Abstract Generalized hydrodynamics (GHD) has successfully addressed several open questions regarding finite-temperature ( T > 0) transport in one-dimensional (1D) integrable models, including the mechanisms behind anomalous superdiffusive transport. However, disagreements persist concerning finite-temperature charge transport in the 1D Hubbard model at zero chemical potential ( µ = 0). These disagreements do not arise from the GHD framework itself but rather from differing treatments of the model’s global symmetry at the h = μ = 0 point, where h is the magnetic field. Specifically, the controversy concerns whether finite-temperature charge transport at µ = 0 is anomalous superdiffusive, as predicted using GHD when only the SU (2) × SU (2) symmetries are considered, or instead normal diffusive, as found when employing a representation that fully accounts for the model’s global [ SU (2) × SU (2) × U (1)]/ Z 2 2 symmetry at h = μ = 0 . This work aims to contribute to resolving this issue and advancing the understanding of transport in such systems. We begin by briefly reviewing GHD and other methods predictions regarding anomalous superdiffusive spin transport at finite temperature in gapless 1D integrable models such as the spin- 1 / 2 XXX chain at h = 0 and the 1D Hubbard model at h = μ = 0 . We then review T > 0 transport results in gapped 1D integrable models, focusing on charge transport in the 1D Hubbard model at h = μ = 0 and spin transport in the spin- 1 / 2 XXZ chain at h = 0 with anisotropy Δ > 1 . We identify and discuss the origin of the aforementioned discrepancies, showing how their resolution leads to a deeper understanding of anomalous superdiffusion.
The one-dimensional (1D) Hubbard model at zero magnetic field, h=0, and zero chemical potential, μ=0, and, thus, at the h=μ=0[SO(4)×U(1)]/Z2-symmetry point, is the paradigmatic quantum system for low-dimensional strongly correlated electron 1D Mott-Hubbard insulators. Studies relying on hydrodynamic theory and Kardar-Parisi-Zhang (KPZ) scaling have found that for the 1D Hubbard model at the h=μ=0 point, both spin and charge transports are for all temperatures T>0 that are anomalous superdiffusive. In this paper, we review recent results that correct the hydrodynamic theory and KPZ scaling prediction concerning the finite-temperature charge transport, as it is normal diffusive for all finite temperatures T>0. This failure of the hydrodynamic theory and KPZ scaling stems from the misleading assumption that the h=μ=0[SO(4)×U(1)]/Z2-symmetry point rather is a h=μ=0SO(4)-symmetry point. Such an assumption has ignored the quantum effects associated with the τ-translational U(1) symmetry beyond SO(4).
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.
It is well established that at zero magnetic field
h=0, the spin diffusion constant of the spin-
21 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-
21 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-
21 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 $V/J\ensuremath{\ge}1$.
The use of hydrodynamic transport theory seems to indicate that the charge diffusion constant
D of the one-dimensional (1D) half-filled Hubbard model, whose Drude weight vanishes, diverges for temperature
T>0, which would imply anomalous superdiffusive charge transport. Here the leading term of that constant is derived for low finite temperatures ${k}_{B}T/{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\eta}}\ensuremath{\ll}1$ where ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\eta}}$ is the Mott-Hubbard gap. It only diverges in the $T\ensuremath{\rightarrow}0$ limit, being finite and decreasing upon increasing
T within the ${k}_{B}T/{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\eta}}\ensuremath{\ll}1$ regime. Our exact results both provide valuable physical information on a complex quantum problem and bring about the interesting unsolved issue of how charge transport evolves from normal diffusive for ${k}_{B}T/{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\eta}}\ensuremath{\ll}1$ to anomalous superdiffusive in the ${k}_{B}T\ensuremath{\rightarrow}\ensuremath{\infty}$ limit.
Most studies on the one-dimensional (1D) Hubbard model with transfer integral
t and on-site repulsion
U have assumed that for
u=U/4t>0 and $h=\ensuremath{\mu}=0$, where
h denotes the magnetic field and $\ensuremath{\mu}$ the chemical potential, its global symmetry is $\mathrm{SO}(4)=[\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)]/{Z}_{2}$. However, both in 1D and for the
U>0 Hubbard model on any bipartite lattice it is actually larger and given by $[\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)]/{Z}_{2}^{2}=[\mathrm{SO}(4)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)]/{Z}_{2}$. The studies of this paper account for the irreducible representations of that symmetry to identify both the spin and charge carriers that populate each of the energy eigenstates that span that model full Hilbert space. The global $\ensuremath{\tau}$-translational U(1) symmetry beyond
SO(4) is found to describe the relative translational degrees of freedom of such spin and charge carriers, respectively. We show that important finite-temperature transport quantities as the
h>0 spin and $\ensuremath{\mu}>0$ charge stiffnesses and the
h=0 spin and $\ensuremath{\mu}=0$ charge diffusion constants are controlled by microscopic processes associated with the spin and charge elementary currents carried by these spin and charge carriers, respectively. We describe the model finite-temperature transport properties in terms of the microscopic processes associated with the spin and charge elementary currents carried by the corresponding carriers. We expect that the general $\ensuremath{\tau}$- and physical $\ensuremath{\alpha}$-spin representation used in our study is that suitable to address the open problems on finite-temperature transport in the 1D Hubbard model also discussed in this paper.
Abstract The gapped spin-
21 XXZ chain with spin anisotropy
Δ>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
Δ>1 , and diverges in the limit
Δ→1 . Consequently, at zero magnetic field, finite-temperature spin transport is diffusive for
Δ>1 , but becomes superdiffusive as
Δ→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.