Anomalous low-frequency conductivity in easy-plane XXZ spin chains

Abstract

In the easy-plane regime of XXZ spin chains, spin transport is ballistic, with a Drude weight that has a discontinuous fractal dependence on the value of the anisotropy ∆ = cos πλ at nonzero temperatures. We show that this structure necessarily implies the divergence of the low-frequency conductivity for generic irrational values of λ. Within the framework of generalized hydrodynamics, we show that in the high-temperature limit the low-frequency conductivity at a generic anisotropy scales as σ(ω) ∼ 1/ √ ω; anomalous response occurs because quasiparticles undergo Lévy flights. For rational values of λ, the divergence is cut off at low frequencies and the corrections to ballistic spin transport are diffusive. We also use our approach to recover that at the isotropic point ∆ = 1, spin transport is superdiffusive with σ(ω) ∼ ω −1/3 . We support our results with extensive numerical studies using matrix-product operator methods.

Finite-temperature transport in one-dimensional quantum lattice models

“…The same result has been derived via a GHD-based kinetic picture , and the time-dependent diffusion constant was predicted to scale as D (S) (t) ∝ t 1/3 . This was confirmed in another GHD study (Agrawal et al, 2019), yet without invoking the Kardar-Parisi-Zhang (KPZ) scaling mechanism (Kardar et al, 1986;Ljubotina et al, 2019aLjubotina et al, , 2017Weiner et al, 2020).…”

“…Combined with the expectation that the frequencyintegrated conductivity should be continuous everywhere due to sum rules, this hints at the possibility of superdiffusive corrections away from the commensurate points. This conjecture was put onto firmer grounds in (Agrawal et al, 2019). Let us consider a value ∆ = cos(πλ ∞ ) where λ ∞ is a generic irrational number.…”

“…GHD allows one to obtain the exponents associated with the superdiffusive correction (Agrawal et al, 2019): The low-frequency conductivity behaves as σ(ω) ∝ ω −α with α = 1/2 for generic values of ∆. This divergence is cut-off at the rational points, leading to a diffusive correction.…”

“…This divergence is cut-off at the rational points, leading to a diffusive correction. Furthermore, a qualitative picture emerges from GHD: the subleading correction arises from scattering of charged quasiparticles off neutral quasiparticles and an interpretation in terms of a Lévy flight has been put forward (Agrawal et al, 2019;.…”

“…The approach allows to compute Drude weights (Ilievski and De Nardis, 2017b), diffusion constants (De Nardis et al, 2018) and can provide the full temperature dependence of both quantities. Moreover, subleading corrections to transport coefficients can be extracted such as, e.g., diffusice or superdiffusive corrections in the presence of a Drude weight (Agrawal et al, 2019). Most importantly, GHD often allows for developing an intuition and interpretation as it is based on a kinetic theory of the characteristic excitations of integrable models.…”