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Advanced series in mathematical physics

Twisted Boundary Conditions and Effective Mass in Heisenberg-Ising and Hubbard Rings

B. Sriram Shastry, Bill Sutherland

Published May 1, 1994• 0 citations
Publisher

Abstract

We identify the boundary energy of a many-body system of fermions on a lattice under twisted boundary conditions as the inverse of the effective charge-carrying mass, or the stiffness, renormalizing nontrivially under interactions due to the absence of Galilean invariance. We point out that this quantity is a sensitive and direct probe of the metal-insulator transitions possible in these systems, i.e., the Mott-Hubbard transition or Density-wave formation. We calculate exactly the stiffness, or the effective mass, in the 1D Heisenberg-Ising ring and the 1D Hubbard model by using the ansatz of Bethe. For the Hubbard ring we also calculate a spin stiffness by extending the nested ansatz of Bethe-Yang to this case.

DOI
10.1142/9789812798268_0024

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We identify the boundary energy of a many-body system of fermions on a lattice under twisted boundary conditions as the inverse of the effective charge-carrying mass, or the stiffness, renormalizing nontrivially under interactions due to the absence of Galilean invariance. We point out that this quantity is a sensitive and direct probe of the metal-insulator transitions possible in these systems, i.e., the Mott-Hubbard transition or Density-wave formation. We calculate exactly the stiffness, or the effective mass, in the 1D Heisenberg-Ising ring and the 1D Hubbard model by using the ansatz of Bethe. For the Hubbard ring we also calculate a spin stiffness by extending the nested ansatz of Bethe-Yang to this case.
We identify the boundary energy of a many-body system of fermions on a lattice under twisted boundary conditions as the inverse of the effective charge-carrying mass, or the stiffness, renormalizing nontrivially under interactions due to the absence of Galilean invariance. We point out that this quantity is a sensitive and direct probe of the metal-insulator transitions possible in these systems, i.e., the Mott-Hubbard transition or Density-wave formation. We calculate exactly the stiffness, or the effective mass, in the 1D Heisenberg-Ising ring and the 1D Hubbard model by using the ansatz of Bethe. For the Hubbard ring we also calculate a spin stiffness by extending the nested ansatz of Bethe-Yang to this case.
The dc conductivity for the t-J model is exactly obtained at t=J and for the Hubbard model in one dimension. Our calculation is carried out for arbitrary band filling by use of the finite-size-scaling method based on the Bethe-ansatz solution under twisted boundary conditions. The effective transport mass defined as the inverse of the conductivity serves as a sensitive probe of the metal-insulator transition. The effective mass is extremely enhanced near half-filling both for the t-J model at t=J and the repulsive Hubbard model, which is a direct evidence of the metal-insulator transition due to the electron correlation. We observe that the large antiferromagnetic coupling J in the t-J model increases the effective mass. On the other hand, the conductivity in the attractive Hubbard model exhibits quite different properties from the repulsive models.
We consider the Ising phase of the antiferromagnetic XXZ Heisenberg chain on a finite-size lattice with N sites. We compute the large-NNN behavior of the spin stiffness, obtaining the correlation length $\ensuremath{\xi}.$ We use our results to discuss the scaling behavior of the metal-insulator transitions in one-dimensional systems, taking into account the mapping between the XXZ Heisenberg chain and the spinless fermion model, and known results for the Hubbard model. We study the scaling properties of both the Hubbard model and the XXZ Heisenberg chain by solving numerically the Bethe ansatz equations. We find that for some range of values of $\ensuremath{\xi}/N,$ the scaling behavior may be observed for the Hubbard model but not for the XXZ Heisenberg chain. We show how $\ensuremath{\xi}$ can be obtained from the scaling properties of the spin stiffness for small system sizes. This method can be applied to models having not an exact solution, illuminating their transport properties.
The one-dimensional attractive lattice fermion gas equivalent to the Heisenberg-Ising spin-1/2 chain is studied for a ring geometry threaded by magnetic flux. We find that for charged fermions having interaction strength \ensuremath{\Delta}=cos(\ensuremath{\pi}/p) with p noninteger, the adiabatic ground state is periodic in the magnetic flux threading the chain, with period two flux quanta, as found by Shastry and Sutherland for the repulsive case. We find that, at particular values of the threading field, a sequence of initially zero-energy bound states form at the Fermi surface during the adiabatic process, the largest containing [p] (the integer part of p) fermions. This largest bound state moves rapidly around to the other Fermi point where it sequentially unbinds. We find Berry's phase for the whole process to be [p]\ensuremath{\pi}. For p integer, as \ensuremath{\Phi} increases, eventually all the particles in the system go into bound states of size [p]. The period in this case is of order the size of the system. The charge-carrying mass of the fermions is calculated by finding the energy of the adiabatic ground state with twisted boundary conditions.
Thermal transport in condensed matter systems is traditionally formulated as a response to a background gravitational field. In this work, we seek a twisted-boundary-condition formalism for thermal transport in analogy to the U(1) twisted boundary condition for electrical transport. Specifically, using the transfer matrix formalism, we introduce what we call the energy-twisted boundary condition, and study the response of the system to the boundary condition. As specific examples, we obtain the thermal Meissner stiffness of (1+1)-dimensional CFT, the Ising model, and disordered fermion models. We also identify the boost deformation of integrable systems as a bulk counterpart of the energy-twisted boundary condition. We show that the boost deformation of the free fermion chain can be solved explicitly by solving the inviscid Burgers equation. We also discuss the boost deformation of the XXZ model, and its nonlinear thermal Drude weights, by studying the boost-deformed Bethe ansatz equations.