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Physical Review Letters

Quantum Spherical Models for Dirty Phase Transitions

Yuhai Tu, Peter B. Weichman

Published Jul 4, 1994• 35 citations
Publisher

Abstract

We construct large-$n$ (spherical) limits for a series of interesting quantum phase transitions in disordered systems, including quantum ferromagnets and spin glasses, superconducting thin films with and without an external magnetic field, the dirty boson problem, the fractional quantum Hall effect, and Nelson's model of flux lines in high-temperature superconductors. The spherical limit always produces a random matrix inversion problem with self-consistency conditions, which then must be solved numerically. We present preliminary results for the dirty boson problem in two spatial dimensions.

DOI
10.1103/physrevlett.73.6

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We construct large-nnn (spherical) limits for a series of interesting quantum phase transitions in disordered systems, including quantum ferromagnets and spin glasses, superconducting thin films with and without an external magnetic field, the dirty boson problem, the fractional quantum Hall effect, and Nelson's model of flux lines in high-temperature superconductors. The spherical limit always produces a random matrix inversion problem with self-consistency conditions, which then must be solved numerically. We present preliminary results for the dirty boson problem in two spatial dimensions.
The spherical limit of strongly commensurate dirty bosons is studied perturbatively at weak disorder and numerically at strong disorder in two dimensions (2D). We argue that disorder is not perfectly screened by interactions and consequently that the ground state in the effective Anderson localization problem always remains localized. As a result there is only a gapped Mott insulator phase in the theory. Comparisons with other studies and the parallel with disordered fermions in 2D are discussed. We conjecture that while for the physical cases N = 2 (XY) and N = 1 (Ising) the theory should have the ordered phase, it may not for N = 3 (Heisenberg).
Superfluid and superconducting phase transitions of interacting bosons are observed in diverse physical systems such as helium, superconducting materials, ultra-cold atomic gases, etc. For all these systems one-, two-, and three-dimensional (1D, 2D, and 3D) realizations are experimentally accessible. However, the current experimental focus is shifting to the low dimensional systems (e.g., films and wires) due to facilitated measurement techniques and a constant demand of technological applications. From a theoretical standpoint large fluctuations in 1D and 2D systems make them a challenging problem, especially when disorder is present. Here we study peculiarities associated with disorder in low-dimensional cases. Specifically, we are interested in the class of the systems where superfluid or superconducting transitions can be mapped to the finite temperature 2D superfluid-normal Kosterlitz-Thouless (KT) transition: (a) 2D bosonic system with a strong diagonal disorder, (b) 1D system of lattice bosons with off-diagonal disorder in the limit of large integer filling factor at zero temperature, and (c) 3D layered superconductors with no tunnelling between layers in zero external magnetic field.
A classical spherical random-field Hamiltonian with long-range (power-law) interactions is investigated by means of the replica theory. Both ferromagnetic and anti-ferromagnetic interactions are considered. The use of continuous variables instead of Ising variables in the spherical version of the model allows one to calculate the free energy exactly. The existence of an equilibrium phase transition is investigated based on the replica-symmetric solution. The results are applied to the Coulomb-glass model of interacting localized electrons in a disordered solid. This model is shown not to have an equilibrium phase transition for spatial dimensions D<or=4. For D>4 the model has a phase transition to an ordered phase; however, it does not have a phase transition to a 'glassy' phase.
We analyze finite-temperature classical phase transitions that a two-dimensional collection of directed line objects placed in various external potentials may undergo. We find the conditions for a spin-glass-like transition in a disordered material and show that the collection of flux lines in dirty two-dimensional superconductors always stays in a vortex-glass phase. In addition, we resolve some of the controversies on the subject existing in the literature. We also look at a related problem of zero-temperature quantum phase transitions that a one-dimensional many-body system placed in various external potentials may undergo. In particular we predict that short-range repulsive bosons with sufficiently strong interaction will be pinned by a single impurity while translational symmetry is not broken for weakly repulsive particles; there is a phase transition between the indicated regimes. Taking into account the effects of commensurability, we construct the phase diagram for the conductive properties of a one-dimensional quantum liquid in the presence of a single impurity.
The spherical model for spins describes ferromagnetic phase transitions well, but it fails at low temperatures. A quantum version of the spherical model is proposed. It does not induce qualitative changes near the phase transition. However, it produces a physical low temperature behavior. The entropy is non-negative. Model parameters can be adapted to the description of real quantum spins. Several applications are discussed. Zero-temperature quantum phase transitions are analyzed for a ferromagnet and a spin glass in a transverse field. Their crossover exponents are presented.
The large-NNN commensurate dirty boson model, in both the weakly and strongly commensurate cases, is considered via a perturbative renormalization-group treatment. In the weakly commensurate case, there exists a fixed line under renormalization-group flow, with varying amounts of disorder along the line. Including 1/N1/N1/N corrections causes the system to flow to strong disorder, indicating that the model does not have a phase transition perturbatively connected to the Mott insulator-superfluid (MI-SF) transition. I discuss the qualitative effects of instantons on the low-energy density of excitations. In the strongly commensurate case, a fixed point found previously is considered and results are obtained for higher moments of the correlation functions. To lowest order, correlation functions have a log-normal distribution. Finally, I prove two interesting theorems for large-NNN vector models with disorder, relevant to the problem of replica symmetry breaking and frustration in such systems.