Boundary effects on the mass and coupling constant in the compactified Ginzburg–Landau model: The boundary dependent critical temperature

Abstract

We consider the Euclidean D-dimensional N-component ͉͉ 4 ͑Ͼ0͒ model with d ͑d Յ D͒ compactified dimensions. Introducing temperature by means of the Ginzburg-Landau prescription in the mass term of the Hamiltonian, this model can be interpreted as describing a second-order phase transition for a system in a region of the D-dimensional space, limited by d pairs of parallel planes, orthogonal to the coordinates axis x 1 , x 2 , ... ,x d. The planes in each pair are separated by distances L 1 , L 2 , ... ,L d. Making the appropriate boundary corrections to the coupling constant, we obtain in the large-N limit the transition temperature as a function of the size of the system, T c ͕͑L i ͖͒, i =1,2, ... ,d. For D = 3 we particularize this formula, taking L 1 = L 2 =¯= L d = L for the physically interesting cases d =1 ͑a film͒, d =2 ͑an infinitely long wire having a square cross section͒, and d =3 ͑a cubic grain͒.

Phase transition in the massive Gross-Neveu model in toroidal topologies

“…This allows us to consider field theoretical models with spatial constraints, at zero or finite temperature, by using generating functionals with a path-integral formalism on the topology [29][30][31][32]. These ideas have been established recently on a firm foundation [33,34] and applied in different physical situations, for example: for spontaneous symmetry breaking in the compactified φ 4 model [37][38][39]; for second-order phase transitions in superconducting films, wires and grains [40][41][42]; for the Casimir effect for bosons and fermions [43][44][45][46][47][48]; for size effects in the NJL model [49][50][51][52][53]; and, for electrodynamics with an extra dimension [54].…”

Advances in Quantum Field Theory

“…with C 1 , C 2 and C 3 as before and where E 2 and E 3 are constants, resulting from sums involving the Bessel functions [Malbouisson et al (2009)]. We see that the critical temperature has the same kind of dependence on the size extension L for d = 1, 2, 3, only constants differ in each case.…”

“…In the last few decades, this generalized Matsubara formalism has been employed in many instances of condensed-matter and particle physics. Some of them are: (1) the Casimir effect, studied in various geometries, topologies, fields, and physical boundary conditions [Bordag et al (2001); Milonni (1993); Mostepanenko & Trunov (1997)], in a diversity of subjects ranging from nanodevices to cosmological models [Bordag et al (2001); Boyer (2003); Levin & Micha (1993); Milonni (1993); Mostepanenko & Trunov (1997); Seife (1997)]; (2) the confinement/deconfinement phase transition of hadronic matter, in the Gross-Neveu and Nambu-Jona-Lasinio models as effective theories for quantum chromodynamics [Abreu et al (2009); Khanna et al (2010); Malbouisson et al (2002)]; (3) quantum electrodynamics with one extra compactified dimension, which leads to estimates of the size of extra dimensions compatible with present-day experimental data [Ccapa Tira et al (2010)]; (4) the study of superconductors in the form of films, wires and grains [Abreu et al (2003;2005); Khanna et al (2009); Linhares et al (2006;; Malbouisson (2002); Malbouisson et al (2009)], in which the Ginzburg-Landau model for phase transitions is defined on a three-dimensional Euclidean space with one, two or three dimensions compactified.…”

A note on the infrared behavior of the compactified Ginzburg--Landau model in a magnetic field

“…Physical manifestations of this type of topology include, for instance, the vacuum-energy fluctuations giving rise to the Casimir effect (see for instance [10] and other references therein). In the study of phase transitions, the dependence of the critical temperature on the compactification parameters is found in several situations of condensedmatter physics [10,[14][15][16][17][18]. Also, this kind of formalism has been employed in the investigation of the confining phase transition in effective theories for Quantum Chromodynamics [19][20][21][22][23].…”

“…The compactification formalism described above has been applied to field-theoretical models in arbitrary dimension with compactification of any subspace [17,18,24]. This formalism has also been developed from a pathintegral approach in [13].…”