A microscopic approach to phase transitions in quantum systems

Abstract

We present a new theoretical approach for the study of the phase diagram of interacting quantum particles: bosons, fermions or spins. In the neighborhood of a phase transition, the expected renormalization group structure is recovered both near the upper and lower critical dimension. Information on the microscopic hamiltonian is also retained and no mapping to effective field theories is needed. A simple approximation to our formally exact equations is studied for the spin-S Heisenberg model in three dimensions where explicit results for critical exponents, critical temperature and coexistence curve are obtained.Several physical systems, ranging from magnets to superfluids and superconductors, display rich phase diagrams in a temperature regime where quantum effects cannot be neglected. Different scenarios, characterized by competing order parameters and zero temperature phase transitions have been recently advocated also in the framework of high temperature superconductivity where antiferromagnetic order, Cooper pairing and, possibly, phase separation are at play in the same region of the phase diagram [1]. A satisfactory understanding of phase transitions in quantum models has been attained years ago through the seminal work by Hertz [2] who showed that, at low energy and long wavelengths, quantum models may be described by a suitable classical action. However, a quantitative theory of the thermodynamic behavior is still lacking and we mostly rely on mean field approaches or weak coupling renormalization group (RG) calculations [3], applied to quantum systems via the mapping to the appropriate effective field theory. In particular, the interplay between thermal and quantum fluctuations is expected to give rise to crossover phenomena whose extent strongly depends on the microscopic features of the system. Even for the most extensively studied models, like the Heisenberg antiferromagnet, our knowledge of the phase diagram is in fact limited, and the first precise finite temperature simulation attempting to fill this gap has become available only recently [4]. By contrast, in classical models, numerical simulations are quite efficient even in the neighborhood of critical points [5] and, from the analytical side, microscopic approaches especially devised for the quantitative description of the phase diagram of classical fluids and magnets are available. For instance, the hierarchical reference theory of fluids (HRT) [6] has proven quite accurate in locating the phase transition lines both in lattice and in continuous models.In this Letter we sketch the derivation of the quantum hierarchical reference theory of fluids (QHRT) which we then apply to the Heisenberg antiferromagnet. We will demonstrate that the known renormalization group equations near four and near two dimensions are naturally recovered within our approach, which therefore unifies two complimentary techniques. On approaching the critical point, the spin velocity vanishes according to the expected dynamical critical exponent for an antiferromagnet. Fi...