key: cord-0616303-wyvold7p
authors: Nardis, Jacopo De; Medenjak, Marko; Karrasch, Christoph; Ilievski, Enej
title: Universality classes of spin transport in one-dimensional isotropic magnets: the onset of logarithmic anomalies
date: 2020-01-17
journal: nan
DOI: nan
sha: 623b4446bf549a34a63dfa5f3b8402313cafaff1
doc_id: 616303
cord_uid: wyvold7p

We report a systematic study of finite-temperature spin transport in quantum and classical one-dimensional magnets with isotropic spin interactions, including both integrable and non-integrable models. Employing a phenomenological framework based on a generalized Burgers' equation in a time-dependent stochastic environment, we identify four different universality classes of spin fluctuations. These comprise, aside from normal spin diffusion, three types of superdiffusive transport: the KPZ universality class and two distinct types of anomalous diffusion with multiplicative logarithmic corrections. Our predictions are supported by extensive numerical simulations on various examples of quantum and classical chains. Contrary to common belief, we demonstrate that even non-integrable spin chains can display a diverging spin diffusion constant at finite temperatures.

Introduction. Obtaining a theoretical framework that is able to explain how macroscopic laws of transport emerge from the microscopic deterministic dynamics presents one of the central challenges of condensed matter physics. This transcends purely academic interest, as many problems of quantum transport remain unresolved in the presence of strong interactions [1, 2] . One viable strategy to improve our understanding of transport phenomena is to identify universality classes and study certain representative instances which can either be solved exactly, or at least simulated numerically in an efficient manner [3] [4] [5] [6] . In this respect interacting many-particle systems confined to one spatial dimension, in the realms of both quantum and classical models, take a special role as they often exhibit anomalous features [7, 8] . One of the prominent examples of a nonequilibrium universality class is given by the Kardar-Parisi-Zhang (KPZ) equation [9] which is widespread in the area of growing onedimensional interfaces [10, 11] . The KPZ and Lévy universality classes which also occur in systems of classical particles can be understood in the scope of the nonlinear fluctuating hydrodynamics [12] [13] [14] [15] [16] .

In recent years, the advent of the generalized hydrodynamics [17, 18] , studies of quantum chaos and its relation to transport [19] [20] [21] [22] [23] , and of noisy quantum systems [24] [25] [26] , reinvigorated the field of transport laws in spin chain models. In integrable quantum chains a closed-form universal expression for the conductivity matrix was found, comprising both the Drude weights [27] [28] [29] [30] and diffusion constants [20, [31] [32] [33] [34] , together with a myriad of other applications [35] [36] [37] [38] [39] [40] [41] . This provided a coherent picture for earlier numerical results (see, e.g., [42] [43] [44] [45] [46] and references therein). In spite of these developments, the discovery of superdiffusive spin transport and KPZ scaling (cf. Eq. (4)) in integrable spin chains with isotropic inter-actions, originally discovered numerically in the Heisenberg spin-1/2 chain in [47, 48] and further surveyed in [49] [50] [51] [52] [53] , came as a surprise. Although recent numerical works [52, [54] [55] [56] , in combination with scaling arguments explaining the dynamical exponent [50] , constitute convincing evidence in support of the KPZ universality, a rigorous analytical account of this phenomenon is still lacking. Most recent studies suggest that anomalous spin transport occurs only in integrable systems invariant under non-Abelian (SU (2) or SO(3)) Lie groups. In contrast, normal spin diffusion is expected to be immediately recovered upon breaking integrability [52, 56] . The occurrence of normal diffusion in non-integrable symmetric chains was also suggested in the numerical study [57] (see also [53, 56] ), after a long-lasting controversy [58] [59] [60] [61] [62] [63] . This perspective has been further reinforced in [53] which proposes a phenomenological explanation of the KPZ-type superdiffusion. At this stage, two key questions remain unanswered:

(i) Do all rotationally invariant integrable spin chains display superdiffusive spin transport?

(ii) Do all non-integrable homogeneous spin chains display normal diffusive spin transport?

To address these questions, we carry out a systematic study of magnetization transport in classical and quantum spin systems in the non-magnetized sector of thermal equilibrium states where the global rotational symmetry remains unbroken. Aiming at complete classification of admissible transport laws, we build on a recent work by V. B. Bulchandani [53] and devise a simple model that help us to single out two novel nonequilibrium universality classes of spin transport. We corroborate our findings with extensive numerical simulations of classical arXiv:2001.06432v2 [cond-mat.stat-mech] 30 Jan 2020 and quantum chains. Quite unexpectedly, we find that the answers to questions (i) and (ii) are both negative: the presence of global non-Abelian symmetries alongside integrability is not a sufficient condition for an anomalous spin transport of the KPZ-type. Perhaps even more surprisingly, despite the lack of integrability in the classical isotropic Heisenberg chain, the spin diffusion constant is found to diverge logarithmically in time, thus refuting a widely held belief that non-integrable models cannot display infinite diffusion constants at finite temperatures. From our perspective, the results summarized in diagram 1 should provide a comprehensive classification of magnetization transport in homogeneous rotationally invariant quantum and classical spin models with short-range interactions.

We proceed by first introducing the formalism of an effective spin field theory which we relate to the KPZ equation in a time-dependent noisy environment. To systematically test our predictions, we subsequently concentrate on a number of simple representative examples.

Spin-field theory of isotropic magnets. In order to capture various universality classes of spin superdiffusion, the task at hand is to devise an effective theory for finite-temperature magnetization dynamics in quantum and classical spin systems with isotropic interactions valid on large spatio-temporal scales. Here we propose an effective description which is inherently classical in nature, by employing the continuum theory for a classical spin-field which is in turn treated within a hydrodynamic approximation. Microscopic details are included implicitly through an appropriate phenomenological noise.

As our starting point we consider the most general form of a manifestly SO(3)-symmetric Hamiltonian equation of motion, specified by some functional F involving scalar and vector products of the spin-field and derivatives thereof. We can include classical lattice models as well, which are analyzed through their continuum counterparts. To additionally incorporate quantum spin chains we first perform a mean-field average [64] of the microscopic spin Hamiltonian. It has to be stressed that such a correspondence cannot retain all quantitative features of spin dynamics. Nonetheless, we shall argue, in the spirit of [53] , that correspondence is still meaningful to capture the correct large-time transport behaviour. The outlined effective theory applies in thermal equilibrium in the non-magnetized sector (i.e. at half-filling) where the global rotational invariance of the underlying invariant measure is unbroken. This is of paramount importance for the anomalous character of magnetization dynamics, as the addition of finite chemical potential (or external magnetic field), which dynamically breaks the non-Abelian symmetry, leads to restoration of normal spin diffusion (accompanied in integrable systems by a finite spin Drude weight [65] or ballistic cur- characterizing the time-asymptotic behavior of the spin dynamical correlations S(x, t)· S(0, 0) ∼ t −1/z log −b (t), as predicted by the generalized noisy Burgers' equation for the hydrodynamic evolution of the torsion component τt + (τ n + Dτx + γ(t) η)x = 0 with a time-dependent noisy environment γ(t) ∼ t −ζ . rent). On large spatio-temporal scales, the evolution of a spin-field is accurately captured by a 'hydrodynamic soft mode' carrying a negligible energy density, which can be conveniently described in terms of two intrinsic geometric quantities, curvature κ = S x · S x 1/2 and torsion τ = κ −2 S · ( S x × S xx ). The soft mode pertains to long-wavelength (k ∼ O(1/ )) limit of the spin procession (about a distinguished axis fixed by the perturbation which breaks the gauge invariance, assumed subsequently to be the z-axis) at constant latitude S z = h, with S x (x, t) ± iS y (x, t) = √ 1 − h 2 exp [±i(k x + w t)], frequency w(k) = −k 2 h and dispersion E(k) = 1 2 k 2 (1 − h 2 ), which in terms of the curve describes helices with constant (on scale ∼ −1 ) κ = √ 1 − h 2 k and τ = h k [66] . Hydrodynamic modulation on a characteristic scale , leading to scaling κ ∼ τ ∼ O(1/ ), indicates that the energy density E = κ 2 /2 ∼ −2 is suppressed compared to the dynamics of torsion τ (see also [53] ). Further noticing that energy fluctuations are diffusive (known to hold in both integrable and non-integrable systems [33, 67] ), they can be effectively decoupled from the fluctuations of the torsion field provided the latter are superdiffusive. Based on this we can conclude that torsion τ remains the only relevant scalar field at large times and that, since τ ∼ h at small h, the finite-temperature spin-spin fluctuations S(x, t) · S(0, 0) are proportional to fluctuations of the torsional mode τ (x, t)τ (0, 0) . We shall assume that such a decoupling mechanism holds generically for equations of the form (1) .

Generalized noisy Burgers' equation. To account for thermal fluctuations we invoke the standard arguments of the nonlinear fluctuating hydrodynamics (NLFHD) [14] , where the microscopic degrees of freedom of the underlying Hamiltonian dynamics are effectively taken into account through an appropriate stochastic term and effective diffusion (i.e. dissipation). This brings us to the generalized noisy Burgers' equation of the form τ t + (τ n + D τ x + √ γ η) x = 0, (2) where D = γ/χ τ is the phenomenological diffusion constant, χ τ is static susceptibility of τ , η(x, t) a white noise with unit variance, γ is the effective variance of the noisy environment and parameter n ≥ 2 specifies the degree of nonlinearity. From the general scaling relations τ (x, t) t −1/2z f(x t −1/z ) and τ (x, t)τ (0, 0) t −1/z g(x t −1/z ) (here and below the bracket refers to the average with respect to the canonical invariant measure) one however deduces that z = (n + 1)/2, which implies that nonlinearities of degree n ≥ 4 (with z > 2) are subdiffusive and thus irrelevant at large times. Although NLFHD has no predictive power in this case, one generically expects to find normal spin transport.

The final key ingredient is to impose the structure of the noise, reflecting the nature of fluctuating modes which are relevant on a hydrodynamic scale. In the spirit of conventional NLFHD (see e.g. [14] for application to anharmonic chains), in the presence of long-lived ballistically propagating normal modes of Euler hydrodynamics, we adopt a time-independent white noise γ = γ 0 . The same applies to integrable systems which exhibit extensively many local conserved fields, as previously suggested in [53] . On the contrary, generic spin systems do not support ballistic modes and excitations dissipate through the system. In this case the variance γ is expected to obey the diffusive scaling and decay with time γ ≡ γ(t) ∼ (t 0 /(t 0 + t)) ζ with ζ = 1/2, (3) where t 0 denotes an unknown model-and temperaturedependent scale. The picture behind this is that fluctuations excited by the spatio-temporal variation of the chemical potential should dissipate away diffusively as their density decays to zero with exponent ζ = 1/2.

For n = 2, Eq. (2) is just the ordinary noisy Burgers' equation equivalent to the KPZ equation, up to a change of variable [9] . A recent work [68] examined the properties of such KPZ equations with time-dependent noise term of the form (3), finding non-universal largetime behavior for ζ > 1/2 and universal KPZ dynamics (with modified dynamical exponents) for ζ ≤ 1/2. Exactly at the 'critical' point ζ = 1/2, corresponding precisely to diffusive spreading of microscopic excitations, in [68] the authors deduce a modified diffusive scaling x ∼ t 1/2 log 2/3 (t/t 0 ) (for the particular case the scaling should be understood as a lower bound and not a rigorous statement [69] , as numerics are not able to distinguish slightly different exponents, see also additional numerical data in [70] ). Keeping this in mind, the statistics of spin fluctuations in this case is expected to exhibit a crossover from an effective KPZ dynamics at short-intermediate times t t 0 , S(x, t) · S(0, 0)

to the asymptotic scaling of the form S(x, t) · S(0, 0) G Γ G x t −1/2 log −2/3 (t/t 0 ) t 1/2 log 2/3 (t/t 0 ) , (5) with Gaussian profile G(x) e −x 2 in the limit t t 0 . The anomalous form (5) implies a divergent behavior D(t) ∼ [log (t)] 4/3 .

Integrable isotropic magnets. Our main example is the Heisenberg continuous magnet H (2) = 1 2 dx S x · S x (using standard notation S x = ∂ x S etc. for partial derivatives), also known as the isotropic Landau-Lifshitz model [71, 72] , which is a paradigmatic example of an integrable classical field theory. The time-evolution is governed by the nonlinear PDE S t = F and H (4) 

Invoking the 'decoupling hypothesis' and the phenomenological noisy environment, the torsional mode in each H (n) is governed by the generalized Burgers' equations (2) with non-linearity of degree n (see also [70] for the details). Adopting a constant value for γ, the dynamics falls into the KPZ class at the lowest order n = 2. This is however no longer the case for n > 2 where the quadratic nonlinearity is absent. For n ≥ 4 the non-linearity is dominated by diffusive processes and Hamiltonians H (n>3) thus do not display any enhancement of normal diffusion. The cubic n = 3 case is however marginally irrelevant in the dynamic RG sense. This type of nonlinearity has been previously examined in the study of Toom interface [73] [74] [75] and argued to result in a logarithmic-type correction. We shall corroborate on this scenario later on.

The quantum Heisenberg hierarchy. We proceed by examining the spin diffusion constant in integrable quantum spin-S Heisenberg chains,Ĥ (2) = jˆ S j ·ˆ S j+1 , together with its hierarchy of local conservation lawŝ H (n) = jĥ (n) j with n-site densitiesĥ (n) j (hats denotes quantum operators). These models, we believe, exhaust all homogeneous SU (2)-symmetric integrable quantum spin chains with short-range interactions (excluding cases symmetric under higher-rank Lie groups).

Taking full advantage of quantum integrability and the underlying quasi-particle picture [17, 18] , we have computed the exact spin diffusion constant, being the dominant contribution to spin transport in the zero magnetization sector where the Drude weigh vanishes. The details of this computations are spelled out in [70] , where we show that D (n≥4) < ∞. For the marginal case n = 3 we however find a logarithmic divergence D (3) (t) ∼ log (t). This is our first example of a logarithmically enhanced diffusion, labelled by D (3) log in Fig. 1 . As a proof of principle, the same conclusion can be independently reached by following the lines of our phenomenological programme, using that the appropriate classical mean-field continuous limits of the HamiltoniansĤ (n) are the higher Landau-

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shown for a few representative classical non-integrable lattice spin models at infinite temperature. The dashed curves are fitting lines with log (t). The Heisenberg and next-to-nearest neighbour Hamiltonians belong to the D (2) log class, whereas the non-integrable lattice discretizations of H (n) (denoted by H (n) lattice ) exhibit normal diffusion D for n ≥ 3.

Lifshitz Hamiltonians H (n) (cf. [76, 77] ), which are subsequently reduced to effective Burgers' equations of the form (2) .

Non-integrable classical and quantum spin chains. To demonstrate how non-integrable chains also fall in the classification scheme of Fig. 1 , we next examine the non-integrable classical Heisenberg chain H Heis = j S j · S j+1 , where the most recent works align in favour of normal spin diffusion [57] (see also [53, 56, 78] ). Performing numerical simulations with the method of [57] (which conserves exactly energy density and | S j | at all times, see [70] for more details on the numerical simulations) and checking in addition the next-nearest neighbour scalar interaction, H NNN = j S j · S j+1 + 0.8 S j · S j+2 along with the non-integrable lattice analogues of H (3) and H (4) flows (denoted by H yielding (contrary to claims in refs. [19, 57] ) a logarithmically divergent spin diffusion constant (D (2) log universality class in Fig. 1 ). The continuous counterpart of non-integrable Hamiltonian H (3) lattice is instead described by a time-inhomogeneous cubic n = 3 Burgers' equation. Despite the lack of theoretical prediction for the latetime dynamics in this case, our numerics suggests that spin dynamics is most likely purely diffusive. Finally, the simulation of H (4) lattice nicely conforms with normal diffusion as expected from Eq. (2) with a higher-degree of nonlinearity.

As an independent test, we additionally considered the anisotropic model H δ = j S j · S j+1 + δŜ z j ·Ŝ z j+1 , with interaction anisotropy δ acting as a 'regulator' which restores normal diffusion, see also additional numerical data in [70] . By computing the diffusion constant D(δ) and monitoring its value by approaching δ → 0 + , we find behavior (cf. Fig. 3 ) compatible with a mild divergence

with an exponent d ≈ 1. A reliable extraction of transport coefficient in quantum spin chains is a very demanding task obstructed by a rapid growth of entanglement entropy which renders tDMRG simulations with fixed bond dimension uncertain at large times. Despite inherent issues, a recent numerical study of isotropic spin-S chains [52] concluded in favour of normal spin diffusion (z = 2) in nonintegrable spin chains (S ≥ 1). Here we prefer to facilitate a direct comparison with classical spin chains. To this end, we carried out tDMRG simu-lations [79] [80] [81] of the quantum anisotropic spin-1 chain H δ and, restricting ourselves to only moderately small δ, numerically extracted the spin diffusion constant from the time-dependent DC conductivity with a diffusive tail σ(t) (χ/T )D+c t −1/2 with spin susceptibility χ and fitting parameters D and c, see additional numerical data in [70] . The data shown in Fig. 3 indicates that the spin dynamics in the non-integrable spin-1 chain mirrors that of its classical counterparts and thus experiences the same divergence (6) (D (2) log class). Notice that in the quantum integrable S = 1/2 chain the divergence in the δ → 0 + limit is different as it diverges polynomially as D ∼ δ −1/2 [20, 32] , signalling the onset of the KPZ dynamical exponent at δ = 0.

Conclusions. We have proposed a phenomenological description of finite-temperature spin transport in one-dimensional quantum and classical systems with isotropic interactions. We have predicted four different classes, including in particular two distinct types of logarithmically enhanced diffusion which have not been previously disclosed in the context of many-body deterministic systems. We conjecture that in homogeneous SU (2) or SO(3) spin systems with short-range interactions this list is exhaustive (cf. Fig. 1 ). While the outlined approach can adequately capture the qualitative features of spin dynamic (dynamical exponents and logarithmic corrections) it does not give an access to exact values of transport coefficients or couplings of the effective hydrodynamical equations (2) . Important refinements in this direction are left to future works.

Our findings shine some light on the puzzling observations in our previous work [51] which discusses spin dynamics in the context of Haldane antiferromagnets. The observed short-time behavior with approximate exponent z = 3/2 (numerically detected also in [82] ), later argued in [52] to be merely a pronounced transient regime which crossovers into normal diffusion, indeed plays nicely with the expected transient scenario assisted by an effective time-dependent noise. Nonetheless, we argue now that despite the broken integrability the spin diffusion constant does not saturate at asymptotically large times. This can be reconciled with the predictions of [51] based on the effective low-energy quantum field theory provided that the transient is regulated by a temperaturedependent time-scale diverging in the T → 0 limit, as a consequence of the diverging effective lifetime t 0 of the microscopic degrees of freedom in (3), despite different mechanisms resembling the situation in gapless onedimensional systems [83, 84] .

Finally, It is reasonable to expect that the divergence (6) could be seen in a real experimental setting and we hope that this can be successfully addressed in the near future.

Universality classes of spin transport in one-dimensional isotropic magnets: the onset of logarithmic anomalies Appendix A: On the numerical simulation of classical spin dynamics

We have carried out numerical simulations of classical spin dynamics using the method introduced in [57] . The latter proves to be superior compared to more standard Euler or Runge-Kutta integration schemes (as e.g. employed in [78] ), owing to exact conservation of both total energy and spin magnitude | S j | = 1 at each lattice site. The method employs the fact that any equation of motion of type

with vector B j depending in general on the spins S j+1 , S j−1 . . . , S j+n , S j−n can be analytically integrated with aid of the Rodrigues' rotational formula, yielding

where φ = | B i ∆t| andB = B/| B|. By evolving first S j , S j+n , . . . S j+2n and then S j+1 , S j+1+n , . . . and so on, energy and | S j | are conserved to all orders in ∆t, while all other local conserved quantities fluctuate within a range of order (∆t) 3 even on time-scales of the order t ∼ 3000, see Fig. 4 and 5. In order to simulate dynamics at infinite temperature, we have computed an average over 5 × 10 5 − 10 6 random initial states, which also reduces the error at large times, see Fig. 6 . Moreover we stress that a large number of states is of fundamental importance in order to well recognise the logarithmic corrections at large times. In order to reduce the noise, we have additionally performed the ergodic time-average t −1 t 0 dt of the spin-spin correlations C j (t).

We stress that our numerical results do not differ from the ones reported in [57] , where the presence of logarithmic corrections to diffusion was however missed in the analysis of the data. We instead believe that the numerical data at infinite temperature in [78] are incorrect, as they show normal diffusion, probably due to the Runge-Kutta integration schemes or lack of proper averaging on initial conditions. The results in [78] for slightly anomalous diffusion at lower temperature are instead in agreement with the presence of logarithmic corrections. 4 . Time-evolution of the approximately conserved magnetizations in the classical (non-integrable) Heisenberg chain with integration step-size ∆t = 0.025 and L = 1000, starting from a single random initial state. We shall employ the toolbox of the generalized hydrodynamics [17, 18] to examine the spin diffusion constant in the quantum Heisenberg hierarchy. For definiteness we shall specialize here to the fundamental spin chains S = 1/2, noticing that integrable spin-S chains can be essentially treated along the same lines. The higher Hamiltonian densities can be obtained by the iterative application of the boost operatorB,

In close analogy to the isotropic Landau-Lifshitz magnet, the 'second Hamiltonian flow' H (3) corresponds to the chiral three-spin interactionĥ

whereas the Hamiltonian density of the 'third flow' is supported on four adjacent lattice siteŝ

Now we turn to the computation of the spin diffusion constant. The total contribution can be conveniently presented as a spectral sum over quasi-particle excitations. The latter form an infinite tower of bound states made out of s constituent magnons carrying s quanta of (bare) magnetization. The spin diffusion constant associated with the n-th Hamiltonian flow can be accordingly decomposed as D (n) = s≥1 D (n) s , with contributions of individual quasi-particle species s given by a closed-form expression [51] D (n)

where the integration is taken over the range of the rapidity variable θ, n s (θ) are Fermi occupation functions of the reference half-filled (i.e. S z = 0) equilibrium background, ε 

and χ h = dx S z (x)S z (0) is the rescaled spin susceptibility at half filling. The task at hand is to isolate the conditions under which D (n) becomes divergent. It is sufficient to inspect the high-temperature limit of the grand-canonical Gibbs ensemble,

where closed-form expressions are available (see [51] ) in the half-filled h → 0 limit. In particular, functions

become independent of the rapidity variable θ. Furthermore, using the exact expressions ε (2) s (θ) = 8θ(s + 1) 1 (4θ 2 + s 2 ) 2 − 1 (4θ 2 + (s + 2) 2 ) 2 ,

and ε (n) s (θ) = ∂ n−2 ε

(2) s (θ) ∂θ n−2 for n > 2,

we deduce that dθ |ε (n)

Based on this we conclude that the sum (B4) is convergent whenever n ≥ 4. We now take a closer look at anomalous cases n = 2, 3 with a divergent spin diffusion constant. For this purpose we introduce the regularized diffusion constant,

where we have imposed a spectral cut-off s which integrates out the 'heavy' quasi-particles. Now we can essentially reiterate the dimensional analysis along the lines of ref. [50] . The key piece of information is the large-s behavior of the dressed velocities v (n)dr s (θ) = ∂ε (n)

where p s (θ) denote momenta of dressed excitations. At large s, these satisfy the algebraic law (to be intended under integration over θ) v (n)dr s (θ) ∼ 1 s n−1 .

(B13)

The associated 'anomalous diffusive length'

can then be converted into the time-domain by comparing it to the time-dependent diffusion constant D (n) (t) via

As previously shown in [50] , for n = 2 one makes the ansatz D (2) (t) ∼ t α and deduces that s ∼ t (1−α)/2 . Inserting this result back to D (2) (s ) ∼ s and comparing it to D (2) (t) ∼ t α yields the superdiffusive exponent α = 1/3 (which translates into the dynamical exponent z = 2/(α + 1) = 3/2).

In the n = 3 case, where

we instead plug in an ansatz D (3) (t) ∼ [log (t)] r(t) . The self-consistent value for r, which follows from the dimensional analysis requires that lim t→∞ r(t) = 1.

We owe to point out the mismatch in comparison to ref. [74] which, using a perturbative analysis at one-loop order, predicts the logarithmic correction of the type D DS (t) ∼ [log (t)] 1/2 . In contrast, our conclusion follows from a non-perturbative calculation based on exact spectrum of thermally-dressed quasi-particle excitations, but it also relies on a scaling analysis that could in principle fail to distinguish different types of logarithmic terms. 

Below we analyze the spin dynamics with aid of the Frenet-Serret apparatus, mapping the spin-field S ∈ S 2 to a dynamical smooth curve in Euclidean space R 3 . To each point on a curve, parametrized by its arclength x, we attach a triad of orthonormal vectors { e i } 3 i=1 , representing the tangent, normal and binormal vectors of the curve. The local change of frame is then generated by a pair of so(3) transformations,

specified by the Darboux and angular-velocity vectors

satisfying compatibility relation Ω t − ω x = Ω × ω. Identifying the spin-field with the tangent vector, e 1 ≡ S, the time-evolution (1) can be cast in the form S t = ( e 1 ) t = ω 3 e 2 − ω 2 e 3 , or in terms of curvature and torsion as κ t = (ω 3 ) x + τ ω 2 , τ t = (ω 1 ) x − κω 2 .

Therefore, we can express S xx = −κ 2 e 1 + κ x e 2 + κτ e 3 , whence we deduce the angular velocities

and accordingly the Frenet-Serret equations

These are also known as the Betchov-Da Rios equations [86, 87] and govern the motion of a vortex filament in a viscous liquid. The higher Hamiltonians H (n≥3) = dx h (n) (x) can be constructed recursively [88] ,

where D ≡ d/dx. In particular, the second (i.e. the third-order) flow is given

and is generated by the chiral interaction of the form

We shall in addition examine the third (i.e. fourth-order) flow

which corresponds to

The total integral of elastic energy density E ≡ h (2) = 1 2 κ 2 is a conserved quantity, obeying the local conservation law E tn + J (n)

with flux densities

and so forth. The angular velocities of the second flow H (3) read

implying the following Frenet-Serret equations for the curvature and torsion κ t3 + κ xx + 1 2 κ 3

x + 3 2 (κ 2 τ 2 ) x κ = 0, (C20)

The above exact dynamical equations are still exact. The next step is to simplify them by taking into account that on a large coarse-graining scale the curvature and torsion components of a hydrodynamically modulated soft mode obey κ ∼ τ ∼ O(1/ ), which in effect allows to neglect the fluxes in Eq. (C13). This means, in particular, that to the leading-order approximation E ∼ O(1/ 2 ) (and likewise κ) can be treated as constant and thus effectively decouple from the torsion dynamics. Additionally, by dropping the dispersive terms in the equation for τ , we are left with the cubic Burgers' equation

The structure of the fourth flow H (4) is slightly more involved, and a lengthy calculation yields

In this case the Frenet-Serret equations are of the form

leading to, after repeating the above logic, a quartic Burgers' equation ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ The numerical data is unable to reliably distinguish between the decay C0(t) ∼ t −1/2 (log t) −1 and C0(t) ∼ t −1/2 (log t) −2/3 , despite the latter is a slightly better fit.

Random Matrices: Theory and Applications

Large Scale Dynamics of Interacting Particles

For spin-S quantum chains linear in spin generators this is achieved by taking the expectation value on the SU (2) spin-coherent states

While in the Landau-Lifshitz hierarchy of commuting flows helices are stationary states of the full nonlinear Hamiltonian dynamics, their fate in generic spin systems is less clear. We nonetheless expect they become stabilized by effective decoupling of the curvature field at large times

Private Comunication

Supplemental Material associated with this manuscript

Hamiltonian Methods in the Theory of Solitons

Acknowledgement. We thank G. Barraquand 

in the classical H δ chain at infinite temperature for different anisotropies δ shown on the log-log scale. The decay of the autocorrelation crosses over from a fast decay to normal diffusive scaling . Spin autocorrelation Cj(t) = S z L/2+j (t)S z L/2 (0) in the anisotropic classical spin chain H δ at infinite temperature, shown for different value of j. The data is consistent with convergence towards anomalous diffusion law S z L/2 (t)S z L/2+j (0) ∼ t −1/2 (log t) −3/2 .