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arXiv:2503.21573 · Physical review. B./Physical review. B

Finite-temperature charge and spin transport in the one-dimensional Hubbard model accounting for its global [ SU ( 2 ) × SU ( 2 ) × U ( 1 ) ]/ Z 2 2 symmetry

J. M. P. Carmelo, J. E. C. Carmelo

Published Mar 26, 2025• 7 citations
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Abstract

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 $\mathrm{SO}(4)$ 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.

DOI
10.1103/physrevb.111.115157

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Most studies on the one-dimensional (1D) Hubbard model with transfer integral ttt and on-site repulsion UUU have assumed that for u=U/4t>0u=U/4t>0u=U/4t>0 and h=\ensuremathμ=0h=\ensuremath{\mu}=0h=\ensuremathμ=0, where hhh denotes the magnetic field and \ensuremathμ\ensuremath{\mu}\ensuremathμ the chemical potential, its global symmetry is SO(4)=[SU(2)\ifmmode×\else\texttimes\fiSU(2)]/Z2\mathrm{SO}(4)=[\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)]/{Z}_{2}SO(4)=[SU(2)\ifmmode×\else\texttimes\fiSU(2)]/Z2​. However, both in 1D and for the U>0U>0U>0 Hubbard model on any bipartite lattice it is actually larger and given by [SU(2)\ifmmode×\else\texttimes\fiSU(2)\ifmmode×\else\texttimes\fiU(1)]/Z22=[SO(4)\ifmmode×\else\texttimes\fiU(1)]/Z2[\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}[SU(2)\ifmmode×\else\texttimes\fiSU(2)\ifmmode×\else\texttimes\fiU(1)]/Z22​=[SO(4)\ifmmode×\else\texttimes\fiU(1)]/Z2​. 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 \ensuremathτ\ensuremath{\tau}\ensuremathτ-translational U(1) symmetry beyond SO(4)\mathrm{SO}(4)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>0h>0h>0 spin and \ensuremathμ>0\ensuremath{\mu}>0\ensuremathμ>0 charge stiffnesses and the h=0h=0h=0 spin and \ensuremathμ=0\ensuremath{\mu}=0\ensuremathμ=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 \ensuremathτ\ensuremath{\tau}\ensuremathτ- and physical \ensuremathα\ensuremath{\alpha}\ensuremathα-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.
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).
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.
Finite-temperature T > 0 transport properties of integrable and nonintegrable one-dimensional (1D) many-particle quantum systems are rather different, showing ballistic and diffusive behavior, respectively. The repulsive 1D Hubbard model is a prominent example of an integrable correlated system. For electronic densities n ≠ 1 (and spin densities m ≠ 0) it is an ideal charge (and spin) conductor, with ballistic charge (and spin) transport for T ⩾ 0. In spite of the fact that it is solvable by the Bethe ansatz, at n = 1 (and m = 0) its T > 0 charge (and spin) transport properties are an issue that remains poorly understood. Here we combine this solution with symmetry and the explicit calculation of current-operator matrix elements between energy eigenstates to show that for on-site repulsion U > 0 and at n = 1 the charge stiffness D η ( T ) vanishes for T > 0 in the thermodynamic limit. A similar behavior is found by such methods for the spin stiffness D s ( T ) for U > 0 and T > 0, which vanishes at m = 0. This absence of finite temperature n = 1 ballistic charge transport and m = 0 ballistic spin transport are exact results that clarify long-standing open problems.
Whether in the thermodynamic limit of lattice length L→∞, hole concentration mηz=−2Sηz/L=1−ne→0, nonzero temperature T>0, and U/t>0 the charge stiffness of the 1D Hubbard model with first neighbor transfer integral t and on-site repulsion U is finite or vanishes and thus whether there is or there is no ballistic charge transport, respectively, remains an unsolved and controversial issue, as different approaches yield contradictory results. (Here Sηz=−(L−Ne)/2 is the η-spin projection and ne=Ne/L the electronic density.) In this paper we provide an upper bound on the charge stiffness and show that (similarly as at zero temperature), for T>0 and U/t>0 it vanishes for mηz→0 within the canonical ensemble in the thermodynamic limit L→∞. Moreover, we show that at high temperature T→∞ the charge stiffness vanishes as well within the grand-canonical ensemble for L→∞ and chemical potential μ→μu where (μ−μu)≥0 and 2μu is the Mott–Hubbard gap. The lack of charge ballistic transport indicates that charge transport at finite temperatures is dominated by a diffusive contribution. Our scheme uses a suitable exact representation of the electrons in terms of rotated electrons for which the numbers of singly occupied and doubly occupied lattice sites are good quantum numbers for U/t>0. In contrast to often less controllable numerical studies, the use of such a representation reveals the carriers that couple to the charge probes and provides useful physical information on the microscopic processes behind the exotic charge transport properties of the 1D electronic correlated system under study.
We study the charge and spin currents transported by the elementary excitations of the one-dimensional (1D) Hubbard model and derive the corresponding current spectra. We present results both for finite-size systems and in the thermodynamic limit. This includes finding the couplings of both the low-energy and finite-energy (string) excitations to external charge and spin probes. At zero magnetic field the general structure of the charge-spin separation survives at all energy scales and the effective charges of both the low-energy and finite-energy charge excitations are studied as functions of the on-site Coulomb interaction U , electronic density n , and applied magnetic field H . In some limits the effective charge of the low-energy excitations equals that of the electrons, whereas that of the finite-energy charge-string excitations of rapidity length γ is found to be 2γ times the electronic charge. At U = ∞ the spin excitations do not contribute to spin transport, whereas the low-energy charge excitations feel an effective flux given by (ϕ ↑ N ↑ -ϕ ↓ N ↓ )/( N ↑ + N ↓ ), where N σ is the number of electrons of spin σ and ϕ σ is a spin-dependent flux. This reveals that at zero magnetic field and U = ∞ there is no spin transport, while at finite magnetic field the low-energy charge excitations also carry spin. In the U >> t limit the spin is carried both by holons and spinons. Finally, we find that the charge- and spin-current spectra can be derived from a semi-classical approach.
Using the framework of generalized hydrodynamics for integrable systems, the authors study finite-temperature transport in the 1D Hubbard model. At strong coupling, the authors characterize transport in the spin-incoherent regime -- where spin degrees of freedom are ``hot'' while charge degrees of freedom are ``cold'' -- and its crossover to the Luttinger liquid regime. Furthermore, using a combination of analytical arguments and numerical simulations, the authors show that when the Hamiltonian features a spin or $\ensuremath{\eta}$-pairing SU(2) symmetry or both, spin or charge transport is superdiffusive with Kardar-Parisi-Zhang scaling.