We investigate chains of interacting spinless fermions subject to a finite external field $F$ (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 $q$ shows ${q}^{4}$ 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 $F$. On the other hand, our study of the Stark model indicates that upon increasing $q$ 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 ${q}^{*}$ which vanishes for $F\ensuremath{\rightarrow}0$.
We investigate chains of interacting spinless fermions subject to a finite external field F (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 \ensuremathΓq of the density modulation with wave vector q shows 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 F. On the other hand, our study of the Stark model indicates that upon increasing q there is a crossover from subdiffusive behavior, \ensuremathΓq\ensuremath∝q4, to the normal diffusive relaxation, \ensuremathΓq\ensuremath∝q2, at the wave vector q∗ which vanishes for F\ensuremath→0.
We study transport in an interacting tilted (Stark) chain. We show that the crossover between diffusive and subdiffusive relaxation is governed by FL, 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.
The t-model represents the Hubbard model in the limit U→∞ and is one of the basic models of strongly correlated electrons. On a one-dimensional chain, the model is integrable, and the charge dynamics corresponds to that of free spinless fermions. However, the sequence of spins is frozen, leading to the Hilbert space fragmentation and nontrivial spin dynamics. We consider integrable and perturbed models with perturbations that break integrability while preserving fragmentation, and show that they exhibit various types of spin dynamics, from ballistic transport to anomalous diffusion in the integrable case, and from diffusion to subdiffusion in the perturbed case. Due to fragmentation, in all cases considered, spin transport is mediated by charge transport, with a particular magnetization dependence, most notably leading to subdiffusion in the grandcanonical average of the perturbed model, with a mechanism distinct from subdiffusion in disordered or dipole-conserving models.
We investigate charge relaxation in the spin-less disordered fermionic Hubbard chain (t−V-model). Our observable is the time-dependent density propagator, Πε(x,t), calculated in windows of different energy density, ε, of the many-body Hamiltonian and at different disorder strengths, W, not exceeding the critical value Wc. The width Δxε(t) of Πε(x,t) exhibits a behavior dlnΔxε(t)/dlnt=βε(t), where the exponent function βε(t)≲1/2 is seen to depend strongly on L at all investigated parameter combinations. (i) We cannot confirm the existence of a region in phase space that exhibits (genuine) subdiffusive dynamics in the sense that βε<1/2 is numerically fixed in the limit of large L. Instead, subdiffusion might possibly be transient, only, finally giving way to conventional diffusive behavior with βε=1/2. (ii) Similarly, we cannot confirm the existence of many-body mobility edges deep in the delocalized phase. (iii) (Transient) subdiffusion 0<βε(t)≲1/2, coexists with an enhanced probability for returning to the origin, Πε(0,t), decaying much slower than 1/Δxε(t). Correspondingly, the spatial decay of Πε(x,t) is far from Gaussian being exponential or even slower. On a phenomenological level, our findings are broadly consistent with effects of strong disorder and (fractal) Griffiths regions.
We give a detailed derivation of the second-order (local) hydrodynamics for Boltzmann equation with an external force by using the renormalization group method. In this method, we solve the Boltzmann equation faithfully to extract the hydrodynamics without recourse to any ansatz. Our method leads to microscopic expressions of not only all the transport coefficients that are of the same form as those in Chapman-Enskog method but also those of the viscous relaxation times τi that admit physically natural interpretations. As an example, we apply our microscopic expressions to calculate the transport coefficients and the relaxation times of the cold fermionic atoms in a quantitative way, where the transition probability in the collision term is given explicitly in terms of the s-wave scattering length as. We thereby discuss the quantum statistical effects, temperature dependence, and scattering-length dependence of the first-order transport coefficients and the viscous relaxation times: It is shown that as the temperature is lowered, the transport coefficients and the relaxation times increase rapidly because Pauli principle acts effectively. On the other hand, as as is increased, these quantities decrease and become vanishingly small at unitarity because of the strong coupling. The numerical calculation shows that the relation τπ=η/P, which is derived in the relaxation-time approximation and used in most of literature without almost any foundation, turns out to be satisfied quite well, while the similar relation for the relaxation time τJ of the heat conductivity is satisfied only approximately with a considerable error.
We investigate charge relaxation in quantum wires of spinless disordered fermions ($t\text{\ensuremath{-}}V$ model). Our observable is the time-dependent density propagator ${\mathrm{\ensuremath{\Pi}}}_{ϵ}(x,t)$, calculated in windows of different energy density ϵ of the many-body Hamiltonian and at different disorder strengths W, not exceeding the critical value Wc. The width ${\mathrm{\ensuremath{\Delta}}x}_{ϵ}(t)$ of ${\mathrm{\ensuremath{\Pi}}}_{ϵ}(x,t)$ exhibits a behavior $d\mathrm{ln}{\mathrm{\ensuremath{\Delta}}x}_{ϵ}(t)/d\mathrm{ln}t={\ensuremath{\beta}}_{ϵ}(t)$, where the exponent function ${\ensuremath{\beta}}_{ϵ}(t)\ensuremath{\lesssim}1/2$ is seen to depend strongly on L at all investigated parameter combinations. (i) We confirm the existence of a region in phase space that exhibits subdiffusive dynamics in the sense that ${\ensuremath{\beta}}_{ϵ}(t)<1/2$ in a large window of times. However, subdiffusion might possibly be transient, only, finally giving way to a conventional diffusive behavior with ${\ensuremath{\beta}}_{ϵ}=1/2$. (ii) We cannot confirm the existence of many-body mobility edges even in regions of the phase diagram that have been reported to be deep in the delocalized phase. (iii) (Transient) subdiffusion $0<{\ensuremath{\beta}}_{ϵ}(t)\ensuremath{\lesssim}1/2$ coexists with an enhanced probability for returning to the origin ${\mathrm{\ensuremath{\Pi}}}_{ϵ}(0,t)$, decaying much slower than $1/{\mathrm{\ensuremath{\Delta}}x}_{ϵ}(t)$. Correspondingly, the spatial decay of ${\mathrm{\ensuremath{\Pi}}}_{ϵ}(x,t)$ is far from Gaussian, being exponential or even slower. On a phenomenological level, our findings are broadly consistent with the effects of strong disorder and (fractal) Griffiths regions.
The t model represents the Hubbard model in the limit U →∞and is one of the basic models of stronglycorrelated electrons. On a one-dimensional chain, the model is integrable, and its charge dynamics is equivalentto that of free spinless fermions. However, within its dynamics the sequence of spins is frozen, implyingHilbert-space fragmentation and nontrivial spin dynamics. We focus primarily on perturbed versions of themodel in which the perturbations break integrability while preserving fragmentation. Considered examplesof such perturbations are spin-dependent hopping and additional spin interaction. We show that these modelsexhibit a variety of spin-transport regimes. In contrast to the integrable model, which displays either ballistictransport or dissipationless diffusion, the perturbed systems exhibit either normal diffusion or subdiffusion. Dueto Hilbert-space fragmentation, spin transport is mediated by charge transport in all cases considered. This leadsto a characteristic dependence on the magnetization, most notably resulting in subdiffusive spin transport in thegrand-canonical ensemble of the perturbed model. The underlying mechanism is distinct from the subdiffusionobserved in disordered systems or in models with dipole conservation.