Quantum diffusion in spin chains with phase space methods

Jonathan Wurtz and Anatoli Polkovnikov

Phys. Rev. E 101, 052120 – Published 15 May 2020

Abstract

Connecting short-time microscopic dynamics with long-time hydrodynamics in strongly correlated quantum systems is one of the outstanding questions. In particular, it is hard to determine various hydrodynamic coefficients such as the diffusion constant or viscosity starting from a microscopic model: exact quantum simulations are limited to either small system sizes or to short times, which are insufficient to reach asymptotic behavior and so various approximations must be applied. We show that these difficulties, at least for particular models, can be circumvented by using the cluster truncated Wigner approximation (CTWA), which maps quantum Hamiltonian dynamics into classical Hamiltonian dynamics in auxiliary high-dimensional phase space. We apply CTWA to XXZ next-nearest-neighbor spin-1/2 chains and XY spin ladders, and find behavior consisting of short-time spin relaxation which gradually crosses over to emergent diffusive behavior at long times. For a random initial state, we show that CTWA correctly reproduces the whole spin spectral function. Necessary in this construction is sampling from properly fluctuating initial conditions: the Dirac mean-field (variational) ansatz, which neglects such fluctuations, leads to incorrect predictions.

Received 4 September 2018
Revised 31 March 2020
Accepted 17 April 2020

DOI:https://doi.org/10.1103/PhysRevE.101.052120

Physics Subject Headings (PhySH)

Diffusion Noise Nonequilibrium statistical mechanics Quantum quench Quantum statistical mechanics Quantum thermodynamics Quantum transport

Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied PhysicsStatistical Physics & Thermodynamics

Authors & Affiliations

Jonathan Wurtz^* and Anatoli Polkovnikov

Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA

^*Corresponding author: jwurtz@bu.edu

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Issue

Vol. 101, Iss. 5 — May 2020

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Images

Figure 1
Nonequal time spin-spin correlation functions of the next-nearest-neighbor XXZ chain of Eq. (3) at infinite temperature. (a) Correlation function for $t^{'} = 10$ , compared to exact results; $(L, N) = (8, 16)$ . (b) Correlation function for $(L, N) = (8, 64)$ , which shows that the correlation function is well captured for offsets $t^{'}$ . The mean field (dashed line) does not capture correctly, emphasizing the importance of fluctuations. (c) Time traces of individual points in phase space for a typical (dashed line) mean-field and (solid line) Gaussian initial condition, and 64 sites. Fluctuations persist at all times for the Gaussian case, but are exponentially small in the cluster size for the mean-field case.
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Figure 2
Diffusive dynamics for the next-nearest-neighbor XXZ chain of Eq. (3) at infinite temperature. (a) The conformal width of the correlation function defined by Eq. (5). The black dashed line is a single-parameter fit for the classical diffusion of Eq. (6). The gray box and inset show a comparison of the exact results for $N = 16$ (solid black line) with CTWA (red solid line) and mean-field (dashed lines) simulations for a larger system $N = 64$ . (b) Scaled values of $C_{X}$ for size-8 clusters, averaged over offsets, which collapses to the form of a Gaussian. Colors are for times $t \in (5, 10, 20)$ . (c) A fit of the diffusion constant as a function of cluster size for $N \approx 64$ .
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Figure 3
Dynamic structure factors of the next-nearest-neighbor XXZ chain of Eq. (3) at infinite temperature. (a) Log-log version of Fig. 1 showing diffusive decay. (b) Momentum-averaged structure factor $S (ω)$ , which is the Fourier transform of (a). (c1)–(c6) Dynamic structure factor $S (k, ω)$ for the first few $k$ . The system size is $N = 64$ ; the dashed black lines are for classical diffusion for $D = 3.75$ .
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Figure 4
Diffusion constant for the XXZ model as derived from $S (k, ω)$ . (a): Diffusion constant as a function of anisotropy $Δ$ and cluster size. The inset is log-log of the same, showing $1 / Δ$ , which is consistent with previous results [30]. Clearly, for $Δ < 1$ , the diffusion constant does not converge, as expected for ballistic behavior (b), while for $Δ = 2$ , it converges to about $1.5 \times$ its “classical” value (c).
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Figure 5
Diffusion constant for the XXZ model as derived from the boundary wall initial condition. (a): Diffusion constant as a function of $Δ$ . For $Δ < 1$ , the diffusion constant diverges, consistent with being nondiffusive. The inset is log-log of the same data; the dashed line is of $Δ^{- 1.5}$ . (b1), (b2): Spin profile for $Δ = 0$ , showing ballistic growth improving with cluster size. Lines are for times [2.5,5,7.5,10]. (b3), (b4): Spin profile for $Δ = 2$ , renormalized by $\sqrt{t}$ showing diffusive collapse. The dashed line is the classical profile for $D = 0.52$ .
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Figure 6
Behavior of symmetric correlator for an XY ladder. Top to bottom are widths of 2, 3, and 4. Left: Gaussian profile at $t = 10$ for different widths. Right: Self-correlator demonstrating diffusive behavior (black dashed line for $D = 2.75$ .). Similarly to the next-nearest-neighbor model, there is short-time quantum behavior plus long-time diffusive behavior.
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