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 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 study of lattice gauge theories with Monte Carlo simulations is hindered by the infamous sign problem that appears under certain circumstances, in particular at non-zero chemical potential. So far, there is no universal method to overcome this problem. However, recent years brought a new class of non-perturbative Hamiltonian techniques named tensor networks, where the sign problem is absent. In previous work, we have demonstrated that this approach, in particular matrix product states in 1+1 dimensions, can be used to perform precise calculations in a lattice gauge theory, the massless and massive Schwinger model. We have computed the mass spectrum of this theory, its thermal properties and real-time dynamics. In this work, we review these results and we extend our calculations to the case of two flavours and non-zero chemical potential. We are able to reliably reproduce known analytical results for this model, thus demonstrating that tensor networks can tackle the sign problem of a lattice gauge theory at finite density.
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.
Two-component Fermi gases model the behavior of many systems in different fields of physics, and one of their interesting features is that they condense into superfluids at low temperatures. Ultracold atoms experiments represent one realization of such Fermi gases, and their great flexibility sparked active research into their phase structure. In the literature, there are studies of the phase structure using functional methods, mean-field approximations, and other approaches. In the present work, we aim to perform ab-initio Monte-Carlo simulations of the system to probe their phase structure for inhomogeneous phases in the presence of spin imbalance. Such simulations generally require a bosonization of the theoretical description of the system that rewrites the theory in terms of an auxiliary bosonic field. For this auxiliary field, many possible choices achieve this, and previous studies often use a field that corresponds to a density of fermions. In the present work, we develop a novel approach to this problem by bosonizing the system in terms of the so-called pairing field which corresponds to the superfluid order parameter. Even in the absence of mass- or spin-imbalance, this approach exhibits a sign problem that presents a problem to many Monte-Carlo methods. To circumvent this sign problem, we base our simulation on the Complex Langevin method. The central question of this thesis is if a simulation of a pairing-field-based formalism using the Complex-Langevin method is suitable to study the phase structure of two-component Fermi gases in the presence of spin imbalance. To this end, we develop a lattice theory based on the pairing field by discretizing the continuous Hamiltonian and performing a rigorous derivation of the path integral from there. We pay special attention to the derivation of lattice derivative operators and rescale the theory to be dimensionless. Beyond that, we develop an efficient notation for lattice theories that makes their handling arguably easier than that of continuum theories. With the obtained theory, we derive the Langevin equation for the system together with the expressions we need to sample observables and develop a numerical simulation on that basis. We use the simulation to study 0+1-dimensional systems as a proof-of-concept. Specifically, we calculate density equations of state and two-point functions for a range of dimensionless couplings and compare them to exact solutions. Additionally, we compare our obtained results for the two-point functions to general discussions of the analytic properties of correlation functions to gain additional insight into the qualitative and numerical behavior of our simulation. We found that our results for both density equations of state and two-point functions are in excellent agreement with the exact solutions. Going forward, we plan to study systems in d>0 spatial dimensions, beginning with simulations of 1+1-dimensional systems. In d=3 dimensions the pairing field's correspondence to the superfluid order parameter may allow us to efficiently study the spontaneous breaking of the U(1) symmetry of the system. On that basis, we can probe the phase diagram of the system for inhomogeneous phases at finite spin imbalance in future studies.
The study of lattice gauge theories with Monte Carlo simulations is hindered by the infamous sign problem that appears under certain circumstances, in particular at non-zero chemical potential.So far, there is no universal method to overcome this problem.However, recent years brought a new class of non-perturbative Hamiltonian techniques named tensor networks, where the sign problem is absent.In previous work, we have demonstrated that this approach, in particular matrix product states in 1+1 dimensions, can be used to perform precise calculations in a lattice gauge theory, the massless and massive Schwinger model.We have computed the mass spectrum of this theory, its thermal properties and real-time dynamics.In this work, we review these results and we extend our calculations to the case of two flavours and non-zero chemical potential.We are able to reliably reproduce known analytical results for this model, thus demonstrating that tensor networks can tackle the sign problem of a lattice gauge theory at finite density.
Quantum field theories with a complex action suffer from a sign problem in stochastic nonperturbative treatments, making many systems of great interest – such as polarized or mass-imbalanced fermions and QCD at finite baryon density – extremely challenging to treat numerically. Another such system is that of bosons at finite angular momentum; experimentalists have successfully achieved vortex formation in supercooled bosonic atoms, and have measured quantities of interest such as the moment of inertia. However, the rotation results in a complex action, making the usual numerical treatments of the theory unusable. This thesis treats systems of nonrelativistic bosons with finite angular momentum using two approaches. One approach is to determine the virial coefficients using a semi-classical lattice approximation (SCLA). Through this approach, we are able to compute the thermodynamic equation of state of the bosons for finite trapping frequency, rotation, and inter-particle interaction. The second approach uses the complex Langevin (CL) method – a method which employs an extension of the Langevin equation to complex space and circumvents the sign problem to compute the full quantum behavior of a low energy system of interacting, trapped, and rotating bosons. We examine the density and angular momentum of the system using all three methods, but the CL method in principal allows us to compute properties unique to rotating superfluids, in particular to show the formation of density singularities (so-called vortex lattices) and compute the circulation of the fluid around those vortices. This work advances our understanding of the quantum effects of rotation on ultracold bosonic gases in two different limits.
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.