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arXiv:1904.09287 · Proceedings of the National Academy of Sciences

Superdiffusive transport of energy in one-dimensional metals

Vir B. Bulchandani, Christoph Karrasch, Joel E. Moore

Published May 26, 2020• 33 citations
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Abstract

Metals in one spatial dimension are described at the lowest energy scales by the Luttinger liquid theory. It is well understood that this free theory, and even interacting integrable models, can support ballistic transport of conserved quantities including energy. In contrast, realistic one-dimensional metals, even without disorder, contain integrability-breaking interactions that are expected to lead to thermalization and conventional diffusive linear response. We argue that the expansion of energy when such a nonintegrable Luttinger liquid is locally heated above its ground state shows superdiffusive behavior (i.e., spreading of energy that is intermediate between diffusion and ballistic propagation), by combining an analytical anomalous diffusion model with numerical matrix-product-state calculations on a specific perturbed spinless fermion chain. Different metals will have different scaling exponents and shapes in their energy spreading, but the superdiffusive behavior is stable and should be visible in time-resolved experiments.

DOI
10.1073/pnas.1916213117

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Metals in one spatial dimension are described at the lowest energy scales by the Luttinger liquid theory. It is well understood that this free theory, and even interacting integrable models, can support ballistic transport of conserved quantities including energy. In contrast, realistic one-dimensional metals, even without disorder, contain integrability-breaking interactions that are expected to lead to thermalization and conventional diffusive linear response. We argue that the expansion of energy when such a nonintegrable Luttinger liquid is locally heated above its ground state shows superdiffusive behavior (i.e., spreading of energy that is intermediate between diffusion and ballistic propagation), by combining an analytical anomalous diffusion model with numerical matrix-product-state calculations on a specific perturbed spinless fermion chain. Different metals will have different scaling exponents and shapes in their energy spreading, but the superdiffusive behavior is stable and should be visible in time-resolved experiments.
The impact of lattice vibrations on the electric conductivity has been considered a settled issue since the proof of the Migdal theorem. The precondition for its validity is a higher velocity of electronic than phononic excitation modes. In some new materials, e. g. weakly coupled spin chains and heavy fermion systems, this precondition is not fulfilled. The heat conductivity of such systems is calculated by means of perturbation theory in this work. Another subject is the interplay of Umklapp scattering and weak disorder in spin chains and one-dimensional correlated metals. Hidden conservation laws lead to a strong coupling between heat current and charge or spin current, respectively. The transport properties are calculated perturbatively within the hydrodynamic memory matrix formalism, which gives a lower bound to the conductivies. To calculate the Lorenz number of one-dimensional metals, we model within this formalism also the influence of phonons, which in physical systems contribute substantially to the heat current. The introduction of ultracold quantum gases in optical lattices in the recent years provides a powerful tool for the modeling of many systems to solid state physics, in particular systems out of equilibrium. The relatively slow dynamics of the cold atoms allows for time resolved imaging of the atoms and thus opens up new perspectives. In the last part of this thesis, a recent experiment, which measures the expansion of a correlated atom cloud in an optical lattice, is compared to a lattice Boltzmann simulation. Depending on the local density, the problem is shown to be described by the crossover from diffusive to ballistic transport.
We study the thermal conductivity of the one-dimensional Fermi-Hubbard model at a finite temperature using a density matrix renormalization group approach. The integrability of this model gives rise to ballistic thermal transport. We calculate the temperature dependence of the thermal Drude weight at half filling for various interaction strengths. The finite-frequency contributions originating from the fact that the energy current is not a conserved quantity are investigated as well. We report evidence that breaking the integrability through a nearest-neighbor interaction leads to vanishing Drude weights and diffusive energy transport. Moreover, we demonstrate that energy spreads ballistically in local quenches with initially inhomogeneous energy density profiles in the integrable case. We discuss the relevance of our results for thermalization in ultracold quantum-gas experiments and for transport measurements with quasi-one-dimensional materials.
Many low-dimensional materials are well described by integrable one-dimensional models such as the Hubbard model of electrons or the Heisenberg model of spins. However, the small perturbations to these models required to describe real materials are expected to have singular effects on transport quantities: integrable models often support dissipationless transport, while weak nonintegrable terms lead to finite conductivities. We use matrix-product-state methods to obtain quantitative values of spin/electrical and thermal conductivities in an almost integrable gapless (XXZ-like) spin chain. At low temperatures, we observe power laws whose exponents are solely determined by the Luttinger liquid parameter. This indicates that our results are independent of the actual model under consideration.
A common wisdom posits that transport of conserved quantities across clean nonintegrable quantum systems at high temperatures is diffusive when probed from the emergent hydrodynamic regime. We show that this empirical paradigm may alter if the strong interaction limit is taken. Using Krylov-typicality and purification matrix-product-state methods, we establish in short-to-intermediate time scales the following observations for the nonintegrable lattice model imitating the experimental Rydberg blockade simulator. Given the strict projection owing to the infinite density-density repulsion V, the Rydberg chain’s energy transport in the presence of a transverse field g is tentatively superdiffusive at infinite temperature featured by an anomalous scaling exponent 34\frac{3}{4}43​ , indicating the potential existence of a novel dynamical universality class. Imposing, in addition, a growing longitudinal field h causes a putative superdiffusion-to-ballistic transport transition at h ≈ g. Interestingly, all the above results persist for large but finite interactions and temperatures, provided that the strongly interacting condition g, h ≪ kBT ≪ V is fulfilled. Our predictions are testable by current Rydberg quantum simulation facilities.
Many low-dimensional materials are well described by integrable one-dimensional models such as the Hubbard model of electrons or the Heisenberg model of spins. However, the small perturbations to these models required to describe real materials are expected to have singular effects on transport quantities: integrable models often support dissipationless transport, while weak nonintegrable terms lead to finite conductivities. We use matrix-product-state methods to obtain quantitative values of spin/electrical and thermal conductivities in an almost integrable gapless (XXZXXZXXZ-like) spin chain. At low temperatures, we observe power laws whose exponents are solely determined by the Luttinger liquid parameter. This indicates that our results are independent of the actual model under consideration.
It has been conjectured that transport in integrable one-dimensional systems is necessarily ballistic. The large diffusive response seen experimentally in nearly ideal realizations of the S=1/2 1D Heisenberg model is therefore puzzling and has not been explained so far. Here, we show that, contrary to common belief, diffusion is universally present in interacting 1D systems subject to a periodic lattice potential. We present a parameter-free formula for the spin-lattice relaxation rate which is in excellent agreement with experiment. Furthermore, we calculate the current decay directly in the thermodynamic limit using a time-dependent density matrix renormalization group algorithm and show that an anomalously large time scale exists even at high temperatures.