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arXiv:2005.11846 · Physical review. D/Physical review. D.

Reliability of the local truncations for the random tensor models renormalization group flow

Vincent Lahoche, Dine Ousmane Samary

Published Sep 3, 2020• 13 citations
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Abstract

The standard nonperturbative approaches of renormalization group for tensor models are generally focused on a purely local potential approximation (i.e., involving only generalized traces and products of them) and are showed to strongly violate the modified Ward identities. This paper is a continuation of our recent contribution [Phys. Rev. D 101, 106015 (2020), intended to investigate the approximation schemes compatible with Ward identities and constraints between $2n$-points observables in the large $N$ limit. We consider separately two different approximations: In the first one, we try to construct a local potential approximation from a slight modification of the Litim regulator, so that it remains optimal in the usual sense, and preserves the boundary conditions in deep UV and deep IR limits. In the second one, we introduce derivative couplings in the truncations and show that the compatibility with Ward identities implies strong relations between $\ensuremath{\beta}$ functions, allowing one to close the infinite hierarchy of flow equations in the nonbranching sector, up to a given order in the derivative expansion. Finally, using an exact relation between correlations functions in large $N$ limit, we show that strictly local truncations are insufficient to reach the exact value for the critical exponent, highlighting the role played by these strong relations between observables taking into account the behavior of the flow; and the role played by the multitrace operators, discussed in the two different approximation schemes. In both cases, we compare our conclusions to the results obtained in the literature and conclude that, at a given order, by taking into account the exact functional relations between observables like Ward identities in a systematic way, we can strongly improve the physical relevance of the approximation for an exact renormalization group equation.

DOI
10.1103/physrevd.102.056002

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The standard nonperturbative approaches of renormalization group for tensor models are generally focused on a purely local potential approximation (i.e., involving only generalized traces and products of them) and are showed to strongly violate the modified Ward identities. This paper is a continuation of our recent contribution [Phys. Rev. D 101, 106015 (2020), intended to investigate the approximation schemes compatible with Ward identities and constraints between 2n2n2n-points observables in the large NNN limit. We consider separately two different approximations: In the first one, we try to construct a local potential approximation from a slight modification of the Litim regulator, so that it remains optimal in the usual sense, and preserves the boundary conditions in deep UV and deep IR limits. In the second one, we introduce derivative couplings in the truncations and show that the compatibility with Ward identities implies strong relations between \ensuremathβ\ensuremath{\beta}\ensuremathβ functions, allowing one to close the infinite hierarchy of flow equations in the nonbranching sector, up to a given order in the derivative expansion. Finally, using an exact relation between correlations functions in large NNN limit, we show that strictly local truncations are insufficient to reach the exact value for the critical exponent, highlighting the role played by these strong relations between observables taking into account the behavior of the flow; and the role played by the multitrace operators, discussed in the two different approximation schemes. In both cases, we compare our conclusions to the results obtained in the literature and conclude that, at a given order, by taking into account the exact functional relations between observables like Ward identities in a systematic way, we can strongly improve the physical relevance of the approximation for an exact renormalization group equation.
The nonperturbative renormalization group has been considered as a solid framework to investigate fixed point and critical exponents for matrix and tensor models, expected to correspond with the so-called double scaling limit. In this paper, we focus on matrix models and address the question of the compatibility between the approximations used to solve the exact renormalization group equation and the modified Ward identities coming from the regulator. We show in particular that standard local potential approximation strongly violates the Ward identities, especially in the vicinity of the interacting fixed point. Extending the theory space including derivative couplings, we recover an interacting fixed point with a critical exponent not so far from the exact result, but with a nonzero value for derivative couplings, evoking a strong dependence concerning the regulator. Finally, we consider a modified regulator, allowing to keep the flow not so far from the ultralocal region and recover the results of the literature up to a slight improvement.
Conformal invariance is expected to be an emergent property of many systems in their critical regime. However, approximation schemes generically spoil this property. This is in particular the case of the derivative expansion, a widely used approximation scheme in the framework of the functional renormalization group. In this article, we consider Ward identities associated with conformal invariance in the 3-d Ising universality class with truncations at order 4 (next-to-next-to-leading order) in the derivative expansion, with Z2Z_2Z2​ invariant composite operators. Our results confirm that the regulating functions which yield a small breaking of conformal invariance also present a small sensitivity of the universal critical exponents with the choice of this regulating function. We also show that, in the vicinity of regulator-parameter values for which the conformal constraints are best satisfied, the breaking of conformal invariance reduces as the order of the derivative expansion is increased, providing a new indication of the convergence of this approximation scheme.
We consider the non-perturbative renormalization group (RG) equations, obtained as approximations of the exact Wetterich RG flow equation within the Blaizot–Mendez–Wschebor (BMW) truncation scheme. For the first time, we derive explicit RG flow equations for the scalar model at the arbitrary order of truncation. Moreover, we consider original, as well as modified, approximations, used to obtain a set of closed equations. We compare these equations at the s = 2 order of truncation with those recently derived in J. Phys. A: Math. Theor. 53, 415002 (2020) within a new truncation scheme and find a striking similarity. Namely, the first-order equations of the latter scheme, those of the original BMW scheme, and those of the modified BMW scheme (at s = 2) differ only in one term. We solved these equations by a recently proposed and tested method of semi-analytic approximations. Thus, the critical exponents η, ν, and ω were evaluated, recovering also the known results of the original BMW scheme. In addition, we estimated the subleading correction-to-scaling exponent ω2 for the three equations considered. To the best of our knowledge, this exponent has not yet been extracted from the Wetterich equation beyond the local potential (the zeroth order) approximation. Our current estimate for the 3D Ising model is ω2 = 2.02 (40), where the error bars include the expected truncation error in the BMW scheme.
Continuous phase transitions provide a natural setting for understanding universality. Near criticality, the correlation length diverges, microscopic details become increasingly irrelevant, and physical observables obey scaling laws characterized by universal critical exponents. Within the Functional Renormalization Group (FRG), this scale dependence is described through the Effective Average Action, which progressively incorporates fluctuations as the momentum scale is lowered. A first part of this thesis is developed within the Local Potential Approximation (LPA), where we study multicritical fixed points of scalar field theories. Their universal properties are compared with results obtained at higher orders in the derivative expansion. Particular attention is devoted to the relation between global fixed-point potentials and their local representations through finite polynomial truncations. We show that the convergence of such expansions is constrained by the analytic structure of the global solution, and that expanding around a suitable nonzero stationary point significantly improves the reconstruction of multicritical branches. Beyond the structure of individual fixed points, a central part of this thesis concerns RG irreversibility. In two dimensions, Zamolodchikov’s c-theorem describes the flow from the ultraviolet to the infrared through a monotonically decreasing c-function. Within the FRG framework, we develop a variational procedure to reconstruct RG trajectories between fixed points and integrate the corresponding c-function along them. The resulting quantity displays monotonic behaviour and becomes increasingly stable as the truncation order is raised, providing a manifestation of RG irreversibility within the approximation considered. Finally, close to the upper critical dimensions, the linearized FRG flow around the Gaussian fixed point is naturally organized in terms of Hermite polynomials. Using the epsilon-expansion, we derive a curvature-based candidate c-function and obtain an analytic explanation for the alternating curvature of successive multicritical fixed points and for their progressive accumulation around the Gaussian fixed point.
The behavior of many critical phenomena at large distances is expected to be invariant under the full conformal group, rather than only isometries and scale transformations. When studying critical phenomena, approximations are often required, and the framework of the nonperturbative, or functional, renormalization group is no exception. The derivative expansion is one of the most popular approximation schemes within this framework, due to its great performance on multiple systems, as evidenced in the past few decades. Nevertheless, it has the downside of breaking conformal symmetry at a finite order. This breaking is not observed at the leading order of the expansion, denoted the local potential approximation, and it only appears once one considers, at least, the next-to-leading order of the derivative expansion [O(∂^{2})] when including composite operators. In this work, we study the constraints arising from conformal symmetry for the O(N) models using the derivative expansion at order O(∂^{2}). We explore various values of N and minimize the breaking of conformal symmetry to fix the nonphysical parameters of the approximation procedure. We compare our prediction for the critical exponents with those coming from a more usual procedure, known as the principle of minimal sensitivity.
This paper, as a continuation of our previous investigation [Nucl. Phys. B 1005 (2024) 116582] aims to study the glassy random matrices with quenched Wigner disorder. In this previous work, we have constructed a renormalization group based on the effective deterministic kinetic spectrum emerging from large N limit, and we extended approximate solutions using standard vertex expansion, at the leading order of the derivative expansion. Now in the following work, by introducing the non-trivial Ward identities which come from the symmetry broken of the effective kinetic action, we provide in the deep IR the explicit solution of the functional renormalization group for a model with quartic coupling by solving the Hierarchy to all orders in the local sector, which in particular imply the vanishing of the anomalous dimension. The numerical investigations confirm the first-order phase transition discovered in the vertex expansion framework, both in the active and passive schemes. Finally, we extend the discussion to hermitian matrices.