Sign problem revisited

For the one band Hubbard model, the Determinantal Monte Carlo simulations described before can be carried out at half-filling without problems since the product $\det M^{+}\det M^{-}$ in Eq.(315) is positive (it can be shown that $\det M^{+}=A\times \det M^{-}$ for any configuration of the Hubbard-Stratonovich spin fields, where $A$ is a positive number [16]). However, in the case of an arbitrary density $\langle n\rangle \neq 1$ this is no longer true for the repulsive Hubbard model (other models like the attractive Hubbard model can still be simulated at all densities). Then, the ``probability'' of a given spin configuration is no longer positive definite. In this situation, to obtain results using this technique it is convenient to separate the product of the determinants into its absolute value and its sign i.e. $\det M^{+}\det M^{-}=sign\times \vert\det M^{+}\det M^{-}\vert$ for each spin configuration. Using this trick, the expectation value of any operator ${\hat{O}}$ can be written as

$$\langle {\hat{O}}\rangle ={\frac{{\langle \langle {\hat{O}}sign\rangle \rangle }}{{\langle \langle sign\rangle \rangle }}},$$ (321)

where $\langle \langle ...\rangle \rangle$ denotes an expectation value obtained using a probability proportional to $\vert\det M^{+}\det M^{-}\vert$. Similar tricks can be applied to cases where the determinant becomes complex as it occurs in problems of lattice gauge theory in the context of particle physics [19]. Although Eq.(324) is an exact identity, in practice the denominator can become very small if the number of spin configurations with positive and negative determinants are similar. Regretfully, this is the case for the Hubbard model in some regime of couplings and densities, and at low temperatures: the sign decreases rapidly when the temperature is reduced, specially at densities close to half-filling[18]. Actually, it has been shown that $\langle \langle sign\rangle \rangle$ converges exponentially to zero as the temperature decreases [20]. This effect imposes severe constraints on the temperatures that can be reached using Monte Carlo techniques in simulations of the Hubbard model away from half-filling.

The study of the ``sign-problem'', and the possibility of finding a cure for this malice, is a very important subject in the context of simulations of correlated electrons. Some time ago, considerable excitement was generated by a paper by Sorella et al. [21] where it was claimed that using a projector Monte Carlo algorithm, and an appropriate trial wave function $\vert\phi _{0}\rangle$, the mean value of the sign converges to a nonzero constant as $\beta \rightarrow \infty$. In such a case it was argued that some physical quantities could be calculated simply by neglecting the signs of the determinants. Regretfully, these conclusions were somewhat premature as discussed later by Loh et al. [20], where it was shown that the expectation value of the sign actually decreases exponentially with $\beta$. Then, neglecting the signs of the determinants leads to an uncontrolled approximation. Loh et al. [20] showed that some physical quantities related with superconducting correlations present a qualitatively different behavior with and without the signs included in the averages.

It is also important to clarify that the ``sign-problem'' is not only caused by the sign that appear due to fermionic anticommutations. For example, consider the case of spin-1/2 problems with nearest and next-nearest neighbors interactions, which can be simulated using Random Walk Monte Carlo methods [22]. In this technique, matrix elements of the interactions are used as probability in the Monte Carlo algorithm. Regretfully, it is not possible to write these matrix elements in a positive definite way for an arbitrary value of the couplings in the Hamiltonian.

Several techniques have been proposed to alleviate the sign-problem. One method is based on the possibility that the operators used to describe, e.g., hole excitations in Hubbard and $\mathrm{t-J}$ models are ``poor'', in the sense that they are a bad approximation to the actual ``dressed'' quasiparticle operators that create real holes in these models. Having proper quasiparticle operators alleviates the sign problem since in Projector or Green's function Monte Carlo methods an initial state is selected upon which $e^{-\Delta \tau {\hat{H}}}$ acts repeatedly till convergence is reached ($\Delta \tau$ being a small number), and thus if the initial Ansatz is very good, it may occur that the sign problem destroys the statistics only after a good convergence is observed (at least in the ground state energy). A method to systematically construct better operators was discussed by Dagotto and Schrieffer [32], Boninsegni and Manousakis [33], and Furukawa and Imada [34], with good results for the cases of one and two holes in the $\mathrm{t-J}$ model, and the weak coupling Hubbard model. In this technique the information gathered using Lanczos methods is very useful to guide the construction of the variational states.

The Fixed Node Monte Carlo [23] is a GFMC variant that restricts the random walkers to move into regions of the phase space in such a way that they cannot cross nodal boundaries of the wave function, where its sign changes. The nodes are provided by a suitable variational trial state. Although this method has represented a remarkable improvement in controlling the sign, the obtained results keep being variational in nature due to the constrain imposed.

A recent attemp to overcome this ``stigma'' in the study of strongly correlated electron models have had some notorious success in the case of frustrated antiferromagnets [24,25], and the t-J model [26]. Instead of constraining the region where the walkers can move, this new original approach consists in performing a ``stochastic reconfiguration'' of the distribution of random walkers at regular intervals of the simulation, mapping them stochastically into new ones that correct the sign instabilities.

Then, the sign-problem is a widely extended plague that affects several areas of theoretical physics, not only strongly correlated electrons. The study of the sign-problem continues attracting considerable attention. Other attempts to fight it can be found in [27], [28], [29], [30], [31], [35]; and references therein.