arXiv subjects

Jun Nishimura

Publications and source records attributed to Jun Nishimura.

At least 73 records · Page 4Linked to original sources

arXiv2017

Unification of the complex Langevin method and the Lefschetz-thimble method

Recently there has been remarkable progress in solving the sign problem, which occurs in investigating statistical systems with a complex weight. The two promising methods, the complex Langevin method and the Lefschetz thimble method, share the idea of complexifying the dynamical variables, but their relationship has not been clear. Here we propose a unified formulation, in which the sign problem is taken care of by both the Langevin dynamics and the holomorphic gradient flow. We apply our formulation to a simple model in three different ways and show that one of them interpolates the two methods by changing the flow time.

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arXiv2017

Combining the complex Langevin method and the generalized Lefschetz-thimble method

The complex Langevin method and the generalized Lefschetz-thimble method are two closely related approaches to the sign problem, which are both based on complexification of the original dynamical variables. The former can be viewed as a generalization of the stochastic quantization using the Langevin equation, whereas the latter is a deformation of the integration contour using the so-called holomorphic gradient flow. In order to clarify their relationship, we propose a formulation which combines the two methods by applying the former method to the real variables that parametrize the deformed integration contour in the latter method. Three versions, which differ in the treatment of the residual sign problem in the latter method, are considered. By applying them to a single-variable model, we find, in particular, that one of the versions interpolates the complex Langevin method and the original Lefschetz-thimble method.

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arXiv2017

A new method for probing the late-time dynamics in the Lorentzian type IIB matrix model

The type IIB matrix model has been investigated as a possible nonperturbative formulation of superstring theory. In particular, it was found by Monte Carlo simulation of the Lorentzian version that the 9-dimensional rotational symmetry of the spatial matrices is broken spontaneously to the 3-dimensional one after some "critical time". In this paper we develop a new simulation method based on the effective theory for the submatrices corresponding to the late time. Using this method, one can obtain the results for $N\times N$ matrices by simulating matrices typically of the size $O(\sqrt{N})$. We confirm the validity of this method and demonstrate its usefulness in simplified models.

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arXiv2017

Universality and the dynamical space-time dimensionality in the Lorentzian type IIB matrix model

The type IIB matrix model is one of the most promising candidates for a nonperturbative formulation of superstring theory. In particular, its Lorentzian version was shown to exhibit an interesting real-time dynamics such as the spontaneous breaking of the 9-dimensional rotational symmetry to the 3-dimensional one. This result, however, was obtained after regularizing the original matrix integration by introducing "infrared" cutoffs on the quadratic moments of the Hermitian matrices. In this paper, we generalize the form of the cutoffs in such a way that it involves an arbitrary power ($2p$) of the matrices. By performing Monte Carlo simulation of a simplified model, we find that the results become independent of $p$ and hence universal for $p \gtrsim 1.3$. For $p$ as large as 2.0, however, we find that large-$N$ scaling behaviors do not show up, and we cannot take a sensible large-$N$ limit. Thus we find that there is a certain range of $p$ in which a universal large-$N$ limit can be taken. Within this range of $p$, the dynamical space-time dimensionality turns out to be $(3+1)$, while for $p=2.0$, where we cannot take a sensible large-$N$ limit, we observe a (5+1)d structure.

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arXiv2016

The complex Langevin analysis of spontaneous symmetry breaking induced by complex fermion determinant

In many interesting physical systems, the determinant which appears from integrating out fermions becomes complex, and its phase plays a crucial role in the determination of the vacuum. An example of this is QCD at low temperature and high density, where various exotic fermion condensates are conjectured to form. Another example is the Euclidean version of the type IIB matrix model for 10d superstring theory, where spontaneous breaking of the SO(10) rotational symmetry down to SO(4) is expected to occur. When one applies the complex Langevin method to these systems, one encounters the singular-drift problem associated with the appearance of nearly zero eigenvalues of the Dirac operator. Here we propose to avoid this problem by deforming the action with a fermion bilinear term. The results for the original system are obtained by extrapolations with respect to the deformation parameter. We demonstrate the power of this approach by applying it to a simple matrix model, in which spontaneous symmetry breaking from SO(4) to SO(2) is expected to occur due to the phase of the complex fermion determinant. Unlike previous work based on a reweighting-type method, we are able to determine the true vacuum by calculating the order parameters, which agree with the prediction by the Gaussian expansion method.

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arXiv2016

The argument for justification of the complex Langevin method and the condition for correct convergence

The complex Langevin method is a promising approach to the complex-action problem based on a fictitious time evolution of complexified dynamical variables under the influence of a Gaussian noise. Although it is known to have a restricted range of applicability, the use of gauge cooling made it applicable to various interesting cases including finite density QCD in certain parameter regions. In this paper, we revisit the argument for justification of the method. In particular, we point out a subtlety in the use of time-evolved observables, which play a crucial role in the previous argument. This requires that the probability of the drift term should fall off exponentially or faster at large magnitude. We argue that this is actually a necessary and sufficient condition for the method to be justified. Using two simple examples, we show that our condition tells us clearly whether the results obtained by the method are trustable or not. We also discuss a new possibility for the gauge cooling, which can reduce the magnitude of the drift term directly.

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arXiv2016

Spontaneous symmetry breaking induced by complex fermion determinant --- yet another success of the complex Langevin method

In many interesting systems, the fermion determinant becomes complex and its phase plays a crucial role in the determination of the vacuum. For instance, in finite density QCD at low temperature and high density, exotic fermion condensates are conjectured to form due to such effects. When one applies the complex Langevin method to such a complex action system naively, one cannot obtain the correct results because of the singular-drift problem associated with the appearance of small eigenvalues of the Dirac operator. Here we propose to avoid this problem by adding a fermion bilinear term to the action and extrapolating its coefficient to zero. We test this idea in an SO(4)-invariant matrix model with a Gaussian action and a complex fermion determinant, whose phase is expected to induce the spontaneous breaking of the SO(4) symmetry. Our results agree well with the previous results obtained by the Gaussian expansion method.

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arXiv2016

On the condition for correct convergence in the complex Langevin method

The complex Langevin method (CLM) provides a promising way to perform the path integral with a complex action using a stochastic equation for complexified dynamical variables. It is known, however, that the method gives wrong results in some cases, while it works, for instance, in finite density QCD in the deconfinement phase or in the heavy dense limit. Here we revisit the argument for justification of the CLM and point out a subtlety in using the time-evolved observables, which play a crucial role in the argument. This subtlety requires that the probability distribution of the drift term should fall off exponentially or faster at large magnitude. We demonstrate our claim in some examples such as chiral Random Matrix Theory and show that our criterion is indeed useful in judging whether the results obtained by the CLM are trustable or not.

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arXiv2016

Large-scale computation of the exponentially expanding universe in a simplified Lorentzian type IIB matrix model

The type IIB matrix model is a conjectured nonperturbative formulation of superstring theory. Recent studies on the Lorentzian version of the model have shown that only three out of nine spatial directions start to expand after some critical time. On the other hand, due to the unbounded action of the Lorentzian model, one has to introduce infrared (IR) cutoffs in order to make the partition function finite. In this work we investigate whether the effects of the IR cutoffs disappear in the infinite volume limit. For that purpose, we study a simplified model with large matrix size up to $N=256$ by Monte Carlo simulation. First we confirm the exponentially expanding behavior of the "universe". Then we generalize the form of the IR cutoffs by one parameter, and find that the results become universal in some region of the parameter. It is suggested that the effects of IR cutoffs disappear in this region, which is confirmed also from the studies of Schwinger-Dyson equations.

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arXiv2016

Gauge cooling for the singular-drift problem in the complex Langevin method --- an application to finite density QCD

We study full QCD at finite density and low temperature with light quark mass using the complex Langevin method. Since the singular drift problem turns out to be mild on a $4^3 \times 8$ lattice we use, the gauge cooling is performed only to control the unitarity norm in this exploratory study. We report on our preliminary data obtained from the complex Langevin simulation up to certain Langevin time. While the data are still noisy due to lack of statistics, the onset of the baryon number density seems to occur at larger $μ$ than half the pion mass, which is the value for the phase quenched QCD. The validity of our simulation is tested by the recently proposed criterion based on the probability distribution of the drift term.

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arXiv2016

Gauge cooling for the singular-drift problem in the complex Langevin method --a test in Random Matrix Theory for finite density QCD

Recently, the complex Langevin method has been applied successfully to finite density QCD either in the deconfinement phase or in the heavy dense limit with the aid of a new technique called the gauge cooling. In the confinement phase with light quarks, however, convergence to wrong limits occurs due to the singularity in the drift term caused by small eigenvalues of the Dirac operator including the mass term. We propose that this singular-drift problem should also be overcome by the gauge cooling with different criteria for choosing the complexified gauge transformation. The idea is tested in chiral Random Matrix Theory for finite density QCD, where exact results are reproduced at zero temperature with light quarks. It is shown that the gauge cooling indeed changes drastically the eigenvalue distribution of the Dirac operator measured during the Langevin process. Despite its non-holomorphic nature, this eigenvalue distribution has a universal diverging behavior at the origin in the chiral limit due to a generalized Banks-Casher relation as we confirm explicitly.

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arXiv2016

Numerical tests of the gauge/gravity duality conjecture for D0-branes at finite temperature and finite N

According to the gauge/gravity duality conjecture, the thermodynamics of gauge theory describing D-branes corresponds to that of black branes in superstring theory. We test this conjecture directly in the case of D0-branes by applying Monte Carlo methods to the corresponding gauge theory, which takes the form of the BFSS matrix quantum mechanics. In particular, we take the continuum limit by extrapolating the UV cutoff to infinity. First we perform simulations at large N so that string loop corrections can be neglected on the gravity side. Our results for the internal energy exhibit the temperature dependence consistent with the prediction including the α' corrections. Next we perform simulations at small N but at lower temperature so that the α' corrections can be neglected on the gravity side. Our results are consistent with the prediction including the leading string loop correction, which suggests that the conjecture holds even at finite N.

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arXiv2015

Understanding the problem with logarithmic singularities in the complex Langevin method

In recent years, there has been remarkable progress in theoretical justification of the complex Langevin method, which is a promising method for evading the sign problem in the path integral with a complex weight. There still remains, however, an issue concerning occasional failure of this method in the case where the action involves logarithmic singularities such as the one appearing from the fermion determinant in finite density QCD. In this talk, we point out that this failure is due to the breakdown of the relation between the complex weight which satisfies the Fokker-Planck equation and the probability distribution generated by the stochastic process. In fact, this kind of failure can occur in general when the stochastic process involves a singular drift term. We show, however, in simple examples, that there exists a parameter region in which the method works although the standard reweighting method is hardly applicable.

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arXiv2015

Testing a generalized cooling procedure in the complex Langevin simulation of chiral Random Matrix Theory

The complex Langevin method has been attracting much attention as a solution to the sign problem since the method was shown to work in finite density QCD in the deconfined phase by using the so-called gauge cooling procedure. Whether it works also in the confined phase with light quarks is still an open question, though. In order to shed light on this question, we apply the method to the chiral Random Matrix Theory, which describes the epsilon regime of finite density QCD. Earlier works reported that a naive implementation of the method fails to reproduce the known exact results and that the problem can be solved by choosing a suitable coordinate. In this work we stick to the naive implementation, and show that a generalized gauge cooling procedure can be used to avoid the problem.

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arXiv2015

Monte Carlo studies of dynamical compactification of extra dimensions in a model of nonperturbative string theory

The IIB matrix model has been proposed as a non-perturbative definition of superstring theory. In this work, we study the Euclidean version of this model in which extra dimensions can be dynamically compactified if a scenario of spontaneously breaking the SO(10) rotational symmetry is realized. Monte Carlo calculations of the Euclidean IIB matrix model suffer from a very strong complex action problem due to the large fluctuations of the complex phase of the Pfaffian which appears after integrating out the fermions. We employ the factorization method in order to achieve effective sampling. We report on preliminary results that can be compared with previous studies of the rotational symmetry breakdown using the Gaussian expansion method.

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arXiv2015

Justification of the complex Langevin method with the gauge cooling procedure

Recently there has been remarkable progress in the complex Langevin method, which aims at solving the complex action problem by complexifying the dynamical variables in the original path integral. In particular, a new technique called the gauge cooling was introduced and the full QCD simulation at finite density has been made possible in the high temperature (deconfined) phase or with heavy quarks. Here we provide a rigorous justification of the complex Langevin method including the gauge cooling procedure. We first show that the gauge cooling can be formulated as an extra term in the complex Langevin equation involving a gauge transformation parameter, which is chosen appropriately as a function of the configuration before cooling. The probability distribution of the complexified dynamical variables is modified by this extra term. However, this modification is shown not to affect the Fokker-Planck equation for the corresponding complex weight as far as observables are restricted to gauge invariant ones. Thus we demonstrate explicitly that the gauge cooling can be used as a viable technique to satisfy the convergence conditions for the complex Langevin method. We also discuss the "gauge cooling" in 0-dimensional systems such as vector models or matrix models.

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arXiv2015

New insights into the problem with a singular drift term in the complex Langevin method

The complex Langevin method aims at performing path integral with a complex action numerically based on complexification of the original real dynamical variables. One of the poorly understood issues concerns occasional failure in the presence of logarithmic singularities in the action, which appear, for instance, from the fermion determinant in finite density QCD. We point out that the failure should be attributed to the breakdown of the relation between the complex weight that satisfies the Fokker-Planck equation and the probability distribution associated with the stochastic process. In fact, this problem can occur in general when the stochastic process involves a singular drift term. We show, however, in a simple example that there exists a parameter region in which the method works although the standard reweighting method is hardly applicable.

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arXiv2015

Power-law expansion of the Universe from the bosonic Lorentzian type IIB matrix model

Recent studies on the Lorentzian version of the type IIB matrix model show that (3+1)D expanding universe emerges dynamically from (9+1)D space-time predicted by superstring theory. Here we study a bosonic matrix model obtained by omitting the fermionic matrices. With the adopted simplification and the usage of a large-scale parallel computer, we are able to perform Monte Carlo calculations with matrix size up to $N=512$, which is twenty times larger than that used previously for the studies of the original model. When the matrix size is larger than some critical value $N_{\rm c}\simeq 110$, we find that (3+1)D expanding universe emerges dynamically with a clear large-$N$ scaling property. Furthermore, at sufficiently late times, we observe a power-law behavior $t^{1/2}$ of the spatial extent with respect to time $t$, which is reminiscent of the expanding behavior of the Friedmann-Robertson-Walker universe in the radiation dominated era. We discuss possible implications of this result on the original model including fermionic matrices.

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