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Tensor renormalization group approach to higher-dimensional lattice field theories

秋山, 進一郎 筑波大学 DOI:10.15068/0002005504

2022.11.17

概要

The quantum field theory (QFT) is a powerful framework to describe various types of physical phe- nomena in high-energy systems and many-body cooperative phenomena. Within this framework, one of the most important problems is to consider the path integral for a theory defined by the Lagrangian or the Hamiltonian. In high-energy physics, such a theory is defined on a continuum, which corre- sponds to the four-dimensional spacetime. Therefore, the path integral results in multiple integrals with uncountably infinite degrees of freedom, that is a mathematically ill-defined object. Related to this issue, the QFT in high-energy physics is usually defined via the perturbation theory, so there is an essential difficulty when we discuss non-perturbative physics. A typical example is the quantum chromodynamics (QCD), which describes the strong interaction between quarks and gluons. Although we can understand the short-range physics in the QCD with the perturbative discussion thanks to the asymptotic freedom, we can no longer investigate the long-range physics, which is outside of the scope of perturbation theory.

　As a successful way to access the non-perturbative regimes, we have the lattice field theory pio- neered by K. G. Wilson [Wil74]. In the lattice field theory, a continuum spacetime is replaced by a discrete lattice that allows us to consider a theory characterized by countably infinite, or finite, de- grees of freedom and its continuum limit defines the quantum field theory in a mathematically rigorous manner. Moreover, the lattice field theory has a deep connection with the statistical physics [Kog79]. The procedure of the renormalization is well understood by the renormalization group, which is an important tool to understand not only high-energy physics but also the critical phenomena [WK74]. This is of great significance because once we move on to the lattice theory, we have a chance to evaluate various observables via the philosophy of the statistical ensemble. One of the most efficient methods to investigate the field theory on a lattice is the Monte Carlo (MC) simulation, which is a stochastic numerical method based on the idea of sampling. The MC method is widely applied in the study of high-energy physics. Lattice QCD calculation is a typical example of the application of the MC method and it has been playing an essentially important role to understand the non-perturbative physics in the QCD.

　However, the MC simulation faces a serious difficulty when one cannot assume the probabilistic interpretation for the given Boltzmann weight. This difficulty is known as the sign problem, a serious obstacle to the numerical study of lattice field theories. A notable example where the sign problem takes place is in the systems at finite density. In particular, high-density systems such as the internal states of neutron stars, are extremely difficult to understand experimentally, and thus the expectation for numerical approaches is very high. However, due to the sign problem, many physical phenomena peculiar to finite-density systems are still not understood from the fundamental theory.

　In view of this situation, we need a numerical methodology that potentially enables us to investigate the regimes where it is difficult to access with the standard MC simulation. This thesis focuses on the tensor renormalization group (TRG) from this point of view. The TRG approach is a type of the tensor network method, where we express a physical object, such as a (ground) state, observable, or path integral, by a tensor contraction, which is numerically evaluated via a certain tensor network algorithm. The idea of the tensor network was originally developed in the field of statistical physics and one of the most famous applications is the density matrix renormalization group (DMRG). The DMRG provides one with an extraordinary accurate numerical simulation for a certain class of one-dimensional quantum system [Whi92, Whi93]. Recently, the tensor network method has been in the spotlight, also in the context of the quantum information theory. From such a perspective, the various numerical algorithms are reviewed in Refs. [Oru´19, ONU21]. The application of the tensor network method is a recent hot topic also in particle physics [BC20, BBC+20], including the TRG study of lattice field theories [MSUY20]. The TRG has many advantages over the standard MC method. Firstly, the TRG, and also other tensor network algorithms, are free from the sign problem, because they do not resort to any probabilistic interpretation for the given Boltzmann weight. Secondly, the computational cost of the TRG algorithm scales logarithmically with respect to the lattice volume. This benefit comes from the fact that the TRG algorithm is based on the idea of a real-space renormalization group, which allows us to enlarge the system size toward the thermodynamic limit as a result of iterating the renormalization-group transformations. Thirdly, we can deal with the Grassmann fields directly within the TRG approach. In other words, we do not have to introduce any bosonic auxiliary fields to rewrite fermions. This is another practical benefit because such a bosonic auxiliary field sometimes introduces non-local interactions, which require more computational cost to simulate. Fourthly, the TRG enables us to numerically evaluate the partition function or the path integral itself. Indeed, with the MC method, we cannot directly calculate it, because it is not regarded as any expectation value. Since the partition function or the path integral is a generating functional, we are then allowed to derive various thermodynamic functions in principle once we obtain it. In addition, there is a superior feature in the TRG approach, compared with other tensor network algorithms, which is the applicability to the higher-dimensional, more than two-dimensional, systems on the thermodynamic lattice. Originally, the TRG was proposed to investigate the two-dimensional spin systems in Ref. [LN07], which has been followed by the various extensions with respect to the higher-dimensional application and the improvement of the accuracy. Therefore, it can be said that the TRG approach is a powerful candidate to investigate the regime whose investigation with the MC method is seriously hindered by numerical obstacles. The aim of this thesis is to study such a regime in various types of lattice field theory, exploiting these advantages of the TRG approach.

　This thesis is organized as follows. In Sec. 2, we review how to construct a tensor network rep- resentation for a path integral. After that, we review the several TRG algorithms which are applied in the following sections. An extension of these algorithms to investigate lattice fermions is also demonstrated here, based on the formalism given in Ref. [AK21a]. The first part of Sec. 3 is also devoted to reviewing the numerical accuracy of the TRG algorithms by benchmarking with the Ising model. We then apply the TRG method to the four-dimensional Ising model. This part is based on Refs. [AKYY19], which is the first application of the TRG approach to the four-dimensional system. Sec. 4 explains the TRG study of the lattice ϕ4 theory. Since the ϕ4 theory is a non-compact field theory, its numerical treatment within the TRG approach is nontrivial. We look at the lattice ϕ4 theory from the viewpoint of the relationship to the Ising model and discuss its phase transition. This part is based on Ref. [AKY21b]. In Sec. 5, we demonstrate the efficiency of the TRG approach for the four-dimensional system with the sign problem. In this context, we investigate the complex ϕ4 theory at finite density, whose Silver Blaze phenomenon is a primary target. This section is based on Ref. [AKK+20], which is the first application of the TRG approach to the four-dimensional quantum field theory. Application of the TRG method to lattice fermions is discussed in Sec. 6, where we study the Hubbard model at finite density. This model is a very fundamental model of the strongly correlated electrons. Moving on to the path-integral formalism, we are allowed to investigate the models with the TRG approach, whose efficiency at finite density is discussed. These TRG studies are based on Refs. [AK21b, AKY21a]. In Sec. 7, the TRG approach is extended to investigate the four-dimensional Nambu–Jona-Lasinio model at finite density, which is an effective theory of the QCD at finite density. Our research target is to numerically investigate the chiral-symmetry restoration in the cold and dense regime, where a naive MC simulation is prohibited by the serious sign problem. This section is based on Ref. [AKYY20], which is the first application of the TRG method to the four-dimensional lattice fermions. Finally, Sec. 8 is devoted to the conclusion.

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