Vlasov response, fluctuation-dissipation, and collective 1/f fluctuations in mean-field classical Heisenberg spin systems beyond second-order transitions

For the mean-field classical Heisenberg spin model with single-spin anisotropy, critical exponents describing the response to a small external field derived within the Vlasov (collisionless mean-field) dynamics differ sharply from the static equilibrium predictions ($\gamma^-=\beta/(n+1)$, $\delta=(n+2)/(n+1)$ instead of $\gamma^-=1$, $\delta=3$), because the infinite family of Casimir invariants makes the Vlasov dynamics nonergodic. The 2019 source paper left three linked directions open: extending the dynamical response theory to reference-state families that exhibit first-order (discontinuous) nonequilibrium phase transitions; establishing the validity and form of the Kubo fluctuation-dissipation theorem for these spin models under Vlasov evolution with Casimir constraints; and characterizing collective long-time $1/f$ fluctuations generated by pseudo-Casimir invariants. None of these three linked directions has been resolved in the subsequent literature.

Background

Systems with long-range interactions, in which every pair of elements couples with a strength that decays slowly or not at all with distance, are intrinsically non-additive and cannot be analyzed by the equilibrium statistical mechanics of short-range systems (Levin, Pakter, Rizzato, Teles and Benetti, Phys. Rep. 535, 1 (2014)). A minimal laboratory is the mean-field classical Heisenberg spin model studied by Yamaguchi, Das and Gupta (Phys. Rev. E 100, 032131 (2019)):

$H_N=-\frac{J}{2N}\sum_{i,j=1}^{N}\mathbf{S}_i\cdot\mathbf{S}_j+D\sum_{i=1}^{N}S_{iz}^{2n}-\mathbf{h}(t)\cdot\sum_{i=1}^{N}\mathbf{S}_i,\qquad |\mathbf{S}_i|=1,$

with ferromagnetic coupling $J>0$ scaled by $1/N$ (the Kac Prescription, making the energy extensive), local anisotropy $D>0$ with even exponent $2n$ (the model with exponent $2n$ is called Model-$n$; Model-0 is the isotropic Heisenberg model), and a small external field $\mathbf{h}(t)$ switched on at an instant $t_0$. In the thermodynamic limit, the one-spin phase-space distribution $f(\phi,p,t)$ on the unit sphere obeys the Vlasov equation, the collisionless mean-field kinetic equation also used for plasmas and self-gravitating systems. The Vlasov dynamics conserves an infinite family of Casimir invariants $C_G[f]=\iint G(f)\,d\phi\,dp$; as a consequence a finite-$N$ system remains trapped for times diverging with $N$ in quasistationary states (QSSs) — Vlasov-stationary distributions that are typically not Boltzmann-Gibbs equilibrium states — and the dynamics is nonergodic, so equilibrium ensembles cannot be used to predict response. A classic observation is that constants of motion suppress susceptibilities; the Casimir constraints realize this suppression dynamically.

A purely static analysis of the stationary phase-space distribution (e.g. of the thermal-equilibrium family $f\propto G[H_0(\phi,p)/T]$) predicts the usual mean-field exponents $\beta=1/2$, $\gamma^{\pm}=1$, $\delta=3$ for the second-order phase transition at a critical value $T_c$ of the family parameter, where $\beta$ governs the spontaneous magnetization $m_0\sim(T_c-T)^{\beta}$, $\gamma^{\pm}$ the susceptibility $\chi\sim|T-T_c|^{-\gamma^{\pm}}$ above and below $T_c$, and $\delta$ the critical-isotherm field scaling $m\sim h^{1/\delta}$. Yamaguchi, Das and Gupta instead derived the response within the Vlasov dynamics itself and found, for reference families with second-order transitions: $\beta=1/2$, $\gamma^+=1$, $\gamma^-=\beta/(n+1)$, $\delta=(n+2)/(n+1)$ for $n\ge1$, while the isotropic Model-0 shows no Vlasov response at all ($\gamma^{\pm}=0$, $\delta=1$). Thus the Heisenberg spin system on the unit sphere lies in a different universality class from the Hamiltonian mean-field (HMF) model — the XY analogue whose one-particle phase space is a cylinder, for which the Vlasov exponents are $\gamma^+=2\beta$, $\gamma^-=\beta/2$, $\delta=3/2$ (Ogawa and Yamaguchi, Phys. Rev. E 91, 062108 (2015); earlier evidence of dynamics-versus-statics discrepancy in Ogawa, Patelli and Yamaguchi, Phys. Rev. E 89, 032131 (2014)) — although the scaling relation $\gamma^-=\beta(\delta-1)$ survives in both. Two steps of the derivation rest on unverified physical assumptions: the neglect of a transient term $\mathcal{L}_K f$ in the perturbation evolution, and the estimate of the coefficient $L_2(T)$ that controls the ordered-phase susceptibility, which assumes that the dominant contribution to a phase-space integral comes from the neighborhood of the point $O=(1,0,0)$ where the single-spin Hamiltonian $H_0$ is minimized (with local expansion $H_0\simeq Dp^{2n}+(m_0/2)\phi^2$), because explicit angle-action variables — the canonical coordinates in which the integrable single-particle motion is uniform — are unavailable for the spin Hamiltonian.

Two fluctuation questions arise in the same setting. First, the Kubo fluctuation-dissipation theorem (FDT) — the equilibrium identity relating the linear response to an external field to the spontaneous equilibrium fluctuations of the conjugate variable — is derived for ergodic short-range systems; its validity for these spin models prepared in thermal-equilibrium reference states but evolving under Vlasov dynamics, with the Casimir constraints blocking relaxation to the forced equilibrium, is unknown (for the HMF model, finite-size equilibrium fluctuations had shown 'strange scaling', Yamaguchi, Phys. Rev. E 94, 012133 (2016)). Second, Yamaguchi and Kaneko (Phys. Rev. E 98, 020201(R) (2018)) showed that in long-range Hamiltonian systems the finite-$N$ breaking of the Casimir invariants into pseudo-invariants generates collective $1/f$ fluctuations of macroscopic variables, but this mechanism has not been studied for the mean-field spin systems here, nor has its feedback on response and relaxation been quantified. For the HMF model it is moreover known that special families of Vlasov-stationary states (e.g. waterbag-type states, distributions uniform on a compact region of phase space) exhibit first-order nonequilibrium phase transitions and a tricritical point (Antoniazzi, Fanelli, Ruffo and Yamaguchi, Phys. Rev. Lett. 99, 040601 (2007); Teles, Benetti, Pakter and Levin, Phys. Rev. Lett. 109, 230601 (2012)), whereas the spin-model analysis covered only second-order-transition families.

Problem Statement

For the mean-field classical Heisenberg spin model with Hamiltonian $H_N=-(J/2N)\sum_{ij}\mathbf{S}_i\cdot\mathbf{S}_j+D\sum_i S_{iz}^{2n}-\mathbf{h}(t)\cdot\sum_i\mathbf{S}_i$ on the unit sphere, extend the Vlasov-dynamical theory of response and fluctuations beyond the second-order-transition reference families analyzed by Yamaguchi, Das and Gupta (2019), by resolving three linked questions:

1. First-order-transition families. Classes of Vlasov-stationary reference states of this model other than the second-order-transition families studied (in the HMF model, waterbag-type stationary states are known to produce first-order nonequilibrium phase transitions and a tricritical point) should be considered, and the response to a small external field within the Vlasov dynamics characterized: what replaces the susceptibility divergence and the field-scaling laws at a discontinuous transition, and what are the corresponding jump laws or exponent analogues obtained dynamically?

2. Kubo fluctuation-dissipation theorem. Determine whether the Kubo fluctuation-dissipation theorem, valid for short-range systems in thermal equilibrium, holds — and in what form — for these spin models prepared in thermal-equilibrium reference states but evolving under Vlasov dynamics and subject to small external fields; that is, clarify the fluctuation-response relation in the presence of the Casimir constraints, in quantitative comparison with the HMF-model finite-size fluctuation results of Yamaguchi (2016).

3. Collective 1/f fluctuations. Characterize the collective long-time $1/f$ fluctuations generated by the pseudo-Casimir invariants in these spin models (the mechanism demonstrated for long-range systems by Yamaguchi and Kaneko (2018)).

These three deliverables jointly constitute one coherent objective: a Vlasov response-and-fluctuation theory of mean-field spin systems covering discontinuous transitions and the fluctuation side of the response, complementing the already-verified second-order exponent results.

References

1. Yoshiyuki Y. Yamaguchi, Debraj Das, and Shamik Gupta (2019). Critical exponents in mean-field classical spin systems. Physical Review E 100, 032131. DOI: 10.1103/PhysRevE.100.032131. arXiv:1905.08970. https://arxiv.org/abs/1905.08970
2. Yoshiyuki Y. Yamaguchi, Debraj Das, and Shamik Gupta (2020). Erratum: Critical exponents in mean-field classical spin systems [Phys. Rev. E 100, 032131 (2019)]. Physical Review E 102, 039901. DOI: 10.1103/PhysRevE.102.039901. https://doi.org/10.1103/PhysRevE.102.039901
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