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arXiv:hep-th/9710241 · Physical Review Letters

Relative Entropy in 2D Quantum Field Theory, Finite-Size Corrections, and Irreversibility of the Renormalization Group

José Gaite

Published Oct 26, 1998• 16 citations
arXivHTMLPDF

Abstract

The relative entropy in two-dimensional field theory is studied for its application as an irreversible quantity under the renormalization group, relying on a general monotonicity theorem for that quantity previously established. In the cylinder geometry, interpreted as finite-temperature field theory, one can define from the relative entropy a monotonic quantity similar to Zamolodchikov's $c$ function. On the other hand, the one-dimensional quantum thermodynamic entropy also leads to a monotonic quantity, with different properties. The relation of thermodynamic quantities with the complex components of the stress tensor is also established and hence the entropic $c$ theorems are proposed as analogs of Zamolodchikov's $c$ theorem for the cylinder geometry.

DOI
10.1103/physrevlett.81.3587

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The relative entropy in two-dimensional field theory is studied for its application as an irreversible quantity under the renormalization group, relying on a general monotonicity theorem for that quantity previously established. In the cylinder geometry, interpreted as finite-temperature field theory, one can define from the relative entropy a monotonic quantity similar to Zamolodchikov's ccc function. On the other hand, the one-dimensional quantum thermodynamic entropy also leads to a monotonic quantity, with different properties. The relation of thermodynamic quantities with the complex components of the stress tensor is also established and hence the entropic ccc theorems are proposed as analogs of Zamolodchikov's ccc theorem for the cylinder geometry.
The relative entropy in two-dimensional field theory is studied on a cylinder geometry, interpreted as finite-temperature field theory. The width of the cylinder provides an infrared scale that allows us to define a dimensionless relative entropy analogous to Zamolodchikov's c function. The one-dimensional quantum thermodynamic entropy gives rise to another monotonic dimensionless quantity. I illustrate these monotonicity theorems with examples ranging from free field theories to interacting models soluble with the thermodynamic Bethe ansatz. Both dimensionless entropies are explicitly shown to be monotonic in the examples that we analyze.
We consider relative entropy in Field Theory as a well defined (non-divergent) quantity of interest. We establish a monotonicity property with respect to the couplings in the theory. As a consequence, the relative entropy in a field theory with a hierarchy of renormalization group fixed points ranks the fixed points in decreasing order of criticality. We argue from a generalized HHH theorem that Wilsonian RG flows induce an increase in entropy and propose the relative entropy as the natural quantity which increases from one fixed point to another in more than two dimensions.
We establish the irreversibility of renormalization group flows on a pointlike defect inserted in a d-dimensional Lorentzian conformal field theory. We identify the impurity entropy g with the quantum relative entropy in two equivalent ways. One involves a null deformation of the Cauchy surface, and the other is given in terms of a local quench protocol. Positivity and monotonicity of the relative entropy imply that g decreases monotonically along renormalization group flows, and provides a clear information-theoretic meaning for this irreversibility.
Abstract We study conformal field theories with boundaries, and their boundary renormalization group (RG) flows, using methods from quantum information theory. Positivity of the relative entropy, together with unitarity and Lorentz invariance, give rise to bounds that characterize the irreversibility of such flows. This generalizes the recently proved entropic g-theorem to higher dimensions. In 2 + 1 dimensions with a boundary, we prove the entropic b-theorem — the decrease of the two-dimensional Weyl anomaly under boundary RG flows. In higher dimensions, the bound implies that the leading area coefficient of the entanglement entropy induced by the defect decreases along the flow. Our proof unifies these properties, and provides an information-theoretic interpretation in terms of the distinguishability between the short distance and long distance states. Finally, we establish a sum rule for the change in the area term in theories with boundaries, which could have implications for models with localized gravity.
The trace over the degrees of freedom located in a subset of the space transforms the vacuum state into a mixed density matrix with nonzero entropy. This is usually called entanglement entropy, and it is known to be divergent in quantum field theory (QFT). However, it is possible to define a finite quantity F(A,B) for two given different subsets A and B which measures the degree of entanglement between their respective degrees of freedom. We show that the function F(A,B) is severely constrained by the Poincaré symmetry and the mathematical properties of the entropy. In particular, for one component sets in two-dimensional conformal field theories its general form is completely determined. Moreover, it allows to prove an alternative entropic version of the c-theorem for (1+1)-dimensional QFT. We propose this well-defined quantity as the meaningfull entanglement entropy and comment on possible applications in QFT and the black hole evaporation problem.
We prove an entropic ccc-theorem in two dimensions by constructing a ccc-function from the mutual information between spacelike separated wedges, defined through the relative entropy. The proof follows Casini and Huerta's, with strong subadditivity of entanglement entropy replaced by increasing differences of the mutual information, which we prove directly in the continuum from the split property and standard properties of relative entropy. The benefit of relative entropy is that every quantity we use is finite and no cutoff ever needs to be introduced. The assumptions of the theorem are only the standard axioms of algebraic quantum field theory, the split property with finite mutual information across a gap (both implied by modular ppp-nuclearity), and a finite ultraviolet bound on the resulting ccc-functions. We also note that, formally, the ccc-function is proportional to the minimal averaged null energy of a state that reproduces the vacuum correlations between the two wedges.