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arXiv:2012.00782 · SciPost Physics

Analyticity of critical exponents of the $O(N)$ models from nonperturbative renormalization

Andrzej Chlebicki, Pawel Jakubczyk

Published Jun 7, 2021• 20 citations
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Abstract

We employ the functional renormalization group framework at the second order in the derivative expansion to study the O(N) O ( N ) models continuously varying the number of field components N N and the spatial dimensionality d d . We in particular address the Cardy-Hamber prediction concerning nonanalytical behavior of the critical exponents \nu ν and \eta η across a line in the (d,N) ( d , N ) plane, which passes through the point (2,2) ( 2 , 2 ) . By direct numerical evaluation of \eta(d,N) η ( d , N ) and \nu^{-1}(d,N) ν − 1 ( d , N ) as well as analysis of the functional fixed-point profiles, we find clear indications of this line in the form of a crossover between two regimes in the (d,N) ( d , N ) plane, however no evidence of discontinuous or singular first and second derivatives of these functions for d>2 d > 2 . The computed derivatives of \eta(d,N) η ( d , N ) and \nu^{-1}(d,N) ν − 1 ( d , N ) become increasingly large for d\to 2

DOI
10.21468/scipostphys.10.6.134

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Analyticity of critical exponents of the O(N)O(N)O(N) models from nonperturbative renormalization
We employ the functional renormalization group framework at the second order in the derivative expansion to study the O(N) O ( N ) models continuously varying the number of field components N N and the spatial dimensionality d d . We in particular address the Cardy-Hamber prediction concerning nonanalytical behavior of the critical exponents \nu ν and \eta η across a line in the (d,N) ( d , N ) plane, which passes through the point (2,2) ( 2 , 2 ) . By direct numerical evaluation of \eta(d,N) η ( d , N ) and \nu^{-1}(d,N) ν − 1 ( d , N ) as well as analysis of the functional fixed-point profiles, we find clear indications of this line in the form of a crossover between two regimes in the (d,N) ( d , N ) plane, however no evidence of discontinuous or singular first and second derivatives of these functions for d>2 d > 2 . The computed derivatives of \eta(d,N) η ( d , N ) and \nu^{-1}(d,N) ν − 1 ( d , N ) become increasingly large for d\to 2
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≤2 and N≤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 η. 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 favor 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 η(d,N) and ν^{-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.
We compute the critical exponents ν, η and ω of O(N) models for various values of N by implementing the derivative expansion of the nonperturbative renormalization group up to next-to-next-to-leading order [usually denoted O(∂^{4})]. We analyze the behavior of this approximation scheme at successive orders and observe an apparent convergence with a small parameter, typically between 1/9 and 1/4, compatible with previous studies in the Ising case. This allows us to give well-grounded error bars. We obtain a determination of critical exponents with a precision which is similar or better than those obtained by most field-theoretical techniques. We also reach a better precision than Monte Carlo simulations in some physically relevant situations. In the O(2) case, where there is a long-standing controversy between Monte Carlo estimates and experiments for the specific heat exponent α, our results are compatible with those of Monte Carlo but clearly exclude experimental values.
We develop a nonperturbative functional renormalization group approach for the random-field O(N) model that allows us to investigate the ordering transition in any dimension and for any value of N including the Ising case. We show that the failure of dimensional reduction and standard perturbation theory is due to the nonanalytic nature of the zero-temperature fixed point controlling the critical behavior, nonanalyticity, which is associated with the existence of many metastable states. We find that this nonanalyticity leads to critical exponents differing from the dimensional reduction prediction only below a critical dimension dc(N)<6, with dc(N=1)>3.
We apply the derivative expansion of the effective action in the exact renormalization group equation up to fourth order to the Z_{2} and O(N) symmetric scalar models in d=3 Euclidean dimensions. We compute the critical exponents ν, η, and ω using polynomial expansion in the field. We obtain our predictions for the exponents employing two regulators widely used in exact renormalization group computations. We apply Wynn's epsilon algorithm to improve the predictions for the critical exponents, extrapolating beyond the next-to-next-to-leading order prediction of the derivative expansion.
The determination of the critical exponents by means of the Exact Renormalizion Group approach is still a topic of debate. The general flow equation is by construction scheme independent, but the use of the truncated derivative expansion generates a model depencence in the determination of the universal quantities. We derive new nonperturbative flow equations for the one-component, Z_2 symmetric scalar field to the next-to-leading order of the derivative expansion by means of a class of proper time regulators. The critical exponents eta, nu and omega for the Wilson-Fisher fixed point are computed by numerical integration of the flow equations, without resorting to polynomial truncations. We show that by reducing the width of the cut-off employed, the critical exponents become rapidly insensitive to the cut-off width and converge to values that are in good agreement with the results of entirely different approaches.
The critical behavior of the O(n)O(n)O(n) classical Heisenberg model in ddd dimensions close to n=d=2n=d=2n=d=2 is discussed, assuming analyticity in nnn and ddd of the renormalization-group equations. In the (ddd,nnn) plane there is a line passing through (2,2) across which the critical exponents are nonanalytic. For $d=2 \mathrm{and} \ensuremath{-}2<~n<~2$, a conjecture is presented for the exact form of the leading and subdominant thermal eigenvalues.