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arXiv:cond-mat/9401040 · Physical review. B, Condensed matter

Finite-temperature properties of quantum antiferromagnets in a uniform magnetic field in one and two dimensions

Subir Sachdev A, T. Senthil, R. Shankar

Published Jul 1, 1994• 190 citations
arXivHTMLPDF

Abstract

Consider a d-dimensional antiferromagnet with a quantum disordered ground state and a gap to bosonic excitations with nonzero spin. In a finite external magnetic field, this antiferromagnet will undergo a phase transition to a ground state with nonzero magnetization, describable as the condensation of a dilute gas of bosons. The finite-temperature properties of the Bose gas in the vicinity of this transition are argued to obey a hypothesis of zero scale-factor universality for d2, with logarithmic violations in d=2. Scaling properties of various experimental observables are computed in an expansion in \ensuremath{\epsilon}=2-d, and exactly in d=1.

DOI
10.1103/physrevb.50.258

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Consider a d-dimensional antiferromagnet with a quantum disordered ground state and a gap to bosonic excitations with nonzero spin. In a finite external magnetic field, this antiferromagnet will undergo a phase transition to a ground state with nonzero magnetization, describable as the condensation of a dilute gas of bosons. The finite-temperature properties of the Bose gas in the vicinity of this transition are argued to obey a hypothesis of zero scale-factor universality for d2, with logarithmic violations in d=2. Scaling properties of various experimental observables are computed in an expansion in \ensuremath{\epsilon}=2-d, and exactly in d=1.
We study frustrated, two-dimensional, quantum antiferromagnets in the vicinity of a quantum transition from a non-collinear, magnetically-ordered ground state to a quantum disordered phase. The general scaling properties of this transition are described. A detailed study of a particular field-theoretic model of the transition, with bosonic spin-1/2 spinon fields, is presented. Explicit universal scaling forms for a variety of observables are obtained and the results are compared with numerical data on the spin-1/2 triangular antiferromagnet. Universal properties of an alternative field-theory, with confined spinons, are also briefly noted.
We investigate the thermodynamic properties of the frustrated bilayer quantum Heisenberg antiferromagnet at low temperatures in the vicinity of the saturation magnetic field. The low-energy degrees of freedom of the spin model are mapped onto a hard-square gas on a square lattice. We use exact diagonalization data for finite spin systems to check the validity of such a description. Using a classical Monte Carlo method we give a quantitative description of the thermodynamics of the spin model at low temperatures around the saturation field. The main peculiarity of the considered two-dimensional Heisenberg antiferromagnet is related to a phase transition of the hard-square model on the square lattice, which belongs to the two-dimensional Ising model universality class. It manifests itself in a logarithmic (low-) temperature singularity of the specific heat of the spin system observed for magnetic fields just below the saturation field.
The thermodynamic properties of a Bose gas of particles with nonzero spin and magnetic moment in states with and without a Bose-Einstein condensate are studied as a function of the external magnetic field. It is shown that there exists a temperature interval where a transition into a state with a Bose-Einstein condensate occurs in a Bose gas with constant temperature and particle number density when the magnetic field is increasing and reaches a critical magnitude. The temperature and magnetic field derivatives of the specific heat at constant volume and the magnetic susceptibility undergo a jump at such a transition.
We investigate a simple model antiferromagnet which shows a Quantum Phase Transition between a conventional Neel antiferromagnetic phase and a dimerized ‘Valence Bond Solid’ phase. Both the critical temperature and the ground state magnetization in the Neel phase approach zero at the critical point, and are found to scale according to power laws, with exponents 0.5, as expected from general considerations. 1. Introduction The subject of Quantum Phase Transitions (QPT’s) in condensed matter systems remains a frontier area of research [1]. These are transitions, at temperature T = 0, in the nature of the ground state of a strongly correlated quantum system. At QPT’s large quantum fluctuations play the same role as do large thermal fluctuations at normal phase transitions, and analogous universal scaling laws are to be expected. In real materials QPT’s can be induced by pressure, by applied magnetic fields, or by disorder. Theoretical studies of QPT’s are largely based on various simplified models, particularly low dimensional antiferromagnets, where the system can be tuned through a QPT by varying a particular parameter g in the Hamiltonian. Most of the models hereto studied have been two-dimensional. Examples include antiferromagnets with strong and weak bonds, with or without frustration, and bilayer systems, where the QPT separates a conventional Neel antiferromagnetic phase from a dimerized phase with only short-range correlations and no magnetic order. In such systems the magnetic order present in the ground state does not extend to finite temperatures. In the present work we study a three-dimensional model where magnetic order persists to some critical temperature Tc(g). We expect Tc(g) to vanish as g → gc, and we are interested in comparing the vanishing of Tc with the vanishing of the ground state magnetization M0. The magnetization is expected to vanish as (gc – g) β , where the critical exponent β is expected to have the value 0.5, corresponding to a classical thermally driven transition in four spatial dimensions, which is mean-field like (with possible logarithmic corrections). Does Tc(g) scale with (gc – g) in the same way? 2. The model and results
We present closed-form, analytical expressions for the thermodynamic properties of an ideal, two-dimensional (2D) charged Fermi or Bose gas in the presence of a uniform magnetic field of arbitrary strength. We consider both the homogeneous quantum gas (in which case our expressions are exact) and the inhomogeneous gas within the local-density approximation. Our results for the Fermi gas are relevant to the current-density-functional theory of low-dimensional electronic systems in magnetic fields. For a 2D charged Bose gas (CBG) in a homogeneous magnetic field, we show that the uniform system undergoes a sharp transition at a critical temperature ${T}_{c}^{\ensuremath{\star}},$ below which there is a macroscopic occupation of the lowest Landau level. An examination of the one-body density matrix, however, reveals the absence of long-range order, thereby indicating that the transition cannot be interpreted to a Bose-Einstein condensate. Nevertheless, for $T<{T}_{c}^{\ensuremath{\star}}$ and weak magnetic fields, the system still exhibits magnetic properties which are practically indistinguishable from those of a condensed, superconducting CBG. We therefore conclude that while a condensate is a sufficient condition for the ideal CBG to exhibit a superconducting state, it may not be a necessary condition.