Quantum critical scaling and the Gross-Neveu model in 2+1 dimensions

Abstract

-The quantum critical behavior of the (2 + 1)-dimensional Gross-Neveu model in the vicinity of its zero-temperature critical point is considered. The model is known to be renormalisable in the large-N limit, which offers the possibility to obtain expressions for various thermodynamic functions in closed form. We have used the concept of finite-size scaling to extract information about the leading temperature behavior of the free energy and the mass term, defined by the fermionic condensate and determined the crossover lines in the coupling (g)-temperature (T ) plane. These are given by T ∼ |g − gc|, where gc denotes the critical coupling at zero temperature. According to our analysis no spontaneous symmetry breaking survives at finite temperature. We have found that the leading temperature behavior of the fermionic condensate is proportional to the temperature with the critical amplitudeπ. The scaling function of the singular part of the free energy is found to exhibit a maximum at Introduction. -Quantum phase transitions take place at zero temperature by varying some non-thermal parameter, say g, such as composition or pressure, and are driven by genuine quantum fluctuations. At rather small (when compared to characteristic excitations in the system) temperature the singularities of the thermodynamic quantities are altered. It is then expected that the leading T dependence of all thermodynamic observables is specified by the properties of the zero-temperature critical points, which take place in quantum systems. In the close vicinity of a second-order quantum phase transition the coupling of statics and dynamics introduces an effective dimensionality which depends upon (imaginary) time in addition to space [1]. In this case the inverse temperature acts as a finite size in the imaginary time direction for the quantum system at its critical point. This allows the

Casimir force in theO(n→∞)model with free boundary conditions

“…Among them are versions of Bose gas [80,[82][83][84]. Let us also mention the large-n limit of the so-called 2+1 Gross-Neveu model [85], representative of a broader class of four fermionic models, which lead to mathematics very similar to that of the three dimensional spherical model and to a Casimir amplitude that is exactly equal and opposite to the Casimir amplitude of the three-dimensional spherical model subject to antiperiodic boundary conditions [26]. The methods utilized here for the treatment of the spherical model with free boundary conditions may well point the way to progress in the investigation of some of the above-mentioned quantum systems subject to similar boundary conditions; in the references above these models are usually studied in their thermodynamic limit or subject to periodic boundary conditions.…”

Critical Casimir Effect: Exact Results

“…Among them are certain versions of the Bose gas [220,278,512,514]. We also mention the large-n limit of the so-called 2+1 Gross-Neveu model [515], which represents a broader class of four fermionic models, which are mathematically very similar to the threedimensional spherical model. The Casimir amplitude is oppositely equal to the Casimir amplitude of the threedimensional spherical model subject to antiperiodic boundary conditions [211].…”

“…The interested reader can find the corresponding information in books [316,[525][526][527][528], and in specialized reviews [313-315, 527, 529-535]. Here, we summarize only certain basic concepts and facts, which are needed in order to be able to present results related to the Casimir effect in such systems [9,13,317,515,533,536,537].…”

“…model (GN) [515], which represents a broader class of four fermionic models, is mathematically very similar to that of the three-dimensional spherical model. The corresponding Casimir amplitude ∆ GN Cas which is exactly equal in absolute values but opposite in sign to the Casimir amplitude of the three-dimensional spherical model, subject to antiperiodic boundary conditions [211] If d = σ = 2, the "temporal" Casimir amplitude ∆ t Cas reduces to the "normal" Casimir amplitude ∆ Cas (ρ = 0), given by Eq.…”

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

“…Also, a rich phase structure, including BCSand BEC-like phases, in strongly interacting matter has been obtained with an extended NJL model having six-fermion interaction [25]. Besides, such kind of interaction might be of relevance to systems in condensed matter where GN and NJL models are used, like graphene [26] and quantum phase transitions [27]. In all these cases, fluctuations due to finite-size play an important role in the phase structures.…”