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arXiv:2112.02216 · Physical Review Letters

Vanishing and Nonvanishing Persistent Currents of Various Conserved Quantities

Hirokazu Kobayashi, Haruki Watanabe

Published Oct 21, 2022• 3 citations
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Abstract

For every conserved quantity written as a sum of local terms, there exists a corresponding current operator that satisfies the continuity equation. The expectation values of current operators at equilibrium define the persistent currents that characterize spontaneous flows in the system. In this Letter, we consider quantum many-body systems on a finite one-dimensional lattice and discuss the scaling of the persistent currents as a function of the system size. We show that, when the conserved quantities are given as the Noether charges associated with internal symmetries or the Hamiltonian itself, the corresponding persistent currents can be bounded by a correlation function of two operators at a distance proportional to the system size, implying that they decay at least algebraically as the system size increases. In contrast, the persistent currents of accidentally conserved quantities can be nonzero even in the thermodynamic limit and even in the presence of the time-reversal symmetry. We discuss "the current of energy current" in S=1/2 XXZ spin chain as an example and obtain an analytic expression of the persistent current.

DOI
10.1103/physrevlett.129.176601

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For every conserved quantity written as a sum of local terms, there exists a corresponding current operator that satisfies the continuity equation. The expectation values of current operators at equilibrium define the persistent currents that characterize spontaneous flows in the system. In this Letter, we consider quantum many-body systems on a finite one-dimensional lattice and discuss the scaling of the persistent currents as a function of the system size. We show that, when the conserved quantities are given as the Noether charges associated with internal symmetries or the Hamiltonian itself, the corresponding persistent currents can be bounded by a correlation function of two operators at a distance proportional to the system size, implying that they decay at least algebraically as the system size increases. In contrast, the persistent currents of accidentally conserved quantities can be nonzero even in the thermodynamic limit and even in the presence of the time-reversal symmetry. We discuss "the current of energy current" in S=1/2 XXZ spin chain as an example and obtain an analytic expression of the persistent current.
The nonequilibrium dynamics of isolated many-body quantum systems leads generically to the emergence of standard thermal ensembles. However, some models, denoted as integrable, can display nontrivial steady states with a large memory of the initial state. This is due to the presence of conserved quantities constraining the unitary dynamics. Here, the authors show that in interacting integrable spin chains there exist also conserved operators that are responsible for ballistic spin transport. Such conserved operators highly influence both the equilibrium and nonequilibrium physics: at equilibrium they are responsible for a nonvanishing spin Drude weight even at half filling, while in the nonequilibrium case they are responsible for out-of-equilibrium steady states characterized by persistent spin currents or dynamically expanding magnetic domains.
In integrable one-dimensional quantum systems an infinite set of local conserved quantities exists which can prevent a current from decaying completely. For cases like the spin current in the XXZ\mathit{XXZ}XXZ model at zero magnetic field or the charge current in the attractive Hubbard model at half filling, however, the current operator does not have overlap with any of the local conserved quantities. We show that in these situations transport at finite temperatures is dominated by a diffusive contribution with the Drude weight being either small or even zero. For the XXZ\mathit{XXZ}XXZ model we discuss in detail the relation between our results, the phenomenological theory of spin diffusion, and measurements of the spin-lattice relaxation rate in spin chain compounds. Furthermore, we study the Haldane-Shastry model where a conserved spin current exists.
The Bloch theorem is a general theorem restricting the persistent current associated with a conserved U(1) charge in a ground state or in a thermal equilibrium. It gives an upper bound of the magnitude of the current density, which is inversely proportional to the system size. In a recent paper, Else and Senthil applied the argument for the Bloch theorem to a generalized Gibbs ensemble, assuming the presence of an additional conserved charge, and predicted a nonzero current density in the nonthermal steady state [Else and Senthil, Phys. Rev. B 104, 205132 (2021)]. In this paper, we provide a complementary derivation based on the canonical ensemble, given that the additional charge is strictly conserved within the system by itself. Furthermore, using the example where the additional conserved charge is the momentum operator, we discuss that the persistent current tends to vanish when the system is in contact with an external momentum reservoir in the comoving frame of the reservoir.
The presence and character of local integrals of motion---quasilocal operators that commute with the Hamiltonian---encode valuable information about the dynamics of a quantum system. In particular, strongly disordered many-body systems can generically avoid thermalization when there are extensively many such operators. In this work, we explicitly construct local conserved operators in one-dimensional spin chains by directly minimizing their commutator with the Hamiltonian. We demonstrate the existence of an extensively large set of local integrals of motion in the many-body localized phase of the disordered XXZ spin chain. These operators are shown to have exponentially decaying tails, in contrast to the ergodic phase where the decay is (at best) polynomial in the size of the subsystem. We study the algebraic properties of localized operators and confirm that in the many-body localized phase, they are well described by ``dressed'' spin operators.
The nonrelativistic current algebra with conserved current consisting of ρ(x), the particle number density, and J(x), the flux density of particles, is studied. The Hamiltonian for any time reversal invariant system of spinless particles, interacting via a two-body interaction potential, is expressed as a Hermitian form in the currents. This leads to a functional equation for the generating functional, which is the ground state expectation value of exp[i ∫d xρ(x)f(x)]. In the N / V limit an expression for the generating functional in terms of correlation functions is given. Representations of the exponentiated current algebra which are translation invariant satisfy the cluster decomposition property, and those which have different Hamiltonians are shown to be unitarily inequivalent.
We study the effect of conservation laws on the finite-temperature transport properties in one-dimensional integrable quantum many-body systems. We show that the energy current is closely related to the first conservation law in these systems and therefore the thermal transport coefficients are anomalous. Using an inequality on the time decay of current correlations we show how the existence of conserved quantities implies a finite charge stiffness (weight of the zero-frequency component of the conductivity) and so ideal conductivity at finite temperatures.