The auxiliary-field quantum Monte Carlo method is reviewed. The Hubbard-Stratonovich transformation converts an interacting Hamiltonian into a non-interacting Hamiltonian in a time-dependent stochastic field, allowing calculation of the resulting functional integral by Monte Carlo methods. The method is presented in a sufficiently general form to be applicable to any Hamiltonian with one- and two-body terms, with special reference to the Heisenberg model and one- and many-band Hubbard models. Many physical correlation functions can be related to correlation functions of the auxiliary field; general results are given here. Issues relating to the choice of auxiliary fields are addressed; operator product identities change the relative dimensionalities of the attractive and repulsive parts of the interaction. Frequently the integrand is not positive-definite, rendering numerical evaluation unstable. If the auxiliary field violates time-reversal invariance, the integrand is complex and this sign problem becomes a phase problem. The origin of this sign or phase is examined from a number of geometrical and other viewpoints and illustrated by simple examples: the phase problem by the spin (1/2) Heisenberg model, and the sign problem by the attractive SU(N) Hubbard model on a triangular molecule with negative hopping integrals. In the latter case, widely studied in the Jahn-Teller literature, the sign is due neither to fermions nor spin, but to frustration. This system is used to illustrate a number of suggested interpretations of the sign problem.
The auxiliary-field quantum Monte Carlo method is reviewed. The Hubbard-Stratonovich transformation converts an interacting Hamiltonian into a non-interacting Hamiltonian in a time-dependent stochastic field, allowing calculation of the resulting functional integral by Monte Carlo methods. The method is presented in a sufficiently general form to be applicable to any Hamiltonian with one- and two-body terms, with special reference to the Heisenberg model and one- and many-band Hubbard models. Many physical correlation functions can be related to correlation functions of the auxiliary field; general results are given here. Issues relating to the choice of auxiliary fields are addressed; operator product identities change the relative dimensionalities of the attractive and repulsive parts of the interaction. Frequently the integrand is not positive-definite, rendering numerical evaluation unstable. If the auxiliary field violates time-reversal invariance, the integrand is complex and this sign problem becomes a phase problem. The origin of this sign or phase is examined from a number of geometrical and other viewpoints and illustrated by simple examples: the phase problem by the spin (1/2) Heisenberg model, and the sign problem by the attractive SU(N) Hubbard model on a triangular molecule with negative hopping integrals. In the latter case, widely studied in the Jahn-Teller literature, the sign is due neither to fermions nor spin, but to frustration. This system is used to illustrate a number of suggested interpretations of the sign problem.
In the functional-integral technique an auxiliary field, coupled to appropriate operators such as spins, linearizes the interaction term in a quantum many-body system. The partition function is then averaged over this time-dependent stochastic field. Quantum Monte Carlo methods evaluate this integral numerically, but suffer from the sign (or phase) problem: the integrand may not be positive definite (or not real). It is shown that, in certain cases that include the many-band Hubbard model and the Heisenberg model, the sign problem is inevitable on fundamental grounds. Here, Monte Carlo simulations generate a distribution of incompatible operators---a Wigner function---from which expectation values and correlation functions are to be calculated; in general no positive-definite distribution of this form exists. The distribution of time-averaged auxiliary fields is the convolution of this operator distribution with a Gaussian of variance proportional to temperature, and is interpreted as a Boltzmann distribution exp(-\ensuremath{\beta}Veff) in classical configuration space. At high temperatures and large degeneracies this classical effective Hamiltonian Veff tends to the static approximation as a classical limit. In the low-temperature limit the field distribution becomes a Wigner function, the sign problem occurs, and Veff is complex. Interpretations of the distributions, and a criterion for their positivity, are discussed. The theory is illustrated by an exact evaluation of the Wigner function for spin s and the effective classical Hamiltonian for the spin-1/2 van der Waals model. The field distribution can be negative here, more noticeably if the number of spins is odd.
The sign problem poses a fundamental barrier to numerically exact quantum Monte Carlo simulations, severely hindering our understanding of quantum many-body physics across disciplines ranging from condensed matter to particle and nuclear physics. Geometrical frustration makes this barrier especially acute for interacting bosons. Here, we develop a bosonic auxiliary-field quantum Monte Carlo (BAFQMC) method and rigorously prove that time-reversal symmetry and reflection positivity each provide a general sufficient condition for sign-problem-free simulations. Crucially, under either criterion, BAFQMC evades the sign problem even on frustrated lattices where conventional world-line and stochastic-series-expansion methods fail. Benchmarking our approach on a two-flavor Bose-Hubbard model on the triangular lattice yields excellent quantitative agreement with finite-temperature exact diagonalization. By establishing unifying principles to solve the sign problem in BAFQMC, our work substantially expands the reach of unbiased, numerically exact simulations for frustrated bosonic matter.
Auxiliary field quantum Monte Carlo simulations of interacting fermions require sampling over a Hubbard-Stratonovich field h introduced to decouple the interactions. The weight for a given configuration involves the products of the determinant of matrices ${M}_{\ensuremath{\sigma}}(h)$, where $\ensuremath{\sigma}$ labels the species, and hence is typically not positive definite. Indeed, the average sign $\ensuremath{\langle}\mathcal{S}\ensuremath{\rangle}$ of the determinants goes to zero exponentially with increasing spatial size and decreasing temperature for most Hamiltonians of interest. This statement, however, does not explicitly separate two possible origins for the vanishing of $\ensuremath{\langle}\mathcal{S}\ensuremath{\rangle}$. Does $\ensuremath{\langle}\mathcal{S}\ensuremath{\rangle}\ensuremath{\rightarrow}0$ because randomly chosen field configurations have det[M(h)]<0, or does the sign problem arise because the specific subset of configurations chosen by the weighting function have a greater preponderance of negative values? In the latter case, the process of weighting the configurations with ∣det[M(h)]∣ might steer the simulation to a region of configuration space of h where positive and negative determinants are equally likely, even though randomly chosen h would preferentially have determinants with a single dominant sign. In this paper, we address the relative importance of these two mechanisms for the vanishing of $\ensuremath{\langle}\mathcal{S}\ensuremath{\rangle}$ in quantum simulations.
Auxiliary field Quantum Monte Carlo simulations of interacting fermions require sampling over a Hubbard-Stratonovich field x introduced to decouple the interactions. The weight for a given configuration involves the products of the determinant of matrices Mσ(x), where σ labels the species, and hence is typically not positive definite. Indeed, the average sign ⟨S⟩ of the determinants goes to zero exponentially with increasing spatial size and decreasing temperature for most Hamiltonians of interest. This statement, however, does not explicitly separate two possible origins for the vanishing of ⟨S⟩. Does ⟨S⟩→0 because {\it randomly} chosen field configurations have det(M(x))<0, or does the `sign problem' arise as a result of {\it importance sampling} in the simulations? In the latter case, the process of weighting the configurations with ∣det(M(x))∣ might steer the simulation to a region of configuration space of x where positive and negative determinants are equally likely, even though randomly chosen x would preferentially have determinants with a single dominant sign. In this paper we address the relative importance of these two mechanisms for the vanishing of ⟨S⟩ in quantum simulations.
Abstract This paper describes a different Q11C method for solving the Schrödinger equation, one unlike variational, diffusion, and Green’s function methods. It can be applied to many-body systems to produce solutions for ground states. For systems of fermions, antisymmetrization can be imposed to overcome the problem of node location, but the instability of the method is a serious shortcoming, so that the method as described is limited to treating relatively simple systems. Nevertheless, this auxiliary field quantum 1,fonte Carlo (AFMC) method was demonstrated to be successful for test cases reported in this paper, and in time has been improved to allow treatment of realistic systemsa. The method makes use of the imaginary-time propagation of an initial wavefunction toward the ground-state function. It requires use of a trial wavefunction which is the product of single-particle orbitals for bosons or an antisymmetrized product for fermions. A Hubbard-Stratonovich transformation of the propagator changes the many-body problem to a problem of a collection of single particles interacting with a fluctuating (auxiliary) field which replaces the many-body interactions. The authors developed an integral expression for the energy which, after discretization, could be evaluated by a Metropolis Monte Carlo integration forward in time. The algorithm was tested for one-dimensional systems of particles: 6 to 20 bosons with varied interactions, 2 to 6 neutron-proton pairs, others with repulsive potentials. Importance sampling was found to be critical and computational requirements to be large. For spin systems on a lattice the AFMC method was considered feasible, but for general multidimensional systems the method appeared to be limited by its computation requirements.
This thesis presents the path-integral formulation of the nuclear shell model using the Hubbard-Stratonovich transformation, which linearizes the two-body interaction by auxiliary fields. The path-integral was evaluated via Monte Carlo. The method scales favorably with valence-nucleon number and shell-model basis: full-basis calculations can be done up to the rare-earth region, which cannot be treated by other methods. Observables are calculated for the ground state and in a thermal ensemble. Dynamical correlations are obtained, from which strength functions are extracted through the Maximum Entropy method. Examples in the s-d shell, where exact diagonalization can be carried out, compare well with exact results. The "sign problem", which is generic to fermion Monte Carlo calculations, is proved to be absent in a wide class of interactions including the attractive pairing-plus-multipole interactions. The formulation is general for interacting fermion systems and is well suited for parallel computation. The method has been implemented on the Intel Touchstone Delta System, achieving better than 99% parallelization.