Transport and Approximate Conservation Laws in Low Dimensional Systems

Peter Jung

Published Jan 1, 2007• 0 citations

Abstract

We set up a framework in which controlled lower bounds to conductivities of strongly correlated quantum systems may be calculated perturbatively in situations where the conductivities are infinite in the unperturbed model. This framework is then applied to study the transport properties of almost integrable systems, in general, and heat and spin conductivities in the integrable XXZ Heisenberg spin chain perturbed by various integrability breaking couplings, in particular. Furthermore, we systematically develop a microscopic description of the contribution of spin chains to experimentally measured heat conductivity in spin chain compounds, with an emphasis on the effect of weak disorder.

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OpenAlex

Abstract

Related papers

Transport in Almost Integrable Models: Perturbed Heisenberg Chains

Peter Jung, R. W. Helmes, Achim Rosch

The heat conductivity $\ensuremath{\kappa}(T)$ of integrable models, like the one-dimensional spin- $1/2$ nearest-neighbor Heisenberg model, is infinite even at finite temperatures as a consequence of the conservation laws associated with integrability. Small perturbations lead to finite but large transport coefficients which we calculate perturbatively using exact diagonalization and moment expansions. We show that there are two different classes of perturbations. While an interchain coupling of strength ${J}_{\ensuremath{\perp}}$ leads to $\ensuremath{\kappa}(T)\ensuremath{\propto}1/{J}_{\ensuremath{\perp}}^{2}$ as expected from simple golden-rule arguments, we obtain a much larger $\ensuremath{\kappa}(T)\ensuremath{\propto}1/{J}^{\ensuremath{'}4}$ for a weak next-nearest-neighbor interaction ${J}^{\ensuremath{'}}$. This can be explained by a new approximate conservation law of the $J\mathrm{\text{\ensuremath{-}}}{J}^{\ensuremath{'}}$ Heisenberg chain.The heat conductivity $\ensuremath{\kappa}(T)$ of integrable models, like the one-dimensional spin-$1/2$ nearest-neighbor Heisenberg model, is infinite even at finite temperatures as a consequence of the conservation laws associated with integrability. Small perturbations lead to finite but large transport coefficients which we calculate perturbatively using exact diagonalization and moment expansions. We show that there are two different classes of perturbations. While an interchain coupling of strength ${J}_{\ensuremath{\perp}}$ leads to $\ensuremath{\kappa}(T)\ensuremath{\propto}1/{J}_{\ensuremath{\perp}}^{2}$ as expected from simple golden-rule arguments, we obtain a much larger $\ensuremath{\kappa}(T)\ensuremath{\propto}1/{J}^{\ensuremath{'}4}$ for a weak next-nearest-neighbor interaction ${J}^{\ensuremath{'}}$. This can be explained by a new approximate conservation law of the $J\mathrm{\text{\ensuremath{-}}}{J}^{\ensuremath{'}}$ Heisenberg chain.

arXiv:cond-mat/0509615Physical Review LettersFeb 14, 2006123 citations

Spin conductivity in almost integrable spin chains

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Jan 1, 20101 citation

Scaling of electrical and thermal conductivities in an almost integrable chain

Joel E. Moore, Christoph Karrasch, Roni Ilan

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.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.

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Jesko Sirker, Rodrigo G. Pereira, Ian K. Affleck

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 $\mathit{XXZ}$ 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 $\mathit{XXZ}$ 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.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 $\mathit{XXZ}$ 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 $\mathit{XXZ}$ 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.

arXiv:1011.1354Physical Review BJan 18, 2011204 citations

The heat conductivity $\ensuremath{\kappa}(T)$ of integrable models, like the one-dimensional spin-

1/2

nearest-neighbor Heisenberg model, is infinite even at finite temperatures as a consequence of the conservation laws associated with integrability. Small perturbations lead to finite but large transport coefficients which we calculate perturbatively using exact diagonalization and moment expansions. We show that there are two different classes of perturbations. While an interchain coupling of strength ${J}_{\ensuremath{\perp}}$ leads to $\ensuremath{\kappa}(T)\ensuremath{\propto}1/{J}_{\ensuremath{\perp}}^{2}$ as expected from simple golden-rule arguments, we obtain a much larger $\ensuremath{\kappa}(T)\ensuremath{\propto}1/{J}^{\ensuremath{'}4}$ for a weak next-nearest-neighbor interaction ${J}^{\ensuremath{'}}$. This can be explained by a new approximate conservation law of the $J\mathrm{\text{\ensuremath{-}}}{J}^{\ensuremath{'}}$ Heisenberg chain.

The spin conductivity in the integrable spin-

1∕2

XXZ

chain is known to be infinite at finite temperatures

T

for anisotropies $\ensuremath{-}1<\ensuremath{\Delta}<1$. Perturbations, which break integrability, e.g., a next-nearest neighbor coupling ${J}^{\ensuremath{'}}$, render the conductivity finite. We construct numerically a nonlocal conserved operator ${J}_{\ensuremath{\parallel}}$ which is responsible for the finite spin Drude weight of the integrable model and calculate its decay rate for small ${J}^{\ensuremath{'}}$. This allows us to obtain a lower bound for the spin conductivity ${\ensuremath{\sigma}}_{s}\ensuremath{\geqslant}c(T)∕{{J}^{\ensuremath{'}}}^{2}$, where

c(T)

is finite for ${J}^{\ensuremath{'}}\ensuremath{\rightarrow}0$. We discuss the implication of our result for the general question how nonlocal conservation laws affect transport properties.

We show how one can obtain a lower bound for the electrical, spin, or heat conductivity of correlated quantum systems described by Hamiltonians of the form

H={H}_{0}+g{H}_{1}

. Here,

{H}_{0}

is an interacting Hamiltonian characterized by conservation laws which lead to an infinite conductivity for

g=0

. The small perturbation

g{H}_{1}

, however, renders the conductivity finite at finite temperatures. For example,

{H}_{0}

could be a continuum field theory, where momentum is conserved, or an integrable one-dimensional model, while

{H}_{1}

might describe the effects of weak disorder. In the limit $g\ensuremath{\rightarrow}0$, we derive lower bounds for the relevant conductivities and show how they can be improved systematically using the memory matrix formalism. Furthermore, we discuss various applications and investigate under what conditions our lower bound may become exact.

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

\mathit{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

\mathit{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.