Hydrodynamics in long-range interacting systems with center-of-mass conservation

Letter

Hydrodynamics in long-range interacting systems with center-of-mass conservation

Alan Morningstar, Nicholas O'Dea, and Jonas Richter

Phys. Rev. B 108, L020304 – Published 24 July 2023

Abstract

In systems with a conserved density, the additional conservation of the center of mass (dipole moment) has been shown to slow down the associated hydrodynamics. At the same time, long-range interactions generally lead to faster transport and information propagation. Here, we explore the competition of these two effects and develop a hydrodynamic theory for long-range center-of-mass-conserving systems. We demonstrate that these systems can exhibit a rich dynamical phase diagram containing subdiffusive, diffusive, and superdiffusive behaviors, with continuously varying dynamical exponents. We corroborate our theory by studying quantum lattice models whose emergent hydrodynamics exhibit these phenomena.

Received 28 April 2023
Accepted 12 July 2023

DOI:https://doi.org/10.1103/PhysRevB.108.L020304

Physics Subject Headings (PhySH)

Anomalous diffusion Quantum transport

Quantum chaotic systems Quantum many-body systems

Hydrodynamics Kinetically constrained models

Statistical Physics & ThermodynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Alan Morningstar ¹, Nicholas O'Dea ¹, and Jonas Richter ^1,2

¹Department of Physics, Stanford University, Stanford, California 94305, USA
²Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany

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Issue

Vol. 108, Iss. 2 — 1 July 2023

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Images

Figure 1
(a) We consider models involving long-range interactions and c.m. conservation. The dynamics are governed by pairs of equal and opposite currents (or the underlying “pair-hopping” processes depicted) acting on lattice sites separated by the distances $s \geq 0$ and $r \geq 1$ (the “interaction” and “hopping” distances). The strengths of the currents are suppressed by power laws ${(s + 1)}^{- α} r^{- β}$ , so $α$ controls the range of interactions and $β$ the range of hopping. (b), (c) Schematic cuts through the dynamical phase diagram for the limiting cases $r \to 1 (β \to \infty)$ and $s \to 0 (α \to \infty)$ .
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Figure 2
The dynamical exponent $z$ . (a) Contour plot of ${log}_{2} (γ_{2 k} / γ_{k}) ≅ \hat{z}$ on the $(α, β)$ plane, numerically evaluated using $k = 2 π / 3200$ . The dashed box marks $z = 4$ subdiffusion. The dotted line along $β = 2 + α$ marks where the behavior of $z$ changes from $z = β - 1$ to $z = α + 1$ . (b) Demonstration of the boundary behavior $\hat{z} (k) = 4 - ln {(1 / k)}^{- 1}$ where the horizontal axis is $L = 2 π / k$ , the lattice sizes used in the estimates. The vertical shift between the two curves is from different subleading additive contributions. (c), (d) Cuts along the top and right edges of the contour plot. Colors (light to dark) represent estimates using $k \geq 2 π / 3200$ , i.e., lattices of size $L \leq 3200$ .
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Figure 3
Scaled density profiles obtained from time evolving $n (x, 0) = δ_{x, 0}$ under Eq. (2) (red), and comparison with $F_{z} (ξ)$ . The system size is $L = 3200$ and the time is $t \sim γ_{2 π / L}^{- 1}$ for each panel. Position $x$ extends from $- \frac{L}{2}$ to $\frac{L}{2}$ but we show $x \geq 0$ . We set $α = \infty$ , and $β$ varies over (a)–(f). The annotated values of $z$ are used for the scaling of the axes. The slowly vanishing logarithmic correction to $z = 4 [\hat{z} (k) = 4 - ln {(1 / k)}^{- 1}]$ that is applicable to (d) is small enough to ignore. Data at other late times collapse well onto the same curves, but we show curves from one time for clarity. Data are in red, and $D^{- 1 / z} F_{z}$ is plotted in faint blue for comparison, where $D$ is extracted from $n (0, t) = D t^{1 / z}$ , but these are directly behind the data curves due to excellent agreement.
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Figure 4
Infinite-temperature autocorrelation function for a long-range c.m.-conserving spin-1 chain as in Eq. (1) with $J_{s r} \propto s^{- α / 2} r^{- β / 2}$ and system size $L = 16$ . (a) Varying $α$ for $β \to \infty$ . (b) Varying $β$ for $α \to \infty$ . The dashed curves indicate power laws $\propto t^{- 1 / z}$ with $z$ set according to the prediction of our hydrodynamic theory. The thin gray curves are data for smaller $L = 14$ .
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