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arXiv:2208.01446 · Journal of Physics A Mathematical and Theoretical

Popcorn Drude weights from quantum symmetry

Enej Ilievski

Published Dec 9, 2022• 10 citations

Abstract

Abstract Integrable models provide emblematic examples of non-ergodic phenomena. One of their most distinguished properties are divergent zero-frequency conductivities signalled by finite Drude weights. Singular conductivities owe to long-lived quasiparticle excitations that propagate ballistically through the system without any diffraction. The case of the celebrated quantum Heisenberg chain, one of the best-studied many-body paradigms, turns out to be particularly mysterious. About a decade ago, it was found that the spin Drude weight in the critical phase of the model assumes an extraordinary, nowhere continuous, dependence on the anisotropy parameter in the shape of a ‘popcorn function’. This unprecedented discovery has been afterwards resolved at the level of the underlying deformed quantum symmetry algebra which helps explaining the erratic nature of the quasiparticle spectrum at commensurate values of interaction anisotropy. This work is devoted to the captivating phenomenon of discontinuous Drude weights, with the aim to give a broader perspective on the topic by revisiting and reconciling various perspectives from the previous studies. Moreover, it is argued that such an anomalous non-ergodic feature is not exclusive to the integrable spin chain but can be instead expected in a number of other integrable systems that arise from realizations of the quantum group , specialized to unimodular values of the quantum deformation parameter q. Our discussion is framed in the context of gapless anisotropic quantum chains of higher spin and the sine-Gordon quantum field theory in two space-time dimensions.

DOI: 10.1088/1751-8121/acaa77

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Abstract Integrable models provide emblematic examples of non-ergodic phenomena. One of their most distinguished properties are divergent zero-frequency conductivities signalled by finite Drude weights. Singular conductivities owe to long-lived quasiparticle excitations that propagate ballistically through the system without any diffraction. The case of the celebrated quantum Heisenberg chain, one of the best-studied many-body paradigms, turns out to be particularly mysterious. About a decade ago, it was found that the spin Drude weight in the critical phase of the model assumes an extraordinary, nowhere continuous, dependence on the anisotropy parameter in the shape of a ‘popcorn function’. This unprecedented discovery has been afterwards resolved at the level of the underlying deformed quantum symmetry algebra which helps explaining the erratic nature of the quasiparticle spectrum at commensurate values of interaction anisotropy. This work is devoted to the captivating phenomenon of discontinuous Drude weights, with the aim to give a broader perspective on the topic by revisiting and reconciling various perspectives from the previous studies. Moreover, it is argued that such an anomalous non-ergodic feature is not exclusive to the integrable spin chain but can be instead expected in a number of other integrable systems that arise from realizations of the quantum group , specialized to unimodular values of the quantum deformation parameter q. Our discussion is framed in the context of gapless anisotropic quantum chains of higher spin and the sine-Gordon quantum field theory in two space-time dimensions.

Nonergodic dynamical systems display anomalous transport properties. Prominent examples are integrable quantum systems, whose exceptional properties are diverging dc conductivities. In this Letter, we explain the microscopic origin of ideal conductivity by resorting to the thermodynamic particle content of a system. Using group-theoretic arguments we rigorously resolve the long-standing controversy regarding the nature of spin and charge Drude weights in the absence of chemical potentials. In addition, by employing a hydrodynamic description, we devise an efficient computational method to calculate exact Drude weights from the stationary currents generated in an inhomogeneous quench from bipartitioned initial states. We exemplify the method on the anisotropic Heisenberg model at finite temperatures for the entire range of anisotropies, accessing regimes that are out of reach with other approaches. Quite remarkably, spin Drude weight and asymptotic spin current rates reveal a completely discontinuous (fractal) dependence on the anisotropy parameter.

The anomalous nature of spin transport in the XXZ quantum spin chain has been a topic of theoretical interest for some time. Here, the integrability of the underlying dynamics leads to a ballistic component of the spin current, characterized by a spin Drude weight, which measures the degree of divergence of the zero-frequency spin conductivity. However, this quantity had previously proven to be beyond the reach of standard Bethe ansatz techniques. Here, the authors show that a recently developed hydrodynamic formalism for quantum integrable models may be used to compute the spin Drude weight. They also propose a numerical scheme to obtain hydrodynamic predictions for finite-time energy transport. This suggests that the hydrodynamic approach captures completely the ballistic component that dominates transport at long times and distances in the gapless regime of the XXZ model.

Nondecaying dc currents in systems showing ballistic transport are characterized by a constant, known as the Drude weight. Here, the authors obtain analytical results for the spin Drude weight in the interacting anisotropic Heisenberg spin-

\frac{1}{2}

chain at low temperatures. The obtained analytical formula reveals that the Drude weight for any finite temperature exhibits a nowhere-continuous dependence on the spin anisotropy of the model.

We study the Heisenberg-Kitaev spin chain in order to uncover the interplay between two qualitatively different integrable points in the physics of heat transport in one dimension. Focusing on high temperatures and using analytical as well as numerical approaches within linear response theory, we explore several directions in parameter space including exchange-coupling ratios, anisotropies, and external magnetic fields. We show the emergence of purely ballistic energy transport at all integrable points, manifest in pronounced Drude weights and low-frequency suppression of regular-conductivity contributions. Moreover, off integrability, we find extended quantum chaotic regions with vanishing Drude weights and well-defined dc conductivities. In the vicinity of the Kitaev point, we observe clear signatures of the topological gap in the response function. This gap coexists with a nonzero Drude weight in the Kitaev chain.

Integrable models such as the spin-1/2 Heisenberg chain, the Lieb-Liniger, or the one-dimensional Hubbard model are known to avoid thermalization, which was also demonstrated in several quantum-quench experiments. Another dramatic consequence of integrability is the zero-frequency anomaly in transport coefficients, which results in ballistic finite-temperature transport, despite the presence of strong interactions. While this aspect of nonergodic dynamics has been known for a long time, there has so far not been any unambiguous experimental realization thereof. We make a concrete proposal for the observation of ballistic transport via local quantum-quench experiments in fermionic quantum-gas microscopes. Such an experiment would also unveil the coexistence of ballistic and diffusive transport channels in one and the same system and provide a means of measuring finite-temperature Drude weights. The connection between local quenches and linear-response functions is established via time-dependent Einstein relations.

In the absence of frustration, interacting bosons in their ground state in one or two dimensions exist either in the superfluid or insulating phases. Superfluidity corresponds to frictionless flow of the matter field, and in optical conductivity is revealed through a distinct δ-functional peak at zero frequency with the amplitude known as the Drude weight. This characteristic low-frequency feature is instead absent in insulating phases, defined by zero static optical conductivity. Here we demonstrate that bosonic particles in disordered one dimensional chains can also exist in a conducting, non-superfluid, phase when their hopping is of the dipolar type, often viewed as short-ranged in one dimension. This phase is characterized by finite static optical conductivity, followed by a broad anti-Drude peak at finite frequencies. Off-diagonal correlations are also unconventional: they feature an integrable algebraic decay for arbitrarily large values of disorder. These results do not fit the description of any known quantum phase, and strongly suggest the existence of an unusual conducting state of bosonic matter in the ground state.