Nonlinear Fluctuating Hydrodynamics for the Classical XXZ Spin Chain

  • Avijit DasEmail author
  • Kedar Damle
  • Abhishek Dhar
  • David A. Huse
  • Manas Kulkarni
  • Christian B. Mendl
  • Herbert Spohn


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.


Hydrodynamics Dynamical correlations Heisenberg spin chain 



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.


  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). CrossRefGoogle 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). CrossRefADSGoogle 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). CrossRefADSGoogle 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). CrossRefADSGoogle Scholar
  5. 5.
    Giardinà, C., Livi, R., Politi, A., Vassalli, M.: Finite thermal conductivity in 1d lattices. Phys. Rev. Lett. 84, 2144–2147 (2000). CrossRefADSGoogle 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). CrossRefADSGoogle 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). CrossRefADSMathSciNetzbMATHGoogle 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). CrossRefGoogle Scholar
  12. 12.
    Faddeev, L., Takhtajan, L.: Hamiltonian Methods in the Theory of Solitons. Springer, Berlin (2007). CrossRefzbMATHGoogle 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). CrossRefADSGoogle 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). CrossRefADSGoogle 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). CrossRefGoogle Scholar
  16. 16.
    Mendl, C.B., Spohn, H.: Equilibrium time-correlation functions for one-dimensional hard-point systems. Phys. Rev. E 90, 012147 (2014). CrossRefADSGoogle 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].
  18. 18.
    Mendl, C.B., Spohn, H.: Current fluctuations for anharmonic chains in thermal equilibrium. J. Stat. Mech. 2015, P03007 (2015). CrossRefMathSciNetGoogle Scholar
  19. 19.
    Zhao, H.: Identifying diffusion processes in one-dimensional lattices in thermal equilibrium. Phys. Rev. Lett. 96, 140602 (2006). CrossRefADSGoogle Scholar
  20. 20.
    Kundu, A., Dhar, A.: Equilibrium dynamical correlations in the toda chain and other integrable models. Phys. Rev. E 94, 062130 (2016). CrossRefADSMathSciNetGoogle Scholar
  21. 21.
    Prosen, T., Žunkovič, B.: Macroscopic diffusive transport in a microscopically integrable hamiltonian system. Phys. Rev. Lett. 111, 040602 (2013). CrossRefADSGoogle 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). CrossRefADSGoogle Scholar
  24. 24.
    Das, A., Kulkarni, M., Dhar, A., Huse, D.A.: In preparation, (2019)Google Scholar
  25. 25.
    Prähofer, M., Spohn, H.: Exact scaling functions for one-dimensional stationary Kardar-Parisi-Zhang growth. J. Stat. Phys. 115, 255–279 (2004). CrossRefADSzbMATHGoogle Scholar
  26. 26.
    Frank, J., Huang, W., Leimkuhler, B.: Geometric integrators for classical spin systems. J. Comput. Phys. 133, 160–172 (1997). CrossRefADSMathSciNetzbMATHGoogle 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). CrossRefADSGoogle 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). CrossRefGoogle 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). CrossRefADSGoogle 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). CrossRefADSMathSciNetGoogle Scholar
  32. 32.
    Hu, G.Y., O’Connell, R.F.: Analytical inversion of symmetric tridiagonal matrices. J. Phys. A 29, 1511–1513 (1996). CrossRefADSMathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.International Centre for Theoretical SciencesTata Institute of Fundamental ResearchBengaluruIndia
  2. 2.Tata Institute of Fundamental ResearchMumbaiIndia
  3. 3.Department of PhysicsPrinceton UniversityPrincetonUSA
  4. 4.Institute of Scientific ComputingTechnische Universität DresdenDresdenGermany
  5. 5.Department of Informatics and Institute for Advanced StudyTechnische Universität MünchenGarchingGermany
  6. 6.Zentrum Mathematik and Physik DepartmentTechnische Universität MünchenGarchingGermany

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