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Register a new userMisanthropic Entropy and Renormalization as a Communication Channel
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A central physical question is the extent to which infrared (IR) observations are sufficient to reconstruct a candidate ultraviolet (UV) completion. We recast this question as a problem of communication, with messages encoded in field configurations of the UV being transmitted to the IR degrees of freedom via a noisy channel specified by renormalization group (RG) flow, with noise generated by coarse graining / decimation. We present an explicit formulation of these considerations in terms of lattice field theory, where we show that the misanthropic entropy---the mutual information obtained from decimating neighbors---encodes the extent to which information is lost in marginalizing over / tracing out UV degrees of freedom. Our considerations apply both to statistical field theories as well as density matrix renormalization of quantum systems, where in the quantum case, the statistical field theory analysis amounts to a leading order approximation. As a concrete example, we focus on the case of the 2D Ising model, where we show that the misanthropic entropy detects the onset of the phase transition at the Ising model critical point.
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We investigate the analogy between the renormalization group (RG) and deep neural networks, wherein subsequent layers of neurons are analogous to successive steps along the RG. In particular, we quantify the flow of information by explicitly computing the relative entropy or Kullback-Leibler divergence in both the one- and two-dimensional Ising models under decimation RG, as well as in a feedforward neural network as a function of depth. We observe qualitatively identical behavior characterized by the monotonic increase to a parameter-dependent asymptotic value. On the quantum field theory side, the monotonic increase confirms the connection between the relative entropy and the c-theorem. For the neural networks, the asymptotic behavior may have implications for various information maximization methods in machine learning, as well as for disentangling compactness and generalizability. Furthermore, while both the two-dimensional Ising model and the random neural networks we consider exhibit non-trivial critical points, the relative entropy appears insensitive to the phase structure of either system. In this sense, more refined probes are required in order to fully elucidate the flow of information in these models.
We investigate the supersymmetric Sachdev-Ye-Kitaev (SYK) model, $N$ Majorana fermions with infinite range interactions in $0+1$ dimensions. We have found that, close to the ground state $E approx 0$, discrete symmetries alter qualitatively the spectral properties with respect to the non-supersymmetric SYK model. The average spectral density at finite $N$, which we compute analytically and numerically, grows exponentially with $N$ for $E approx 0$. However the chiral condensate, which is normalized with respect the total number of eigenvalues, vanishes in the thermodynamic limit. Slightly above $E approx 0$, the spectral density grows exponential with the energy. Deep in the quantum regime, corresponding to the first $O(N)$ eigenvalues, the average spectral density is universal and well described by random matrix ensembles with chiral and superconducting discrete symmetries. The dynamics for $E approx 0$ is investigated by level fluctuations. Also in this case we find excellent agreement with the prediction of chiral and superconducting random matrix ensembles for eigenvalues separations smaller than the Thouless energy, which seems to scale linearly with $N$. Deviations beyond the Thouless energy, which describes how ergodicity is approached, are universality characterized by a quadratic growth of the number variance. In the time domain, we have found analytically that the spectral form factor $g(t)$, obtained from the connected two-level correlation function of the unfolded spectrum, decays as $1/t^2$ for times shorter but comparable to the Thouless time with $g(0)$ related to the coefficient of the quadratic growth of the number variance. Our results provide further support that quantum black holes are ergodic and therefore can be classified by random matrix theory.
The relation between the spectral density of the QCD Dirac operator at nonzero baryon chemical potential and the chiral condensate is investigated. We use the analytical result for the eigenvalue density in the microscopic regime which shows oscillations with a period that scales as 1/V and an amplitude that diverges exponentially with the volume $V=L^4$. We find that the discontinuity of the chiral condensate is due to the whole oscillating region rather than to an accumulation of eigenvalues at the origin. These results also extend beyond the microscopic regime to chemical potentials $mu sim 1/L$.
We renormalize the SU(N) Gross-Neveu model in the modified minimal subtraction (MSbar) scheme at four loops and determine the beta-function at this order. The theory ceases to be multiplicatively renormalizable when dimensionally regularized due to the generation of evanescent 4-fermi operators. The first of these appears at three loops and we correctly take their effect into account in deriving the renormalization group functions. We use the results to provide estimates of critical exponents relevant to phase transitions in graphene.
The microscopic spectral density of the QCD Dirac operator at nonzero baryon chemical potential for an arbitrary number of quark flavors was derived recently from a random matrix model with the global symmetries of QCD. In this paper we show that these results and extensions thereof can be obtained from the replica limit of a Toda lattice equation. This naturally leads to a factorized form into bosonic and fermionic QCD-like partition functions. In the microscopic limit these partition functions are given by the static limit of a chiral Lagrangian that follows from the symmetry breaking pattern. In particular, we elucidate the role of the singularity of the bosonic partition function in the orthogonal polynomials approach. A detailed discussion of the spectral density for one and two flavors is given.
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