Exact Analytic Continuation with Respect to the Replica Number in the Discrete Random Energy Model of Finite System Size

An expression for the moment of partition function valid for any finite system size $N$ and complex power $n (\Re(n)>0)$ is obtained for a simple spin glass model termed the {\em discrete random energy model} (DREM). We investigate the behavior of the moment in the thermodynamic limit $N \to \infty$ using this expression, and find that a phase transition occurs at a certain real replica number when the temperature is sufficiently low, directly clarifying the scenario of replica symmetry breaking of DREM in the replica number space {\em without using the replica trick}. The validity of the expression is numerically confirmed.Comment: 31 pages, 8 eps figure

Observing how deep neural networks understand physics through the energy spectrum of one-dimensional quantum mechanics

We investigate how neural networks (NNs) understand physics using 1D quantum mechanics. After training an NN to accurately predict energy eigenvalues from potentials, we used it to confirm the NN's understanding of physics from four different aspects. The trained NN could predict energy eigenvalues of different kinds of potentials than the ones learned, predict the probability distribution of the existence of particles not used during training, reproduce untrained physical phenomena, and predict the energy eigenvalues of potentials with an unknown matter effect. These results show that NNs can learn physical laws from experimental data, predict the results of experiments under conditions different from those used for training, and predict physical quantities of types not provided during training. Because NNs understand physics in a different way than humans, they will be a powerful tool for advancing physics by complementing the human way of understanding.Comment: 31 pages, 19 figure

Critical Exponents of O(N) Scalar Model at Temperatures below the Critical Value using Auxiliary Mass Method

We investigate a phase transition of the O(N) invariant scalar model using the auxiliary mass method. We determine the critical exponent $\beta$ by calculating an effective potential below the critical temperature. This work follows that of a previous paper.Comment: 6 pages, 3 EPS figures, typeset PTP-Tex, published versio

End-point of the Electroweak Phase Transition using the auxiliary mass method

We study the end-point of the Electroweak phase transition using the auxiliary mass method. The end point is $m_H\sim40$ (GeV) in the case $m_t=0$ (GeV) and strongly depends on the top quark mass. A first order phase transition disappears at $m_t\sim 160$ (GeV). The renormalization effect of the top quark is significant.Comment: 10 pages, 5 EPS figures, typeset using REV-Te

Non-perturbative approach to the effective potential of the $\lambda\phi^{4} theory at finite temperature

We construct a non-perturbative method to investigate the phase structure of the scalar theory at finite temperature. The derivative of the effective potential with respect to the mass square is expressed in terms of the full propagator. Under a certain approximation this expression reduces to the partial differential equation for the effective potential. We numerically solve the partial differential equation and obtain the effective potential non-perturbatively. It is found that the phase transition is of the second order. The critical exponents calculated in this method are consistent with the results obtained in Landau approximation.Comment: 17page, Latex, 9 figure

The Auxiliary Mass Method beyond the Local Potential Approximation

We show that the evolution equation of the effective potential in the auxiliary mass method corresponds to a leading approximation of a certain series. This series is derived from an evolution equation of an effective action using a derivative expansion. We derived an expression of the next-to-leading approximation of the evolution equation, which is a simultaneous partial differential equation.Comment: 23 pages, 3 EPS figure

Temperature phase transition and an effective expansion parameter in the O(N)-model

The temperature phase transition in the N-component scalar field theory with spontaneous symmetry breaking is investigated in the perturbative approach. The second Legendre transform is used together with the consideration of the gap equations in the extrema of the free energy. Resummations are performed on the super daisy level and beyond. The phase transition turns out to be weakly of first order. The diagrams beyond the super daisy ones which are calculated correspond to next-to-next-to-leading order in 1/N. It is shown that these diagrams do not alter the phase transition qualitatively. In the limit N goes to infinity the phase transition becomes second order. A comparison with other approaches is done.Comment: 28 pages, 5 figures, corrected for some misprints, unnecessary section remove

Non-perturbative Evaluation of the Effective Potential of λϕ4\lambda\phi^4 Theory at Finite Temperature under the Super-Daisy Approximation

We calculate the effective potential of the scalar theory at finite temperature under the super-daisy approximation, after expressing its derivative with respect to mass square in terms of the full propagator. This expression becomes the self-consistent equation for the derivative of the effective potential. We find the phase transition is first order with this approximation. We compare our result with others.Comment: 12 page, 8 figure

Investigation into O(N) Invariant Scalar Model Using Auxiliary-Mass Method at Finite Temperature

Using auxiliary-mass method, O(N) invariant scalar model is investigated at finite temperature. This mass and an evolution equation allow us to calculate an effective potential without an infrared divergence. Second order phase transition is indicated by the effective potential. The critical exponents are determined numerically.Comment: LaTex 8 pages with 3 eps figure