The Hubbard chain at finite temperatures: ab initio calculations of Tomonaga-Luttinger liquid properties

doi:10.1016/S0550-3213(98)00256-9

Nuclear Physics B

Volume 522, Issue 3, 6 July 1998, Pages 471-502

https://doi.org/10.1016/S0550-3213(98)00256-9 Get rights and content

Abstract

We present a novel treatment of finite temperature properties of the one-dimensional Hubbard model. Our approach is based on a Trotter-Suzuki mapping utilizing Shastry's classical model and a subsequent investigation of the quantum transfer matrix. We derive non-linear integral equations for three auxiliary functions which have a clear physical interpretation of elementary excitations of spin type and charge excitations in lower and upper Hubbard bands. This allows for a transparent analytical study of certain limiting cases as well as for precise numerical investigations. We present data for the specific heat, magnetic and charge susceptibilities for various particle densities and coupling strengths U. The structure exposed by these curves is discussed in terms of the elementary charge and spin excitations. Special emphasis is placed on the study of the low-temperature behavior within our ab initio approach confirming the scaling predictions by Tomonaga-Luttinger liquid theory. In addition we make contact with the “dressed energy” formalism established for the analysis of ground-state properties.

References (52)

D. Uglov et al.
Phys. Lett. A
(1994)
S. Murakami et al.
Phys. Lett. A
(1997)
N. Kawakami et al.
Phys. Lett. A
(1989)
G. Jüttner et al.
Euro. Phys. Lett.
(1997)
G. Jüttner et al.
J. Phys. A
(1997)
B.S. Shastry
J. Stat. Phys.
(1988)
E.H. Lieb et al.
Phys. Rev. Lett.
(1968)
A. Ovchinnikov
Soviet Phys. JETP
(1970)
C. Coll
Phys. Rev. B
(1974)
F. Woynarovich
J. Phys. A
(1982)

K. Tsunetsugu

J. Phys. Soc. Japan

(1991)

Cited by (98)

Canonical formulation for the thermodynamics of sl<inf>n</inf>-invariant integrable spin chains
2024, Nuclear Physics B
Integrable quantum spin chains display distinctive physical properties making them a laboratory to test and assess different states of matter. The study of the finite temperature properties is possible by use of the thermodynamic Bethe ansatz, however at the expense of dealing with non-linear integral equations for, in general, infinitely many auxiliary functions. The definition of an alternative finite set of auxiliary functions allowing for the complete description of their thermodynamic properties at finite temperature and fields has been elusive. Indeed, in the context of $s l_{n}$ -invariant models satisfactory auxiliary functions have been established only for $n \leq 4$ . In this paper we take a step further by proposing a systematic approach to generate finite sets of auxiliary functions for $s l_{n}$ -invariant models. We refer to this construction as the canonical formulation. The numerical efficiency is illustrated for $n = 5$ , for which we present some of the thermodynamic properties of the corresponding spin chain.
The one-dimensional Hubbard model in a magnetic field: Density profiles and ground-state phase diagram
2023, Physics Letters, Section A: General, Atomic and Solid State Physics
We study the single-band one-dimensional Hubbard model in an arbitrary magnetic field using the cumulant Green's functions method (CGFM) with clusters containing 5, 6, 7, and 8 sites. We calculate the occupation numbers as functions of the chemical potential in a positive magnetic field and identify a partially filled region with negative magnetization. We obtain the ground-state phase diagram in chemical potential vs. magnetic field coordinates. We include a straightforward application of the CGFM by using a cluster as a correlated quantum dot connected to correlated leads to construct a single-electron transistor that presents a spin-polarised conductance.
Density matrix renormalization group approach to the low temperature thermodynamics of correlated 1D fermionic models
2022, Journal of Magnetism and Magnetic Materials
The low temperature thermodynamics of correlated 1D fermionic models with spin and charge degrees of freedom is obtained by exact diagonalization (ED) of small systems and followed by density matrix renormalization group (DMRG) calculations that target the lowest hundreds of states ${E (N)}$ at system size $N$ instead of the ground state. Progressively larger $N$ reaches $T < 0.05 t$ in correlated models with electron transfer $t$ between first neighbors and bandwidth $4 t$ . The size dependence of the many-fermion basis is explicitly included for arbitrary interactions by scaling the partition function. The remaining size dependence is then entirely due to the energy spectrum ${E (N)}$ of the model. The ED/DMRG method is applied to Hubbard and extended Hubbard models, both gapped and gapless, with $N_{e} = N$ or $N / 2$ electrons and is validated against exact results for the magnetic susceptibility $χ (T)$ and entropy $S (T)$ per site. Some limitations of the method are noted. Special attention is given to the bond-order-wave phase of the extended Hubbard model with competing interactions and low $T$ thermodynamics sensitive to small gaps.
Pseudoparticle approach to 1D integrable quantum models
2018, Physics Reports
Over the last three decades a large number of experimental studies on several quasi one-dimensional (1D) metals and quasi 1D Mott–Hubbard insulators have produced evidence for distinct spectral features identified with charge-only and spin-only fractionalized particles. They can be also observed in ultra-cold atomic 1D optical lattices and quantum wires. 1D exactly solvable models provide nontrivial tests of the approaches for these systems relying on field theories. Different schemes such as the pseudofermion dynamical theory (PDT) and the mobile quantum impurity model (MQIM) have revealed that the 1D correlated models high-energy physics is qualitatively different from that of a low-energy Tomonaga–Luttinger liquid (TLL). This includes the momentum dependence of the exponents that control the one- and two-particle dynamical correlation functions near their spectra edges and in the vicinity of one-particle singular spectral features.
On the one hand, the low-energy charge-only and spin-only fractionalized particles are usually identified with holons and spinons, respectively. On the other hand, “particle-like” representations in terms of pseudoparticles, related PDT pseudofermions, and MQIM particles are suitable for the description of both the low-energy TLL physics and high-energy spectral and dynamical properties of 1D correlated systems.
The main goal of this review is to revisit the usefulness of pseudoparticle and PDT pseudofermion representations for the study of both static and high-energy spectral and dynamical properties of the 1D Lieb–Liniger Bose gas, spin- $1 ∕ 2$ isotropic Heisenberg chain, and 1D Hubbard model. Moreover, the relation between the PDT and the MQIM is clarified. The fractionalized particles and related composite pseudoparticles/pseudofermions emerging within such non-perturbative 1D correlated systems are qualitatively different from the Fermi-liquid quasiparticles. In contrast to the holons and spinons, the relation to the electron creation and annihilation operators of the operators associated with the 1D Hubbard model three fractionalized particles is uniquely defined. The occupancy configurations of such fractionalized particles generate all energy and momentum eigenstates of that model. Both the static and dynamical properties of the three models under review are shown to be controlled at all energy scales by pseudofermion phase shifts associated with only zero-momentum forward scattering. The corresponding microscopic processes are much simpler than those of the underlying particles non-perturbative interactions.
Absence of ballistic charge transport in the half-filled 1D Hubbard model
2018, Nuclear Physics B
Whether in the thermodynamic limit of lattice length $L \to \infty$ , hole concentration $m_{η}^{z} = - 2 S_{η}^{z} / L = 1 - n_{e} \to 0$ , 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 $S_{η}^{z} = - (L - N_{e}) / 2$ is the η-spin projection and $n_{e} = N_{e} / L$ 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 $m_{η}^{z} \to 0$ within the canonical ensemble in the thermodynamic limit $L \to \infty$ . Moreover, we show that at high temperature $T \to \infty$ the charge stiffness vanishes as well within the grand-canonical ensemble for $L \to \infty$ and chemical potential $μ \to μ_{u}$ where $(μ - μ_{u}) \geq 0$ and $2 μ_{u}$ 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.
Study of local conserved quantities in the one-dimensional Hubbard model
2024, arXiv

View all citing articles on Scopus

^∗: Permanent address: Institute of Physics, University of Tokyo at Komaba.

View full text

The Hubbard chain at finite temperatures: ab initio calculations of Tomonaga-Luttinger liquid properties

Abstract

Phys. Lett. A

Phys. Lett. A

Phys. Lett. A

Euro. Phys. Lett.

J. Phys. A

J. Stat. Phys.

Phys. Rev. Lett.

Soviet Phys. JETP

Phys. Rev. B

J. Phys. A

Z. Physik B

Phys. Rev. B

Phys. Rev. B

Phys. Rev. Lett.

Phys. Rev. Lett.

Phys. Rev. B

Phys. Rev. B

J. Phys. A

Int. J. Mod. Phys. B

Prog. Theor. Phys.

J. Phys. Soc. Japan

Phys. Rev. B

Prog. Theor. Phys.

Prog. Theor. Phys.

J. Phys. Soc. Japan

J. Phys. Soc. Japan