We develop a renormalization group for weak Harris-marginal disorder in otherwise strongly interacting quantum critical theories, focusing on systems which have emergent conformal invariance. Using conformal perturbation theory, we argue that previously proposed random lines of fixed points with Lifshitz scaling in fact flow toward other universal fixed points, and this flow is captured by a ``one-loop'' analysis. Our approach appears best controlled in theories with only a few operators with low scaling dimension. In this regime, we compare our predictions for the flow of disorder to holographic models, and find complete agreement.
We develop a renormalization group for weak Harris-marginal disorder in otherwise strongly interacting quantum critical theories, focusing on systems which have emergent conformal invariance. Using conformal perturbation theory, we argue that previously proposed random lines of fixed points with Lifshitz scaling in fact flow toward other universal fixed points, and this flow is captured by a ``one-loop'' analysis. Our approach appears best controlled in theories with only a few operators with low scaling dimension. In this regime, we compare our predictions for the flow of disorder to holographic models, and find complete agreement.
We investigate the entanglement entropy of a two-dimensional disordered system holographically. In particular, we study the evolution of the entanglement entropy along renormalization group flows for a conformal theory at the UV fixed point, that is perturbed by weak disorder into a Lifshitz theory at the IR fixed point. Through numerical fitting, we find that the disorder correlations lead to a subleading power-law term in the entanglement entropy that vanishes at the IR fixed point. Interestingly, the exponent that controls the power-law vanishing of the subleading term seems to be almost universal as it depends very weakly on the strength of the disorder. We show that our results can be put in the context of the c-theorem by defining an effective central charge that decreases along the RG flow. We also investigate disorder induced long-range correlations between the two subsystems by studying the holographic mutual information.
In this paper, we analyze the renormalization group (RG) flow of field theories with quenched disorder, in which the couplings vary randomly in space.We analyze both classical (Euclidean) disorder and quantum disorder, emphasizing general properties rather than specific cases.The RG flow of the disorderaveraged theories takes place in the space of their coupling constants and also in the space of distributions for the disordered couplings, and the two mix together.We write down a generalization of the Callan-Symanzik equation for the flow of disorder-averaged correlation functions.We find that local operators can mix with the response of the theory to local changes in the disorder distribution and that the generalized Callan-Symanzik equation mixes the disorder averages of several different correlation functions.For classical disorder, we show that this can lead to new types of anomalous dimensions and to logarithmic behavior at fixed points.For quantum disorder, we find that the RG flow always generates a rescaling of time relative to space, which at a fixed point generically leads to Lifshitz scaling.The dynamical scaling exponent z behaves as an anomalous dimension (as in other nonrelativistic RG flows), and we compute it at leading order in perturbation theory in the disorder for a general theory.Our results agree with a previous perturbative computation by Boyanovsky and Cardy, and with a holographic disorder computation of Hartnoll and Santos.We also find in quantum disorder that local operators mix with nonlocal (in time) operators under the RG, and that there are critical exponents associated with the disorder distribution that have not previously been discussed.In large-N theories, the disorder averages may be computed exactly, and we verify that they are consistent with the generalized Callan-Symanzik equations.
In this article, we study quantum critical phenomena in surfaces of symmetry-protected topological matter, i.e., surface topological quantum criticality. A generic phase boundary of gapless surfaces in a symmetry-protected state shall be a codimension one manifold in an interaction parameter space of dimension Dp (where p refers to the parameter space) where the value of Dp further depends on bulk topologies. In the context of fermionic topological insulators that we focus on, Dp depends on the number of half-Dirac cones N. We construct such manifolds explicitly for a few interaction parameter spaces with various Dp values. Most importantly, we further illustrate that in cases with Dp=3 and 6, there are submanifolds of fixed points that dictate the universalities of surface topological quantum criticality. These infrared stable manifolds are associated with emergent symmetries in the renormalization-group-equation flow naturally appearing in the loop expansion. Unlike in the usual order-disorder quantum critical phenomena, typically governed by an isolated Wilson-Fisher fixed point, we find in the one-loop approximation surface topological quantum criticalities are naturally captured by conformal manifolds where the number of marginal operators uniquely determines their codimensions. Isolated strong-coupling fixed points also appear, usually as the endpoints in the phase boundary of surface topological quantum phases. However, their extreme infrared instabilities along multiple directions suggest that they shall be related to multicritical surface topological quantum critical phenomena rather than generic surface topological quantum criticality. We also discuss and classify higher-loop symmetry-breaking effects, which can either distort the conformal manifolds or further break the conformal manifolds down to a few distinct fixed points.
We study the renormalization group flow in general quantum field theories with quenched disorder, focusing on random quantum critical points. We show that in disorder-averaged correlation functions the flow mixes local and nonlocal operators. This leads to a new critical exponent related to the disorder (as in classical disorder). We show that the time coordinate is rescaled at each renormalization group step, leading to anisotropic spacetime scaling at critical points. We write a universal formula for the dynamical scaling exponent z for weak disorder.
In this article, we propose the realization of conformal manifolds, which are smooth manifolds of scale-conformal invariant interacting Hamiltonians in two-dimensional quantum many-body systems. Such phenomena can occur in various interacting systems, including topological surfaces or 2D bulks. Building on previous observations, we demonstrate that a conformal manifold can emerge as an exact solution when the number of fermion colors, Nc, becomes infinite. We identify distinct exact marginal deformation operators uniquely associated with the conformal manifolds. By considering Nc as finite but large, we show that quantum fluctuations induce a fermion field renormalization that results in mildly infrared relevant or irrelevant renormalization-group (RG) flow within a conformal manifold, producing standard isolated infrared stable Wilson-Fisher fixed points. These can be grouped with ultraviolet stable fixed points into a discrete manifold due to the spontaneous symmetry breaking of an emergent SO(N) dynamical symmetry in the RG flow as Nc→∞. Additionally, we find a correlation between the direction of the RG flow within the manifold and an increase in EPR-like entanglement entropy. The infrared-stable Wilson-Fisher fixed points, induced by quantum fluctuations, are linked to theories on the conformal manifold where interaction operators are maximally entangled in flavor space. Our studies provide an effective framework for addressing topological quantum critical points with high-dimensional interaction parameter spaces, potentially housing many fixed points of various stabilities. They also highlight the central role of entangled conformal operators and their entropy in shaping universality classes of surface topological quantum phase transitions. We conclude with open questions and possible future directions.
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 k has occupation numbers n k ∼ k − γ , are well known in nonlinear physics. RG flow in such states is qualitatively different from that in the vacuum—a positive γ 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.