Low-temperature behavior of the $O(N)$ models below two dimensions

Abstract

We investigate the critical behavior and the nature of the low-temperature phase of the $O(N)$ models treating the number of field components $N$ and the dimension $d$ as continuous variables with a focus on the $d\le 2$ and $N\le 2$ quadrant of the $(d,N)$ plane. We precisely chart a region of the $(d,N)$ plane where the low-temperature phase is characterized by an algebraic correlation function decay similar to that of the Kosterlitz-Thouless phase but with a temperature-independent anomalous dimension $\eta$. We revisit the Cardy-Hamber analysis leading to a prediction concerning the nonanalytic behavior of the $O(N)$ models' critical exponents and emphasize the previously not broadly appreciated consequences of this approach in $d<2$. In particular, we discuss how this framework leads to destabilization of the long-range order in favour of the quasi long-range order in systems with $d<2$ and $N<2$. Subsequently, within a scheme of the nonperturbative renormalization group we identify the low-temperature fixed points controlling the quasi long-range ordered phase and demonstrate a collision between the critical and the low-temperature fixed points upon approaching the lower critical dimension. We evaluate the critical exponents $\eta (d,N)$ and ${\nu }^{-1}(d,N)$ and demonstrate a very good agreement between the predictions of the Cardy-Hamber type analysis and the nonperturbative renormalization group in $d<2$.

arXiv comment10 pages, 10 figures, 1 table

References in corpus (22)

• Exact evolution equation for the effective potential
• Non-Perturbative Renormalization Flow in Quantum Field Theory and Statistical Physics
• The nonperturbative functional renormalization group and its applications
• Elastic Theory of pinned flux lattices
• Quantum simulation of an extra dimension
• Emergent Photons and New Transitions in the O(3) Sigma Model with Hedgehog Suppression
• Nonperturbative renormalization group approach to the Ising model: a derivative expansion at order ${\partial }^4$
• Conformal Field Theories in Fractional Dimensions
• Precision calculation of critical exponents in the $O(N)$ universality classes with the nonperturbative renormalization group
• Kosterlitz-Thouless Phase Transition In The Two Dimensional Linear Sigma Model
• The critical O(N) CFT: Methods and conformal data
• A unified picture of ferromagnetism, quasi-long range order and criticality in random field models
• Quasi-long range order in glass states of impure liquid crystals, magnets, and superconductors
• Quantum simulation of non-trivial topology
• Deligne Categories in Lattice Models and Quantum Field Theory, or Making Sense of $O(N)$ Symmetry with Non-integer $N$
• Reexamination of the nonperturbative renormalization-group approach to the Kosterlitz-Thouless transition
• Random-field Ising and $O(N)$ models: Theoretical description through the functional renormalization group
• Critical exponents at the superconductor-insulator transition in dirty-boson systems
• Conformal invariance in the nonperturbative renormalization group: a rationale for choosing the regulator
• Critical behavior at superconductor-insulator phase transitions near one dimension
• Dimensional crossovers and Casimir forces for the Bose gas in anisotropic optical lattices
• Analyticity of critical exponents of the $O(N)$ models from nonperturbative renormalization