We investigate the large-distance asymptotic behavior of the static density-density and field-field correlation functions in the one-dimensional Bose gas at finite temperature. The asymptotic expansions of the Bose gas correlators are obtained performing a specific continuum limit in the similar low-temperature expansions of the longitudinal and transversal correlation functions of the
XXZ spin chain. In the lattice system the correlation lengths are computed as ratios of the largest and next-largest eigenvalues of the
XXZ spin chain quantum transfer matrix. In both cases, lattice and continuum, the correlation lengths are expressed in terms of solutions of Yang-Yang type [C. N. Yang and C. P. Yang, J. Math. Phys. 10, 1151 (1969)] nonlinear integral equations which are easily implementable numerically.
We consider the low-temperature limit of the long-distance asymptotic behavior of the finite-temperature density–density correlation function in the one-dimensional Bose gas derived recently in the algebraic Bethe ansatz framework. Our results confirm the predictions based on the Luttinger liquid and conformal field theory approaches. We also demonstrate that the amplitudes arising in this asymptotic expansion at low temperature coincide with the amplitudes associated with the so-called critical form factors.
We consider the low-temperature limit of the long-distance asymptotic behavior of the finite-temperature density–density correlation function in the one-dimensional Bose gas derived recently in the algebraic Bethe ansatz framework. Our results confirm the predictions based on the Luttinger liquid and conformal field theory approaches. We also demonstrate that the amplitudes arising in this asymptotic expansion at low temperature coincide with the amplitudes associated with the so-called critical form factors.
We investigate analytically the magnetization of Heisenberg ferromagnetic lattices in one and two dimensions, and we derive approximate expressions that are valid at high and low temperatures. In the case of the spin-
21 Heisenberg XX chain in a transverse field, we show that, when the applied magnetic field
h exceeds its critical value
hc=J, where
J is the exchange coupling constant, the magnetization per site deviates at low temperatures from its saturation value,
21, following a power series with terms involving the power laws
T1/2,
T3/2,
T5/2 ..etc. When
h<hc, the zero temperature magnetization per spin turns out to be equal to
π1arcsin(hch). In this case, the temperature dependence of the magnetization is given by a series expansion with power laws of the form
T,
T3,
T5,...etc. In both cases, the coefficients of the expansions are temperature-dependent and are explicitly derived. Using he properties of the Eulerian polynomials, we show that, because of the fast convergence of the derived series for the Fermi-Dirac and the Bose-Einstein distributions, it is possible (in particular in strong magnetic fields) to express the magnetization of the Heisenberg model in a simple analytical form. Furthermore, the analytical results are compared with the exact numerical ones.
Recently a new integral equation describing the thermodynamics of the 1D Heisenberg model was discovered by Takahashi. Using the integral equation we have succeeded in obtaining the high-temperature expansion of the specific heat and the magnetic susceptibility up to O[(J/T)(100)]. This is much higher than those obtained so far by the standard methods such as the linked-cluster algorithm. Our results will be useful to examine various approximation methods to extrapolate the high-temperature expansion to the low temperature region.
This thesis contains two results for the low temperature behavior of quantum spin systems. First, we present a lower bound for the spin-1 XXZ chain in finite volumes in terms of the gap of the two-site Hamiltonian. The estimate is derived by a method developed by Nachtergaele in (cond-mat/9410110) called the Martingale Method. Our bound relies on an assumption which we have, as yet, been unable to verify analytically in all cases. We present numerical evidence that strongly indicates our assumption is valid. The second result is a proof that the spin-1/2, d-dimensional XY model in the presence of an external magnetic field does not undergo a phase transition at low temperature, provided that the strength of the field is great enough. Using a contour expansion inspired by Kennedy, we show that the weights of contours satisfy a condition of Kotecky and Preiss which allows us to express the free energy of the system as a cluster expansion. As part of the setup we give a simple proof that the all-spin-up state is the unique ground state when the external magnetic field has strength at least 2d.