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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Approaching the critical region of two-dimensional φ4 quantum field theory with post-Gaussian approximations

M Funke, Ulrich B. Kaulfuss, Hermann G. Kümmel

Published Jan 15, 1987• 52 citations

Abstract

We investigate the vacuum state of (1+1)-dimensional ${\ensuremath{\varphi}}^{4}$ quantum field theory utilizing a modification of the powerful coupled cluster method by the additional maximum-overlap condition. This permits us to construct the ground state of that field theory for nearly all values of the coupling strength. Only a small region has to be excluded where our method still fails. This is most probably due to critical behavior showing up in a change of the order parameter of the model. Our procedure predicts a behavior of the (${\ensuremath{\varphi}}^{4}$${)}_{2}$ model in complete agreement with some rigorous mathematical statements which is not possible in the case of a Gaussian approximation only. Perhaps somewhat unexpectedly, the symmetry-breaking Hamiltonian does not have any critical point.

DOI: 10.1103/physrevd.35.621

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We investigate the vacuum state of (1+1)-dimensional

{\ensuremath{\varphi}}^{4}

quantum field theory utilizing a modification of the powerful coupled cluster method by the additional maximum-overlap condition. This permits us to construct the ground state of that field theory for nearly all values of the coupling strength. Only a small region has to be excluded where our method still fails. This is most probably due to critical behavior showing up in a change of the order parameter of the model. Our procedure predicts a behavior of the (

{\ensuremath{\varphi}}^{4}$${)}_{2}

model in complete agreement with some rigorous mathematical statements which is not possible in the case of a Gaussian approximation only. Perhaps somewhat unexpectedly, the symmetry-breaking Hamiltonian does not have any critical point.

The boson field theory with ${\mathrm{\ensuremath{\Phi}}}^{4}$ and ${\mathrm{\ensuremath{\Phi}}}^{6}$ coupling in (1+1) dimensions is investigated. Rigorous duality relations covering the whole range of coupling constants are derived. Vacuum, one-, and two-particle states are computed via the coupled cluster method in the four-particle approximations. Outside a critical region around second-order phase transitions the method works very well even in regions with first-order phase transitions and bound states.

Recent applications of the coupled-cluster method to the (${\ensuremath{\varphi}}^{4}$

{)}_{2}

quantum field theory showed the necessity of a detailed analysis of its convergence behavior. The anharmonic oscillator has always served as a good test case for different approximation schemes. We treat the model both in the maximum-overlap condition and in the Hartree approximation. The ground-state energy is reproduced very well for all values of the coupling strength and already for a low-order truncation scheme. The expansion in correlation amplitudes can be carried out in extremely high order and shows a divergent tendency of the amplitudes. Introducing temperature dependence allows us to select the stable ground state out of a variety of solutions of the hierarchy of equations.

We defend the Fock-space Hamiltonian truncation method, which allows us to calculate numerically the spectrum of strongly coupled quantum field theories, by putting them in a finite volume and imposing a UV cutoff. The accuracy of the method is improved via an analytic renormalization procedure inspired by the usual effective field theory. As an application, we study the two-dimensional ${\ensuremath{\phi}}^{4}$ theory for a wide range of couplings. The theory exhibits a quantum phase transition between the symmetry-preserving and symmetry-breaking phases. We extract quantitative predictions for the spectrum and the critical coupling and make contact with previous results from the literature. Future directions to further improve the accuracy of the method and enlarge its scope of applications are outlined.

We develop a variational framework for addressing two-dimensional non-integrable quantum field theories through the exact structure of their integrable counterparts. Concentrating on the

\varphi^4

Landau-Ginzburg model, we use the analytical Vacuum Expectation Values and Form Factors of local operators in the sinh-Gordon theory as the foundation of a variational ansatz. In this way, we obtain controlled estimates of central physical quantities of the

\varphi^4

theory - such as the finite-volume ground-state energy and the physical mass as a function of the coupling constant. The strengths of the variational methods are leveraged in combination with the Hamiltonian truncation techniques and the LeClair-Mussardo formula, which also allow to probe the accuracy of the variational approximation varying the system size. Within the weak-coupling regime, a detailed numerical analysis reveals the behaviour of the finite-volume spectrum, the ground-state energy, and the elastic part of the scattering matrix, showing how the rigorous machinery of integrable models can serve as a guiding light into the complex landscape of non-integrable quantum field dynamics.

Starting from the Gaussian-variational approximation the state of the single sine-Gordon soliton is constructed using the coupled cluster method. The quantum contributions to the soliton mass and those to the space dependence of the soliton are computed and found to be non-negligible. If the coupling constant ${\ensuremath{\beta}}^{2}$ approaches the critical value $4\ensuremath{\pi}$ the mass becomes classical, but in other respects the soliton remains a quantum object.