑇^−𝑑/2 (𝜓=𝑑/2). This feature agrees with the RG predictions for the same quantum spin model near and below four dimensions [27].

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 𝜒_⟂(𝐤,𝜔)=−𝐺_⟂(𝐤,𝜔) for small values of the arguments. Indeed, from (33)-(34) close to the (𝑇=0)-CP we find𝜒_⟂(𝐤,𝜔)≃2𝑆𝜉²_⟂𝑆𝐽^⟂𝑘𝜉_⟂²+1−𝜔𝜉²_⟂⁻¹, (79) where 𝜉_⟂≃(𝑆𝐽^⟂)^1/2(ℎ−ℎ_𝑐)^−1/2∝𝜒_⟂^1/2 defines the transverse correlation length at zero temperature for ℎ≥ℎ_𝑐. Then, comparing (79) with the general dynamic scaling relation 𝜒(𝐤,𝜔)≃𝜉^2−𝜂𝑊(𝑘𝜉,𝜔𝜉^𝑧) we immediately get  𝜂=0 and 𝑧=2 for the Fisher 𝜂 and dynamic 𝑧 critical exponents, and 𝑊(𝑥,𝑦)=2𝑆[𝑆𝐽^⟂(𝑥²+1)−𝑦]⁻¹.

In conclusion, our results are in agreement with the statements made in [21] for the CTIM chain and in [28] for 𝑑>1. 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.

Appendices

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 (𝜈=𝑟) and advanced (𝜈=𝑎) GF’s framework in classical statistical mechanics in a form strictly parallel to the quantum counterpart [40–44]. For two arbitrary dynamical variables 𝐴 and 𝐵, they are defined as [29]𝐺^(𝜈)_𝐴𝐵𝑡,𝑡′=𝜃_𝜈𝑡−𝑡^′𝐴(𝑡),𝐵𝑡^′≡𝐴(𝑡);𝐵𝑡^′_𝜈,(𝜈=𝑟,𝑎), (A.1) where 𝜃_𝑟(𝑡−𝑡^′)=𝜃(𝑡−𝑡^′), 𝜃_𝑎(𝑡−𝑡^′)=−𝜃(𝑡^′−𝑡), 𝜃(𝑥) is the usual step function,  ⟨⋯⟩ denotes an equilibrium ensemble average, and {𝐴,𝐵} is the Poisson bracket of 𝐴 and 𝐵.

In (A.1), the dynamical variables 𝐴 and 𝐵 depend on time via the conjugate canonical coordinates (𝑞(𝑡),𝑝(𝑡))≡(𝑞₁(𝑡),…,𝑞_𝒩(𝑡);𝑝₁(𝑡),…,𝑝_𝒩(𝑡)), (𝒩 is the number of degrees of freedom of the classical system under study), 𝑋(𝑡)=𝑒^𝑖𝐿𝑡𝑋(0) with 𝑋=𝐴,𝐵,𝑞,𝑝, 𝐿=𝑖{ℋ,…} is the Liouville operator, and ℋ is the Hamiltonian of the system and 𝑋(0)≡𝑋(𝑞(0),𝑝(0)) at the initial time 𝑡=0. Of course, the time evolution of the generic dynamical variable 𝑋(𝑡) is governed by the well-known Liouville equation of motion (EM):𝑑𝑋(𝑡)={𝑋(𝑡)}𝑑𝑡,ℋ. (A.2) One can easily prove that the two-time GFs (A.1) depends on times 𝑡, 𝑡^′only through the difference 𝑡−𝑡^′, that is𝐺^(𝜈)_𝐴𝐵𝑡−𝑡^′=𝐴𝑡−𝑡^′;𝐵_𝜈=𝐴;𝐵𝑡^′−𝑡_𝜈, (A.3) and the two-time correlation function 𝐹_𝐴𝐵(𝑡,𝑡^′)=𝐹_𝐴𝐵(𝑡−𝑡^′)=⟨𝐴(𝑡)𝐵(𝑡^′)⟩=⟨𝐴(𝜏)𝐵⟩=⟨𝐴𝐵(−𝜏)⟩, with 𝜏=𝑡−𝑡^′, is related to the classical GFs (A.3) by the following relation [29]:𝐺^(𝜈)_𝐴𝐵(𝜏)=𝛽𝜃_𝜈(𝜏)𝑑⟨𝐴(𝜏)𝐵⟩𝑑𝜏=𝛽𝜃_𝜈(𝜏)⟨{𝐴(𝜏),ℋ}𝐵⟩, (A.4) where 𝛽=(𝐾_𝐵𝑇)⁻¹, 𝑇 is the temperature, and 𝐾_𝐵 is the Boltzmann constant (we assume 𝐾_𝐵=1). In particular, we have also𝐴(𝜏)𝑑⟨{,𝐵}⟩=𝛽⟨𝐴(𝜏)𝐵⟩𝐴(𝜏)}𝐵⟩.𝑑𝜏=𝛽⟨{,ℋ (A.5) For 𝐺^(𝜈)_𝐴𝐵(𝜏) and 𝐹_𝐴𝐵(𝜏) one can introduce the Fourier transforms:𝐺^(𝜈)_𝐴𝐵(𝜏)=^+∞_−∞𝑑𝜔𝐺2𝜋^{(𝜈)𝐴𝐵}(𝜔)𝑒^{−𝑖𝜔𝜏},𝐹_𝐴𝐵(𝜏)=^+∞_−∞𝑑𝜔𝐹2𝜋^𝐴𝐵(𝜔)𝑒^{−𝑖𝜔𝜏}, (A.6) where 𝐺^(𝜈)_𝐴𝐵(𝜔)=⟨⟨𝐴(𝜏);𝐵⟩⟩_𝜈,𝜔 and 𝐹_𝐴𝐵(𝜔)=⟨𝐴(𝜏)𝐵⟩_𝜔 are called the 𝜈-GF of 𝐴 and 𝐵 in the 𝜔-representation and the classical spectral intensity of the time-dependent correlation function 𝐹_𝐴𝐵(𝜏), respectively, with 𝑓(𝜔)=∫^+∞_−∞𝑑𝜏𝑒^𝑖𝜔𝜏𝑓(𝜏). Then, using (A.4) and the integral representations 𝜃(𝜏)=𝑖^+∞_−∞𝑑𝑥𝑒2𝜋𝑥+𝑖^{−𝑖𝑥𝜏}𝜀,𝜀⟶0⁺;𝛿(𝑥)=^+∞_−∞𝑑𝜏𝑒2𝜋^𝑖𝑥𝜏, (A.7) for the step function and the Dirac 𝛿-function, 𝐺^(𝜈)_𝐴𝐵(𝜔) can be expressed in terms of the corresponding spectral intensity as𝐺^(𝜈)_𝐴𝐵(𝜔)=^+∞_−∞𝑑𝜔^′2𝜋𝛽𝜔^′𝐹_𝐴𝐵𝜔^′𝜔−𝜔^′+(−1)^𝜈𝑖𝜀,𝜀⟶0⁺, (A.8) where the symbol (−1)^𝜈 means +1 if 𝜈=𝑟 and −1 if 𝜈=𝑎. It is interesting to compare (A.8) with the quantum corresponding expression for two operators 𝐴 and 𝐵 [40–44]:𝐺^(𝜈)_𝐴𝐵(𝜔)=^+∞_−∞𝑑𝜔^′2𝜋1+𝜂𝑒^{−𝛽ℏ𝜔′}𝐹_𝐴𝐵𝜔^′𝜔−𝜔^′+(−1)^𝜈𝑖𝜀,𝜀⟶0⁺, (A.9) where 𝜂=−1 and 𝜂=+1 by definition of quantum two-time GFs with commutator or anticommutator, respectively, and ℏ is the reduced Planck constant. Notice that formally, as expected for internal consistency, the function 𝐶(𝜔)=𝛽𝜔 or 𝑄_𝜂(𝜔)=(1+𝜂𝑒^{−𝛽ℏ𝜔})/ℏ characterizes the classical or quantum nature of the problem under study, respectively.

In analogy with the quantum case [45–47], we now introduce the time-dependent classical spectral density (CSD) for 𝐴 and 𝐵 [31, 37, 48, 49]:Λ_𝐴𝐵(𝜏)=𝑖⟨{𝐴(𝜏),𝐵}⟩, (A.10) with the Fourier transform:Λ_𝐴𝐵(𝜔)=𝑖⟨{𝐴(𝜏),𝐵}⟩_𝜔=^+∞_−∞𝑑𝜏𝑒^𝑖𝜔𝜏Λ_𝐴𝐵(𝜏)=𝛽𝜔𝐹_𝐴𝐵(𝜔). (A.11) Hence, from (A.8), one immediately obtains the spectral representation:𝐺^(𝜈)_𝐴𝐵(𝜔)=^+∞_−∞𝑑𝜔^′Λ2𝜋^𝐴𝐵𝜔^′𝜔−𝜔^′+(−1)^𝜈𝑖𝜀,𝜀⟶0⁺, (A.12) for the two-time GFs (A.1) in terms of the corresponding CSD Λ_𝐴𝐵(𝜔) in the 𝜔-representation. Also the dynamical correlation function ⟨𝐴(𝜏)𝐵⟩ can be easily expressed in terms of Λ_𝐴𝐵(𝜔). From (A.5), (A.10), and (A.11), we obtain indeed (classical spectral theorem)⟨𝐴(𝜏)𝐵⟩≡⟨𝐵𝐴(𝜏)⟩=^+∞_−∞𝑑𝜔Λ2𝜋^𝐴𝐵(𝜔)𝑒𝛽𝜔^{−𝑖𝜔𝜏}. (A.13) From (A.10)–(A.13) some formally exact results can be easily obtained. First, (A.10) and (A.11) yield^+∞_−∞𝑑𝜔Λ2𝜋^𝐴𝐵(𝜔)=𝑖⟨{𝐴,𝐵}⟩. (A.14) Besides, from (A.13), it follows^+∞_−∞𝑑𝜔Λ2𝜋^𝐴𝐵(𝜔)=⟨⟩.𝛽𝜔𝐴𝐵 (A.15) The relations (A.14) and (A.15) constitute useful examples of the so-called sum rules of the CSD, Λ_𝐴𝐵(𝜔), which have great relevance for physical consistency of practical calculations and approximations. Combining now (A.12) and (A.14), one can easily obtain another general result which plays an important role for calculation of the GFs. As 𝜔→∞ we have indeed [33]𝐺^(𝜈)_𝐴𝐵(𝜔)=𝑖⟨{𝐴,𝐵}⟩𝜔∼𝜔⁻¹,if⟨{𝐴,𝐵}⟩≠0,∼𝜔^−𝛼(𝛼≥2),if⟨{𝐴,𝐵}⟩=0, (A.16) which provide a relevant boundary condition for the 𝜈-GFs.

Let us come back now to the relations (A.12) for classical retarded and advanced GFs in the 𝜔-representation. As in the quantum counterpart [40–42], one can prove that 𝐺^(𝑟)_𝐴𝐵(𝜔) and 𝐺^(𝑎)_𝐴𝐵(𝜔), analytically continued in the 𝜔-complex plane, are analytical functions in the upper and lower half-plane, respectively. Then, combining these two analytical functions, one can construct a single function  𝐺_𝐴𝐵(𝜔)=∫^+∞_−∞𝑑𝑡𝑒^𝑖𝜔𝑡𝐺_𝐴𝐵(𝑡) of complex 𝜔 such that𝐺_𝐴𝐵(𝜔)=𝐺^(𝑟)_𝐴𝐵(𝜔)𝐺,Im𝜔>0,_{(𝑎)𝐴𝐵}(𝜔),Im𝜔<0. (A.17)

Hence, (A.12) provides for 𝐺_𝐴𝐵(𝜔) the spectral representation:𝐺_𝐴𝐵(𝜔)=^+∞_−∞𝑑𝜔′Λ2𝜋^𝐴𝐵𝜔^′𝜔−𝜔^′. (A.18) This function is analytical in the whole complex 𝜔-plane with a cut along the real axis where singularities for 𝐺_𝐴𝐵(𝜔) may occur. It is worth noting that, in terms of Λ_𝐴𝐵(𝜔), no formal differences exist for the spectral representations of 𝐺^(𝜈)_𝐴𝐵(𝜔) and 𝐺_𝐴𝐵(𝜔) in the classical and quantum context. Hence, all the developments already known in the quantum framework remain formally valid for the classical one. In particular, one has the important exact relationΛ_𝐴𝐵(𝜔)=𝑖𝐺_𝐴𝐵(𝜔+𝑖𝜀)−𝐺_𝐴𝐵(𝜔−𝑖𝜀), (A.19) which expresses the CSD in terms of the related two-time GFs in the 𝜔-representation. This allows us to state also that the cut for 𝐺_𝐴𝐵(𝜔) along the real axis in 𝜔-complex plane is determined by (A.19) and its singularities are the points of the real axis where the condition Λ_𝐴𝐵(𝜔)≠0 is satisfied. For the spectral intensity of classical systems, (A.11) and (A.19) yield𝐹_𝐴𝐵(𝜔)=⟨𝐴(𝜏)𝐵⟩_𝜔𝐺=𝑖^𝐴𝐵(𝜔+𝑖𝜀)−𝐺_𝐴𝐵(𝜔−𝑖𝜀).𝛽𝜔 (A.20) Of course, other known quantum relations are formally valid for classical many-body theory. Besides, when Λ_𝐴𝐵(𝜔) is real, the classical Kramers-Kronig relations (classical dispersion relations) between the real and imaginary parts of 𝐺^(𝜈)_𝐴𝐵(𝜔) are true:Re𝐺^(𝜈)_𝐴𝐵(𝜔)=(−1)^(𝜈)𝜋℘^+∞_−∞𝑑𝜔^′Im𝐺^(𝜈)_𝐴𝐵𝜔^′𝜔^′,−𝜔 (A.21) where the symbol ℘ denotes the main part of the integral. We have alsoΛ_𝐴𝐵(𝜔)=−2(−𝑖)^𝜈Im𝐺^(𝜈)_𝐴𝐵(𝜔), (A.22) and, in particular,Λ_𝐴𝐵(𝜔)=−2Im𝐺^(𝑟)_𝐴𝐵(𝜔). (A.23) Differentiating (A.3) with respect to 𝜏=𝑡−𝑡^′, with EM (A.2) for dynamical variables and 𝑑𝜃_𝜈(𝜏)/𝑑𝜏=𝛿(𝜏), yields𝑑𝐴(𝜏)𝑑𝜏⟨⟨;𝐵⟩⟩_𝜈=𝛿(𝜏)⟨{𝐴,𝐵}⟩+⟨⟨{𝐴(𝜏),ℋ};𝐵⟩⟩_𝜈, (A.24) which is the basic EM for the GF ⟨⟨𝐴(𝜏);𝐵⟩⟩_𝜈. This, however, is not a closed differential equation since in the right-hand side of (A.24) a new higher-order 𝜈-GF occurs involving Poisson brackets of a greater number of dynamical variables. Then, one needs to consider a new EM for the two-time 𝜈-GF ⟨⟨{𝐴(𝜏),ℋ};𝐵⟩⟩_𝜈. The 𝜏-derivative of this function provides an additional equation, formally identical to (A.24) with 𝐴(𝜏) replaced by {𝐴(𝜏),ℋ}, the right-hand side of which contains the new 𝜈-GF ⟨⟨{{𝐴(𝜏),ℋ},ℋ};𝐵⟩⟩_𝜈. By iteration of this procedure, we obtain the infinite chain of coupled EMs for GFs of increasing order:𝑑ℒ𝑑𝜏^𝑚ℋ𝐴(𝜏);𝐵_𝜈=𝛿(𝜏)ℒ^𝑚_ℋ+ℒ𝐴,𝐵_ℋ𝑚+1𝐴(𝜏);𝐵_𝜈(𝑚=0,1,2,…). (A.25) Here ℒ_ℋ=𝑖𝐿={…,ℋ} and ℒ^𝑚_ℋ𝐴 means ℒ⁰_ℋ𝐴=𝐴, ℒ¹_ℋ𝐴={𝐴,ℋ}, ℒ²_ℋ𝐴={{𝐴,ℋ},ℋ}, and so on. Notice that the chain of EMs (A.25) is formally the same for different types of GFs and hence one can eliminate the index 𝜈 when the physical context is clear.

In the practical calculations it is more convenient to work in the 𝜔-Fourier space. With 𝑖∫^+∞_−∞𝑑𝜏𝑒^𝑖𝜔𝜏(𝑑𝑓(𝜏)/𝑑𝜏)=𝜔𝑓(𝜔), the chain of equations, in the 𝜔-representation, assumes the following form:𝜔ℒ^𝑚_ℋ𝐴(𝜏);𝐵_𝜈,𝜔=𝑖ℒ^𝑚_ℋℒ𝐴,𝐵+𝑖_ℋ^𝑚+1𝐴(𝜏);𝐵_𝜈,𝜔(𝑚=0,1,2,…), (A.26) which has to be solved with appropriate boundary conditions (A.17). Since an exact solution is, of course, impossible, in practical calculations one must resort to decoupling procedures, and hence to approximate methods, to reduce the infinite chain of coupled equations to a finite closed one. Unfortunately, systematic and controllable decouplings are not easy to find and one must check for the reliability of a given approximation for each specific problem, by comparing the results with experiments, simulations, or other types of approaches.

B. Callen-Like Approach for Magnetization: A Classical Moment Problem

In this appendix we present an instructive method to solve the differential equation (25) for Ω(𝑎) via a classical moment problem. The differential equation is the following:Ω^′′(𝑎)1+2Φ+𝑎⁻¹Ω^′(𝑎)−𝑆²Ω(𝑎)=0, (B.1) for which the initial condition Ω(0)=1 is valid by definition. This is, of course, insufficient to find the physical solution of (25) and one should add a supplementary condition to be searched properly. Unfortunately, there is not classical analogue of the operatorial identity Π^𝑆_𝑝=−𝑆(𝑆^𝑧−𝑝)=0 which is the key ingredient of the Callen approach for the quantum HM [35]. In the following, we will show that, at our level of approximation, the additional conditionΩ(𝑎)=^𝑆_−𝑆𝑑𝑆^𝑧𝑓𝑆^𝑧𝑒^𝑎𝑆𝑧, (B.2) which follows formally from the definition of the canonical ensemble average of the dynamical variable 𝑒^𝑎𝑆𝑧, combined with Ω(0)=1, allows to determine Ω(𝑎) as a classical moment problem [50–53]. In view of the structure of the differential equation (25) for Ω(𝑎), it is convenient to define𝑓𝑆^𝑧=𝑔𝑆^𝑧𝑒^𝑆𝑧/Φ, (B.3) so that, we can writeΩ(𝑎)=^𝑆_−𝑆𝑑𝑆^𝑧𝑔𝑆^𝑧𝑒^{(1/Φ+𝑎)𝑆𝑧}=𝑆¹₋₁𝑑𝑥𝑔(𝑆𝑥)𝑒^{𝑆(1/Φ+𝑎)𝑥}. (B.4) Replacing this expression and its derivatives in (25), we have for 𝑔(𝑆^𝑧) the following:¹₋₁𝑑𝑥𝑦𝑥²+2𝑥−𝑦𝑔(𝑆𝑥)𝑒^𝑦𝑥=0, (B.5) where 𝑦=𝑆(1/Φ+𝑎). Then, with 𝑒^𝑦𝑥=∑^∞_𝑛=0(𝑦^𝑛/𝑛!)𝑥^𝑛, (B.5) provides^∞_𝑛=0𝑦^𝑛𝑦𝐼𝑛!^𝑛+2−𝐼_𝑛+2𝐼_𝑛+1=0, (B.6) with𝐼_𝑘=¹₋₁𝑑𝑥𝑔(𝑆𝑥)𝑥^𝑘. (B.7) It is now simple to show that the moments (B.7) of 𝑔(𝑆𝑥)≡𝒢(𝑥) are determined by the recursion relations:𝐼₁=0,…,𝐼_𝑛+1=𝑛𝐼𝑛+2^𝑛−1,𝑛=1,2,…. (B.8) These provide 𝐼_𝑘=𝐼₀𝑘+1,𝑘=2𝑛,0,𝑘=2𝑛+1,𝑛≥0, (B.9) where, at this stage, the moment 𝐼₀ is unknown. From (B.9), it is immediate to see that the solution of the moment problem for 𝑔(𝑆𝑥) is given by 𝑔(𝑆𝑥)=𝐼₀/2.

However, for the univocal determination of Ω(𝑎) it is not necessary to have the explicit form of 𝑔(𝑆𝑥)=𝑔(𝑆^𝑧) but rather its moments (B.9). From the integral representation (B.4), we have indeedΩ(𝑎)=𝑆^∞_𝑘=0𝑦^𝑘𝑘!¹₋₁𝑑𝑥𝑔(𝑆𝑥)𝑥^𝑘=𝑆𝐼₀𝑦^∞_𝑛=0𝑦^2𝑛+1()!=2𝑛+1𝑆𝐼₀𝑦sinh𝑦, (B.10) or, explicitly,Ω(𝑎)=𝑆𝐼₀sinh[𝑆(1/Φ+𝑎)]𝑆().1/Φ+𝑎 (B.11) Finally, the boundary condition Ω(0)=1 yields 𝐼₀=(1/Φ)(1/sinh(𝑆/Φ)) so that we obtain the following solution:Ω(𝑎)=1/Φ[𝑆(1/Φ+𝑎sinh1/Φ+𝑎)]().sinh𝑆/Φ (B.12) This is the central result of this appendix which constitutes the classical analogue [34] of the quantum Callen formula [35]. It provides the required expression for 𝑚, which is valid for any 𝑑, 𝑇, and ℎ.

Acknowledgments

A. Cavallo acknowledges the MIUR (Italian Ministry of Research) for financial support within the program “Incentivazione alla mobilità di studiosi stranieri e italiani residenti all’estero.”