Narrow Results

We investigate the critical short-time scaling of the two-dimensional lattice ϕ⁴ field theory with a Langevin dynamics. Starting from a "hot" initial configuration but with a small magnetization, the critical initial increase of the magnetization is observed, through which we determine the critical point. From the short-time relaxation dynamics of various quantities at the critical point obtained, the dynamic critical exponents θ, z and the static exponent β/ν are evaluated. In executing a Langevin simulation, an appropriate discretization method with respect to the time degree of freedom becomes essentially important to be adopted and we show that the well-known second-order form of discretized Langevin equation works well to investigate the short-time dynamics.

We present a perturbative calculation of finite-size effects near T_c of the φ⁴ lattice model in a d-dimensional cubic geometry of size L with periodic boundary conditions for d>4. The structural differences between the φ⁴ lattice theory and the φ⁴ field theory found previously in the spherical limit are shown to exist also for a finite number of components of the order parameter. The two-variable finite-size scaling functions of the field theory are nonuniversal whereas those of the lattice theory are independent of the nonuniversal model parameters. One-loop results for finite-size scaling functions are derived. Their structure disagrees with the single-variable scaling form of the lowest-mode approximation for any finite ξ/L where ξ is the bulk correlation length. At T_c, the large-L behavior becomes lowest-mode like for the lattice model but not for the field-theoretic model. Characteristic temperatures close to T_c of the lattice model, such as T_max(L) of the maximum of the susceptibility χ, are found to scale asymptotically as T_c-T_max(L) ~L^-d/2, in agreement with previous Monte Carlo (MC) data for the five-dimensional Ising model. We also predict χ_max~L^d/2 asymptotically. On a quantitative level, the asymptotic amplitudes of this large-L behavior close to T_c have not been observed in previous MC simulations at d=5 because of nonnegligible finite-size terms ~L^(4-d)/2 caused by the inhomogeneous modes. These terms identify the possible origin of a significant discrepancy between the lowest-mode approximation and previous MC data. MC data of larger systems would be desirable for testing the magnitude of the L^(4-d)/2 and L^4-d terms predicted by our theory.

We study the ${\varphi }^4$ field theory in d = 4. Using bold diagrammatic Monte Carlo method, we solve the Schwinger–Dyson equations and find the spectral density function of the theory beyond the weak coupling regime. We then compare our result with the one obtained from the perturbation theory. At the end, we utilize our Monte Carlo result to find the vertex function as the basis for the computation of the physical scattering amplitudes.

We studied the effects of nonextensivity on the phase transition for the system of finite volume $V$ in the ${\varphi }^4$ theory in the Tsallis nonextensive statistics of entropic parameter $q$ and temperature $T$, when the deviation from the Boltzmann–Gibbs (BG) statistics, $|q-1|$, is small. We calculated the condensate and the effective mass to the order $q-1$ with the normalized $q$-expectation value under the free particle approximation with zero bare mass. The following facts were found. The condensate $\mathrm{\Phi }$ divided by $v$, $\mathrm{\Phi }/v$, at $q$ ($v$ is the value of the condensate at $T=0$) is smaller than that at $q^{\prime }$ for $q>q^{\prime }$ as a function of $T_{\mathrm{\text{ph}}}/v$ which is the physical temperature $T_{\mathrm{\text{ph}}}$ divided by $v$. The physical temperature $T_{\mathrm{\text{ph}}}$ is related to the variation of the Tsallis entropy and the variation of the internal energies, and $T_{\mathrm{\text{ph}}}$ at $q=1$ coincides with $T$. The effective mass decreases, reaches minimum, and increases after that, as $T_{\mathrm{\text{ph}}}$ increases. The effective mass at $q>1$ is lighter than the effective mass at $q=1$ at low physical temperature and heavier than the effective mass at $q=1$ at high physical temperature. The effects of the nonextensivity on the physical quantity as a function of $T_{\mathrm{\text{ph}}}$ become strong as $|q-1|$ increases. The results indicate the significance of the definition of the expectation value, the definition of the physical temperature, and the constraints for the density operator, when the terms including the volume of the system are not negligible.