Strong-coupling-expansion analysis of the false-vacuum decay rate of the lattice ϕ⁴model in 1 + 1 dimensions

Abstract

Strong-coupling expansion is performed for the lattice φ 4 model in 1+1 dimensions. Because the strong-coupling limit itself is not solvable, we employed numerical calculations so as to set up unperturbed eigensystems. Restricting the number of Hilbert-space bases, we performed linked-cluster expansion up to eleventh order. We carried out alternative simulation by means of the density-matrix renormalization group. Thereby, we confirmed that our series-expansion data with a convergenceacceleration trick are in good agreement with the simulation result. Through the analytic continuation to the domain of negative biquadratic interaction, we obtain the false-vacuum decay rate. Contrary to common belief that tunnelling phenomenon lies out of perturbative treatments, our series expansion reproduces the instanton-theory behaviour for high potential barrier. For shallow barrier, on the contrary, our result tells that the relaxation is no more described by instanton, but the decay rate acquires notable enhancement.

A variational sinc collocation method for strong-coupling problems

“…As a last example of application of our method we consider a lattice φ 4 in 1 + 1 dimensions. This model has been studied by Nishiyama in [10] and corresponds to the hamiltonian…”

“…The fields obey the canonical commutation relations [φ i , π j ] = iδ ij and [φ i , φ j ] = [π i , π j ] = 0. Following [10] we perform a rescaling of the fields φ → g −1/6 φ and π → g 1/6 π and obtain…”

“…The solid curve corresponds to the perturbative expansion of eq. ( 6) of [10], to order λ 11 , obtained with the LCE. Our result compares quite favorably also with the results obtained with the DMRG -see Fig.…”

Stiffening of fluid membranes due to thermal undulations: Density-matrix renormalization-group study

“…Therefore, the Monte-Carlo technique has been employed in order to simulate soft matters. However, after the advent of the density-matrix renormalization group [11,12,13], the difficulty was removed, and now, soft matters (elastic systems) have come under through investigations by means of the diagonalization method; the examples are lattice vibrations [15,16], collection of oscillators as a heat bath [17], string meandering motions [18,19], and lattice scalar field theory [20]. In essence, the technique allows us to discard "irrelevant states," and hence, the number of states are truncated so as to save the computer-memory space.…”

Direct observation of the effective bending moduli of a fluid membrane: Free-energy cost due to the reference-plane deformations

“…The method was invented, originally, so as to investigate the highly-correlated systems such as the Hubbard models and the spin chains. Later on, it was extended to the field of soft materials such as the lattice vibrations [21,22,23], the quantum string [24,25], and the bosonic systems [26,27]. We repeat densitymatrix renormalization one after another so as to reach sufficiently long transfer-matrix strip length [12].…”