Green Function Monte Carlo with Stochastic Reconfiguration

A new method for the stabilization of the sign problem in the Green Function Monte Carlo technique is proposed. The method is devised for real lattice Hamiltonians and is based on an iterative ”stochastic reconfiguration” scheme which introduces some bias but allows a stable simulation with constant sign. The systematic reduction of this bias is in principle possible. The method is applied to the frustrated J1 − J2 Heisenberg model, and tested against exact diagonalization data. Evidence of a finite spin gap for J2/J1 >∼ 0.4 is found in the thermodynamic limit. 02.70.Lq,75.10.Jm,75.40.Mg Typeset using REVTEX 1 As well known the Green Function Monte Carlo method (GFMC) allows to obtain the exact ground state properties of a many body hamiltonian with a statistical method. One of the most severe restriction is that only positive definite Green function GF can be sampled, otherwise the method is facing the well known ”sign problem”. Approximate techniques like the fixed node approximation (FN) have been developed to circumvent the sign problem but at the very least they cannot be systematically improved to achieve the exact answer within statistical errors. This property has severely limited the applications of GFMC to fermions and frustrated boson models. In this letter I propose a new approach to stabilize the sign problem, the GFMC with stochastic reconfiguration (GFMCSR), which will be shortly described below, revisiting also the basic steps of the standard GFMC on a lattice. [1,2] In order to filter out the ground state of a given lattice hamiltonian H the standard power method may be applied iteratively : ψn+1(x ) = ∑ x (Λδx′,x −Hx′,x)ψn(x) (1) where x represents conventionally the index of a complete basis |x > , Hx′,x being the corresponding matrix elements of the hamiltonian which in the following are assumed real, and Λ is a positive constant that allows the convergence of ψn to the ground state ψ0(x), for large n. In numerical calculations of interesting lattice hamiltonians the dimension of the basis grows exponentially with the size and the particle number, though the matrix itself is very sparse and all its elements Hx′,x , for given x, can be generally computed even for large system size. In this case an exact application of (1) is impossible unless for few steps. A way out is to use a stochastic approach , like GFMC ,which is particularly simple on a lattice. In order to implement stochastically the iteration (1) the corresponding lattice GF Gx′,x = Λδx′,x −Hx′,x (2) may be decomposed in the following way: 2 Gx′,x = sx′,xpx′,xbx (3) where px′,x is a normalized stochastic matrix, bx ≥ 0 is a normalization constant and the matrix s takes into account the sign of the GF. The typical choice is to take px′,x = |Gx′,x|/bx , bx = ∑ x |Gx′,x| and sx′,x = sgnGx′,x, which is identically one if there is no sign problem. In the GFMC method the so called ”walker“ is defined by a weight w and a configuration x.. At a given iteration n the walker is assumed to sample statistically the state ψn(x) in Eq.(1), in the sense that the probability Pn(w, x) to have the walker with weight w (not restricted to be positive) in a given configuration x satisfies: ∫ dwPn(w, x)w = ψn(x). Then the matrix multiplication (1) can be implemented statistically , in the precise sense that