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arXiv:2310.01862 · Physical review. B./Physical review. B

Emergent dipole moment conservation and subdiffusion in tilted chains

Sourav Nandy, J. Herbrych, Zala Lenarčič, Aleksander Głódkowski, Peter Prelovšek, Marcin Mierzejewski

Published Mar 12, 2024• 12 citations
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Abstract

We study transport in an interacting tilted (Stark) chain. We show that the crossover between diffusive and subdiffusive relaxation is governed by $F\sqrt{L}$, where $F$ is the strength of the field, and $L$ is the wavelength of the excitation. While the subdiffusive behavior persists for large fields, the corresponding transport coefficient is exponentially suppressed with $F$ so that the finite-time dynamics appears almost frozen. We (i) explain the crossover scale between the diffusive and subdiffusive transport by bounding the dynamics of the dipole moment for arbitrary initial state, (ii) prove its emergent conservation at infinite temperature for $F\ensuremath{\gg}1/L$, and (iii) argue that the numerical data for the tilted chain are consistent with the hydrodynamics of fractons.

DOI
10.1103/physrevb.109.115120

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We study transport in an interacting tilted (Stark) chain. We show that the crossover between diffusive and subdiffusive relaxation is governed by FLF\sqrt{L}FL​, where FFF is the strength of the field, and LLL is the wavelength of the excitation. While the subdiffusive behavior persists for large fields, the corresponding transport coefficient is exponentially suppressed with FFF so that the finite-time dynamics appears almost frozen. We (i) explain the crossover scale between the diffusive and subdiffusive transport by bounding the dynamics of the dipole moment for arbitrary initial state, (ii) prove its emergent conservation at infinite temperature for F\ensuremath≫1/LF\ensuremath{\gg}1/LF\ensuremath≫1/L, and (iii) argue that the numerical data for the tilted chain are consistent with the hydrodynamics of fractons.
We investigate chains of interacting spinless fermions subject to a finite external field FFF (also called Stark chains) and focus on the regime where the charge thermalization follows the subdiffusive hydrodynamics. First, we study reduced models conserving the dipole moment and derive an explicit Einstein relation which links the subdiffusive transport coefficient with the correlations of the dipolar current. This relation explains why the decay rate ${\mathrm{\ensuremath{\Gamma}}}_{q}$ of the density modulation with wave vector qqq shows q4{q}^{4}q4 dependence. In the case of the Stark model, a similar Einstein relation is also derived and tested using various numerical methods. They confirm an exponential reduction of the transport coefficient with increasing FFF. On the other hand, our study of the Stark model indicates that upon increasing qqq there is a crossover from subdiffusive behavior, ${\mathrm{\ensuremath{\Gamma}}}_{q}\ensuremath{\propto}{q}^{4}$, to the normal diffusive relaxation, ${\mathrm{\ensuremath{\Gamma}}}_{q}\ensuremath{\propto}{q}^{2}$, at the wave vector q∗{q}^{*}q∗ which vanishes for $F\ensuremath{\rightarrow}0$.
The presence of global conserved quantities in interacting systems generically leads to diffusive transport at late times. Here, we show that systems conserving the dipole moment of an associated global charge, or even higher-moment generalizations thereof, escape this scenario, displaying subdiffusive decay instead. Modeling the time evolution as cellular automata for specific cases of dipole- and quadrupole conservation, we numerically find distinct anomalous exponents of the late time relaxation. We explain these findings by analytically constructing a general hydrodynamic model that results in a series of exponents depending on the number of conserved moments, yielding an accurate description of the scaling form of charge correlation functions. We analyze the spatial profile of the correlations and discuss potential experimentally relevant signatures of higher-moment conservation.
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.
We investigate transport in several translationally invariant spin-12\frac{1}{2}21​ chains in the limit of high temperatures. We concretely consider spin transport in the anisotropic Heisenberg chain, the pure Heisenberg chain within an alternating field, and energy transport in an Ising chain which is exposed to a tilted field. Our approach is essentially based on a connection between the evolution of the variance of an inhomogeneous nonequilibrium density and the current-autocorrelation function at finite times. Although this relationship is not restricted to the case of diffusive transport, it allows to extract a quantitative value for the diffusion constant in that case. By means of numerically exact diagonalization we indeed observe diffusive behavior in the considered spin chains for a range of model parameters and confirm the diffusion coefficients which were obtained for these systems from nonequilibrium bath scenarios.
We study the transport and equilibration properties of a classical Heisenberg chain, whose couplings are random variables drawn from a one-parameter family of power-law distributions. The absence of a scale in the couplings makes the system deviate substantially from the usual paradigm of diffusive spin hydrodynamics and exhibit a regime of subdiffusive transport with an exponent changing continuously with the parameter of the distribution. We propose a solvable phenomenological model that correctly yields the subdiffusive exponent, thereby linking local fluctuations in the coupling strengths to the long-time, large-distance behavior. It also yields the finite-time corrections to the asymptotic scaling, which can be important in fitting the numerical data. We show how such exponents undergo transitions as the distribution of the coupling gets wider, marking the passage from diffusion to a regime of slow diffusion, and finally to subdiffusion.
Hilbert space fragmentation is an ergodicity-breaking phenomenon, in which the Hamiltonian shatters into exponentially many dynamically disconnected sectors. In many fragmented systems, these sectors can be labeled by statistically localized integrals of motion, which are nonlocal operators. We study the paradigmatic nearest-neighbor pair hopping model exhibiting the so-called strong fragmentation. We show that this model hosts local integrals of motion (LIOMs), which correspond to frozen density modes with long wavelengths. The latter modes become subdiffusive when longer-range pair hoppings are allowed. Finally, we make a connection with a tilted (Stark) chain. Contrary to the dipole-conserving effective models, the tilted chain is shown to support either a Hamiltonian or dipole moment as an LIOM. Numerical results are obtained from a numerical algorithm, in which finding LIOMs is reduced to a data compression problem.