Nonlinear Fluctuating Hydrodynamics for the Classical XXZ Spin Chain

Abstract

Using the framework of nonlinear fluctuating hydrodynamics (NFH), we examine equilibrium spatio-temporal correlations in classical ferromagnetic spin chains with nearest neighbor interactions. In particular, we consider the classical XXZ-Heisenberg spin chain (also known as Lattice Landau–Lifshitz or LLL model) evolving deterministically and chaotically via Hamiltonian dynamics, for which energy and z-magnetization are the only locally conserved fields. For the easy-plane case, this system has a low-temperature regime in which the difference between neighboring spin’s angular orientations in the XY plane is an almost conserved field. According to the predictions of NFH, the dynamic correlations in this regime exhibit a heat peak and propagating sound peaks, all with anomalous broadening. We present a detailed molecular dynamics test of these predictions and find a reasonably accurate verification. We find that, in a suitable intermediate temperature regime, the system shows two sound peaks with Kardar-Parisi-Zhang (KPZ) scaling and a heat peak where the expected anomalous broadening is less clear. In high temperature regimes of both easy plane and easy axis case of LLL, our numerics show clear diffusive spin and energy peaks and absence of any sound modes, as one would expect. We also simulate an integrable version of the XXZ-model, for which the ballistic component instead moves with a broad range of speeds rather than being concentrated in narrower peaks around the sound speed.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

References

  1. 1.

    Aoki, K., Kusnezov, D.: Bulk properties of anharmonic chains in strong thermal gradients: non-equilibrium \(\varphi 4\) theory. Phys. Lett. A 265, 250–256 (2000). https://doi.org/10.1016/S0375-9601(99)00899-3

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Sirker, J., Pereira, R.G., Affleck, I.: Conservation laws, integrability, and transport in one-dimensional quantum systems. Phys. Rev. B 83, 035115 (2011). https://doi.org/10.1103/PhysRevB.83.035115

    Article  Google Scholar 

  3. 3.

    Mendl, C.B., Spohn, H.: Dynamic correlators of Fermi-Pasta-Ulam chains and nonlinear fluctuating hydrodynamics. Phys. Rev. Lett. 111, 230601 (2013). https://doi.org/10.1103/PhysRevLett.111.230601

    Article  Google Scholar 

  4. 4.

    Das, S.G., Dhar, A., Saito, K., Mendl, C.B., Spohn, H.: Numerical test of hydrodynamic fluctuation theory in the Fermi-Pasta-Ulam chain. Phys. Rev. E 90, 012124 (2014). https://doi.org/10.1103/PhysRevE.90.012124

    Article  Google Scholar 

  5. 5.

    Giardinà, C., Livi, R., Politi, A., Vassalli, M.: Finite thermal conductivity in 1d lattices. Phys. Rev. Lett. 84, 2144–2147 (2000). https://doi.org/10.1103/PhysRevLett.84.2144

    Article  Google Scholar 

  6. 6.

    Gendelman, O.V., Savin, A.V.: Normal heat conductivity of the one-dimensional lattice with periodic potential of nearest-neighbor interaction. Phys. Rev. Lett. 84, 2381–2384 (2000). https://doi.org/10.1103/PhysRevLett.84.2381

    Article  Google Scholar 

  7. 7.

    Yang, L., Grassberger, P.: Are there really phase transitions in 1-d heat conduction models? (2003) arXiv:cond-mat/0306173

  8. 8.

    Das, S.G., Dhar, A.: Role of conserved quantities in normal heat transport in one dimenison (2014) arXiv:1411.5247

  9. 9.

    Spohn, H.: Fluctuating hydrodynamics for a chain of nonlinearly coupled rotators (2014) arXiv:1411.3907

  10. 10.

    Spohn, H.: Nonlinear fluctuating hydrodynamics for anharmonic chains. J. Stat. Phys. 154, 1191–1227 (2014). https://doi.org/10.1007/s10955-014-0933-y

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Spohn, H.: Fluctuating hydrodynamics approach to equilibrium time correlations for anharmonic chains. In: Lepri, S. (ed.) Thermal Transport in Low Dimensions: From Statistical Physics to Nanoscale Heat Transfer (Lecture Notes in Physics), pp. 107–158. Springer, New York (2016). https://doi.org/10.1007/978-3-319-29261-8

    Google Scholar 

  12. 12.

    Faddeev, L., Takhtajan, L.: Hamiltonian Methods in the Theory of Solitons. Springer, Berlin (2007). https://doi.org/10.1007/978-3-540-69969-9

    Book  MATH  Google Scholar 

  13. 13.

    Kulkarni, M., Lamacraft, A.: Finite-temperature dynamical structure factor of the one-dimensional bose gas: From the Gross-Pitaevskii equation to the Kardar-Parisi-Zhang universality class of dynamical critical phenomena. Phys. Rev. A 88, 021603 (2013). https://doi.org/10.1103/PhysRevA.88.021603

    Article  Google Scholar 

  14. 14.

    Kulkarni, M., Huse, D.A., Spohn, H.: Fluctuating hydrodynamics for a discrete Gross-Pitaevskii equation: Mapping onto the Kardar-Parisi-Zhang universality class. Phys. Rev. A 92, 043612 (2015). https://doi.org/10.1103/PhysRevA.92.043612

    Article  Google Scholar 

  15. 15.

    Mendl, C.B., Spohn, H.: Low temperature dynamics of the one-dimensional discrete nonlinear Schrödinger equation. J. Stat. Mech. 2015, P08028 (2015). https://doi.org/10.1088/1742-5468/2015/08/P08028

    Article  Google Scholar 

  16. 16.

    Mendl, C.B., Spohn, H.: Equilibrium time-correlation functions for one-dimensional hard-point systems. Phys. Rev. E 90, 012147 (2014). https://doi.org/10.1103/PhysRevE.90.012147

    Article  Google Scholar 

  17. 17.

    Zagorodny, J.: Dynamics of vortices in the two-dimensional anisotropic heisenberg model with magnetic fields, Ph.D. thesis, Universität Bayreuth, Bayreuth, Germany, (2004). [Online]. https://epub.uni-bayreuth.de/id/eprint/955

  18. 18.

    Mendl, C.B., Spohn, H.: Current fluctuations for anharmonic chains in thermal equilibrium. J. Stat. Mech. 2015, P03007 (2015). https://doi.org/10.1088/1742-5468/2015/03/P03007

    MathSciNet  Article  Google Scholar 

  19. 19.

    Zhao, H.: Identifying diffusion processes in one-dimensional lattices in thermal equilibrium. Phys. Rev. Lett. 96, 140602 (2006). https://doi.org/10.1103/PhysRevLett.96.140602

    Article  Google Scholar 

  20. 20.

    Kundu, A., Dhar, A.: Equilibrium dynamical correlations in the toda chain and other integrable models. Phys. Rev. E 94, 062130 (2016). https://doi.org/10.1103/PhysRevE.94.062130

    MathSciNet  Article  Google Scholar 

  21. 21.

    Prosen, T., Žunkovič, B.: Macroscopic diffusive transport in a microscopically integrable hamiltonian system. Phys. Rev. Lett. 111, 040602 (2013). https://doi.org/10.1103/PhysRevLett.111.040602

    Article  Google Scholar 

  22. 22.

    Das, A., Kulkarni, M., Spohn, H., Dhar, A.: Kardar-parisi-zhang scaling for the faddeev-takhtajan classical integrable spin chain, (2019) arXiv:1906.02760

  23. 23.

    Das, A., Chakrabarty, S., Dhar, A., Kundu, A., Huse, D.A., Moessner, R., Ray, S.S., Bhattacharjee, S.: Light-cone spreading of perturbations and the butterfly effect in a classical spin chain. Phys. Rev. Lett. 121, 024101 (2018). https://doi.org/10.1103/PhysRevLett.121.024101

    Article  Google Scholar 

  24. 24.

    Das, A., Kulkarni, M., Dhar, A., Huse, D.A.: In preparation, (2019)

  25. 25.

    Prähofer, M., Spohn, H.: Exact scaling functions for one-dimensional stationary Kardar-Parisi-Zhang growth. J. Stat. Phys. 115, 255–279 (2004). https://doi.org/10.1023/B:JOSS.0000019810.21828.fc

    Article  MATH  Google Scholar 

  26. 26.

    Frank, J., Huang, W., Leimkuhler, B.: Geometric integrators for classical spin systems. J. Comput. Phys. 133, 160–172 (1997). https://doi.org/10.1006/jcph.1997.5672

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Zhong, Y., Zhang, Y., Wang, J., Zhao, H.: Normal heat conduction in one-dimensional momentum conserving lattices with asymmetric interactions. Phys. Rev. E 85, 060102 (2012). https://doi.org/10.1103/PhysRevE.85.060102

    Article  Google Scholar 

  28. 28.

    Das, S.G., Dhar, A., Narayan, O.: Heat conduction in the \(\alpha \)-\(\beta \) fermi-pasta-ulam chain. J. Stat. Phys. 154, 204–213 (2014). https://doi.org/10.1007/s10955-013-0871-0

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Wang, L., Hu, B., Li, B.: Validity of fourier’s law in one-dimensional momentum-conserving lattices with asymmetric interparticle interactions. Phys. Rev. E 88, 052112 (2013). https://doi.org/10.1103/PhysRevE.88.052112

    Article  Google Scholar 

  30. 30.

    Dhar, A., Kundu, A., Lebowitz, J.L., Scaramazza, J.A.: Transport properties of the classical toda chain: effect of a pinning potential (2018) arXiv:1812.11770

  31. 31.

    Cintio, P.D., Iubini, S., Lepri, S., Livi, R.: Transport in perturbed classical integrable systems: the pinned toda chain. Chaos Solitons Fractals 117, 249–254 (2018). https://doi.org/10.1016/j.chaos.2018.11.003

    MathSciNet  Article  Google Scholar 

  32. 32.

    Hu, G.Y., O’Connell, R.F.: Analytical inversion of symmetric tridiagonal matrices. J. Phys. A 29, 1511–1513 (1996). https://doi.org/10.1088/0305-4470/29/7/020

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

AD would like to thank the support from the grant EDNHS ANR-14-CE25-0011 of the French National Research Agency (ANR) and from Indo-French Centre for the Promotion of Advanced Research (IFCPAR) under project 5604-2. MK gratefully acknowledges the Ramanujan Fellowship SB/S2/RJN-114/2016 from the Science and Engineering Research Board (SERB), Department of Science and Technology, Government of India. MK also acknowledges support the Early Career Research Award, ECR/2018/002085 from the Science and Engineering Research Board (SERB), Department of Science and Technology, Government of India. MK would like to acknowledge support from the project 6004-1 of the Indo-French Centre for the Promotion of Advanced Research (IFCPAR). KD acknowledges the ICTS-TIFR program “Nonequilibrium Statistical Physics 2015” (NESP2015) which interested him in the questions addressed here, and the generous hospitality of ISSP Tokyo during the completion of this work. DAH was supported in part by (USA) DOE grant DE-SC0016244. The numerical simulations were done on Mowgli, Mario and Tetris clusters of ICTS-TIFR and Gaggle and Pride clusters of DTP-TIFR.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Avijit Das.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We thank Joel Lebowitz for being the very special person that he is: always a pleasure to be with and to collaborate with, and always so helpful to so many people in so many ways.

Communicated by Ivan Corwin.

Appendices

A Coupling coefficients for nonlinear fluctuating hydrodynamics

Given the magic identity (3.20), one can follow [15] to obtain the couplings of the quadratic nonlinearities of NFH. Our simulations are for \(\nu = 0\). Restricting to this case, the averages below, denoted by \(\langle \cdot \rangle \), \(\langle \cdot ; \cdot \rangle \), \(\langle \langle \cdot ; \cdot \rangle \rangle \), refer to fixed \(h,\beta \) at \(\nu = 0\). One obtains

$$\begin{aligned} G^0 = \frac{c_{\mathrm{s}}}{2 \beta }\, \langle \langle e_0 - h\,s_0; e_0 - h\,s_0 \rangle \rangle ^{-1/2} \begin{pmatrix} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \end{pmatrix} \end{aligned}$$
(A.1)

and

$$\begin{aligned} G^{+} = \frac{c_{\mathrm{s}}}{2 \beta }\, \langle \langle e_0 - h\,s_0; e_0 - h\,s_0 \rangle \rangle ^{-1/2} \left( \varUpsilon \begin{pmatrix} -1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 3 \end{pmatrix} + \begin{pmatrix} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 \end{pmatrix} \right) ,\quad G^{-} = -(G^{+})^{\mathcal {T}}. \end{aligned}$$
(A.2)

Here \(c_{\mathrm{s}}\) denotes the speed of sound determined through

$$\begin{aligned} c_{\mathrm{s}} = \frac{1}{\beta }\, (\varGamma \langle \langle r_0; r_0 \rangle \rangle )^{-1/2}\, \langle \langle e_0 - h\,s_0; e_0 - h\,s_0 \rangle \rangle ^{1/2}, \end{aligned}$$
(A.3)

where \(\varGamma = \langle \langle s_0; s_0 \rangle \rangle \langle \langle e_0; e_0 \rangle \rangle - \langle \langle s_0; e_0 \rangle \rangle ^2\) and we have introduced the shorthand notation

$$\begin{aligned} \langle \langle f_0; g_0 \rangle \rangle = \sum _{j=1}^N \langle f_j; g_0 \rangle \end{aligned}$$
(A.4)

with \(\langle f; g \rangle = \langle f g \rangle - \langle f \rangle \langle g \rangle \) denoting the second cumulant. Also \({}^\mathcal {T}\) denotes the transpose relative to the anti-diagonal and

$$\begin{aligned} \varUpsilon = - \langle \langle s_0; e_0 - h\,s_0 \rangle \rangle (2 \varGamma )^{-1/2}. \end{aligned}$$
(A.5)

The thermodynamic averages and cumulants can be obtained as appropriate derivatives of the free energy \(F(h,\nu ,\beta )\) defined in (3.17) with \(\nu \)-derivative at \(\nu = 0\),

$$\begin{aligned}&\langle \langle r_0; r_0 \rangle \rangle = -\beta ^{-1}\, \partial _{\nu }^2 F,\qquad \qquad \langle \langle s_0; s_0 \rangle \rangle = -\beta ^{-1}\, \partial _h^2 F, \nonumber \\&\langle \langle e_0 - h\,s_0; e_0 - h\,s_0 \rangle \rangle = - \partial _{\beta }^2\, (\beta F),\qquad \langle \langle s_0; e_0 - h\,s_0\rangle \rangle = \partial _{\beta } \partial _{h} F\,. \end{aligned}$$
(A.6)

As second order Taylor coefficients, the G-matrices are symmetric.

The free energy has to be numerically evaluated. An efficient method is to use transfer operator techniques [15]. Inserting our simulation parameters yields (rounded to 4 digits):

\(\varDelta = 0.5\), \(\beta = 10\), \(h = 0.3\), speed of sound \(c_{\mathrm{s}} = 0.85217\),

$$\begin{aligned} G^{+} = \begin{pmatrix} 0.01179 &{} 0 &{} -0.01179 \\ 0 &{} 0 &{} 0.4079 \\ -0.01179 &{} 0.4079 &{} -0.03536 \\ \end{pmatrix}, \qquad G^0 = \begin{pmatrix} -0.4079 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0.4079 \\ \end{pmatrix}. \end{aligned}$$
(A.7)

\(\varDelta = 0.5\), \(\beta = 10\), \(h = 0\), speed of sound \(c_{\mathrm{s}} = 0.92837\),

$$\begin{aligned} G^{+} = \begin{pmatrix} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0.4488 \\ 0 &{} 0.4488 &{} 0 \\ \end{pmatrix}, \qquad G^0 = \begin{pmatrix} -0.4488 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0.4488 \\ \end{pmatrix}. \end{aligned}$$
(A.8)

In TABLE I and II these results are compared with data coming from molecular dynamics.

B Low temperature approximation

It is instructive to work out the prefactors of G-matrices (A.1), (A.2) in the harmonic approximation. In principle, also the next order correction could be computed, compare with [15]. We start from the hamiltonian

$$\begin{aligned} H = \sum _{j=-N/2}^{N/2-1} \left[ -\sqrt{1 - s_j^2} \sqrt{1 - s_{j+1}^2} \cos (r_j) - \varDelta s_j s_{j+1} - h s_j \right] \end{aligned}$$
(B.1)

with \(s_j \in [-1,1]\), phase difference \(r_j = \phi _{j+1} - \phi _j\), and external field h.

The minimum of H is assumed for \(r_j = 0\) and \(s_j = \frac{h}{2 (1 - \varDelta )}\) for all j. We expand H in a Taylor series around this minimum and set \(t_j = s_j - \frac{h}{2 (1 - \varDelta )}\). Neglecting boundary terms one obtains

$$\begin{aligned} H = N e_{\mathrm{g}} + H_0 + V + \dots \end{aligned}$$
(B.2)

with the ground state energy \(e_{\mathrm{g}} = -1 - \frac{h^2}{4 (1 - \varDelta )}\), the quadratic contribution

$$\begin{aligned} H_0 = \frac{1}{2} \sum _j \left[ a \, t_j^2 - \tfrac{1}{2} b \, t_j (t_{j-1} + t_{j+1}) + c \, r_j^2 \right] , \end{aligned}$$
(B.3)

and the cubic correction

$$\begin{aligned} V = \frac{h}{4 (1 - \varDelta )}\sum _j \Big [ \frac{1}{c^2} t_j^2 \, ( - t_{j-1} + 2 t_j - t_{j+1}) - r_j^2\, ( t_j + t_{j+1} ) \Big ]. \end{aligned}$$
(B.4)

Here we introduced the shorthands

$$\begin{aligned} a = \frac{2}{c}, \quad b = a - 2 (1 - \varDelta ), \quad c = 1 - \left( \frac{h}{2 (1 - \varDelta )} \right) ^2. \end{aligned}$$
(B.5)

We write \(H_0 = \frac{1}{2} \langle x, A x \rangle \) with \(x = (t_{-N/2}, \dots , t_{N/2-1}, r_{-N/2}, \dots , r_{N/2-1}) \in \mathbb {R}^{2N}\) and a block-diagonal matrix A consisting of two blocks \(A_t\) and \(A_r = c\mathbb {I}_N\), with tridiagonal

$$\begin{aligned} A_t = \begin{pmatrix} a &{} -\frac{b}{2} &{} &{} &{} \\ -\frac{b}{2} &{} a &{} -\frac{b}{2} &{} &{} \\ &{} \ddots &{} \ddots &{} \ddots &{} \\ &{} &{} -\frac{b}{2} &{} a &{} -\frac{b}{2} \\ &{} &{} &{} -\frac{b}{2} &{} a \\ \end{pmatrix}. \end{aligned}$$
(B.6)

The partition function of the quadratic part plus \(N e_{\mathrm{g}}\) is defined as

$$\begin{aligned} Z_0^{(N)}(h, \beta ) = \mathrm {e}^{-\beta N e_{\mathrm{g}}} \int _{([-1,1] \times [-\pi ,\pi ])^N} \mathrm {e}^{-\beta H_0} \prod _j \mathrm {d} s_j \mathrm {d} r_j. \end{aligned}$$
(B.7)

For large \(\beta \), the integration domain of \(s_j\) and \(r_j\) can be approximately extended to \(\mathbb {R}\), which yields

$$\begin{aligned} \int _{\mathbb {R}^{2N}} \mathrm {e}^{-\beta H_0} \mathrm {d}^{2N} x = \int _{\mathbb {R}^{2N}} \mathrm {e}^{-\frac{1}{2} \langle x, \beta A x \rangle } \mathrm {d}^{2N} x = \left( \det \frac{\beta A}{2 \pi }\right) ^{-1/2}. \end{aligned}$$
(B.8)

The determinant of the tridiagonal matrix \(A_t\) is available in closed form [32], such that

$$\begin{aligned} \det A = (\det A_t) (\det A_r) =\left( \frac{b c}{2}\right) ^N \frac{\sinh ((N+1)\lambda )}{\sinh (\lambda )} \quad \text {with } \cosh (\lambda ) = \frac{a}{b}. \end{aligned}$$
(B.9)

Inserting these relations results in the free energy

$$\begin{aligned} F_0^{(N)}(h, \beta )\equiv & {} -\frac{1}{\beta N} \log \big (Z_0^{(N)}(h, \beta )\big ) = e_{\mathrm{g}} + \frac{1}{\beta } \log \Big (\frac{\beta }{2\pi }\sqrt{b c/2}\Big ) \nonumber \\&+ \frac{1}{2 \beta N} \log \Big (\frac{\sinh ((N+1)\lambda )}{\sinh (\lambda )}\Big ). \end{aligned}$$
(B.10)

In the thermodynamic limit \(N \rightarrow \infty \),

$$\begin{aligned} F_0(h, \beta )&= \lim _{N \rightarrow \infty } F_0^{(N)}(h, \beta ) = e_{\mathrm{g}} + \frac{1}{\beta } \log \Big (\frac{\beta }{2\pi }\sqrt{b c/2}\Big ) + \frac{\lambda }{2 \beta } \nonumber \\&= e_{\mathrm{g}} + \frac{1}{\beta } \left[ \frac{1}{2}\mathrm {arccosh}\left( \frac{a}{b}\right) + \log (\beta \sqrt{b c /2}) - \log (2 \pi ) \right] . \end{aligned}$$
(B.11)

With this input we work out the free energy derivatives (A.6) to leading order in \(1/\beta \) and obtain

$$\begin{aligned} -\beta ^{-1}\, \partial _{\nu }^2 F= & {} \frac{1}{\beta c}, \qquad -\beta ^{-1}\, \partial _h^2 F = 0,\qquad - \partial _{\beta }^2\, (\beta F) = 0, \nonumber \\ \partial _{\beta } (\beta \,F_0(h,\beta ))= & {} e_{\mathrm{g}} + \frac{1}{\beta },\qquad \partial _\beta \partial _{\nu } F = 0, \qquad \partial _\beta \partial _{h} F = 0. \end{aligned}$$
(B.12)

Therefore the speed of sound is given by

$$\begin{aligned} c_{\mathrm{s}} = \sqrt{(a - b) c} \end{aligned}$$
(B.13)

For \(h=0\) and \(\varDelta =0.5\), we get \(c_s = 1\).

For the G matrices (at \(h=0\)) we arrive at

$$\begin{aligned} G^{+} = \frac{c_{\mathrm{s}}}{2} \begin{pmatrix} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 \\ \end{pmatrix}, \qquad G^0 = \frac{c_{\mathrm{s}}}{2} \begin{pmatrix} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ \end{pmatrix}. \end{aligned}$$
(B.14)

which for our values of parameter (\(\varDelta =0.5\)) gives,

$$\begin{aligned} G^{+} =\begin{pmatrix} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0.5 \\ 0 &{} 0.5 &{} 0 \\ \end{pmatrix}, \qquad G^0 = \begin{pmatrix} -0.5 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0.5 \\ \end{pmatrix}. \end{aligned}$$
(B.15)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Das, A., Damle, K., Dhar, A. et al. Nonlinear Fluctuating Hydrodynamics for the Classical XXZ Spin Chain. J Stat Phys 180, 238–262 (2020). https://doi.org/10.1007/s10955-019-02397-y

Download citation

Keywords

  • Hydrodynamics
  • Dynamical correlations
  • Heisenberg spin chain