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 amplitude √ 5 3 π.The scaling function of the singular part of the free energy is found to exhibit a maximum at ln 2 2π corresponding to one of the crossover lines.The critical amplitude of the singular part of the free energy is given by the universal number 1 3 1 2π ζ(3) -Cl2 π 3 = -0.274543 . .., where ζ(z) and Cl2(z) are the Riemann zeta and Clausen's functions, respectively.Interpreted in terms of the thermodynamic Casimir effect, this result implies an attractive Casimir "force".This study is expected to be useful in shedding light on a broader class of four fermionic models.
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 amplitude √ 5 3 π.The scaling function of the singular part of the free energy is found to exhibit a maximum at ln 2 2π corresponding to one of the crossover lines.The critical amplitude of the singular part of the free energy is given by the universal number 1 3 1 2π ζ(3) -Cl2 π 3 = -0.274543 . .., where ζ(z) and Cl2(z) are the Riemann zeta and Clausen's functions, respectively.Interpreted in terms of the thermodynamic Casimir effect, this result implies an attractive Casimir "force".This study is expected to be useful in shedding light on a broader class of four fermionic models.
We study the thermodynamics of the relativistic quantum O(N) model in two space dimensions. In the vicinity of the zero-temperature quantum critical point (QCP), the pressure can be written in the scaling form $P(T)=P(0)+N({T}^{3}/{c}^{2}){\mathcal{F}}_{N}(\ensuremath{\Delta}/T)$, where c is the velocity of the excitations at the QCP and $|\ensuremath{\Delta}|$ a characteristic zero-temperature energy scale. Using both a large-N approach to leading order and the nonperturbative renormalization group, we compute the universal scaling function FN. For small values of N ($N\ensuremath{\lesssim}10$) we find that FN(x) is nonmonotonic in the quantum critical regime ($|x|\ensuremath{\lesssim}1$) with a maximum near x=0. The large-N approach---if properly interpreted---is a good approximation both in the renormalized classical ($x\ensuremath{\lesssim}\ensuremath{-}1$) and quantum disordered ($x\ensuremath{\gtrsim}1$) regimes, but fails to describe the nonmonotonic behavior of FN in the quantum critical regime. We discuss the renormalization-group flows in the various regimes near the QCP and make the connection with the quantum nonlinear sigma model in the renormalized classical regime. We compute the Berezinskii-Kosterlitz-Thouless transition temperature in the quantum O(2) model and find that in the vicinity of the QCP the universal ratio ${T}_{\mathrm{BKT}}/{\ensuremath{\rho}}_{s}(0)$ is very close to $\ensuremath{\pi}/2$, implying that the stiffness ${\ensuremath{\rho}}_{s}({T}_{\mathrm{BKT}}^{\ensuremath{-}})$ at the transition is only slightly reduced with respect to the zero-temperature stiffness ${\ensuremath{\rho}}_{s}(0)$. Finally, we briefly discuss the experimental determination of the universal function F2 from the pressure of a Bose gas in an optical lattice near the superfluid--Mott-insulator transition.
Two-dimensional Dirac semimetals near a quantum critical point can be described by Gross--Neveu--Yukawa models. In view of recent experimental advances exhibiting a transition from Dirac semimetal to insulator in highly-tunable van-der-Waals heterostructures, a better understanding of finite-temperature effects is mandatory. Here, we study the Gross--Neveu--Yukawa phase diagram of the chiral Ising model with a non-perturbative field-theory approach at zero and finite temperature, both in the semimetallic phase and in the insulating phase with spontaneously broken Z2 symmetry. At zero temperature, we find a quantum critical point with critical exponents that are close to the ones of the chiral Ising universality class, and show that relativistic symmetry is emergent close to the quantum critical point. At finite temperature, the ordered phase survives up to a finite critical temperature, at which we observe a classical phase transition into the disordered phase. We confirm that this transition lies in the two-dimensional Ising universality class. Finally, we determine the extent and scaling properties of the quantum critical fan, and the behavior of the quasiparticle weight, therein. In summary, we present a unified field-theoretical framework for the phase diagram of the chiral Ising model in the surroundings of its quantum critical point.
We briefly review recent results of exact calculations of critical Casimir forces of the O(n) ϕ 4 model as n→∞ on a three-dimensional strip bounded by two planar free surfaces at a distance L. This model has long-range order below the critical temperature Tc of the bulk phase transition only in the limit L→∞ but remains disordered for all T > 0 for an arbitrary finite strip width L < ∞. A proper description of the system scaling behavior near T c turns out to be a quite challenging problem because in addition to bulk, boundary, and finite-size critical behaviors, a nontrivial dimensional crossover must be handled. The model admits an exact solution in the limit n → ∞ in terms of the eigenvalues and eigenenergies of a self-consistent Schrödinger equation. This solution contains a potential v(z) with the near-boundary singular behavior v(z → 0+) ≈ −1/(4z 2)+ 4m/(π 2 z), where m = 1/ξ+(|t|) is the inverse bulk correlation length and t ~ (T − T c )/T c , and a corresponding singularity at the second boundary plane. In recent joint work with colleagues, the potential v(z), the excess free energy, and the Casimir force were obtained numerically with high precision. We explain how these numerical results can be complemented by exact analytic ones for several quantities (series expansion coefficients of v(z), the scattering data of v(z) in the semi-infinite case L = ∞ for all m >/< 0, and the low-temperature asymptotic behavior of the residual free energy and the Casimir force) by a combination of boundary-operator and short-distance expansions, proper extensions of the inverse scattering theory, new trace formulas, and semiclassical expansions.
The canonical Gross-Neveu model for N two-component Dirac fermions in 2+1 dimensions suffers a continuous phase transition at a critical interaction gc1∼1/N at large N , at which its continuous symmetry SO(2N) is preserved and a discrete (Ising) symmetry becomes spontaneously broken. A recent mean-field calculation, however, points to an additional transition at a different critical gc2∼−Ngc1 , at which SO(2N)→SO(N)×SO(N) . To study the latter phase transition we rewrite the Gross-Neveu interaction g(ψ¯ψ)2 in terms of three different quartic terms for the single ( L=1 ) 4N -component real (Majorana) fermion, and then extend the theory to L>1 . This allows us to track the evolution of the fixed points of the renormalization group transformation starting from L≫1 , where one can discern three distinct critical points which correspond to continuous phase transitions into (1) SO(2N) -singlet mass-order-parameter, (2) SO(2N) -symmetric-tensor mass-order-parameters, and (3) SO(2N) -adjoint nematic-order-parameters, down to L=1 value that is relevant to the standard Gross-Neveu model. Below the critical value of Lc(N)≈0.35N for N≫1 only the Gross-Neveu critical point (1) still implies a diverging susceptibility for its corresponding ( SO(2N) -singlet) order parameter, whereas the two new critical points that existed at large L ultimately become equivalent to the Gaussian fixed point at L=1 . We interpret this metamorphosis of the SO(2N) -symmetric-tensor fixed point from critical to spurious as an indication that the transition at gc2 in the original Gross-Neveu model is turned first-order by fluctuations. Published by the American Physical Society 2024
A detailed analysis of the finite-size effects on the bulk critical behaviour of the d -dimensional mean spherical model confined to a film geometry with finite thickness L is reported. Along the finite direction different kinds of boundary conditions are applied: periodic ( p ), antiperiodic ( a ) and free surfaces with Dirichlet ( D ), Neumann ( N ) and a combination of Neumann and Dirichlet ( ND ) on both surfaces. A systematic method for the evaluation of the finite-size corrections to the free energy for the different types of boundary conditions is proposed. The free energy density and the equation for the spherical field are computed for arbitrary d . It is found, for 2 < d < 4, that the singular part of the free energy has the required finite-size scaling form at the bulk critical temperature only for ( p ) and ( a ). For the remaining boundary conditions the standard finite-size scaling hypothesis is not valid. At d = 3, the critical amplitude of the singular part of the free energy (related to the so-called Casimir amplitude) is estimated. We obtain Δ ( p ) = −2ζ(3)/(5π) = −0.153 051..., Δ ( a ) = 0.274 543... and Δ ( ND ) = 0.019 22..., implying a fluctuation-induced attraction between the surfaces for ( p ) and repulsion in the other two cases. For ( D ) and ( N ) we find a logarithmic dependence on L .
A second-order phase transition for the three-dimensional Gross-Neveu model is established for one fermion species, N=1. This transition breaks a paritylike discrete symmetry. It constitutes its peculiar universality class with critical exponent $\ensuremath{\nu}=0.63$ and scalar and fermionic anomalous dimensions ${\ensuremath{\eta}}_{\ensuremath{\sigma}}=0.31$ and ${\ensuremath{\eta}}_{\ensuremath{\psi}}=0.11,$ respectively. We also compute critical exponents for other N. Our results are based on exact renormalization-group equations.