Abstract
Renyi entropies S_q are useful measures of quantum entanglement; they can be calculated from traces of the reduced density matrix raised to power q, with q>=0. For (d+1)-dimensional conformal field theories, the Renyi entropies across S^{d-1} may be extracted from the thermal partition functions of these theories on either (d+1)-dimensional de Sitter space or R x H^d, where H^d is the d-dimensional hyperbolic space. These thermal partition functions can in turn be expressed as path integrals on branched coverings of the (d+1)-dimensional sphere and S^1 x H^d, respectively. We calculate the Renyi entropies of free massless scalars and fermions in d=2, and show how using zeta-function regularization one finds agreement between the calculations on the branched coverings of S^3 and on S^1 x H^2. Analogous calculations for massive free fields provide monotonic interpolating functions between the Renyi entropies at the Gaussian and the trivial fixed points. Finally, we discuss similar Renyi entropy calculations in d>2.
Highlights
The Renyi entropies [1, 2] have recently emerged as powerful diagnostics of long-range entanglement in many-body quantum ground states in d ≥ 2 spatial dimensions [3,4,5,6,7,8,9,10,11,12,13,14,15]
Where ρ is the reduced density matrix obtained after tracing over the degrees of freedom in the complement of the entangling region
Numerical evaluations of the entropies have allowed characterization of many distinct types of ground states: gapped states with topological order [4,5,6,7,8], quantum critical points [9], Fermi liquids [10, 11], non-Fermi liquids [12], and Goldstone phases with broken symmetries [13,14,15]
Summary
The Renyi entropies [1, 2] have recently emerged as powerful diagnostics of long-range entanglement in many-body quantum ground states in d ≥ 2 spatial dimensions [3,4,5,6,7,8,9,10,11,12,13,14,15]. Numerical evaluations of the entropies have allowed characterization of many distinct types of ground states: gapped states with topological order [4,5,6,7,8], quantum critical points [9], Fermi liquids [10, 11], non-Fermi liquids [12], and Goldstone phases with broken symmetries [13,14,15] It appears that the Renyi entropies carry distinct signatures of the known quantum many-body states and are amenable to evaluation by convenient algorithms.
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