Y Tanizaki

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^{3}.

The Lefschetz-thimble approach to path integrals is applied to a one-site model of electrons, i.e., the one-site Hubbard model. Since the one-site Hubbard model shows a non-analytic behavior at the zero temperature and its path integral expression has the sign problem, this toy model is a good testing ground for an idea or a technique to attack the sign problem. Semiclassical analysis using complex saddle points unveils the significance of interference among multiple Lefschetz thimbles to reproduce the non-analytic behavior by using the path integral. If the number of Lefschetz thimbles is insufficient, we found not only large discrepancies from the exact result, but also thermodynamic instabilities. Analyzing such singular behaviors semiclassically, we propose a criterion to identify the necessary number of Lefschetz thimbles. We argue that this interference of multiple saddle points is a key issue to understand the sign problem of the finite-density quantum chromodynamics.

Article funded by SCOAP^{3}.

Understanding strongly correlated quantum many-body systems has been an ultimate goal in contemporary physics. Numerical simulation formulated on the discretized spacetime, especially lattice quantum Monte Carlo method, is a powerful

In many quantum systems of great interest, however, Monte Carlo simulation suffers from the so-called sign problem [

The sign problem of the finite-density QCD is known to become too severe when the quark chemical potential exceeds half of the pion mass [

There have been many attempts to attack the sign problem. The idea of complexification of the integration variables is recently developing, and the complex Langevin method [

In this paper, we apply the Lefschetz-thimble approach to the one-site model of electrons. This toy model can be regarded as an extreme limit of strong couplings because it can be obtained by neglecting hopping terms in the Hubbard model. The Hamiltonian of the one-site model can be easily diagonalized and thus we can calculate any expectation value exactly. However, since this model has the severe sign problem in its path-integral expression, it is hard to calculate expectation values by the conventional Monte Carlo method. This toy model provides us a good playground to study theoretical structures of the Lefschetz-thimble approach. In a previous study [

Based on the semiclassical analysis, we find that interference of complex phases among multiple saddle points is important to reproduce the step-function behavior of the density. The fermion spectrum at a complex saddle point resembles the quark spectrum of phase quenched finite-density QCD, which would suggest the interference of multiple saddle points might also occur at finite-density QCD. We will discuss this point in detail later. This study will help us to understand the sign problem in QCD.

The outline of this paper is as follows. In section

The Hubbard model [

In the strong coupling limit

We have explicitly seen that the one-site Hubbard model can be analytically solved by using the number eigenstates. However, if one changes the basis of the Hilbert space for taking trace, the sign problem emerges and thus the Hubbard model in the strong coupling provides us a good lesson.

Let us first derive the path integral expression of the partition function (

Here we take an approximation for small

In order to explicitly show that the path integral (

Let us start with a multiple integration that gives the partition function^{4}

There is another approach to the sign problem also based on the idea of complexification, the complex Langevin method [

_{σ}:

There are two possible origins where the sign problem reappears in the Lefschetz-thimble method. One is a complex phase coming from the Jacobian of the integration measure

Let us apply the Picard–Lefschetz theory to the path integral (

The logarithmic function has branch singularities at fermionic Matsubara modes _{m} form a pair. If

Behaviors of the downward flow equation (_{*} contributes in the Lefschetz-thimble decomposition. Other dual thimbles shown with dashed green lines do not intersect with the original integration cycle _{*}, the number density is given as

Behaviors of Morse’s downward flow equation for

In the following, we concentrate on the case where the sign problem is severe, _{m} contribute to the partition function, so that the interference among them may not be negligible. This interference requires a careful treatment of the semiclassical analysis in order to solve the sign problem. The big difference between figures

Let us denote the classical action at the saddle point

Let us compute the partition function _{m} is neglected because it only gives an unimportant overall factor at low temperatures. Indeed, within our approximation,

Let us analyze the non-analytic behavior (

Let us apply the same analysis to the phase quenched partition function. We define the phase quenched semiclassical partition function by

In order to investigate this failure of the phase quenched approximation more deeply, we analyze the fermion spectrum (

Let us consider one-, three-, and five-thimble approximations in order to analyze the importance of the interference among multiple Lefschetz thimbles. We consider the case where the sign problem is severe, i.e.,

Behaviors of the number density

The one-thimble approximation, which is shown with the solid red line in figure

The result of the three-thimble approximation provides us a useful lesson for an application of the Lefschetz-thimble approach to the sign problem. The number density diverges at several chemical potentials around a rapid crossover, although the result is improved around each plateau. Let us analyze this divergence by using the semiclassical analysis. For that purpose, we introduce the semiclassical partition function with the

According to figure

Let us discuss this behavior from another point of view. Since

We studied the one-site Hubbard model as a toy model of the sign problem, and the Silver Blaze problem. We elucidate that the interference of complex phases among multiple saddle points is important to analyze these problems. In this section, we discuss the analogy of the Silver Blaze problem in the one-site Hubbard model with that in finite-density QCD.

Let us pay attention to the behavior of the baryon number density _{B} for finite-density QCD at _{B} must be zero for any _{N} is the nucleon mass and _{B} must be a key first step to correctly perform the lattice QCD simulation at arbitrary temperatures and baryon chemical potentials. In [

Schematic illustration of the Silver Blaze problem in the finite-density QCD. The baryon number density jumps at

So far, the baryon Silver Blaze problem has been well understood only for _{B} = 0 for

In contrast, as computed in equation (

For

We speculate that this decomposition of Lefschetz thimbles and accompanying interference of complex phases also play an important role in finite-density QCD, in particular, in the Baryon Silver Blaze problem and in the sign problem beyond half of the pion mass. For _{B} until

Because of the complexity of QCD, we cannot show these statements rigorously so far. Future study of QCD-like models based on Lefschetz thimbles and justifying our speculation will be crucial to develop our understanding on the baryon Silver Blaze problem and on the sign problem of the finite-density QCD beyond half of the pion mass.

One-site repulsive Hubbard model shows a non-analytic behavior by changing the chemical potential at the zero temperature

In this paper, the sign problem in the Lefschetz-thimble approach is carefully studied. Topological structures of the Lefschetz thimbles are analyzed based on the semiclassical analysis, and their relation to the fermion spectrum is investigated. If one picks up only the most relevant Lefschetz thimble, the mean-field approximation is almost recovered. The significance of interference among multiple Lefschetz thimbles is identified in order to explain the non-analytic behavior. As we lower the temperature, there exist regions where the necessary number of Lefschetz thimbles increases linearly in

If the number of Lefschetz thimbles is insufficient, we found not only large discrepancies from the exact result, but also thermodynamic instabilities due to artifact of the approximation. This means that the Lefschetz-thimble decomposition does not manifestly ensure the thermodynamic stability, or the positivity condition on the partition function, although its reality is always true with a reasonable condition [

Since the one-site Hubbard model is quite a simple model, it is not clear whether interference of multiple Lefschetz thimbles is relevant for studying thermodynamic systems with complicated interactions. Nevertheless, we discuss the speculation on the baryon Silver Blaze problem of finite-density QCD based on its mathematical similarity to the one-site Hubbard model. We conjecture that the statistically dominant gauge-field configurations are decomposed into multiple Lefschetz thimbles when the quark chemical potential exceeds half of the pion mass. To justify our conjecture on finite-density QCD and QCD-like models is significantly important to solve the baryon Silver Blaze problem and to numerically simulate the finite-density QCD beyond the half of the pion mass.

In order to understand strongly correlated electron systems with the sign problem, applying the Lefschetz-thimble method to the two-site Hubbard model is a nice straightforward extension of this study. This enables us to take into account the effect of the hopping term. If the atomic potential is sufficiently strong, localization of fermions is a natural scenario and thus two-site models will become benchmark. We believe that semiclassical analysis with complex saddle points provides us a deeper understanding of the sign problem, and developing that technique in general cases is an important future study.

Y T is supported by Grants-in-Aid for the fellowship of Japan Society for the Promotion of Science (JSPS) (No.25-6615). YH is partially supported by JSPS KAKENHI Grants Numbers 15H03652. This work was partially supported by the RIKEN interdisciplinary Theoretical Science (iTHES) project, and by the Program for Leading Graduate Schools of Ministry of Education, Culture, Sports, Science, and Technology (MEXT), Japan.