A Fixed Node Monte Carlo Method for Lattice Fermions


An important numerical method for studying the properties of correlated systems is the Monte Carlo method. In this method, one calculates statistical properties of a system by sampling the configuration space stochastically. One can often think of the stochastic moves as the motion of a random walker in this configuration space. In this picture, the transition probabilities or jump probabilities are determined by the microscopic model of the system, e.g., by the Hamiltonian.

For classical systems, this procedure is very powerful. For quantum systems, there is an additional well known problem: if one formulates the quantum statistical physics of a quantum system as a stochastic problem, the weights of the walkers that one uses to sample the system, do not necessarily stay positive. This can happen both for bosons as well as for fermions, but the problem is particularly severe for fermions: For these, the ground state wave function has both positive and negative signs as a result of the Pauli principle. This usually entails that one tries to determine the average of a quantity which takes on both negative and positive values. Stronger still: when one considers the contributions from the positive and negative terms separately, the contributions from each of these groups diverge exponentially for large system sizes and low temperatures, while the total sum of the negative and positive terms, that determines the average value, remains finite. This problem is often refered to as the sign problem. It severely hampers the application of Monte Carlo methods to many quantum problems.

In passing, it is worth stressing that the sign problem does not always occur. E.g., when one considers the half-filled Hubbard model, sign changes always occur in pairs, and as a result the sign problem does not arise. But it reappears if one goes away from half-filling. Likewise, a trivial transformation makes the sign problem disappear if one formulates the antiferromagnetic Heisenberg problem as a hard core boson problem, but such a transformation does not exist for the J1-J2 Heisenberg model with antiferromagnetic nearest-neighbor and next-nearest-neighbor interactions. So one can not get rid of the sign problem for this model.

A few years ago, my colleague Hans van Leeuwen was motivated by earlier developments by Alder and Reynolds to attempt to develop a lattice fixed node method. The original fixed node method was developed by Alder and coworkers for continuum systems, like when one determines the ground state electronic energy of a molecule. In this case, the configuration space is continous, and the random walkers that one uses to sample the system perform a continuous random walk in configuration space. The idea of a fixed node approximation for such systems is that one then starts with an approximate ground state wavefunction for this system, e.g., one that is obtained from a mean field approximation. This wave function has regions where it is positive and where it is negative, and these domains are separated by nodal surfaces where the wavefunction vanishes. The continuum formulation of the fixed node method now does a Monte Carlo sampling under the constraint that the positions of the nodal surfaces does not change. In other words, it finds the lowest energy of the system under the contraint that the fermionic wave function that one obtains, has the same sign structure as the approximate one. The contraint is easily implemented by making sure that the random walkers never diffuse across a nodal surface in configuration space.

The above method has been quite successful for molecules, so the idea to extend it to lattice fermions is quite tempting. However, there is an immediate problem: for lattice fermions, the configuration space is discrete, and there are no true nodal surfaces. If one has an approximate ground state wavefunction, it will have positive values at half of the points in configuration space and negative values in the other half. Moreover, the hopping terms in a tight-binding type of Hamiltonian (like that of the Hubbard model) connect many points where the wave function is positive directly to points where the wavefunction is negative - it is more as if one ``jumps over the nodal surface''.

Together with two of our graduate students, Hans van Bemmel and Danny ten Haaf, and a former postdoc, Guozhong An, Hans van Leeuwen and I came up with a way to implement the fixed node idea for lattice fermions. The idea is basically to leave out the moves in configuration space that would amount to a jump ``across the nodal surface'' of an approximate ground state wavefunction, and to compensate for these moves by introducing an additional ``sign flip potential'' in the Hamiltonian. With a clever choice of this potential the method is variational, i.e., it gives an upper bound for the true ground state energy. Since there is no sign problem, the method can be used for large systems. Indeed, we have been able to study quantum domain walls with this method.

The weakness of the method is that the method works well if one has an approximate wave function whose sign structure is about right, but that we have no way to tell beforehand whether this is the case or how to build in the right sign structure in an approximate wave function. Nevertheless, I believe that the full power of the method has not been sufficiently explored. Indeed, there are two recent developments that indicate that in combination with other methods, a fixed node idea can become quite powerfull. Lucas du Croo de Jongh, a graduate student who works in Leiden with Hans van Leeuwen, has been able to combine the fixed node idea with the DMRG (Density Matrix Renormalization Method) of Steve White. Here, the DMRG wave function serves as the wave function that determines the sign structure, and the correlations are then improved with a Fixed Node Monte Carlo method. Secondly, Sorella and Capriotti have recently been able to extend the original formalation of ours and to combine it with a stochastic reconfiguration method. This opens up new directions.

References
D. M. Ceperley and B. J. Alder, Science 555, 231 (1986).
H. J. M. van Bemmel, D. F. B. ten Haaf, W. van Saarloos, J. M. J. van Leeuwen and G. An, Fixed-Node Quantum Monte Carlo Method for Lattice Fermions, Phys. Rev. Lett. 72 2442-2445 (1994).
D. F. B. ten Haaf, H. J. M. van Bemmel, J. M. J. van Leeuwen and W. van Saarloos and D. M. Ceperley, Proof for an upper bound in fixed-node Monte Carlo for lattice fermions, Phys. Rev. B 51, 13039 (1995).
H. J. M. van Bemmel, W. van Saarloos and D. F. B. ten Haaf, Fixed-Node Monte Carlo Calculations for the 1d Kondo Lattice Model, Physica A 251, 143-161 (1998).
M. S. L. du Croo de Jongh, Density Matrix Renormalization Group Variants for Spin Systems (thesis, Universiteit Leiden, 1999; reprints available from the author, van Leeuwen or van Saarloos).
S. Sorella and L. Capriotti, Green function Monte Carlo with Stochastic Reconfiguration: an effective remedy for the sign problem disease, cond-mat/9902211.

July 12, 1999

[Correlated systems] [Wim van Saarloos] [Instituut-Lorentz]