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arXiv:1306.1944 · Journal of Statistical Mechanics Theory and Experiment

Systematic effective field theory analysis of the D=2+1 quantum XY model at low temperatures

Christoph P. Hofmann

Published Feb 14, 2014• 7 citations
arXivHTMLPDF

Abstract

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).

DOI
10.1088/1742-5468/2014/02/p02006

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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).
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.
In the previous papers, we studied the bosonic t-J mode and derived an effective field theory, which is a kind of quantum XY model. The bosonic t-J model is expected to be realized by experiments of two-component cold atoms in an optical lattice. In this paper, we consider a similar XY model that describes phase diagram of the t-J model with a mass difference. Phase diagram and critical behavior of the quantum XY model are clarified by means of the Monte-Carlo simulations. Effective field theory that describes the phase structure and low-energy excitations of the quantum XY model is derived. Nambu-Goldstone bosons and the Higgs mode are studied by using the effective field theory and interesting findings are obtained for the system with multiple order, i.e., Bose-Einstein condensations and pseudo-spin symmetry. We also investigate physical properties of the quantum XY model in an effective field that is realized by rotating the optical lattice, etc. We show that low-energy states of the system strongly depend on the strength of the magnetic field. For some specific strength of the field, vortex lattice forms and the correlation function of the bosons exhibits solid like behavior, which is a kind of Bose-Einstein condensation.
Based on a symmetry analysis of the microscopic Hubbard and ttt-JJJ models, a systematic low-energy effective field theory is constructed for hole-doped antiferromagnets on the honeycomb lattice. In the antiferromagnetic phase, doped holes are massive due to the spontaneous breakdown of the SU(2)s\text{SU}{(2)}_{s}SU(2)s​ symmetry, just as nucleons in Quantum Chromodynamics (QCD) pick up their mass from spontaneous chiral symmetry breaking. In the broken phase, the effective action contains a single-derivative term, similar to the Shraiman-Siggia term in the square lattice case. Interestingly, an accidental continuous spatial rotation symmetry arises at leading order. As an application of the effective field theory, we consider one-magnon exchange between two holes and the formation of two-hole bound states. As an unambiguous prediction of the effective theory, the wave function for the ground state of two holes bound by magnon exchange exhibits fff-wave symmetry.
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In the previous papers, we studied the bosonic t-J mode and derived an effective field theory, which is a kind of quantum XY model. The bosonic t-J model is expected to be realized by experiments of two-component cold atoms in an optical lattice. In this paper, we consider a similar XY model that describes phase diagram of the t-J model with a mass difference. Phase diagram and critical behavior of the quantum XY model are clarified by means of the Monte-Carlo simulations. Effective field theory that describes the phase structure and low-energy excitations of the quantum XY model is derived. Nambu-Goldstone bosons and the Higgs mode are studied by using the effective field theory and interesting findings are obtained for the system with multiple order, i.e., Bose-Einstein condensations and pseudo-spin symmetry. We also investigate physical properties of the quantum XY model in an effective magnetic field that is realized by rotating the optical lattice, etc. We show that low-energy states of the system strongly depend on the strength of the "magnetic field". For some specific strength of the magnetic field, vortex lattice forms and the correlation function of the bosons exhibits solid like behavior, which is a kind of Bose-Einstein condensation.
The systematic effective Lagrangian method was first formulated in the context of the strong interaction; chiral perturbation theory (CHPT) is the effective theory of quantum chromodynamics (QCD). It was then pointed out that the method can be transferred to the nonrelativistic domain—in particular, to describe the low-energy properties of ferromagnets. Interestingly, whereas for Lorentz-invariant systems the effective Lagrangian method fails in one spatial dimension (ds=1) , it perfectly works for nonrelativistic systems in ds=1 . In the present brief review, we give an outline of the method and then focus on the partition function for ferromagnetic spin chains, ferromagnetic films, and ferromagnetic crystals up to three loops in the perturbative expansion—an accuracy never achieved by conventional condensed matter methods. We then compare ferromagnets in ds=1 , 2, 3 with the behavior of QCD at low temperatures by considering the pressure and the order parameter. The two apparently very different systems (ferromagnets and QCD) are related from a universal point of view based on the spontaneously broken symmetry. In either case, the low-energy dynamics is described by an effective theory containing Goldstone bosons as basic degrees of freedom.