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arXiv:2107.02454 · Physical review. B./Physical review. B

Coexistence of diffusive and ballistic transport in integrable quantum lattice models

Peter Prelovšek, Marcin Mierzejewski, J. Herbrych

Published Sep 28, 2021• 14 citations
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Abstract

We investigate the high-temperature dynamical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ in two one-dimensional integrable quantum lattice models: the anisotropic XXZ spin chain and the Hubbard chain. The emphasis is on the metallic regime of both models, where besides the ballistic component, the regular part of conductivity might reveal a diffusivelike transport. To resolve the low-frequency dynamics, we upgrade the microcanonical Lanczos method enabling studies of finite-size systems with up to $L\ensuremath{\le}32$ sites for the XXZ spin model with the frequency resolution $\ensuremath{\delta}\ensuremath{\omega}\ensuremath{\sim}{10}^{\ensuremath{-}3}J$. Results for the XXZ chain reveal a fine structure of $\ensuremath{\sigma}(\ensuremath{\omega})$ spectra, which originates from the discontinuous variation of the stiffness, previously found at commensurate values of the anisotropy parameter $\mathrm{\ensuremath{\Delta}}$. Still, we do not find clear evidence for a diffusive component, at least not for commensurate values of $\mathrm{\ensuremath{\Delta}}$, particularly for $\mathrm{\ensuremath{\Delta}}=0.5$, as well as for $\mathrm{\ensuremath{\Delta}}\ensuremath{\rightarrow}0$. Similar is the conclusion for the Hubbard model away from half-filling, where the spectra reveal more universal behavior.

DOI
10.1103/physrevb.104.115163

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We investigate the high-temperature dynamical conductivity \ensuremathσ(\ensuremathω)\ensuremath{\sigma}(\ensuremath{\omega})\ensuremathσ(\ensuremathω) in two one-dimensional integrable quantum lattice models: the anisotropic XXZ spin chain and the Hubbard chain. The emphasis is on the metallic regime of both models, where besides the ballistic component, the regular part of conductivity might reveal a diffusivelike transport. To resolve the low-frequency dynamics, we upgrade the microcanonical Lanczos method enabling studies of finite-size systems with up to L\ensuremath≤32L\ensuremath{\le}32L\ensuremath≤32 sites for the XXZ spin model with the frequency resolution \ensuremathδ\ensuremathω\ensuremath∼10\ensuremath−3J\ensuremath{\delta}\ensuremath{\omega}\ensuremath{\sim}{10}^{\ensuremath{-}3}J\ensuremathδ\ensuremathω\ensuremath∼10\ensuremath−3J. Results for the XXZ chain reveal a fine structure of \ensuremathσ(\ensuremathω)\ensuremath{\sigma}(\ensuremath{\omega})\ensuremathσ(\ensuremathω) spectra, which originates from the discontinuous variation of the stiffness, previously found at commensurate values of the anisotropy parameter \ensuremathΔ\mathrm{\ensuremath{\Delta}}\ensuremathΔ. Still, we do not find clear evidence for a diffusive component, at least not for commensurate values of \ensuremathΔ\mathrm{\ensuremath{\Delta}}\ensuremathΔ, particularly for \ensuremathΔ=0.5\mathrm{\ensuremath{\Delta}}=0.5\ensuremathΔ=0.5, as well as for \ensuremathΔ\ensuremath→0\mathrm{\ensuremath{\Delta}}\ensuremath{\rightarrow}0\ensuremathΔ\ensuremath→0. Similar is the conclusion for the Hubbard model away from half-filling, where the spectra reveal more universal behavior.
We investigate the high-temperature dynamical conductivity σ(ω)\sigma(\omega)σ(ω) in two one-dimensional integrable quantum lattice models: the anisotropic XXZ spin chain and the Hubbard chain. The emphasis is on the metallic regime of both models, where besides the ballistic component, the regular part of conductivity might reveal a diffusive-like transport. To resolve the low-frequency dynamics, we upgrade the microcanonical Lanczos method enabling studies of finite-size systems with up to L≤32L\leq 32L≤32 sites for XXZ spin model with the frequency resolution δω∼10−3J\delta \omega \sim 10^{-3} Jδω∼10−3J. Results for the XXZ chain reveal a fine structure of σ(ω)\sigma(\omega)σ(ω) spectra, which originates from the discontinuous variation of the stiffness, previously found at commensurate values of the anisotropy parameter Δ\DeltaΔ. Still, we do not find a clear evidence for a diffusive component, at least not for commensurate values of Δ\DeltaΔ, particularly for Δ=0.5\Delta =0.5Δ=0.5, as well as for Δ→0\Delta \to 0Δ→0. Similar is the conclusion for the Hubbard model away from half-filling, where the spectra reveal more universal behavior.
We analyze the high-temperature conductivity in one-dimensional integrable models of interacting fermions: the $t\text{\ensuremath{-}}V$ model (anisotropic Heisenberg spin chain) and the Hubbard model, at half-filling in the regime corresponding to insulating ground state. A microcanonical Lanczos method study for finite size systems reveals anomalously large finite-size effects at low frequencies while a frequency-moment analysis indicates a finite dc conductivity. This phenomenon also appears in a prototype integrable quantum system of impenetrable particles, representing a strong-coupling limit of both models. In the thermodynamic limit, the two results could converge to a finite dc conductivity rather than an ideal conductor or insulator scenario.
Finite-temperature T > 0 transport properties of integrable and nonintegrable one-dimensional (1D) many-particle quantum systems are rather different, showing ballistic and diffusive behavior, respectively. The repulsive 1D Hubbard model is a prominent example of an integrable correlated system. For electronic densities n ≠ 1 (and spin densities m ≠ 0) it is an ideal charge (and spin) conductor, with ballistic charge (and spin) transport for T ⩾ 0. In spite of the fact that it is solvable by the Bethe ansatz, at n = 1 (and m = 0) its T > 0 charge (and spin) transport properties are an issue that remains poorly understood. Here we combine this solution with symmetry and the explicit calculation of current-operator matrix elements between energy eigenstates to show that for on-site repulsion U > 0 and at n = 1 the charge stiffness D η ( T ) vanishes for T > 0 in the thermodynamic limit. A similar behavior is found by such methods for the spin stiffness D s ( T ) for U > 0 and T > 0, which vanishes at m = 0. This absence of finite temperature n = 1 ballistic charge transport and m = 0 ballistic spin transport are exact results that clarify long-standing open problems.
We study ballistic transport in integrable quantum lattice models, i.e., in the spin XXZ and Hubbard chains, close to the noninteracting limit. It is by now well established that the stiffnesses of spin and charge currents reveal, at high temperatures, a discontinuous reduction (jump) when the interaction is introduced. We show that the jumps are related to the large degeneracy of the parent noninteracting models and, more generally, can appear in other integrable models with macroscopic degeneracies. These degeneracies are properly captured by the degenerate perturbation calculations which may be performed for large systems. We find that the discontinuities and the quasilocality of the conserved current in this limit can be traced back to the nonlocal character of an effective interaction. From the latter observation we identify a class of observables which show discontinuities.
Finite-temperature T>0 transport properties of integrable and nonintegrable one-dimensional (1D) many-particle quantum systems are rather different, showing in the metallic phases ballistic and diffusive behavior, respectively. The repulsive 1D Hubbard model is an integrable system of wide physical interest. For electronic densities n≠1n\neq1n=1 it is an ideal conductor, with ballistic charge transport for T larger or equal to 0. In spite that it is solvable by the Bethe ansatz, at n=1n=1n=1 its T>0 transport properties are a collective-behavior issue that remains poorly understood. Here we combine that solution with symmetry to show that for on-site repulsion U>0 the charge stiffness D (T) vanishes for T>0 in the thermodynamic limit. This absence of finite-temperature ballistic charge transport is an exact result that clarifies a long-standing open problem.
We present a numerical study on the spin and thermal conductivities of the spin-1 Heisenberg chain in the high temperature limit, in particular, the Drude weight contribution and frequency dependence. We use the exact diagonalization and the recently developed microcanonical Lanczos method; it allows us a finite size scaling analysis by the study of significantly larger lattices. This work, pointing to a diffusive rather than ballistic behavior, is discussed with respect to other recent theoretical and experimental studies.