Monte Carlo calculations in the framework of lattice field theory provide non-perturbative access to the equilibrium physics of quantum fields. When applied to certain fermionic systems, or to the calculation of out-of-equilibrium physics, these methods encounter the so-called sign problem, and computational resource requirements become impractically large. These difficulties prevent the calculation from first principles of the equation of state of quantum chromodynamics, as well as the computation of transport coefficients in quantum field theories, among other things. This thesis details two methods for mitigating or avoiding the sign problem. First, via the complexification of the field variables and the application of Cauchy's integral theorem, the difficulty of the sign problem can be changed. This requires searching for a suitable contour of integration. Several methods of finding such a contour are discussed, as well as the procedure for integrating on it. Two notable examples are highlighted: in one case, a contour exists which entirely removes the sign problem, and in another, there is provably no contour available to improve the sign problem by more than a (parametrically) small amount. As an alternative, physical simulations can be performed with the aid of a quantum computer. The formal elements underlying a quantum computation - that is, a Hilbert space, unitary operators acting on it, and Hermitian observables to be measured - can be matched to those of a quantum field theory. In this way an error-corrected quantum computer may be made to serve as a well controlled laboratory. Precise algorithms for this task are presented, specifically in the context of quantum chromodynamics.
Monte Carlo calculations in the framework of lattice field theory provide non-perturbative access to the equilibrium physics of quantum fields. When applied to certain fermionic systems, or to the calculation of out-of-equilibrium physics, these methods encounter the so-called sign problem, and computational resource requirements become impractically large. These difficulties prevent the calculation from first principles of the equation of state of quantum chromodynamics, as well as the computation of transport coefficients in quantum field theories, among other things. This thesis details two methods for mitigating or avoiding the sign problem. First, via the complexification of the field variables and the application of Cauchy's integral theorem, the difficulty of the sign problem can be changed. This requires searching for a suitable contour of integration. Several methods of finding such a contour are discussed, as well as the procedure for integrating on it. Two notable examples are highlighted: in one case, a contour exists which entirely removes the sign problem, and in another, there is provably no contour available to improve the sign problem by more than a (parametrically) small amount. As an alternative, physical simulations can be performed with the aid of a quantum computer. The formal elements underlying a quantum computation - that is, a Hilbert space, unitary operators acting on it, and Hermitian observables to be measured - can be matched to those of a quantum field theory. In this way an error-corrected quantum computer may be made to serve as a well controlled laboratory. Precise algorithms for this task are presented, specifically in the context of quantum chromodynamics.
Monte Carlo calculations in the framework of lattice field theory provide non- perturbative access to the equilibrium physics of quantum fields. When applied to certain fermionic systems, or to the calculation of out-of-equilibrium physics, these methods encounter the so-called sign problem, and computational resource require- ments become impractically large. These difficulties prevent the calculation from first principles of the equation of state of quantum chromodynamics, as well as the computation of transport coefficients in quantum field theories, among other things. This thesis details two methods for mitigating or avoiding the sign problem. First, via the complexification of the field variables and the application of Cauchy’s integral theorem, the difficulty of the sign problem can be changed. This requires searching for a suitable contour of integration. Several methods of finding such a contour are discussed, as well as the procedure for integrating on it. Two notable examples are highlighted: in one case, a contour exists which entirely removes the sign problem, and in another, there is provably no contour available to improve the sign problem by more than a (parametrically) small amount. As an alternative, physical simulations can be performed with the aid of a quantum computer. The formal elements underlying a quantum computation — that is, a Hilbert space, unitary operators acting on it, and Hermitian observables to be measured — can be matched to those of a quantum field theory. In this way an error-corrected quantum computer may be made to serve as a well controlled laboratory. Precise algorithms for this task are presented, specifically in the context of quantum chromodynamics.
The Monte Carlo evaluation of path integrals is one of a few general purpose methods to approach strongly coupled systems. It is used in all branches of Physics, from QCD/nuclear physics to the correlated electron systems. However, many systems of great importance (dense matter inside neutron stars, the repulsive Hubbard model away from half-filling, dynamical and non-equilibrium observables) are not amenable to the Monte Carlo method as it currently stands due to the so-called "sign-problem". We review a new set of ideas recently developed to tackle the sign problem based on the complexification of field space and the Picard-Lefshetz theory accompanying it. The mathematical ideas underpinning this approach, as well as the algorithms so far developed, are described together with non-trivial examples where the method has already been proved successful. Directions of future work, including the burgeoning use of machine learning techniques, are delineated.
The numerical sign problem is a hindrance for various areas of research that rely on stochastic sampling methods. It occurs when the sign or phase of the weight function in a path integral is not positive-definite. This leads to cancellations in the Monte Carlo sampling, which can significantly slow down convergence and render the simulation infeasible. Calculations of the Hubbard model are affected by the sign problem when the chemical potential is non-zero or when the spatial lattice is not bipartite. The overall objective of this thesis is the reduction of the sign problem in the Hubbard model with methods that may be applied to other models as well. The approach is to deform the integration contour into complex space and find a transformation that efficiently improves the distribution of the complex phase. The first publication of this cumulative thesis explores the simplest case of such a transformation, which is a constant shift of the real plane in imaginary direction. This transformation showed surprising potential and had the numerical advantage of not requiring the calculation of a Jacobian determinant, as opposed to general transformations. However, it was not obvious which offset would yield the best results. We present an algorithm that numerically approaches the optimal offset iteratively. The drawback of this method is that it relies on Markov Chain Monte Carlo sampling, which comes at significant computational cost and is affected by the sign problem itself, leading to reduced reliability. Therefore, we also present two analytic approximations that can be used directly or as a starting point for the iterative algorithm. To test the method, we calculate single particle correlation functions and the charge expectation value of carbon nanostructures, such as graphene sheets and fullerenes. The second publication introduces a refinement of the iterative optimization algorithm, increasing the stability and convergence speed. This enhancement is achieved by fitting a Gaussian curve to the average phase and its first two derivatives. We again present the physical observables, now including a tetrahedron and a larger honeycomb lattice. Additionally we extract single particle energies from the correlation functions, achieving a good agreement with the exactly solvable tetrahedron. The third publication applies a group equivariant neural network as transformation. The goal was to surpass the limitations of the constant shift while keeping the model as efficient as possible. For this purpose, we encoded the physical symmetries into the model architecture. Results show the potential to reduce the sign problem further than the constant shift. A comparison with dense neural networks shows more efficient use of parameters and training data. Tests with transfer learning lead to mixed results, indicating situational applicability. Overall these developments allow for more efficient calculations of sign problem affected systems.
The path integral formulation of quantum mechanical problems including fermions is often affected by a severe numerical sign problem. We show how such a sign problem can be alleviated by a judiciously chosen constant imaginary offset to the path integral. Such integration contour deformations introduce no additional computational cost to the Hamiltonian Monte Carlo algorithm, while its effective sample size is greatly increased. This makes otherwise unviable simulations efficient for a wide range of parameters. Applying our method to the Hubbard model, we find that the sign problem is significantly reduced. Furthermore, we prove that it vanishes completely for large chemical potentials, a regime where the sign problem is expected to be particularly severe without imaginary offsets. In addition to a numerical analysis of such optimized contour shifts, we analytically compute the shifts corresponding to the leading and next-to-leading order corrections to the action. We find that such simple approximations, free of significant computational cost, suffice in many cases. We present a simulation of C60 fullerenes (buckyballs) that are successful over a wide parameter range.
The non-perturbative dynamics plays important roles in quantum field theories and superstring theory.Monte Carlo calculation of the systems is one of the useful methods to study non-perturbative dynamics.There are many successful works.However, the usual Monte Carlo methods are not applicable, for example, in the Lorentzian metric case, theories with a θ term, theories at finite density, and so on.In such cases, since the Boltzmann weight becomes complex, it is not possible to interpret the weight as the probability.This problem is called the sign problem.Some methods to overcome the sign problem have been proposed.The complex Langevin method (CLM) is one of such methods.An advantage of the method compared with others is that the numerical cost for increasing the system size is reasonable.This method was applied to many toy models, in which it worked well.However, the applications to physically interesting models are not so many.In our work, we apply the CLM to the gauge theories with a θ term and the Lorentzian type IIB matrix model.As a validity test of the CLM for the gauge theories with a θ term, we first apply the CLM to the 2d U(1) gauge theory on the torus with a θ term which is solvable with finite lattice spacing and finite volume on an arbitrary manifold.We find that a naive implementation of the method fails because of the topological nature of the θ term.In order to circumvent this problem, we simulate the same theory on a punctured torus, which is equivalent to the original model in the infinite volume limit for |θ| < π.Rather surprisingly, we find that the CLM works and reproduces the exact results for a punctured torus even at large θ, where the link variables near the puncture become very far from being unitary.Since the application of the CLM to the 2d U(1) case is successful, we next apply the CLM to the 4d SU(2) case.There are interesting predictions for the phase structure around θ = π by using 't Hooft anomaly matching condition.Therefore our aim is to investigate the phase structure by the first principle calculation.We apply the CLM to the theory on a torus and find that, in the coarse lattice case, there is no freezing of topological charge.Although the CLM works well even at large θ, we cannot see the 2π periodicity due to the finite lattice spacing effects.Therefore, we need to decrease the lattice spacing.However, there is the freezing problem in the fine lattice case.We try to solve the problem by imposing the periodic boundary conditions We find that imposing the open boundary condition in all spatial directions alleviates the topology freezing sufficiently, and the CLM works well at large θ.However, the finite volume effect occurs as a drawback of the open boundary conditions.We also discuss the possible way to investigate the phase structure.The Lorentzian type IIB matrix model was proposed as a non-perturbative formulation of superstring theory in 1996.The emergence of (3+1)-dimensional expanding space-time in the model is an intriguing phenomenon which was observed in Monte Carlo studies of
The Monte Carlo evaluation of path integrals is one of a few general purpose methods to approach strongly coupled systems. It is used in all branches of physics, from QCD and nuclear physics to the correlated electron systems. However, many systems of great importance (dense matter inside neutron stars, the repulsive Hubbard model away from half filling, and dynamical and nonequilibrium observables) are not amenable to the Monte Carlo method as it currently stands due to the so-called sign problem. A new set of ideas recently developed to tackle the sign problem based on the complexification of field space and the Picard-Lefshetz theory accompanying it is reviewed. The mathematical ideas underpinning this approach, as well as the algorithms developed thus far, are described together with nontrivial examples where the method has already been proved successful. Directions of future work, including the burgeoning use of machine learning techniques, are delineated.