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arXiv:0710.4440 · Europhysics Letters (EPL)

Role of symmetry in the interplay of T = 0 quantum-phase transitions with unconventional T > 0 transport properties in integrable quantum lattice systems

J. M. P. Carmelo, S.-J Gu, Nuno M. R. Peres

Published Mar 15, 2007• 3 citations
arXivHTMLPDF

Abstract

We show that a generalized charge SU (2) symmetry of the one-dimensional (1D) Hubbard model in an infinitesimal flux ϕ generates half-filling states from metallic states which lead to a finite charge stiffness D ( T ) at finite temperature T , whose T -dependence we study. Our results are of general nature for many integrable quantum lattice systems, reveal the microscopic mechanisms behind their exotic T > 0 transport properties and the interplay with T = 0 quantum-phase transitions, and contribute to the further understanding of the transport of charge in systems of interacting ultracold fermionic atoms in 1D optical lattices, quasi-1D compounds, and 1D nanostructures.

DOI
10.1209/0295-5075/78/17005

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Finite-temperature T>0 transport properties of integrable and nonintegrable one-dimensional (1D) many-particle quantum systems are rather different, showing in the metallic phases ballistic and diffusive behavior, respectively. The repulsive 1D Hubbard model is an integrable system of wide physical interest. For electronic densities $n\neq1$ it is an ideal conductor, with ballistic charge transport for T larger or equal to 0. In spite that it is solvable by the Bethe ansatz, at $n=1$ its T>0 transport properties are a collective-behavior issue that remains poorly understood. Here we combine that solution with symmetry to show that for on-site repulsion U>0 the charge stiffness D (T) vanishes for T>0 in the thermodynamic limit. This absence of finite-temperature ballistic charge transport is an exact result that clarifies a long-standing open problem.

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arXiv:1408.1891•Physical Review B•Oct 3, 2014•56 citations
We show that a generalized charge SU (2) symmetry of the one-dimensional (1D) Hubbard model in an infinitesimal flux ϕ generates half-filling states from metallic states which lead to a finite charge stiffness D ( T ) at finite temperature T , whose T -dependence we study. Our results are of general nature for many integrable quantum lattice systems, reveal the microscopic mechanisms behind their exotic T > 0 transport properties and the interplay with T = 0 quantum-phase transitions, and contribute to the further understanding of the transport of charge in systems of interacting ultracold fermionic atoms in 1D optical lattices, quasi-1D compounds, and 1D nanostructures.
Finite-temperature T>0 transport properties of integrable and nonintegrable one-dimensional (1D) many-particle quantum systems are rather different, showing in the metallic phases ballistic and diffusive behavior, respectively. The repulsive 1D Hubbard model is an integrable system of wide physical interest. For electronic densities n≠1n\neq1n=1 it is an ideal conductor, with ballistic charge transport for T larger or equal to 0. In spite that it is solvable by the Bethe ansatz, at n=1n=1n=1 its T>0 transport properties are a collective-behavior issue that remains poorly understood. Here we combine that solution with symmetry to show that for on-site repulsion U>0 the charge stiffness D (T) vanishes for T>0 in the thermodynamic limit. This absence of finite-temperature ballistic charge transport is an exact result that clarifies a long-standing open problem.
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 $h=\ensuremath{\mu}=0$, where hhh 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>0U>0U>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)\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 $\ensuremath{\mu}>0$ charge stiffnesses and the h=0h=0h=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 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.
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.
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.
We study finite-temperature transport properties of the one-dimensional Hubbard model using the density matrix renormalization group. Our aim is twofold: First, we compute both the charge- and the spin-current correlation function of the integrable model at half filling. The former decays rapidly, implying that the corresponding Drude weight is either zero or very small. Second, we calculate the optical charge conductivity ${\ensuremath{\sigma}}_{r}eg(\ensuremath{\omega})$ in the presence of small integrability-breaking next-nearest-neighbor interactions (the extended Hubbard model). The dc conductivity is finite and diverges as the temperature is decreased below the gap. Our results thus suggest that the half-filled, gapped Hubbard model is a normal charge conductor at finite temperatures. As a test bed for our numerics, we compute ${\ensuremath{\sigma}}_{r}eg(\ensuremath{\omega})$ for the integrable XXZ spin chain in its gapped phase.