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arXiv:1602.08036 · Journal of Statistical Mechanics Theory and Experiment

Thermodynamics of the d = 3 + 1 quantum XY model

Christoph P Hofmann

Published Sep 1, 2016• 2 citations
arXivHTMLPDF

Abstract

Abstract Within effective field theory we explore the properties of the d = 3 + 1 quantum XY model at low temperatures and in weak magnetic or staggered fields. For this parameter regime only few results appear to be known, and furthermore are restricted to one-loop order. In the present study we systematically analyze the thermodynamics of the d = 3 + 1 quantum XY model up to three-loop order. In the low-temperature expansion of the free energy density, the free Bose gas term of order T 4 receives corrections of order T 6 and T 8 . The discussion also includes the pressure, (staggered) magnetization and susceptibility. In particular, we show how these quantities are influenced by the spin-wave interaction.

DOI
10.1088/1742-5468/2016/09/093102

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Abstract Within effective field theory we explore the properties of the d = 3 + 1 quantum XY model at low temperatures and in weak magnetic or staggered fields. For this parameter regime only few results appear to be known, and furthermore are restricted to one-loop order. In the present study we systematically analyze the thermodynamics of the d = 3 + 1 quantum XY model up to three-loop order. In the low-temperature expansion of the free energy density, the free Bose gas term of order T 4 receives corrections of order T 6 and T 8 . The discussion also includes the pressure, (staggered) magnetization and susceptibility. In particular, we show how these quantities are influenced by the spin-wave interaction.
The low-temperature properties of the (2 + 1)-dimensional quantum XY model are studied within the framework of effective Lagrangians up to three-loop order. At zero temperature, the system is characterized by a spontaneously broken spin rotation symmetry, O (2) → 1, where the corresponding Goldstone bosons are the spin waves or magnons. Even though there is no spontaneously broken order at finite T , the low-temperature behavior of the system is still governed by the spin waves. The partition function is evaluated and various thermodynamic quantities, including the order parameter, are derived in the presence of a weak external field. In particular, we show that the spin-wave interaction is repulsive at low temperatures, its magnitude depending on the strength of the external field. We compare our results with those for the (2 + 1)-dimensional antiferromagnetic Heisenberg model, which, at T = 0, exhibits the spontaneously broken spin rotation symmetry O (3) → O (2).
The thermodynamic properties of a nonrelativistic free-electron Fermi gas is of fundamental interest in condensed matter physics. Properties previously studied in three-dimensions (3D) in the low- and high-temperature limits include the internal energy, heat capacity, zero-field magnetic spin susceptibility, and pressure. Here we report solutions for the temperature dependence spanning these two temperature regimes of the chemical potential, internal energy, magnetic susceptibility, and the heat capacity at constant volume in 1D, 2D, and 3D. Also calculated are the pressure, enthalpy, heat capacity at constant pressure, isothermal compressibility, and thermal expansion coefficient versus temperature in 2D and 3D. Of primary interest here are the detailed dimension-dependent crossovers of these properties between the degenerate and nondegenerate temperature regime, which are graphically illustrated for each of the above properties.
This article reviews recent advances in the theory of the three-dimensional dilute homogeneous Bose gas at zero and finite temperature. Effective-field-theory methods are used to formulate a systematic perturbative framework that can be used to calculate the properties of a system at T=0.T=0.T=0. The perturbative expansion of these properties is essentially an expansion in the gas parameter na3,\sqrt{{\mathrm{na}}^{3}},na3​, where aaa is the sss-wave scattering length and nnn is the number density. In particular, the leading quantum corrections to the ground-state energy density, the condensate depletion, and long-wavelength collective excitations are rederived in an efficient and economical manner. Nonuniversal effects are also discussed. These effects are higher-order corrections that depend on properties of the interatomic potential other than the scattering length, such as the effective range. The article critically examines various approaches to the dilute Bose gas in equilibrium at finite temperature. These include the Bogoliubov approximation, the Popov approximation, the Hartree-Fock-Bogoliubov approximation, the \ensuremath{\Phi}-derivable approach, optimized perturbation theory, and renormalization-group techniques. The article ends with a look at recent calculations of the critical temperature of the dilute Bose gas, which include 1/N1/N1/N techniques, lattice simulations, self-consistent calculations, and variational perturbation theory.
The selfconsistent high-density expansion for thermodynamic quantities of the quantum spin-polarized fermion lattice gas, derived in the preceding paper, is extended to higher orders. The (1/z)2 terms in the free energy are evaluated by summing two-loop diagrams and the corresponding equation for critical temperature in the classical limit is discussed. Ground state phase diagram of the system for arbitrary electron density is derived in the first and second order of the expansion. Properties of the half-filled, one-dimensional spinless fermion model at T = 0, obtained here in the first three orders in 1/z, are compared with the results of exact calculations and of other approximate methods. This provides a test for the unrenormalized high-density expansion method.
We investigate the low-temperature behavior of the integrable one-dimensional two-component spinor Bose gas using the thermodynamic Bethe ansatz. We find that for strong coupling the characteristics of the thermodynamics at low temperatures are quantitatively affected by the spin ferromagnetic states, which are described by an effective ferromagnetic Heisenberg chain. The free energy, specific heat, susceptibility, and local pair correlation function are calculated for various physical regimes in terms of temperature and interaction strength. These thermodynamic properties reveal spin effects which are significantly different than those of the spinless Bose gas. The zero-field susceptibility for finite strong repulsion exceeds that of a free spin paramagnet. The critical exponents of the specific heat ${c}_{v}\ensuremath{\sim}{T}^{1∕2}$ and the susceptibility $\ensuremath{\chi}\ensuremath{\sim}{T}^{\ensuremath{-}2}$ are indicative of the ferromagnetic signature of the two-component spinor Bose gas. Our analytic results are consistent with general arguments by Eisenberg and Lieb for polarized spinor bosons.
We calculate the quantum corrections of the thermodynamic quantities of a system of confined bosons at finite temperature. Systematically quantum corrections are written in a series of $\ensuremath{\Elzxh},$ which is convergent when kT\mathrm{kT}kT is much larger than the spacing between energy levels of the system. We apply this method to the different physical systems. First we calculate the thermodynamic potential of spin-1 bosons confined in a general power-law potential, and calculate the paramagnetic susceptibility of the gas including the quantum corrections. Local-density approximation for charged bosons in the weak magnetic field is improved by the semiclassical expansion. Diamagnetic susceptibility of a system of confined charged bosons is calculated using this method. Finally we apply this method to calculate analytically the thermodynamic potential of a weakly interacting Bose-gas confined in at three-dimensional harmonic-oscillator potential. For a large number of particles, quantum corrections become small, and contribute to the finite-size corrections to scaling.