arXiv:2301.13840 · SciPost Physics

Finite temperature spin diffusion in the Hubbard model in the strong coupling limit

Oleksandr Gamayun, Arthur Hutsalyuk, Balázs Pozsgay, Mikhail B. Zvonarev

Published Aug 30, 2023• 10 citations

Abstract

We investigate finite temperature spin transport in one spatial dimension by considering the spin-spin correlation function of the Hubbard model in the limiting case of infinitely strong repulsion. We find that in the absence of a magnetic field the transport is diffusive, and derive the spin diffusion constant. Our approach is based on asymptotic analysis of a Fredholm determinant representation. The obtained results are in agreement with Generalized Hydrodynamics approach.

DOI: 10.21468/scipostphys.15.2.073

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Spin subdiffusion in perturbed infinite-U Hubbard chain

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arXiv:2301.13840 · SciPost Physics

Finite temperature spin diffusion in the Hubbard model in the strong coupling limit

Oleksandr Gamayun, Arthur Hutsalyuk, Balázs Pozsgay, Mikhail B. Zvonarev

Published Aug 30, 2023• 10 citations

arXiv HTML PDF

Abstract

DOI: 10.21468/scipostphys.15.2.073

Related papers

Finite-temperature Transport Properties of a Spin Quantum System

J. M. P. Carmelo

Abstract The gapped spin- ${1\over 2}$ XXZ chain with spin anisotropy $\Delta > 1$ is a strongly correlated quantum system of great interest in a variety of physical contexts. It is well established that, at zero magnetic field, the spin-diffusion constant associated with the regular part of the spin conductivity is finite both at very low and at high temperatures. It is therefore expected that the spin-diffusion constant remains finite at all temperatures. However, a direct calculation of the spin-diffusion constant over the full temperature range is typically intractable. In this paper, we review recent results showing that, at zero magnetic field, the spin-diffusion constant is finite for all temperatures when $\Delta > 1$ , and diverges in the limit $\Delta \rightarrow 1$ . Consequently, at zero magnetic field, finite-temperature spin transport is diffusive for $\Delta > 1$ , but becomes superdiffusive as $\Delta \rightarrow 1$ . These results are in agreement with predictions from the generalized hydrodynamics framework. However, the precise values of the spin-diffusion constant remains unknown except in certain temperature limits.Abstract The gapped spin- $${1\over 2}$$ XXZ chain with spin anisotropy $$\Delta > 1$$ is a strongly correlated quantum system of great interest in a variety of physical contexts. It is well established that, at zero magnetic field, the spin-diffusion constant associated with the regular part of the spin conductivity is finite both at very low and at high temperatures. It is therefore expected that the spin-diffusion constant remains finite at all temperatures. However, a direct calculation of the spin-diffusion constant over the full temperature range is typically intractable. In this paper, we review recent results showing that, at zero magnetic field, the spin-diffusion constant is finite for all temperatures when $$\Delta > 1$$ , and diverges in the limit $$\Delta \rightarrow 1$$ . Consequently, at zero magnetic field, finite-temperature spin transport is diffusive for $$\Delta > 1$$ , but becomes superdiffusive as $$\Delta \rightarrow 1$$ . These results are in agreement with predictions from the generalized hydrodynamics framework. However, the precise values of the spin-diffusion constant remains unknown except in certain temperature limits.

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We consider the one-dimensional Hubbard model with the infinitely strong repulsion. The two-point dynamical temperature correlation functions are calculated. They are represented as Fredholm determinants of linear integrable integral operators. 1 Introduction The one-dimensional Hubbard model [1] is one of the most interesting and important model of strongly correlated electrons (see, e.g., book [2]). The hamiltonian of the model with the chemical potential h in the external constant field B describes interacting fermions on the one-dimensional periodic lattice of length L: HU = \\Gamma L X x=1 X ff=";# i c y x;ff c x\\Gamma1;ff + c y x\\Gamma1;ff c x;ff j +h L X x=1 (n x;" + n x;# ) \\Gamma B L X x=1 (n x;" \\Gamma n x;# ) + U L X x=1 n x;" n x;# : (1) Here c x;ff and c y x;ff (x = 1; . . . ; L; ff ="; #; c 0;ff j c L;ff ; c y 0;ff j c y L;ff ) are canonical fermion fields with the anticommutation relations [c x;ff ; c y y;fi ] + = ffi x;y ffi ff;fi ...We consider the one-dimensional Hubbard model with the infinitely strong repulsion. The two-point dynamical temperature correlation functions are calculated. They are represented as Fredholm determinants of linear integrable integral operators. 1 Introduction The one-dimensional Hubbard model [1] is one of the most interesting and important model of strongly correlated electrons (see, e.g., book [2]). The hamiltonian of the model with the chemical potential h in the external constant field B describes interacting fermions on the one-dimensional periodic lattice of length L: HU = \\Gamma L X x=1 X ff=";# i c y x;ff c x\\Gamma1;ff + c y x\\Gamma1;ff c x;ff j +h L X x=1 (n x;" + n x;# ) \\Gamma B L X x=1 (n x;" \\Gamma n x;# ) + U L X x=1 n x;" n x;# : (1) Here c x;ff and c y x;ff (x = 1; . . . ; L; ff ="; #; c 0;ff j c L;ff ; c y 0;ff j c y L;ff ) are canonical fermion fields with the anticommutation relations [c x;ff ; c y y;fi ] + = ffi x;y ffi ff;fi ...

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Studies relying on hydrodynamic theory and Kardar-Parisi-Zhang (KPZ) scaling have found that in the one-dimensional Hubbard model spin and charge transport are for all temperatures T > 0 anomalous superdiffusive at zero magnetic field, h = 0, and zero chemical potential, μ = 0, respectively. However, this contradicts recent exact results that at very low temperature charge transport rather is normal diffusive. In this Letter we identify the mechanisms that control the different types of temperature dependence of the h = 0 spin and μ = 0 charge transport and find that the latter is normal diffusive for all finite temperatures T > 0, in contrast to the hydrodynamic theory and KPZ scaling predictions.Studies relying on hydrodynamic theory and Kardar-Parisi-Zhang (KPZ) scaling have found that in the one-dimensional Hubbard model spin and charge transport are for all temperatures T > 0 anomalous superdiffusive at zero magnetic field, h = 0, and zero chemical potential, μ = 0, respectively. However, this contradicts recent exact results that at very low temperature charge transport rather is normal diffusive. In this Letter we identify the mechanisms that control the different types of temperature dependence of the h = 0 spin and μ = 0 charge transport and find that the latter is normal diffusive for all finite temperatures T > 0, in contrast to the hydrodynamic theory and KPZ scaling predictions.

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Finite-temperature transport in the gapped spin- 1 2 XXZ chain and one-dimensional lattice spinless fermion model

J. M. P. Carmelo, Pedro D Sacramento

It is well established that at zero magnetic field $h=0$ , the spin diffusion constant of the spin- $\frac{1}{2}$ XXZ chain is for anisotropy $\mathrm{\ensuremath{\Delta}}>1$ finite for both low and high temperatures, implying that the type of spin transport is normal diffusive. Although it is expected that this holds for all finite temperatures $T>0$ , the calculation of the spin diffusion constant for all $T$ is typically intractable. Here we consider a class of energy eigenstates that exist both for anisotropies $\mathrm{\ensuremath{\Delta}}=1$ and $\mathrm{\ensuremath{\Delta}}>1$. We show that at the isotropic point their contributions are behind the diffusion constant being infinite, spin transport being anomalous superdiffusive for $T>0$ . That for $\mathrm{\ensuremath{\Delta}}>1$ such states do not contribute to the diffusion constant is shown to imply it is finite, spin transport being normal diffusive for $T>0$ . By combining the connection through a Jordan-Wigner transformation of the spin- $\frac{1}{2}$ XXZ chain to the one-dimensional (1D) lattice spinless fermion model at zero chemical potential $\ensuremath{\mu}=0$, for $V/J\ensuremath{\ge}1$ with its Bethe-ansatz solution, where $V$ is the nearest-neighbor Coulomb repulsion and $J=2t$ is twice the hopping integral $t$ , in this paper we also address the issue of the $T>0$ charge transport of that model at $\ensuremath{\mu}=0$. It is found to be anomalous superdiffusive for $V/J=1$ and normal diffusive for $V/J>1$ . Our results thus open the door to a key advance in the understanding for all finite temperatures $T>0$ of the spin transport in the spin- $\frac{1}{2}$ XXZ chain at $h=0$ for anisotropy $\mathrm{\ensuremath{\Delta}}\ensuremath{\ge}1$ and of the charge transport in the 1D lattice spinless fermion model at $\ensuremath{\mu}=0$ for $V/J\ensuremath{\ge}1$.It is well established that at zero magnetic field $h=0$, the spin diffusion constant of the spin-$\frac{1}{2}$ XXZ chain is for anisotropy $\mathrm{\ensuremath{\Delta}}>1$ finite for both low and high temperatures, implying that the type of spin transport is normal diffusive. Although it is expected that this holds for all finite temperatures $T>0$, the calculation of the spin diffusion constant for all $T$ is typically intractable. Here we consider a class of energy eigenstates that exist both for anisotropies $\mathrm{\ensuremath{\Delta}}=1$ and $\mathrm{\ensuremath{\Delta}}>1$. We show that at the isotropic point their contributions are behind the diffusion constant being infinite, spin transport being anomalous superdiffusive for $T>0$. That for $\mathrm{\ensuremath{\Delta}}>1$ such states do not contribute to the diffusion constant is shown to imply it is finite, spin transport being normal diffusive for $T>0$. By combining the connection through a Jordan-Wigner transformation of the spin-$\frac{1}{2}$ XXZ chain to the one-dimensional (1D) lattice spinless fermion model at zero chemical potential $\ensuremath{\mu}=0$, for $V/J\ensuremath{\ge}1$ with its Bethe-ansatz solution, where $V$ is the nearest-neighbor Coulomb repulsion and $J=2t$ is twice the hopping integral $t$, in this paper we also address the issue of the $T>0$ charge transport of that model at $\ensuremath{\mu}=0$. It is found to be anomalous superdiffusive for $V/J=1$ and normal diffusive for $V/J>1$. Our results thus open the door to a key advance in the understanding for all finite temperatures $T>0$ of the spin transport in the spin-$\frac{1}{2}$ XXZ chain at $h=0$ for anisotropy $\mathrm{\ensuremath{\Delta}}\ensuremath{\ge}1$ and of the charge transport in the 1D lattice spinless fermion model at $\ensuremath{\mu}=0$ for $V/J\ensuremath{\ge}1$.

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Diffusive high-temperature transport in the one-dimensional Hubbard model

Tomaž Prosen, Marko Žnidarič

We consider charge and spin transport in the one-dimensional Hubbard model at infinite temperature, half-filling, and zero magnetization. Implementing matrix-product-operator simulations of the nonequilibrium steady states of boundary-driven open Hubbard chains for up to 100 sites we find clear numerical evidence of diffusive transport for weak driving and any (nonzero and finite) value of the interaction $U$ . In addition, nonequilibrium steady states are characterized by nonzero spin-up spin-down density-density correlations. For maximal driving, on the other hand, we find an insulating behavior and cosine-shaped density profiles.We consider charge and spin transport in the one-dimensional Hubbard model at infinite temperature, half-filling, and zero magnetization. Implementing matrix-product-operator simulations of the nonequilibrium steady states of boundary-driven open Hubbard chains for up to 100 sites we find clear numerical evidence of diffusive transport for weak driving and any (nonzero and finite) value of the interaction $U$. In addition, nonequilibrium steady states are characterized by nonzero spin-up spin-down density-density correlations. For maximal driving, on the other hand, we find an insulating behavior and cosine-shaped density profiles.

arXiv:1203.1727Physical Review BSep 14, 201271 citations

Spin subdiffusion in perturbed infinite-U Hubbard chain

Jakub Rękas, Marcin Mierzejewski, Zala Lenarčič +1 more

The $t$ -model represents the Hubbard model in the limit $U \to \infty$ and is one of the basic models of strongly correlated electrons. On a one-dimensional chain, the model is integrable, and the charge dynamics corresponds to that of free spinless fermions. However, the sequence of spins is frozen, leading to the Hilbert space fragmentation and nontrivial spin dynamics. We consider integrable and perturbed models with perturbations that break integrability while preserving fragmentation, and show that they exhibit various types of spin dynamics, from ballistic transport to anomalous diffusion in the integrable case, and from diffusion to subdiffusion in the perturbed case. Due to fragmentation, in all cases considered, spin transport is mediated by charge transport, with a particular magnetization dependence, most notably leading to subdiffusion in the grandcanonical average of the perturbed model, with a mechanism distinct from subdiffusion in disordered or dipole-conserving models.The $t$-model represents the Hubbard model in the limit $U \to \infty$ and is one of the basic models of strongly correlated electrons. On a one-dimensional chain, the model is integrable, and the charge dynamics corresponds to that of free spinless fermions. However, the sequence of spins is frozen, leading to the Hilbert space fragmentation and nontrivial spin dynamics. We consider integrable and perturbed models with perturbations that break integrability while preserving fragmentation, and show that they exhibit various types of spin dynamics, from ballistic transport to anomalous diffusion in the integrable case, and from diffusion to subdiffusion in the perturbed case. Due to fragmentation, in all cases considered, spin transport is mediated by charge transport, with a particular magnetization dependence, most notably leading to subdiffusion in the grandcanonical average of the perturbed model, with a mechanism distinct from subdiffusion in disordered or dipole-conserving models.

cond-mat.str-elCross-listedarXiv:2603.19743Physical review. B./Physical review. BMar 23, 20260 citations

Abstract The gapped spin-

{1\over 2}

XXZ chain with spin anisotropy

\Delta > 1

is a strongly correlated quantum system of great interest in a variety of physical contexts. It is well established that, at zero magnetic field, the spin-diffusion constant associated with the regular part of the spin conductivity is finite both at very low and at high temperatures. It is therefore expected that the spin-diffusion constant remains finite at all temperatures. However, a direct calculation of the spin-diffusion constant over the full temperature range is typically intractable. In this paper, we review recent results showing that, at zero magnetic field, the spin-diffusion constant is finite for all temperatures when

\Delta > 1

, and diverges in the limit

\Delta \rightarrow 1

. Consequently, at zero magnetic field, finite-temperature spin transport is diffusive for

\Delta > 1

, but becomes superdiffusive as

\Delta \rightarrow 1

. These results are in agreement with predictions from the generalized hydrodynamics framework. However, the precise values of the spin-diffusion constant remains unknown except in certain temperature limits.

It is well established that at zero magnetic field

h=0

, the spin diffusion constant of the spin-

\frac{1}{2}

XXZ chain is for anisotropy $\mathrm{\ensuremath{\Delta}}>1$ finite for both low and high temperatures, implying that the type of spin transport is normal diffusive. Although it is expected that this holds for all finite temperatures

T>0

, the calculation of the spin diffusion constant for all

T

is typically intractable. Here we consider a class of energy eigenstates that exist both for anisotropies $\mathrm{\ensuremath{\Delta}}=1$ and $\mathrm{\ensuremath{\Delta}}>1$. We show that at the isotropic point their contributions are behind the diffusion constant being infinite, spin transport being anomalous superdiffusive for

T>0

. That for $\mathrm{\ensuremath{\Delta}}>1$ such states do not contribute to the diffusion constant is shown to imply it is finite, spin transport being normal diffusive for

T>0

. By combining the connection through a Jordan-Wigner transformation of the spin-

\frac{1}{2}

XXZ chain to the one-dimensional (1D) lattice spinless fermion model at zero chemical potential $\ensuremath{\mu}=0$, for $V/J\ensuremath{\ge}1$ with its Bethe-ansatz solution, where

V

is the nearest-neighbor Coulomb repulsion and

J=2t

is twice the hopping integral

t

, in this paper we also address the issue of the

T>0

charge transport of that model at $\ensuremath{\mu}=0$. It is found to be anomalous superdiffusive for

V/J=1

and normal diffusive for

V/J>1

. Our results thus open the door to a key advance in the understanding for all finite temperatures

T>0

of the spin transport in the spin-

\frac{1}{2}

XXZ chain at

h=0

for anisotropy $\mathrm{\ensuremath{\Delta}}\ensuremath{\ge}1$ and of the charge transport in the 1D lattice spinless fermion model at $\ensuremath{\mu}=0$ for $V/J\ensuremath{\ge}1$.

U

. In addition, nonequilibrium steady states are characterized by nonzero spin-up spin-down density-density correlations. For maximal driving, on the other hand, we find an insulating behavior and cosine-shaped density profiles.

The

t

-model represents the Hubbard model in the limit

U \to \infty

and is one of the basic models of strongly correlated electrons. On a one-dimensional chain, the model is integrable, and the charge dynamics corresponds to that of free spinless fermions. However, the sequence of spins is frozen, leading to the Hilbert space fragmentation and nontrivial spin dynamics. We consider integrable and perturbed models with perturbations that break integrability while preserving fragmentation, and show that they exhibit various types of spin dynamics, from ballistic transport to anomalous diffusion in the integrable case, and from diffusion to subdiffusion in the perturbed case. Due to fragmentation, in all cases considered, spin transport is mediated by charge transport, with a particular magnetization dependence, most notably leading to subdiffusion in the grandcanonical average of the perturbed model, with a mechanism distinct from subdiffusion in disordered or dipole-conserving models.