## This content will become publicly available on March 26, 2020

## 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 self-consistent treatment of this divergence gives superdiffusion, with an effective time-dependent diffusion constant that scales as $$D(t) \sim t^{1/3}$$. This exponent had previously been observed in large-scale numerical simulations, but had not been theoretically explained. We briefly discuss XXZ models with easy-axis anisotropy $$\Delta > 1$$. Our method gives closed-form expressions for the diffusion constant $D$ in the infinite-temperature 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) (SC-22); USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22). 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 0031-9007

- 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.
```

```
@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 self-consistent treatment of this divergence gives superdiffusion, with an effective time-dependent diffusion constant that scales as $D(t) \sim t^{1/3}$. This exponent had previously been observed in large-scale numerical simulations, but had not been theoretically explained. We briefly discuss XXZ models with easy-axis anisotropy $\Delta > 1$. Our method gives closed-form expressions for the diffusion constant $D$ in the infinite-temperature 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}

}