Kinetic Theory of Spin Diffusion and Superdiffusion in $XXZ$ Spin Chains
Abstract
We address the nature of spin transport in the integrable XXZ spin chain, focusing on the isotropic Heisenberg limit. We calculate the diffusion constant using a kinetic picture based on generalized hydrodynamics combined with Gaussian fluctuations: we find that it diverges, and show that a selfconsistent treatment of this divergence gives superdiffusion, with an effective timedependent diffusion constant that scales as $$D(t) \sim t^{1/3}$$. This exponent had previously been observed in largescale numerical simulations, but had not been theoretically explained. We briefly discuss XXZ models with easyaxis anisotropy $$\Delta > 1$$. Our method gives closedform expressions for the diffusion constant $D$ in the infinitetemperature limit for all $$\Delta > 1$$. We find that $D$ saturates at large anisotropy, and diverges as the Heisenberg limit is approached, as $$D \sim (\Delta  1)^{1/2}$$.
 Authors:

 City Univ. (CUNY), NY (United States)
 Univ. of Massachusetts, Amherst, MA (United States)
 Publication Date:
 Research Org.:
 Univ. of Massachusetts, Amherst, MA (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC22); USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC22). Materials Sciences & Engineering Division
 OSTI Identifier:
 1509474
 Grant/Contract Number:
 SC0019168
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Physical Review Letters
 Additional Journal Information:
 Journal Volume: 122; Journal Issue: 12; Journal ID: ISSN 00319007
 Publisher:
 American Physical Society (APS)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY
Citation Formats
Gopalakrishnan, Sarang, and Vasseur, Romain. Kinetic Theory of Spin Diffusion and Superdiffusion in XXZ Spin Chains. United States: N. p., 2019.
Web. doi:10.1103/PhysRevLett.122.127202.
Gopalakrishnan, Sarang, & Vasseur, Romain. Kinetic Theory of Spin Diffusion and Superdiffusion in XXZ Spin Chains. United States. doi:10.1103/PhysRevLett.122.127202.
Gopalakrishnan, Sarang, and Vasseur, Romain. Tue .
"Kinetic Theory of Spin Diffusion and Superdiffusion in XXZ Spin Chains". United States. doi:10.1103/PhysRevLett.122.127202. https://www.osti.gov/servlets/purl/1509474.
@article{osti_1509474,
title = {Kinetic Theory of Spin Diffusion and Superdiffusion in XXZ Spin Chains},
author = {Gopalakrishnan, Sarang and Vasseur, Romain},
abstractNote = {We address the nature of spin transport in the integrable XXZ spin chain, focusing on the isotropic Heisenberg limit. We calculate the diffusion constant using a kinetic picture based on generalized hydrodynamics combined with Gaussian fluctuations: we find that it diverges, and show that a selfconsistent treatment of this divergence gives superdiffusion, with an effective timedependent diffusion constant that scales as $D(t) \sim t^{1/3}$. This exponent had previously been observed in largescale numerical simulations, but had not been theoretically explained. We briefly discuss XXZ models with easyaxis anisotropy $\Delta > 1$. Our method gives closedform expressions for the diffusion constant $D$ in the infinitetemperature limit for all $\Delta > 1$. We find that $D$ saturates at large anisotropy, and diverges as the Heisenberg limit is approached, as $D \sim (\Delta  1)^{1/2}$.},
doi = {10.1103/PhysRevLett.122.127202},
journal = {Physical Review Letters},
number = 12,
volume = 122,
place = {United States},
year = {2019},
month = {3}
}
Web of Science
Works referenced in this record:
Low Temperature Relaxational Dynamics of the Ising Chain in a Transverse Field
journal, March 1997
 Sachdev, Subir; Young, A. P.
 Physical Review Letters, Vol. 78, Issue 11
Thermalization and its mechanism for generic isolated quantum systems
journal, April 2008
 Rigol, Marcos; Dunjko, Vanja; Olshanii, Maxim
 Nature, Vol. 452, Issue 7189
Relaxation and Prethermalization in an Isolated Quantum System
journal, September 2012
 Gring, M.; Kuhnert, M.; Langen, T.
 Science, Vol. 337, Issue 6100
Universal dynamics in an isolated onedimensional Bose gas far from equilibrium
journal, November 2018
 Erne, Sebastian; Bücker, Robert; Gasenzer, Thomas
 Nature, Vol. 563, Issue 7730
Hydrodynamic Diffusion in Integrable Systems
journal, October 2018
 De Nardis, Jacopo; Bernard, Denis; Doyon, Benjamin
 Physical Review Letters, Vol. 121, Issue 16
Superdiffusion in OneDimensional Quantum Lattice Models
journal, December 2018
 Ilievski, Enej; De Nardis, Jacopo; Medenjak, Marko
 Physical Review Letters, Vol. 121, Issue 23
Generalized Gibbs ensemble in integrable lattice models
journal, June 2016
 Vidmar, Lev; Rigol, Marcos
 Journal of Statistical Mechanics: Theory and Experiment, Vol. 2016, Issue 6
Dynamics of the spin $\frac{1}{2}$ Heisenberg chain initialized in a domainwall state
journal, November 2017
 Misguich, Grégoire; Mallick, Kirone; Krapivsky, P. L.
 Physical Review B, Vol. 96, Issue 19
Transport in OutofEquilibrium $XXZ$ Chains: Exact Profiles of Charges and Currents
journal, November 2016
 Bertini, Bruno; Collura, Mario; De Nardis, Jacopo
 Physical Review Letters, Vol. 117, Issue 20
Introduction to ‘Quantum Integrability in Out of Equilibrium Systems’
journal, June 2016
 Calabrese, Pasquale; Essler, Fabian H. L.; Mussardo, Giuseppe
 Journal of Statistical Mechanics: Theory and Experiment, Vol. 2016, Issue 6
Microscopic Origin of Ideal Conductivity in Integrable Quantum Models
journal, July 2017
 Ilievski, Enej; De Nardis, Jacopo
 Physical Review Letters, Vol. 119, Issue 2
Quenching the Anisotropic Heisenberg Chain: Exact Solution and Generalized Gibbs Ensemble Predictions
journal, September 2014
 Wouters, B.; De Nardis, J.; Brockmann, M.
 Physical Review Letters, Vol. 113, Issue 11
Correlations after Quantum Quenches in the $XXZ$ Spin Chain: Failure of the Generalized Gibbs Ensemble
journal, September 2014
 Pozsgay, B.; Mestyán, M.; Werner, M. A.
 Physical Review Letters, Vol. 113, Issue 11
Quench dynamics and relaxation in isolated integrable quantum spin chains
journal, June 2016
 Essler, Fabian H. L.; Fagotti, Maurizio
 Journal of Statistical Mechanics: Theory and Experiment, Vol. 2016, Issue 6
Spin Transport in a OneDimensional Anisotropic Heisenberg Model
journal, May 2011
 Žnidarič, Marko
 Physical Review Letters, Vol. 106, Issue 22
Spin diffusion from an inhomogeneous quench in an integrable system
journal, July 2017
 Ljubotina, Marko; Žnidarič, Marko; Prosen, Tomaž
 Nature Communications, Vol. 8, Issue 1
Relaxation in a Completely Integrable ManyBody Quantum System: An Ab Initio Study of the Dynamics of the Highly Excited States of 1D Lattice HardCore Bosons
journal, February 2007
 Rigol, Marcos; Dunjko, Vanja; Yurovsky, Vladimir
 Physical Review Letters, Vol. 98, Issue 5
BetheBoltzmann hydrodynamics and spin transport in the XXZ chain
journal, January 2018
 Bulchandani, Vir B.; Vasseur, Romain; Karrasch, Christoph
 Physical Review B, Vol. 97, Issue 4
HubbardtoHeisenberg crossover (and efficient computation) of Drude weights at low temperatures
journal, March 2017
 Karrasch, C.
 New Journal of Physics, Vol. 19, Issue 3
Analytic solution of the domainwall nonequilibrium stationary state
journal, February 2018
 Collura, Mario; De Luca, Andrea; Viti, Jacopo
 Physical Review B, Vol. 97, Issue 8
Thermalization near Integrability in a Dipolar Quantum Newton’s Cradle
journal, May 2018
 Tang, Yijun; Kao, Wil; Li, KuanYu
 Physical Review X, Vol. 8, Issue 2
Density dynamics in translationally invariant spin ${\scriptstyle \frac{1}{2}}$ chains at high temperatures: A currentautocorrelation approach to finite time and length scales
journal, November 2009
 Steinigeweg, Robin; Gemmer, Jochen
 Physical Review B, Vol. 80, Issue 18
Anomalous and regular transport in spin $\frac{1}{2}$ chains: ac conductivity
journal, August 2018
 Sánchez, R. J.; Varma, V. K.; Oganesyan, V.
 Physical Review B, Vol. 98, Issue 5
LargeScale Description of Interacting OneDimensional Bose Gases: Generalized Hydrodynamics Supersedes Conventional Hydrodynamics
journal, November 2017
 Doyon, Benjamin; Dubail, Jérôme; Konik, Robert
 Physical Review Letters, Vol. 119, Issue 19
Realtime study of diffusive and ballistic transport in spin ${\scriptstyle \frac{1}{2}}$ chains using the adaptive timedependent density matrix renormalization group method
journal, June 2009
 Langer, S.; HeidrichMeisner, F.; Gemmer, J.
 Physical Review B, Vol. 79, Issue 21
Lower Bounding Diffusion Constant by the Curvature of Drude Weight
journal, August 2017
 Medenjak, Marko; Karrasch, Christoph; Prosen, Tomaž
 Physical Review Letters, Vol. 119, Issue 8
“LightCone” Dynamics After Quantum Quenches in Spin Chains
journal, October 2014
 Bonnes, Lars; Essler, Fabian H. L.; Läuchli, Andreas M.
 Physical Review Letters, Vol. 113, Issue 18
Relaxation after quantum quenches in the spin $\frac{1}{2}$ Heisenberg XXZ chain
journal, March 2014
 Fagotti, Maurizio; Collura, Mario; Essler, Fabian H. L.
 Physical Review B, Vol. 89, Issue 12
Diffusion in Deterministic Interacting Lattice Systems
journal, September 2017
 Medenjak, Marko; Klobas, Katja; Prosen, Tomaž
 Physical Review Letters, Vol. 119, Issue 11
Entanglement and thermodynamics after a quantum quench in integrable systems
journal, July 2017
 Alba, Vincenzo; Calabrese, Pasquale
 Proceedings of the National Academy of Sciences, Vol. 114, Issue 30
Spin dynamics and transport in gapped onedimensional Heisenberg antiferromagnets at nonzero temperatures
journal, April 1998
 Damle, Kedar; Sachdev, Subir
 Physical Review B, Vol. 57, Issue 14
Return probability after a quench from a domain wall initial state in the spin1/2 XXZ chain
journal, October 2017
 Stéphan, JeanMarie
 Journal of Statistical Mechanics: Theory and Experiment, Vol. 2017, Issue 10
Emergent Hydrodynamics in Integrable Quantum Systems Out of Equilibrium
journal, December 2016
 CastroAlvaredo, Olalla A.; Doyon, Benjamin; Yoshimura, Takato
 Physical Review X, Vol. 6, Issue 4
Low Temperature Spin Diffusion in the OneDimensional Quantum $O\left(3\right)$ Nonlinear $\mathit{\sigma}$ Model
journal, February 1997
 Sachdev, Subir; Damle, Kedar
 Physical Review Letters, Vol. 78, Issue 5
Operator growth and eigenstate entanglement in an interacting integrable Floquet system
journal, August 2018
 Gopalakrishnan, Sarang
 Physical Review B, Vol. 98, Issue 6
Time Dependence of Correlation Functions Following a Quantum Quench
journal, April 2006
 Calabrese, Pasquale; Cardy, John
 Physical Review Letters, Vol. 96, Issue 13
Exactly solvable deterministic lattice model of crossover between ballistic and diffusive transport
journal, December 2018
 Klobas, Katja; Medenjak, Marko; Prosen, Tomaž
 Journal of Statistical Mechanics: Theory and Experiment, Vol. 2018, Issue 12
Hydrodynamics of operator spreading and quasiparticle diffusion in interacting integrable systems
journal, December 2018
 Gopalakrishnan, Sarang; Huse, David A.; Khemani, Vedika
 Physical Review B, Vol. 98, Issue 22
The thermodynamic limit of the Whitham equations
journal, May 2003
 El, G. A.
 Physics Letters A, Vol. 311, Issue 45
Soliton Gases and Generalized Hydrodynamics
journal, January 2018
 Doyon, Benjamin; Yoshimura, Takato; Caux, JeanSébastien
 Physical Review Letters, Vol. 120, Issue 4
Conservation laws, integrability, and transport in onedimensional quantum systems
journal, January 2011
 Sirker, J.; Pereira, R. G.; Affleck, I.
 Physical Review B, Vol. 83, Issue 3
Realtime and realspace spin and energy dynamics in onedimensional spin $\frac{1}{2}$ systems induced by local quantum quenches at finite temperatures
journal, February 2014
 Karrasch, C.; Moore, J. E.; HeidrichMeisner, F.
 Physical Review B, Vol. 89, Issue 7
Solvable Hydrodynamics of Quantum Integrable Systems
journal, November 2017
 Bulchandani, Vir B.; Vasseur, Romain; Karrasch, Christoph
 Physical Review Letters, Vol. 119, Issue 22
Open $XXZ$ Spin Chain: Nonequilibrium Steady State and a Strict Bound on Ballistic Transport
journal, May 2011
 Prosen, Tomaž
 Physical Review Letters, Vol. 106, Issue 21
Time Evolution of Local Observables After Quenching to an Integrable Model
journal, June 2013
 Caux, JeanSébastien; Essler, Fabian H. L.
 Physical Review Letters, Vol. 110, Issue 25
Semiclassical theory of front propagation and front equilibration following an inhomogeneous quantum quench
journal, September 2018
 Kormos, Márton; Moca, Cătălin Paşcu; Zaránd, Gergely
 Physical Review E, Vol. 98, Issue 3
Nonequilibrium quantum dynamics and transport: from integrability to manybody localization
journal, June 2016
 Vasseur, Romain; Moore, Joel E.
 Journal of Statistical Mechanics: Theory and Experiment, Vol. 2016, Issue 6
A note on generalized hydrodynamics: inhomogeneous fields and other concepts
journal, January 2017
 Doyon, Benjamin; Yoshimura, Takato
 SciPost Physics, Vol. 2, Issue 2
Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications
journal, November 1990
 Bouchaud, JeanPhilippe; Georges, Antoine
 Physics Reports, Vol. 195, Issue 45
Kinetic Equations and Density Expansions: Exactly Solvable OneDimensional System
journal, March 1967
 Lebowitz, J. L.; Percus, J. K.
 Physical Review, Vol. 155, Issue 1
Microscopic observation of magnon bound states and their dynamics
journal, September 2013
 Fukuhara, Takeshi; Schauß, Peter; Endres, Manuel
 Nature, Vol. 502, Issue 7469
Complete Generalized Gibbs Ensembles in an Interacting Theory
journal, October 2015
 Ilievski, E.; De Nardis, J.; Wouters, B.
 Physical Review Letters, Vol. 115, Issue 15
Drude Weight for the LiebLiniger Bose Gas
journal, January 2017
 Doyon, Benjamin; Spohn, Herbert
 SciPost Physics, Vol. 3, Issue 6
Quasilocal charges in integrable lattice systems
journal, June 2016
 Ilievski, Enej; Medenjak, Marko; Prosen, Tomaž
 Journal of Statistical Mechanics: Theory and Experiment, Vol. 2016, Issue 6
Kinetic Equation for a Dense Soliton Gas
journal, November 2005
 El, G. A.; Kamchatnov, A. M.
 Physical Review Letters, Vol. 95, Issue 20
Universal Relaxational Dynamics of Gapped OneDimensional Models in the Quantum SineGordon Universality Class
journal, October 2005
 Damle, Kedar; Sachdev, Subir
 Physical Review Letters, Vol. 95, Issue 18
Spin Transport in the $XXZ$ Chain at Finite Temperature and Momentum
journal, December 2011
 Steinigeweg, Robin; Brenig, Wolfram
 Physical Review Letters, Vol. 107, Issue 25
EnergyDependent ThreeBody Loss in 1D Bose Gases
journal, January 2019
 Zundel, Laura A.; Wilson, Joshua M.; Malvania, Neel
 Physical Review Letters, Vol. 122, Issue 1
Ballistic transport in the onedimensional Hubbard model: The hydrodynamic approach
journal, August 2017
 Ilievski, Enej; De Nardis, Jacopo
 Physical Review B, Vol. 96, Issue 8
Dynamics of hard rods with initial domain wall state
journal, July 2017
 Doyon, Benjamin; Spohn, Herbert
 Journal of Statistical Mechanics: Theory and Experiment, Vol. 2017, Issue 7
A quantum Newton's cradle
journal, April 2006
 Kinoshita, Toshiya; Wenger, Trevor; Weiss, David S.
 Nature, Vol. 440, Issue 7086
Diffusion and Ballistic Transport in OneDimensional Quantum Systems
journal, November 2009
 Sirker, J.; Pereira, R. G.; Affleck, I.
 Physical Review Letters, Vol. 103, Issue 21
Transport in outofequilibrium XXZ chains: Nonballistic behavior and correlation functions
journal, September 2017
 Piroli, Lorenzo; De Nardis, Jacopo; Collura, Mario
 Physical Review B, Vol. 96, Issue 11