富谷 昭夫

In this paper, we develop the self-learning Monte-Carlo (SLMC) algorithm for non-abelian gauge theory with dynamical fermions in four dimensions to resolve the autocorrelation problem in lattice QCD. We perform simulations with the dynamical staggered fermions and plaquette gauge action by both in HMC and SLMC for zero and finite temperature to examine the validity of SLMC. We confirm that SLMC can reduce autocorrelation time in non-abelian gauge theory and reproduces results from HMC. For finite temperature runs, we confirm that SLMC reproduces correct results with HMC, including higher-order moments of the Polyakov loop and the chiral condensate. Besides, our finite temperature calculations indicate that four flavor QC${}_2$D with $\hat{m} = 0.5$ is likely in the crossover regime in the Colombia plot.

We propose using Normalizing Flows as a trainable kernel within the molecular dynamics update of Hamiltonian Monte Carlo (HMC). By learning (invertible) transformations that simplify our dynamics, we can outperform traditional methods at generating independent configurations. We show that, using a carefully constructed network architecture, our approach can be easily scaled to large lattice volumes with minimal retraining effort. The source code for our implementation is publicly available online at https://github.com/nftqcd/fthmc.

We find that the chiral phase transition (chiral crossover) in QCD at physical point is triggered by big imbalance among three fundamental quantities essential for the QCD vacuum structure: susceptibility functions for the chiral symmetry, axial symmetry, and the topological charge. The balance, dobbed the QCD trilemma, is unavoidably violated when one of the magnitudes among them is highly dominated, or suppressed. Based on a three-flavor Nambu-Jona-Lasinio model, we explicitly evaluate the amount of violation of the QCD trilemma at physical point, and show that the violation takes place not only at vacuum, but even in a whole temperature regime including the chiral crossover epoch. This work confirms and extends the suggestion recently reported from lattice QCD with 2 flavors on dominance of the axial and topological susceptibilities left in the chiral susceptibility at high temperatures. It turns out that the imbalance is essentially due to the flavor symmetry violation of the lightest three flavors. The violation of QCD trilemma and its flavor dependence can be tested by lattice simulations with 2 + 1 flavors in the future, and would also give a new guiding principle to deeper understand the QCD phase structure, such as the Columbia plot, including possible extension with external fields.

We discuss the violation of quark-flavor symmetry at high temperatures, induced from nonperturbative thermal loop corrections and axial anomaly, based on a three-flavor linear-sigma model including an axial-anomaly induced-flavor breaking term. We employ a nonperturbative analysis following the Cornwall-Jackiw-Tomboulis formalism, and show that the model undergoes a chiral crossover with a pseudo-critical temperature, consistently with lattice observations. We find following features regarding the flavor breaking eminent around and above the pseudo-critical temperature: i) up-and down-quark condensates drop faster than the strange quark's toward the criticality, but still keep nonzero value even going far above the critical temperature; ii) the introduced anomaly-related flavor-breaking effect acts as a catalyzer toward the chiral restoration, and reduces the amount of flavor breaking in the up, down and strange quark condensates; iii) a dramatic deformation for the meson flavor mixing structure is observed, in which the anomaly-induced favor breaking is found to be almost irrelevant; iv) the meson spectroscopy gets corrected by the net nonperturbative flavor breaking effects, where the scalar meson mass hierarchy (inverse mass hierarchy) is significantly altered by the presence of the anomaly-related flavor breaking; v) the topological susceptibility significantly gets the contribution from the surviving strange quark condensate, which cannot be dictated by the chiral perturbation theory, and deviates from the dilute instanton gas prediction. There the anomaly-induced flavor breaking plays a role of the destructive interference for the net flavor violation; vi) the U(1)_A breaking is enhanced by the strange quark condensate, which may account for the tension in the effective U(1)_A restoration observed on lattices with two flavors and 2+1 flavors near the chiral limit.

We develop a gauge covariant neural network for four dimensional non-abelian gauge theory, which realizes a map between rank-2 tensor valued vector fields. We find that the conventional smearing procedure and gradient flow for gauge fields can be regarded as known neural networks, residual networks and neural ordinal differential equations for rank-2 tensors with fixed parameters. In terms of machine learning context, projection or normalization functions in the smearing schemes correspond to an activation function in neural networks. Using the locality of the activation function, we derive the backpropagation for the gauge covariant neural network. Consequently, the smeared force in hybrid Monte Carlo (HMC) is naturally derived with the backpropagation. As a demonstration, we develop the self-learning HMC (SLHMC) with covariant neural network approximated action for non-abelian gauge theory with dynamical fermions, and we observe SLHMC reproduces results from HMC.

We investigate the Dirac eigenvalue spectrum ($\rho(\lambda,m_l)$) to study the microscopic origin of axial anomaly in high temperature phase of QCD. We propose novel relations between the derivatives ($\partial^n \rho(\lambda,m_l)/\partial m_l^n$) of the Dirac eigenvalue spectrum with respect to the quark mass ($m_l$) and the $(n+1)$-point correlations among the eigenvalues ($\lambda$) of the massless Dirac operator. Based on these relations, we present lattice QCD results for $\partial^n \rho(\lambda,m_l)/\partial m_l^n$ ($n=1, 2, 3$) with $m_l$ corresponding to pion masses $m_\pi=160-55$ MeV, and at a temperature of about 1.6 times the chiral phase transition temperature. Calculations were carried out using (2+1)-flavors of highly improved staggered quarks and the tree-level Symanzik gauge action with the physical strange quark mass, three lattice spacings $a=0.12, 0.08, 0.06$ fm, and lattices having aspect ratios $4-9$. We find that $\rho(\lambda\to0,m_l)$ develops a peaked structure. This peaked structure, which arises due to non-Poisson correlations within the infrared part of the Dirac eigenvalue spectrum, becomes sharper as $a\to0$, and its amplitude is proportional to $m_l^2$. After continuum and chiral extrapolations, we find that the axial anomaly remains manifested in two-point correlation functions of scalar and pseudo-scalar mesons in the chiral limit. We demonstrate that the behavior of $\rho(\lambda\to0,m_l)$ is responsible for it.

Violation of scale symmetry, scale anomaly, being a radical concept in quantum field theory, is of importance to comprehend the vacuum structure of QCD, and should potentially contribute to the chiral phase transition in thermal QCD, as well as the chiral and U(1) axial symmetry. Though it should be essential, direct evidence of scale anomalies has never been observed in the chiral phase transition. We propose a methodology to detect a scale anomaly in the chiral phase transition, which is an electromagnetically induced scale anomaly: apply a weak magnetic field background onto two-flavor massless QCD with an extremely heavy strange quark, first observe the chiral crossover; second, adjusting the strange quark mass to be smaller and smaller, observe the second-order chiral phase transition, and then the first-order one in the massless-three flavor limit. Thus, the second-order chiral phase transition, observed as the evidence of the quantum scale anomaly, is a new critical endpoint. It turns out that this electromagnetic scale anomaly gets most operative in the weak magnetic field regime, rather than a strong field region. We also briefly address accessibility of lattice QCD, a prospected application to dense matter system, and implications to astrophysical observations, such as gravitational wave productions provided from thermomagnetic QCD-like theories.

We investigate the chiral phase structure of three flavor QCD in a background $U(1)$ magnetic field using the standard staggered action and the Wilson plaquette gauge action. We perform simulations on lattices with a temporal extent of $N_\tau=4$ and four spatial extents of $N_\sigma = 8,16, 20$ and 24. We choose a smaller-than-physical quark mass in lattice spacing as $am = 0.030$ such that there exists a crossover transition at vanishing magnetic fields, and adopt two values of magnetic field strength in lattice spacing $a \sqrt{ e{B } }\simeq 1.5$ and 2. We find that the transition becomes stronger in the presence of a background magnetic field, and turns into a first order as seen from the volume scaling of the order parameter susceptibility as well as the metastable states in the time history of the chiral condensate. On the other hand, the chiral condensate and transition temperature always increase with $B$ even within the regime of a first order phase transition. This suggests that the discrepancy in the behavior of chiral condensates and transition temperature as a function of $B$ between earlier lattice studies using larger-than-physical pion masses with standard staggered fermions and those using physical pions with improved staggered fermions is mainly due to lattice cutoff effects.

We demonstrate that the QCD topological susceptibility nonperturbatively gets a significant contribution signaled by flavor-nonuniversal quark condensates at around the pseudo-critical temperature of the chiral crossover. It implies a remarkable flavor breaking in the axial anomaly as well as the QCD theta vacuum in high temperature QCD, which are almost flavor universal in the vacuum. A nontrivial flavor breaking is triggered by nonperturbative thermal loop corrections at around the chiral crossover, which is different from the trivial flavor violation just scaled by the quark mass ratio, observed at asymptotically high temperatures. This critical flavor violation cannot be dictated by the chiral perturbation theory with that lattice QCD usually compares, or the dilute instanton gas approximation based on that its astrophysical implications have conventionally been made. This would give an impact on the thermal history and the cosmological evolution of QCD axion including the estimate of the relic abundance as a dark matter candidate.

We studied the temporal correlation function of mesons in the pseudo-scalar channel in (2+1)-flavor QCD in the presence of external magnetic fields at zero temperature. The simulations were performed on $32^3 \times 96$ lattices using the Highly Improved Staggered Quarks (HISQ) action with $m_{\pi} \approx $ 230 MeV. The strength of magnetic fields $|eB|$ ranges from 0 to around 3.3 GeV$^2$ ($\sim 60 m_\pi^2$). We found that the masses of neutral pseudo-scalar particles, e.g. neutral pion and kaon, monotonouslly decrease as the magnetic field grows and then saturate at a nonzero value. It is observed that heavier neutral pseudo-scalars are less affected by magnetic fields. Moreover, we found a non-monotonous behavior of charged pion and kaon mass in magnetic field for the first time. In the case of small magnetic field (0 $\leq~|eB| \lesssim$ 0.3 GeV$^2~\sim 6m_\pi^2$ ) the mass of charged pseudo-scalar grows with magnetic field and can be well described by the Lowest Landau Level approximation, while for $|eB|$ larger than 0.3 GeV$^2$ the mass starts to decrease. The possible connection between $|eB|$ dependences of neutral pion mass and the decreasing behavior of pseudo-critical temperature in magnetic field is discussed. Due to the nonzero value of neutral pion mass our simulation indicates that the superconducting phase of QCD does not exist in the current window of magnetic field.

We utilize the eigenvalue filtering technique combined with the stochastic estimate of the mode number to determine the eigenvalue spectrum. Simulations of (2 + 1)-flavor QCD are performed using the Highly Improved Staggered Quarks (HISQ/tree) action on $N_{\tau}$ = 8 lattices with aspect ratios $N_{\sigma}/N_{\tau}$ ranging from 5 to 7. The strange quark mass is fixed to its physical value $m_{s}^{\rm phy}$, and the light quark masses $m_{l}$ are varied from $m_{s}^{\rm phy}/40$ to $m_{s}^{\rm phy}/160$ which correspond to pion mass $m_{\pi}$ ranging from 110 MeV to 55 MeV in the continuum limit. We compute the chiral condensate and $\chi_{\pi} - \chi_{\delta}$ through the eigenvalue spectrum obtained from the the eigenvalue filtering method. We compare these results with those obtained from a direct calculation of the observables which involves inversions of the fermion matrix using the stochastic "noise vector" method. We find that these approaches yield consistent results. Furthermore, we also investigate the quark mass and temperature dependences of the Dirac eigenvalue density at zero eigenvalues to gain more insights about the $U_A(1)$ symmetry breaking in QCD.

We perform a digital quantum simulation of a gauge theory with a topological term in Minkowski spacetime, which is practically inaccessible by standard lattice Monte Carlo simulations. We focus on $1+1$ dimensional quantum electrodynamics with the $\theta$-term known as the Schwinger model. We construct the true vacuum state of a lattice Schwinger model using adiabatic state preparation which, in turn, allows us to compute an expectation value of the fermion mass operator with respect to the vacuum. Upon taking a continuum limit we find that our result in massless case agrees with the known exact result. In massive case, we find an agreement with mass perturbation theory in small mass regime and deviations in large mass regime. We estimate computational costs required to take a reasonable continuum limit. Our results imply that digital quantum simulation is already useful tool to explore non-perturbative aspects of gauge theories with real time and topological terms.

We study the time evolution of the entanglement entropy after quantum quenches in Lifshitz free scalar theories, with the dynamical exponent $z>1$, by using the correlator method. For quantum quenches we consider two types of time-dependent mass functions: end-critical-protocol (ECP) and cis-critical-protocol (CCP). In both cases, at early times the entanglement entropy is independent of the subsystem size. After a critical time ($t_c$), the entanglement entropy starts depending on the subsystem size significantly. This critical time $t_c$ for $z = 1$ in the fast ECP and CCP has been explained well by the fast quasi-particle of the quasi-particle picture. However, we find that for $z > 1$ this explanation does not work and $t_c$ is delayed. We explain why $t_c$ is delayed for $z>1$ based on the quasiparticle picture: in essence, it is due to the competition between the fast and slow quasiparticles. At late times, in the ECP, the entanglement entropy slowly increases while, in the CCP, it is oscillating with a well defined period by the final mass scale, independently of $z$. We give an interpretation of this phenomena by the correlator method. As $z$ increases, the entanglement entropy increases, which can be understood by long-range interactions due to $z$.

We study the phase structure of QCD with three degenerate flavors in the<br /> external magnetic fields using highly improved staggered quarks (HISQ). The<br /> simulations are performed on $16^3\times 6$ lattice. In order to investigate<br /> the quark mass dependence of the chiral transition we choose the values of the<br /> bare quark masses 0.015 and 0.0009375 in the lattice unit, corresponding to<br /> $m_\pi=320$ MeV and 80 MeV in the continuum limit. We found no indication of a<br /> first order phase transition in the current window of quark masses and external<br /> magnetic fields. Unlike to the case with standard staggered fermions inverse<br /> magnetic catalysis is always observed at about the critical temperature. The<br /> microscopic origin of this phenomena are further discussed by looking into the<br /> Dirac eigenvalue spectrum.

We discuss an aspect of neural networks for the purpose of phase transition<br /> detection. To this end, we first train the neural network by feeding<br /> Ising/Potts configurations with labels of temperature so that it can predict<br /> the temperature of input. We do not explicitly supervise if the configurations<br /> are in ordered/disordered phase. Nevertheless, we can identify the critical<br /> temperature from the parameters (weights and biases) of trained neural network.<br /> We attempt to understand how temperature-supervised neural networks capture the<br /> information of phase transition by paying attention to what quantities they<br /> learn. Our detailed analyses reveal that it learns different physical<br /> quantities depending on how well it is trained. Main observation in this study<br /> is how the weights in the trained neural-network can have information of the<br /> phase transition in addition to temperature.

We apply the relation between deep learning (DL) and the AdS/CFT<br /> correspondence to a holographic model of QCD. Using a lattice QCD data of the<br /> chiral condensate at a finite temperature as our training data, the deep<br /> learning procedure holographically determines an emergent bulk metric as neural<br /> network weights. The emergent bulk metric is found to have both a black hole<br /> horizon and a finite-height IR wall, so shares both the confining and<br /> deconfining phases, signaling the cross-over thermal phase transition of QCD.<br /> In fact, a quark antiquark potential holographically calculated by the emergent<br /> bulk metric turns out to possess both the linear confining part and the Debye<br /> screening part, as is often observed in lattice QCD. From this we argue the<br /> discrepancy between the chiral symmetry breaking and the quark confinement in<br /> the holographic QCD. The DL method is shown to provide a novel data-driven<br /> holographic modeling of QCD, and sheds light on the mechanism of emergence of<br /> the bulk geometries in the AdS/CFT correspondence.

We present a deep neural network representation of the AdS/CFT<br /> correspondence, and demonstrate the emergence of the bulk metric function via<br /> the learning process for given data sets of response in boundary quantum field<br /> theories. The emergent radial direction of the bulk is identified with the<br /> depth of the layers, and the network itself is interpreted as a bulk geometry.<br /> Our network provides a data-driven holographic modeling of strongly coupled<br /> systems. With a scalar $\phi^4$ theory with unknown mass and coupling, in<br /> unknown curved spacetime with a black hole horizon, we demonstrate our deep<br /> learning (DL) framework can determine them which fit given response data.<br /> First, we show that, from boundary data generated by the AdS Schwarzschild<br /> spacetime, our network can reproduce the metric. Second, we demonstrate that<br /> our network with experimental data as an input can determine the bulk metric,<br /> the mass and the quadratic coupling of the holographic model. As an example we<br /> use the experimental data of magnetic response of a strongly correlated<br /> material Sm$_{0.6}$Sr$_{0.4}$MnO$_3$. This AdS/DL correspondence not only<br /> enables gravity modeling of strongly correlated systems, but also sheds light<br /> on a hidden mechanism of the emerging space in both AdS and DL.

We study dynamics of quantum entanglement in smooth global quenches with a<br /> finite rate, by computing the time evolution of entanglement entropy in 1 + 1<br /> dimensional free scalar theory with time-dependent masses which start from a<br /> nonzero value at early time and either crosses or approaches zero. The<br /> time-dependence is chosen so that the quantum dynamics is exactly solvable. If<br /> the quenches asymptotically approach a critical point at late time, the<br /> early-time and late-time entropies are proportional to the time and subsystem<br /> size respectively. Their proportionality coefficients are determined by scales:<br /> in a fast limit, an initial correlation length; in a slow limit, an effective<br /> scale defined when adiabaticity breaks down. If the quenches cross a critical<br /> point, the time evolution of entropy is characterized by the scales: the<br /> initial correlation length in the fast limit and the effective correlation<br /> length in the slow limit. The entropy oscillates, and the entanglement<br /> oscillation comes from a coherence between right-moving and left-moving waves<br /> if we measure the entropy after time characterized by the quench rate. The<br /> periodicity of the late-time oscillation is consistent with the periodicity of<br /> the oscillation of zero modes which are zero-momentum spectra of two point<br /> functions of a fundamental field and its conjugate momentum.

In this paper we propose new algorithm to reduce autocorrelation in Markov<br /> chain Monte-Carlo algorithms for euclidean field theories on the lattice. Our<br /> proposing algorithm is the Hybrid Monte-Carlo algorithm (HMC) with restricted<br /> Boltzmann machine. We examine the validity of the algorithm by employing the<br /> phi-fourth theory in three dimension. We observe reduction of the<br /> autocorrelation both in symmetric and broken phase as well. Our proposing<br /> algorithm provides consistent central values of expectation values of the<br /> action density and one-point Green&#039;s function with ones from the original HMC<br /> in both the symmetric phase and broken phase within the statistical error. On<br /> the other hand, two-point Green&#039;s functions have slight difference between one<br /> calculated by the HMC and one by our proposing algorithm in the symmetric<br /> phase. Furthermore, near the criticality, the distribution of the one-point<br /> Green&#039;s function differs from the one from HMC. We discuss the origin of<br /> discrepancies and its improvement.

Lattice simulations for (2+1)-flavor QCD with external magnetic field<br /> demonstrated that the quark mass is one of the important parameters responsible<br /> for the (inverse) magnetic catalysis. We discuss the dependences of chiral<br /> condensates and susceptibilities, the Polyakov loop on the magnetic field and<br /> quark mass in three degenerate flavor QCD. The lattice simulations are<br /> performed using standard staggered fermions and the plaquette action with<br /> spatial sizes Ns = 16 and 24 and a fixed temporal size Nt = 4. The value of the<br /> quark masses are chosen such that the system undergoes a first order chiral<br /> phase transition and crossover with zero magnetic field. We find that in light<br /> mass regime, the quark chiral condensate undergoes magnetic catalysis in the<br /> whole temperature region and the phase transition tend to become stronger as<br /> the magnetic field increases. In crossover regime, deconfinement transition<br /> temperature is shifted by the magnetic field when quark mass ma is less than<br /> 0.4. The lattice cutoff effects are also discussed.

We study the axial U(1) symmetry at a finite temperature in two-flavor lattice QCD. Employing the Mobius domain-wall fermions, we generate gauge configurations slightly above the critical temperature Tc with different lattice sizes L = 2-4 fm. Our action allows frequent topology tunneling while keeping good chiral symmetry close enough to that of overlap fermions. This allows us to recover full chiral symmetry by an overlap/domain-wall reweighting. Above the phase transition, a strong suppression of the low-lying modes is observed in both overlap and domain-wall Dirac spectra. We, however, find a sizable violation of the Ginsparg-Wilson relation in the Mobius domain-wall Dirac eigenmodes, which dominates the signals of the axial U(1) symmetry breaking near the chiral limit. We also find that the use of the overlap fermion only in the valence sector is dangerous since it suffers from the artifacts due to partial quenching. Reweighting the Mobius domain-wall fermion determinant to that of the overlap fermion, we observe the axial U(1) breaking to vanish in the chiral limit, which is stable against the changes of the lattice volume and lattice spacing.

We provide an origin of family replications in the standard model of particle<br /> physics by constructing renormalizable, asymptotically free, four dimensional<br /> local gauge theories that dynamically generate the fifth and sixth dimensions<br /> with magnetic fluxes.

Global quantum quench with a finite quench rate which crosses critical points<br /> is known to lead to universal scaling of correlation functions as functions of<br /> the quench rate. In this work, we explore scaling properties of the<br /> entanglement entropy of a subsystem in a harmonic chain during a mass quench<br /> which asymptotes to finite constant values at early and late times and for<br /> which the dynamics is exactly solvable. When the initial state is the ground<br /> state, we find that for large enough subsystem sizes the entanglement entropy<br /> becomes independent of size. This is consistent with Kibble-Zurek scaling for<br /> slow quenches, and with recently discussed &quot;fast quench scaling&quot; for quenches<br /> fast compared to physical scales, but slow compared to UV cutoff scales.

We design a Convolutional Neural Network (CNN) which studies correlation<br /> between discretized inverse temperature and spin configuration of 2D Ising<br /> model and show that it can find a feature of the phase transition without<br /> teaching any a priori information for it. We also define a new order parameter<br /> via the CNN and show that it provides well approximated critical inverse<br /> temperature. In addition, we compare the activation functions for convolution<br /> layer and find that the Rectified Linear Unit (ReLU) is important to detect the<br /> phase transition of 2D Ising model.

We study the $U(1)_A$ anomaly in two-flavor lattice QCD at finite temperature<br /> with the M\&quot;obius domain-wall Dirac operator. We generate gauge configurations<br /> in the temperature range $(0.9, 1.2) T_c$ on different physical volumes, $L=$<br /> 2--4 fm, and lattice spacings. We measure the difference of the<br /> susceptibilities of the flavor non-singlet scalar ($\chi_\delta$) and<br /> pseudoscalar ($\chi_\pi$) mesons. They are related by an axial $U(1)$<br /> transformation and the difference vanishes if the axial $U(1)$ symmetry is<br /> respected. We identify the source of axial $U(1)$ symmetry breaking at finite<br /> temperature in the lowest eigenmodes, for the observable $\chi_\pi -<br /> \chi_\delta$. We then reweight the M\&quot;obius domain-wall fermion partition<br /> function to that of the overlap-Dirac operator to fully recover chiral<br /> symmetry. Our data show a significant discrepancy in the results coming from<br /> the M\&quot;obius domain-wall valence quarks, the overlap valence quarks on our DWF<br /> configurations and the reweighted ones that have full chiral symmetry. After<br /> recovering full chiral symmetry we conclude that the difference $\chi_\pi -<br /> \chi_\delta$ shows a suppression in the chiral limit that is compatible with an<br /> effective restoration of $U(1)_A$ at $T \gtrsim T_c$ in the scalar meson<br /> channels.

We investigate the effects of the violation of the Ginsparg-Wilson (GW)<br /> relation in the M\&quot;obius domain-wall fermion formulation on the lattice with<br /> finite fifth dimension. Using a decomposion in terms of the eigenmodes of its<br /> four-dimensional effective Dirac operator, we isolate the GW-violating terms<br /> for various physical quantities including the residual mass and the meson<br /> susceptibilities relevant for the effective restoration of the axial U(1)<br /> symmetry at finite temperature. Numerical result shows that the GW-violating<br /> effect is more significant, or even overwhelming, for the quantities that are<br /> dominated by the low-lying eigenmodes.

We investigate a trajectory for the Wilson flow in the theory space. For this<br /> purpose, we determine the coefficient of the plaquette and rectangular terms in<br /> the action for the configurations defined by the solution of the Wilson flow.<br /> The demon method regarded as one of the inverse Monte Carlo methods is used for<br /> the determination of them. Starting from the conventional Wilson plaquette<br /> action of quenched QCD, we find that the coefficient of the plaquette grows<br /> while that of the rectangular tends to negative with the development of the<br /> flow as the known improved actions. We also find that the trajectory forms a<br /> straight line in the two-coupling theory space.

We study the axial U(1)A symmetry of Nf = 2 QCD at finite temperature using<br /> the Dirac eigenvalue spectrum. The gauge configurations are generated employing<br /> the Mobius domain-wall fermion action on 16^3x8 and 32^3x8 lattices. The<br /> physical spatial size of these lattices is around 2 fm and 4 fm, respectively,<br /> and the simulated temperature is around 200 MeV, which is slightly above the<br /> critical temperature of the chiral phase transition. Although the Mobius<br /> domain-wall Dirac operator is expected to have a good chiral symmetry and our<br /> data actually show small values of the residual mass, we observe significant<br /> violation of the Ginsparg-Wilson relation for the low- lying eigenmodes of the<br /> Mobius domain-wall Dirac operator. Using the reweighting technique, we compute<br /> the overlap-Dirac operator spectrum on the same set of configurations and find<br /> a significant difference of the spectrum between the two Dirac operators for<br /> the low-lying eigenvalues. The overlap-Dirac spectrum shows a gap from zero,<br /> which is insensitive to the spacial volume.

We investigate the axial $U(1)$ symmetry restoration at finite temperature in<br /> two flavor QCD. We employ the M\&quot;obius domain-wall formalism that is designed<br /> to achieve good chiral symmetry. We show the measurements of a difference of<br /> meson susceptibilities, sensitive to the $U(1)_A$ symmetry breaking. The signal<br /> is dominated by zero and near-zero modes. By reweighting the measure to that of<br /> overlap fermions we find a suppression of the $U(1)_A$ breaking effects above<br /> the chiral transition temperature.

We show the numerical simulation result for the mass anomalous dimension of<br /> the SU($3$) gauge theory coupled to $N_f = 12$ fundamental fermions. We use two<br /> independent methods, namely the step scaling method and the hyperscaling method<br /> of the Dirac mode number, to determine the anomalous dimension in the vicinity<br /> of the infrared fixed point of the theory. We show the continuum extrapolations<br /> keeping the renormalized coupling constant as a reference in both analyses.<br /> Furthermore, some recent works seems to suggest the lower boundary of the<br /> conformal window of the SU($3$) gauge theory exists between $N_f=8$ and $10$.<br /> We also briefly report our new project, in which the numerical simulation of<br /> the SU($3$) gauge theory coupled to $N_f=9$ fundamental fermions has been<br /> performed.

We discuss the gauge symmetry breaking via the Hosotani mechanism by using<br /> exact results on supersymmetric gauge theories based on the localization<br /> method. We use the theories on S^2 x S^1 Euclidean space, and study how the<br /> effective potential for the Wilson line phase varies by running an imaginary<br /> chemical potential. In order to break the symmetry, we find that large R-charge<br /> is necessary. With such large R-charge, we study the phase structure of the<br /> theory. In addition, we observed that a finite size effect on our curved space<br /> when we take R-charge is not so large.