<p>We study renormalization group (RG) flows in far-from-equilibrium states. The study is made tractable by focusing on states that are spatially homogeneous, time independent, and scale invariant. Such states, in which mode <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>k</mi></math> has occupation numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>n</mi><mi>k</mi></msub><mo>∼</mo><msup><mi>k</mi><mrow><mo>−</mo><mi>γ</mi></mrow></msup></math>, are well known in nonlinear physics. RG flow in such states is qualitatively different from that in the vacuum—a positive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>γ</mi></math> decreases the dimension of an operator, turning marginal interactions into relevant interactions. We compute one-loop beta functions. Depending on the sign of the beta function, backreaction may either cause a minor shift of the state in the IR or completely change the nature of the state. Focusing on nearly marginal interactions, we construct an analog of the epsilon expansion and IR fixed points, with epsilon now set by the scaling of the interaction rather than the spacetime dimension. In the language of RG flow, critical balance scaling—which has applications in fields as varied as astrophysics and ocean waves—corresponds to the state dynamically adjusting itself along the RG flow until the interaction becomes marginal.</p>

United States

Vladimir Rosenhaus

Israel

Michael Smolkin

Renormalization group in far-from-equilibrium states

It is known that for RG flows confined to a two-dimensional defect, where the bulk maintains its conformal nature, the coefficient of the Euler density in the defect’s Weyl anomaly (termed b) cannot increase as the flow progresses from the ultraviolet to the infrared, a principle known as the b-theorem. In this paper, we investigate whether this theorem still holds when the bulk, instead of being critical, also undergoes an RG flow. To address this question, we examine two distinct and perturbatively tractable examples. Our analysis reveals that a straightforward extension of the b-theorem to these cases of RG flows fails.

Tom Shachar

United Kingdom

Ritam Sinha

The defect b-theorem under bulk RG flows

We determine the gravito-magnetic Love numbers of non-rotating black holes in all spacetime dimensions through a novel and direct derivation. The Ishibashi-Kodama master field and its associated field equation are avoided. The matching to the EFT variables is simple. This method allows us to correct the values in the literature. Moreover, we highlight a parity-based selection rule for nonlinear terms that include both electric-type and magnetic-type gravitational field tensors, enabling us to conclude that many of the nonlinear response coefficients in the Schwarzschild black hole effective action vanish.

Tomer Hadad

Barak Kol

Gravito-magnetic polarization of Schwarzschild black hole

We study Krylov complexity in various models of quantum field theory: free massive bosons and fermions on flat space and on spheres, holographic models, and lattice models with a UV-cutoff. In certain cases, we observe asymptotic behavior in Lanczos coefficients that extends beyond the previously observed universality. We confirm that, in all cases, the exponential growth of Krylov complexity satisfies the conjectured inequality, which generalizes the Maldacena-Shenker-Stanford bound on chaos. We discuss the temperature dependence of Lanczos coefficients and note that the relationship between the growth of Lanczos coefficients and chaos may only hold for the sufficiently late, truly asymptotic regime, governed by physics at the UV cutoff. Contrary to previous suggestions, we demonstrate scenarios in which Krylov complexity in quantum field theory behaves qualitatively differently from holographic complexity.

Alexander Avdoshkin

Anatoly Dymarsky

Krylov complexity in quantum field theory, and beyond

Recent work has given a systematic way for studying the kinetics of classical weakly interacting waves beyond leading order, having analogies with renormalization in quantum field theory. An important context is weak wave turbulence, occurring for waves which are small in magnitude and weakly interacting, such as those on the surface of the ocean. Here we continue the work of perturbatively computing correlation functions and the kinetic equation in this far-from-equilibrium state. In particular, we obtain the next-to-leading-order kinetic equation for waves with a cubic interaction. Our main result is a simple graphical prescription for the terms in the kinetic equation, at any order in the nonlinearity.

Daniel Schubring

Md Shuvo

Loop diagrams in the kinetic theory of waves

It has long been known that weakly nonlinear field theories can have a late-time stationary state that is not the thermal state, but a wave turbulent state with a far-from-equilibrium cascade of energy. We go beyond the existence of the wave turbulent state, studying fluctuations about the wave turbulent state. Specifically, we take a classical field theory with an arbitrary quartic interaction and add dissipation and Gaussian-random forcing. Employing the path integral relation between stochastic classical field theories and quantum field theories, we give a prescription, in terms of Feynman diagrams, for computing correlation functions in this system. We explicitly compute the two-point and four-point functions of the field to next-to-leading order in the coupling. Through an appropriate choice of forcing and dissipation, these correspond to correlation functions in the wave turbulent state. In particular, we derive the kinetic equation to next-to-leading order.

Feynman rules for forced wave turbulence

We show there exist UV-complete field-theoretic models in general dimension, including <math><mn>2</mn><mo>+</mo><mn>1</mn></math>, with the spontaneous breaking of a global symmetry, which persists to the arbitrarily high temperatures. Our example is a conformal vector model with the <math><mi>O</mi><mo>(</mo><mi>N</mi><mo>)</mo><mo>×</mo><msub><mi>Z</mi><mn>2</mn></msub></math> symmetry at zero temperature. Using conformal perturbation theory we establish <math><msub><mi>Z</mi><mn>2</mn></msub></math> symmetry is broken at finite temperature for <math><mi>N</mi><mo>&gt;</mo><mn>17</mn></math>. Similar to recent constructions of [N. Chai et al., Phys. Rev. D 102, 065014 (2020)., N. Chai et al., Phys. Rev. Lett. 125, 131603 (2020).], in the infinite <math><mi>N</mi></math> limit our model has a nontrivial conformal manifold, a moduli space of vacua, which gets deformed at finite temperature. Furthermore, in this regime the model admits a persistent breaking of <math><mi>O</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> in <math><mn>2</mn><mo>+</mo><mn>1</mn></math> dimensions, therefore providing another example where the Coleman-Hohenberg-Mermin-Wagner theorem can be bypassed.

Noam Chai

Russia

Model of Persistent Breaking of Discrete Symmetry

Krylov complexity, or K-complexity for short, has recently emerged as a new probe of chaos in quantum systems. It is a measure of operator growth in Krylov space, which conjecturally bounds the operator growth measured by the out of time ordered correlator (OTOC). We study Krylov complexity in conformal field theories by considering arbitrary 2d CFTs, free field, and holographic models. We find that the bound on OTOC provided by Krylov complexity reduces to bound on chaos of Maldacena, Shenker, and Stanford. In all considered examples including free and rational CFTs Krylov complexity grows exponentially, in stark violation of the expectation that exponential growth signifies chaos.

Krylov complexity in conformal field theory

We study perturbative renormalization of the composite operators in the   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ T\overline{T} $$ -deformed two-dimensional free field theories. The pattern of renormalization for the stress-energy tensor is different in the massive and massless cases. While in the latter case the canonical stress tensor is not renormalized up to high order in the perturbative expansion, in the massive theory there are induced counterterms at linear order. For a massless theory our results match the general formula derived recently in [1].

Anshuman Dey

Mikhail Goykhman

Composite operators in   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ T\overline{T} $$ -deformed free QFTs

We study finite N aspects of the O(m) × O(N − m) vector model with quartic interactions in general 2 ≤ d ≤ 6 spacetime dimensions. This model has recently been shown [1, 2] to display the phenomenon of persistent symmetry breaking at a perturbative Wilson-Fisher-like fixed point in d = 4 − ϵ dimensions. The large rank limit of the biconical model displays a conformal manifold and a moduli space of vacua. We find a set of three double trace scalar operators that are respectively irrelevant, relevant and marginal deformations of the conformal manifold in general d. We calculate the anomalous dimensions of the single and multi-trace scalar operators to the first sub-leading order in the large rank expansion. The anomalous dimension of the marginal operator does not vanish in general, indicating that the conformal manifold is lifted at finite N . In the case of equal ranks we are able to derive explicitly the scaling dimensions of various operators as functions of only d.