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arXiv:1307.3871 · Europhysics Letters (EPL)

Multilayer wave functions: A recursive coupling of local excitations

Abolfazl Ramezanpour

Published Oct 1, 2013• 1 citation
arXivHTMLPDF

Abstract

Finding a succinct representation to describe the ground state of a disordered interacting system could be very helpful in understanding the interplay between the interactions that is manifested in a quantum phase transition. In this work we use some elementary states to construct recursively an ansatz of multilayer wave functions, where in each step the higher-level wave function is represented by a superposition of the locally "excited states" obtained from the lower-level wave function. This allows us to write the Hamiltonian expectation in terms of some local functions of the variational parameters, and employ an efficient message-passing algorithm to find the optimal parameters. We obtain good estimations of the ground-state energy and the phase transition point for the transverse Ising model with a few layers of mean-field and symmetric tree states. The work is the first step towards the application of local and distributed message-passing algorithms in the study of structured variational problems in finite dimensions.

DOI
10.1209/0295-5075/104/20003

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Finding a succinct representation to describe the ground state of a disordered interacting system could be very helpful in understanding the interplay between the interactions that is manifested in a quantum phase transition. In this work we use some elementary states to construct recursively an ansatz of multilayer wave functions, where in each step the higher-level wave function is represented by a superposition of the locally "excited states" obtained from the lower-level wave function. This allows us to write the Hamiltonian expectation in terms of some local functions of the variational parameters, and employ an efficient message-passing algorithm to find the optimal parameters. We obtain good estimations of the ground-state energy and the phase transition point for the transverse Ising model with a few layers of mean-field and symmetric tree states. The work is the first step towards the application of local and distributed message-passing algorithms in the study of structured variational problems in finite dimensions.
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We propose a method to construct the ground state \ensuremath{\psi}(\ensuremath{\lambda}) of local lattice Hamiltonians with the generic form H0{\mathit{H}}_{0}H0​+\ensuremath{\lambda}H1{\mathit{H}}_{1}H1​, where \ensuremath{\lambda} is a coupling constant and H0{\mathit{H}}_{0}H0​ is a Hamiltonian with a nondegenerate ground state ${\mathrm{\ensuremath{\psi}}}_{0}$. The method is based on the choice of an exponential ansatz \ensuremath{\psi}(\ensuremath{\lambda})=exp[U(\ensuremath{\lambda})]${\mathrm{\ensuremath{\psi}}}_{0}$, which is a sort of generalized lattice version of a Jastrow wave function. We combine perturbative and variational techniques to get successive approximations of the operator U(\ensuremath{\lambda}). Perturbation theory is used to set up a variational method which in turn produces nonperturbative results. The computation with this kind of ansatz leads us to associate to the original quantum-mechanical problem a statistical-mechanical system defined in the same spatial dimension. In some cases these statistical-mechanical systems turn out to be integrable, which allows us to obtain exact upper bounds to the energy. The general ideas of our method are illustrated in the example of the Ising model in a transverse field.
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