We introduce a one-parameter deformation for one-dimensional quantum lattice models, the hyperbolic deformation, where the scale of the local energy is proportional to cosh λj at the jth site. From the quantum-classical correspondence with a two-dimensional classical system, it is expected that the deformation does not modify the ground state conspicuously, if there is a finite excitation gap. Under this situation, the excited quasi particle is weakly confined around the center of the system. We derive scaling relations among the mean-square width 〈w2〉 of confinement, energy correction to the excitation gap Δ, and deformation parameter λ. This scaling is useful for the extraction of the bulk excitation gap in the limit λ = 0.
We present a way of numerical gap estimation applicable for one-dimensional infinite uniform quantum systems. Using the density matrix renormalization group method for a non-uniform Hamiltonian, which has deformed interaction strength of
j-th bond proportional to
coshλj, the uniform Hamiltonian is analyzed as a limit of
λ→0. As a consequence of the deformation, an excited quasi-particle is weakly bounded around the center of the system, and kept away from the system boundary. Therefore, insensitivity of an estimated excitation gap of the deformed system to the boundary allows us to have the bulk excitation gap
Δ(λ), and shift in
Δ(λ) from
Δ(0) is nearly linear in
λ when
λ≪1. Efficiency of this estimation is demonstrated through application to the S=1 antiferromagnetic Heisenberg chain. Combining the above estimation and another one obtained from the technique of convergence acceleration for finite-size gaps estimated by numerical diagonalizations, we conclude that the Haldane gap is in
[0.41047905, 0.41047931].
We study finite-size effects for the gap of the quasiparticle excitation spectrum in the weakly interacting regime one-dimensional Hubbard model with on-site attraction. Two types of corrections to the result of the thermodynamic limit are obtained. Aside from a power law (conformal) correction due to gapless excitations which behaves as
1/Na, where
Na is the number of lattice sites, we obtain corrections related to the existence of gapped excitations. First of all, there is an exponential correction which in the weakly interacting regime ($|U|\ensuremath{\ll}t$) behaves as
$\mathrm{exp}(\ensuremath{-}{N}_{a}{\ensuremath{\Delta}}_{\ensuremath{\infty}}/4t)$ in the extreme limit of ${N}_{a}{\ensuremath{\Delta}}_{\ensuremath{\infty}}/t\ensuremath{\gg}1$, where
t is the hopping amplitude,
U is the on-site energy, and ${\ensuremath{\Delta}}_{\ensuremath{\infty}}$ is the gap in the thermodynamic limit. Second, in a finite-size system a spin-flip producing unpaired fermions leads to the appearance of solitons with nonzero momenta, which provides an extra (nonexponential) contribution $\ensuremath{\delta}$. For moderate but still large values of ${N}_{a}{\ensuremath{\Delta}}_{\ensuremath{\infty}}/t$, these corrections significantly increase and may become comparable with the
1/Na conformal correction. Moreover, in the case of weak interactions where ${\ensuremath{\Delta}}_{\ensuremath{\infty}}\ensuremath{\ll}t$, the exponential correction exceeds higher-order power law corrections in a wide range of parameters, namely for ${N}_{a}\ensuremath{\lesssim}(8t/{\ensuremath{\Delta}}_{\ensuremath{\infty}})\mathrm{ln}(4t/|U|)$, and so does $\ensuremath{\delta}$ even in a wider range of
Na. For a sufficiently small number of particles, which can be of the order of thousands in the weakly interacting regime, the gap is fully dominated by finite-size effects.
A group of non-uniform quantum lattice Hamiltonians in one dimension is introduced, which is related to the hyperbolic
1+1-dimensional space. The Hamiltonians contain only nearest neighbor interactions whose strength is proportional to
coshjλ, where
j is the lattice index and where
λ≥0 is a deformation parameter. In the limit
λ→0 the Hamiltonians become uniform. Spacial translation of the deformed Hamiltonians is induced by the corner Hamiltonians. As a simple example, we investigate the ground state of the deformed
S=1/2 Heisenberg spin chain by use of the density matrix renormalization group (DMRG) method. It is shown that the ground state is dimerized when
λ is finite. Spin correlation function show exponential decay, and the boundary effect decreases with increasing
λ.
We investigate the effect of a nonuniform deformation applied to one-dimensional (1D) quantum systems, where the local energy scale is proportional to ${g}_{j}=[\mathrm{sin}(j\ensuremath{\pi}/N)]{}^{m}$ determined by a positive integer
m, site index $1\ensuremath{\leqslant}j\ensuremath{\leqslant}N\ensuremath{-}1$, and system size
N. This deformation introduces a smooth boundary to systems with open-boundary conditions. When $m\ensuremath{\geqslant}2$, the leading
1/N correction to the ground-state energy per bond
e0(N) vanishes and one is left with a
1/N2 correction, the same as with periodic boundary conditions. In particular, when
m=2, the value of
e0(N) obtained from the deformed open-boundary system coincides with the uniform system with periodic boundary conditions. We confirm the fact numerically for correlated systems, such as the extended Hubbard model, in addition to 1D free-fermion models.
In this paper we propose an efficient routine to carry out finite size scaling studies of one-dimensional massive lattice models. Unlike massless systems, for which the lattice and continuum models yield the same results for low-lying excitations due to the infinite correlation length, the massive continuum models can at best approximate their lattice counterparts because of the intrinsic length scale $\ensuremath{\xi}\ensuremath{\sim}{\ensuremath{\Delta}}^{\ensuremath{-}1},$ where $\ensuremath{\Delta}$ is the mass gap. On examples of antiferromagnetic gapped spin-1/2
XXZ chains we show explicitly that several relations between physical quantities like the mass gap, spin-wave velocity
v, and the correlation length \ensuremath{\xi}, derived from the continuum models deviate significantly from the numerical results obtained for finite chains, when $\ensuremath{\xi}$ is comparable with the lattice spacing
a0. On the other hand, we find if the dispersion for elementary excitations is modified from the ``Lorentz invariant'' form $\sqrt{{\ensuremath{\Delta}}^{2}{+v}^{2}{p}^{2}}$ to $\sqrt{{\ensuremath{\Delta}}^{2}{+v}^{2}{\mathrm{sin}}^{2}p},$ the spin-wave velocity
v becomes almost size independent, and the appropriately defined correlation length using that dispersion agrees very well with numerical results. In fact, this follows from the Bethe ansatz solution for the spin-1/2
XXZ chains. Based on our previous experience and these considerations we propose scaling equations to obtain the ground-state energy, mass gap, correlation length, spin-wave velocity, and scattering length between the massive elementary excitations. Only three low energy levels are needed in this method. Although the method is illustrated on the gapped
XXZ quantum spins, it is applicable to all one-dimensional lattice models with massive relativistic low energy dispersions. To substantiate our statement we show explicitly that the numerical data for the Ising model in a transverse magnetic field fully agree with the finite-size scaling based on the correlation length newly defined from the exact analytic solution.
A q -deformed SU(2) Heisenberg (3-dimensional) spin model is set up, and the q -deformed spin-wave solution is obtained through the q -analogous Holstein-Primakoff transformation. The result is given for small γ = ln q , which is the quantity characterizing the nonlinearity of the Hamiltonian. A mean-field treatment is arranged to preserve (at least some of) the nonlinearity, and the ordinary ferromagnet ground state is shown as the exact ground state of the new system. Interesting results are obtained for this nonlinear model:(i) There is an energy gap between the ground state and the first excited one, thus the ground state is stable under small perturbation of the background; (ii) the specific heat per volume is modified by a small term proportional to the (1/2)-th power of temperature and the square of γ , which is qualitatively different from the conventional model, and (iii) the magnetization M(T) is modified by a factor that depends on γ .