When a strong magnetic field acts on a one-dimensional chain of weakly charged particles with transverse motion, standard momentum conservation is replaced by conservation of pseudomomentum and, for uniform charges, the sound velocity vanishes. For an exactly solvable harmonic chain with conservative velocity-exchange noise, Tamaki, Sasada and Saito (Phys. Rev. Lett. 119, 110602 (2017)) proved that the equilibrium heat-current correlation then decays with a new exponent β=3/4\beta = 3/4 and measured a nonequilibrium conductivity exponent α=0.375\alpha = 0.375, violating the standard α=1−β\alpha = 1 - \beta phenomenology that holds when sound modes are present. The problem is threefold: (i) derive α\alpha for this non-acoustic class analytically, starting from the equilibrium correlation, the open nonequilibrium system, or a controlled closure (equivalently, derive the cutoff scaling τN∼N3/2\tau_N \sim N^{3/2}); (ii) construct a fluctuating-hydrodynamic theory valid when time-reversal symmetry is broken and the conserved quantity is pseudomomentum, where the standard Euler closure fails, and use it to classify universality classes without sound modes; (iii) characterize the microscopic criteria by which charge structure (uniform versus alternating) determines the exponent. The equilibrium Green-Kubo asymptotics and the weak-noise 5/6-fractional-diffusion limit of the harmonic model are established, and fluid simulations show that pseudomomentum conservation alone does not imply anomalous transport.

Background

In a one-dimensional chain of N particles whose ends are coupled to heat baths with a small temperature difference, the thermal conductivity is defined as κ=JN/ΔT\kappa = JN/\Delta T, with JJ the steady-state heat current per site. In low-dimensional systems κ\kappa generically diverges with system size, κ∼Nα\kappa \sim N^{\alpha} with 0<α<10 < \alpha < 1 (anomalous heat conduction). The divergence is controlled by the equilibrium autocorrelation of the total energy current, C(t)=N−1⟨Jtot(t)Jtot⟩eq∼t−βC(t) = N^{-1}\langle J_{\mathrm{tot}}(t) J_{\mathrm{tot}}\rangle_{\mathrm{eq}} \sim t^{-\beta}, through the Green-Kubo formula κ∼∫0τNC(t) dt\kappa \sim \int_0^{\tau_N} C(t)\,dt, where τN\tau_N is a system-size-dependent cutoff time. When sound waves propagate with finite velocity cc, one phenomenologically takes τN∼N/c\tau_N \sim N/c and obtains α=1−β\alpha = 1 - \beta. Nonlinear fluctuating hydrodynamics (NFH) — the Euler-scale fluctuating theory of the conserved fields (stretch, momentum, energy) developed by Spohn and others — decomposes fluctuations into two sound modes and one heat mode and predicts only two universality classes, β=2/3\beta = 2/3 or 1/21/2, tied to Kardar-Parisi-Zhang fluctuations of the sound modes. The momentum-exchange (ME) model of Basile, Bernardin and Olla — a harmonic chain with stochastic nearest-neighbor velocity exchange that conserves stretch, momentum and energy — made β=1/2\beta = 1/2 rigorous. A non-acoustic momentum-conserving chain (one with no propagating sound modes) studied by Komorowski and Olla was instead shown to conduct diffusively, and mode-coupling arguments had speculated that a nonzero sound velocity is necessary for superdiffusion. Tamaki, Sasada and Saito asked what happens when a magnetic field breaks time-reversal symmetry: for charged particles with transverse motion dominated by the Lorentz force, the standard momentum is not conserved, but the pseudomomentum Pi=vi−eiqi×BP_i = v_i - e_i q_i \times B (with viv_i the velocity, qiq_i the position, eie_i the charge and BB the magnetic field) is. In their exactly solvable model — a harmonic chain with conservative Poissonian velocity-exchange noise in this magnetic dynamics — the dispersion relation has zero sound velocity for uniform charges ei=1e_i = 1 (case I) and a finite one for alternating charges ei=(−1)ie_i = (-1)^i (case II). The exact equilibrium correlation decays as C(t)∼A1t−3/4+A2t−1/2cos⁡(Bt)+A3t−3/2C(t) \sim A_1 t^{-3/4} + A_2 t^{-1/2}\cos(Bt) + A_3 t^{-3/2} in case I (a new exponent β=3/4\beta = 3/4) and as t−1/2t^{-1/2} in case II, while nonequilibrium simulations give α=0.375±0.001\alpha = 0.375 \pm 0.001 in case I versus α=1/2\alpha = 1/2 in case II, so the α=1−β\alpha = 1 - \beta relation fails and the cutoff instead grows as τN∼Nν\tau_N \sim N^{\nu} with ν≈1.5\nu \approx 1.5. The authors identify the mechanism for the failure of the standard theory: with pseudomomentum as the conserved quantity, the Euler equations for the conserved fields are not closed in the standard form, so NFH does not apply.

Problem Statement

Build the missing theory of anomalous heat transport for the non-acoustic universality class created by pseudomomentum conservation in time-reversal-asymmetric (magnetic) low-dimensional systems. The problem has three jointly linked targets. (1) Derive analytically the nonequilibrium conductivity exponent α\alpha for the non-acoustic case (uniform charges, equilibrium exponent β=3/4\beta = 3/4) — equivalently, derive the scaling of the Green-Kubo cutoff, τN∼Nν\tau_N \sim N^{\nu} with ν≈3/2\nu \approx 3/2 — starting from the equilibrium current correlation, from a direct treatment of the nonequilibrium steady state of the open system, or from a controlled hydrodynamic or kinetic closure. The derivation must not assume a finite sound velocity or standard momentum conservation. (2) Construct a precise hydrodynamic description — a fluctuating-hydrodynamics framework — that remains valid when time-reversal symmetry is broken and the conserved quantity is pseudomomentum (where the standard Euler closure fails), and use it to classify the universality classes of heat transport that exist when sound modes are suppressed. (3) Characterize the microscopic mechanism and general criteria by which the charge structure (uniform versus alternating eie_i, and more generally the structure of the conserved fields) determines whether the long-time exponent changes — in particular, explain the rigorous facts that uniform charges give β=3/4\beta = 3/4 while alternating and zero-field systems both give β=1/2\beta = 1/2 — beyond the harmonic stochastic model in which they are established. The three targets are jointly required and form one objective: the hydrodynamic framework of (2) is the natural route to the derivation of (1), and the classification of (3) is the universality statement that framework must produce. The task counts as resolved only when all three are answered — the analytic derivation of the exponent in (1), the framework of (2) together with the universality classification it yields, and the charge-structure criteria of (3) — and no single target alone is a complete answer.

References

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