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arXiv:1803.08529 · Physical Review Letters

Renormalization Group in Field Theories with Quantum Quenched Disorder

Vladimir Narovlansky, Ofer Aharony A

Published Aug 15, 2018• 18 citations
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Abstract

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.

DOI
10.1103/physrevlett.121.071601

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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 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.
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 extend to quenched-disordered systems the variational scheme for real-space renormalization group calculations that we recently introduced for homogeneous spin Hamiltonians. When disorder is present our approach gives access to the flow of the renormalized Hamiltonian distribution, from which one can compute the critical exponents if the correlations of the renormalized couplings retain finite range. Key to the variational approach is the bias potential found by minimizing a convex functional in statistical mechanics. This potential reduces dramatically the Monte Carlo relaxation time in large disordered systems. We demonstrate the method with applications to the two-dimensional dilute Ising model, the random transverse field quantum Ising chain, and the random field Ising in two- and three-dimensional lattices.
A formidable perspective in understanding quantum criticality of a given many-body system is through its entanglement contents. Until now, most progress are only limited to the disorder-free case. Here, we develop an efficient scheme to compute the entanglement entropy of (2+1)(2+1)(2+1)-dimensional quantum critical points with randomness, from a conceptual angle where the quenched disorder can be considered as dimensionally reducible interactions. As a concrete example, we reveal novel entanglement signatures of (2+1)(2+1)(2+1)-dimensional Dirac fermion exposed to a random magnetic field, which hosts a class of emergent disordered quantum critical points. We demonstrate that the entanglement entropy satisfies the area-law scaling, and observe a modification of the area-law coefficient that points to the emergent disordered quantum criticality. Moreover, we also obtain the subleading correction to the entanglement entropy due to a finite correlation length. This subleading correction is found to be a universal function of the correlation length and disorder strength. We discuss its connection to the renormalization group flows of underlying theories.
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Abstract Mean-field theory is usually taken as a first step toward understanding critical phenomena, providing an overview that reveals qualitative behaviour of physical quantities. However, it is necessary to proceed beyond the mean-field theory to better understand the situation, both qualitatively and quantitatively, when fluctuations play vital roles leading to exponents that cannot be explained by dimensional analysis, thus introducing anomalous dimensions. The present chapter explains the basic concepts of the renormalization group and scaling theory, which allow us to analyze critical phenomena with fluctuations systematically taken into account. The essential step in a renormalization group calculation consists of establishing recursion relations between the parameters defining the Hamiltonian of the system. These recursion or renormalization group equations define a flow with well-defined fixed points. Details other than the values of the relevant operators have no influence on the critical exponents and this represents universality.