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Physical Review Letters

New stochastic method for systems with broken time-reversal symmetry: 2D fermions in a magnetic field

Gerardo Ortíz, David M. Ceperley, Richard M. Martin

Published Oct 25, 1993• 175 citations
Publisher

Abstract

We present a stochastic method able to deal with complex Hermitian Hamiltonians where time reversal invariance is broken explicitly. We fix the phase of the wave function and show that the equation for the modulus can be solved by quantum Monte Carlo techniques. Then, any choice for its phase provides a variational upper bound for the ground state energy of the system. We apply the fixed-phase method to the 2D electron gas in the presence of a magnetic field with generalized periodic boundary conditions, where we study the transition between an incompressible \ensuremath{\nu}=1/m Laughlin liquid and a Wigner crystal.

DOI
10.1103/physrevlett.71.2777

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EXPLORING THE QUANTUM WORLD OF COMPLEX STATES

Gerardo Ortíz, M. D. JONES

Solving the fundamental microscopic equations of interacting quantum particles is a goal of many-body physicists. Statistical methods reduce the complexity of the problem by sampling phase space selectively using random-walks and real states. Many interesting physical phenomena (e.g., electrons in external magnetic fields) involve systems whose state functions are inherently complex-valued. The Fixed-Phase method is a stochastic approach to deal with such problems. Its key ingredient is a trial phase that plays the role of gauge function in the transformation that maps the original fermion (or boson) problem to a boson problem for the modulus of the state function. The Released-Phase method relaxes that constraint and allows us to obtain, in principle, the "exact" properties, although it is subjected to the infamous "phase problem." In our tour of the (complex) Quantum World, we will show how these methods have been successfully applied to a wide variety of physical phenomena ranging from quantum Hall topological fluids and Wigner crystals to the study of the core structure of vortices in superfluid 4 He and atomic systems in superstrong magnetic fields found in astrophysical settings.

International Journal of Modern Physics B•Mar 10, 1999•0 citations
We present a stochastic method able to deal with complex Hermitian Hamiltonians where time reversal invariance is broken explicitly. We fix the phase of the wave function and show that the equation for the modulus can be solved by quantum Monte Carlo techniques. Then, any choice for its phase provides a variational upper bound for the ground state energy of the system. We apply the fixed-phase method to the 2D electron gas in the presence of a magnetic field with generalized periodic boundary conditions, where we study the transition between an incompressible \ensuremath{\nu}=1/m Laughlin liquid and a Wigner crystal.
We elaborate on the methodology to simulate bulk systems in the absence of time-reversal symmetry by the phase-fixed path-integral Monte Carlo method under (possibly twisted) periodic boundary conditions. Such systems include two-dimensional electrons in the quantum Hall regime and rotating ultracold Bose and Fermi gases; time-reversal symmetry is broken by an external magnetic field and the Coriolis force, respectively. We provide closed-form expressions in terms of Jacobi elliptic functions for the thermal density matrix (or the Euclidean propagator) of a single particle on a flat torus under very general conditions. We then modify the multislice sampling method in order to sample paths by the magnitude of the complex-valued thermal density matrix. Finally, we demonstrate that these inventions let us study the vortex melting process of a two-dimensional Yukawa gas in terms of the de Boer interaction strength parameter, temperature, and rotation (Coriolis force). The bosonic case is relevant to ultracold Fermi-Fermi mixtures of widely different masses under rotation.
We propose a ground-state trial wave function for a two-dimensional Wigner crystal in a strong perpendicular magnetic field. The wave function includes Laughlin-Jastrow correlations between electron pairs, and may be interpreted as a crystal state of composite fermions or composite bosons. Treating the power mmm of the Laughlin-Jastrow factor as a variational parameter, we use quantum Monte Carlo simulations to compute the energy of these new states. We find that our wave functions have lower energy than existing crystalline wave functions in the lowest Landau level. Our results are consistent with experimental observations of the filling factor at which the transition between the fractional quantum Hall liquid and the Wigner crystal occurs for electron systems. Exchange contributions to the wave functions are estimated quantitatively and shown to be negligible for sufficiently small filling factors.
The two-dimensional Wigner crystals are studied with the variational quantum Monte Carlo method. The close relationship between the ground-state wave function and the collective excitations in the system is illustrated, and used to guide the construction of the ground-state wave function of the strongly correlated solid. Exchange, correlation, and magnetic-field effects all give rise to distinct physical phenomena. In the absence of any external magnetic field, interesting spin orderings are observed in the ground state of the electron crystal in various two-dimensional lattices. In particular, two-dimensional bipartite lattices are shown not to lead necessarily to an antiferromagnetic ground state. In the quantum Hall effect regime, a strong magnetic field introduces energy and length scales. The magnetic field quenches the kinetic energy and poses constraints on how the electrons may correlate with each other. Care is taken to ensure the appropriate translational properties of the wave function when the system is in a uniform magnetic field. We have examined the exchange, intra-Landau-level correlation as well as Landau-level-mixing effects with various variational wave functions. We also determine their dependences on the experimental parameters such as the carrier effective mass at a modulation-doped semiconductor heterojunction. Our results, when combined with some recent calculations for the energy of the fractional quantum Hall liquid including Landau-level-mixing, show quantitatively that in going from n doping to p doping in GaAs/Alx{\mathrm{Al}}_{\mathit{x}}Alx​${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As heterojunction systems, the crossover filling factor from the fractional quantum Hall liquid to the Wigner crystal changes from filling factor \ensuremath{\nu}\ensuremath{\sim}1/5 to \ensuremath{\nu}\ensuremath{\sim}1/3. This lends strong support to the claim that the observed reentrant insulating phases around \ensuremath{\nu}=1/5 for n-doped and around \ensuremath{\nu}=1/3 for p-doped high-mobility samples are primarily caused by electron-electron interaction effects. We discuss the possible implications of our theoretical results for some recent experiments carried out in the quantum Hall regime in search of the electron solid.
On the base of Diffusion Monte-Carlo method it is developed a new Complex Diffusion Monte-Carlo (CDMC) method allowing to simulate the quantum systems with complex wave function. There are no approximations on the calculation of modulus and phase of wave function in contrast to other methods. We find that the averaged value of any quantity in CDMC will have no direct contribution from the phase of the distribution function but only from the phase of the Green function of the diffusion equation. This is most important and crucial point of CDMC. We are testing CDMC by the calculation of the wave function and the ground state energy of two-dimensional electron placed into the external uniform magnetic field. There is an excellent agreement between simulations results and an analytical ones.
Solving the fundamental microscopic equations of interacting quantum particles is a goal of many-body physicists. Statistical methods reduce the complexity of the problem by sampling phase space selectively using random-walks and real states. Many interesting physical phenomena (e.g., electrons in external magnetic fields) involve systems whose state functions are inherently complex-valued. The Fixed-Phase method is a stochastic approach to deal with such problems. Its key ingredient is a trial phase that plays the role of gauge function in the transformation that maps the original fermion (or boson) problem to a boson problem for the modulus of the state function. The Released-Phase method relaxes that constraint and allows us to obtain, in principle, the "exact" properties, although it is subjected to the infamous "phase problem." In our tour of the (complex) Quantum World, we will show how these methods have been successfully applied to a wide variety of physical phenomena ranging from quantum Hall topological fluids and Wigner crystals to the study of the core structure of vortices in superfluid 4 He and atomic systems in superstrong magnetic fields found in astrophysical settings.