Critical line of the Φ^4 scalar field theory on a 4D cubic lattice in the local potential approximation

Abstract

We establish the critical line of the one-component (or Landau-Ginzburg) model on a simple four-dimensional cubic lattice. Our study is performed in the framework of the non-perturbative renormalization group in the local potential approximation with a soft infra-red regulator. The transition is found to be of the second order even in the Gaussian limit where the first order would be expected according to some recent theoretical predictions.

The nonperturbative functional renormalization group and its applications

“…This lattice FRG captures both local and critical fluctuations and therefore enables us to compute nonuniversal quantities such as transition temperatures. It has been applied to the O(N) model defined on a lattice [223][224][225] and to classical [223] and quantum [226,227] spin models as well as the superfluid-Mott transition in the Bose-Hubbard model [171,[228][229][230][231].…”

Critical behavior of the hopping expansion from the Functional Renormalization Group

“…The above technique has been applied to determine κ c (λ ) for φ 4 theories in both D = 3 and D = 4. The shooting technique has been implemented in Mathematica without encountering significant obstructions from stiffness for reasonably large N. Related results have been obtained in [4], [5] in a different LPA formulation but without relation to (1.3) and the hopping expansion's radius of convergence.…”

A general method for calculating lattice green functions on the branch cut

“…The ability to compute LGFs can be useful, for example, for simulations of Hubbard models 1 and for non-perturbative renormalization group studies of scalar boson models. 2,3 For i = j, G ij (ω) can be expressed as closed forms in terms of named special functions (mainly elliptic integrals or generalized hypergeometric functions) for square, bcc, honeycomb, diamond, cubic, hypercubic, triangular, and fcc lattices. [4][5][6][7][8][9][10][11][12][13][14][15][16] For i = j, G ij (ω) can be expressed as closed forms for square, bcc, 14 honeycomb, 14 triangular, 14 kagome, 14 diced, 14 and cubic 10 lattices; spatial recurrence relations exist but are often numerically unstable.…”

“…ultracold atoms in optical lattices [4], electrical networks [5,6], statistical physics [7][8][9][10], and lattice gauge theory [11][12][13][14], as noted in [15]. The ability to compute LGFs can be useful, for example, for simulations of Hubbard models [4] and for non-perturbative renormalization group studies of scalar boson models [13,14].…”

“…ultracold atoms in optical lattices [4], electrical networks [5,6], statistical physics [7][8][9][10], and lattice gauge theory [11][12][13][14], as noted in [15]. The ability to compute LGFs can be useful, for example, for simulations of Hubbard models [4] and for non-perturbative renormalization group studies of scalar boson models [13,14]. For i = j, G ij (ω) can be expressed as closed forms in terms of named special functions (mainly elliptic integrals or generalized hypergeometric functions) for square, bcc, honeycomb, diamond, cubic, hypercubic, triangular, and fcc lattices [15][16][17][18][19][20][21][22][23][24][25][26][27].…”