Spontaneous symmetry breaking in compactified λϕ4 theory

Abstract

We consider the massive vector N-component (λϕ 4 ) D theory in Euclidian space and, using an extended Matsubara formalism we perform a compactification on aThis allows us to treat jointly the effect of temperature and spatial confinement in the effective potential of the model, setting forth grounds for an analysis of phase transitions driven by temperature and spatial boundary. For d = 2, which corresponds to the heated system confined between two parallel planes (separation L), we obtain, in the large N limit at one-loop order, formulas for temperature-and boundary-dependent mass and coupling constant. The equation for the critical curve in the β × L plane is also derived.

Gauge fluctuations and transition temperature for superconducting wires

“…To generate the contributions from gauge fluctuations, we have used the Gaussian effective potential [2][3][4][5], which allows to obtain a gap equation that can be treated with the method of recent developments [8,9]. We have derived a critical equation that describes the changes in the critical temperature due to confinement.…”

“…This generalization has been done in [8] and we briefly describe here its main steps. ¿From the identity, 8) and using the representation for Bessel functions of the third kind, K ν ,…”

“…[10] and has been assumed to be valid for higher orders [11]. Relying on the fact that for Euclidian field theories inverse imaginary time and spatial coordinates are on the same footing, such a topological result has been generalized to treat different physical situations in which fields are confined in higher purely spatial dimensions, considering the Matsubara mechanism on a R D−d × S 11 × S 12 ... × S 1 d topology, describing space confinement in a d-dimensional subspace [8,9,12]. As a consequence the Matsubara formalism can be thought, in a generalized way, as a mechanism to deal with spatial constraints in a field theory model.…”

Finite-size effects on the phase transition in a four- and six-fermion interaction model

“…Since correlation functions should satisfy periodicity conditions on the time coordinate, known as Kubo-Martin-Schwinger (KMS) conditions, the finite-temperature theory is defined on the compactified manifold Γ 1 4 = S 1 ×R 3 , where S 1 is a circumference with length proportional to the inverse of the temperature and R 3 is the Euclidean 3-dimensional space. Compactification of spatial dimensions [5,6] is considered in a similar way. An unified treatment, generalizing various approaches dealing with finite-temperature and spatialcompactification concurrently, has been constructed [7,8,9] These methods have been employed to investigate spontaneous symmetrybreaking induced by temperature and/or spatial constraints in some bosonic and fermionic models describing phase transitions in condensed-matter, statistical and particle physics; for instance, for describing the size-dependence of the transition temperature of superconducting films, wires and grains [10,11]; for investigating size-effects in first-and second-order transitions [12,13,14,15]; and for analyzing size and magnetic-field effects on the Gross-Neveu (GN) [16] and the Nambu-Jona-Lasinio (NJL) [17] models, taken as effective theories [18] for hadronic physics [19,20,21].…”

Modelo de Gross-Neveu e simetrias : soluções analíticas e dinâmica de campos térmicos

“…Comparando potências de u, tem -se 27) A xxt = 0 , (4.28) A xxx = 0 . 30) que tem 10 constantes arbitrárias, dando origem a 10 geradores distintos.…”