Entanglement Entropy in the $\sigma$-Model with the de Sitter Target Space

Abstract

We derive the formula of the entanglement entropy between the left and right oscillating modes of the $\sigma$-model with the de Sitter target space. To this end, we study the theory in the \emph{cosmological gauge} in which the non-vanishing components of the metric on the two-dimensional base space are functions of the expansion parameter of the de Sitter space. The model is embedded in the causal north pole diamond of the Penrose diagram. We argue that the cosmological gauge is natural to the $\sigma$-model as it is compatible with the canonical quantization relations. In this gauge, we obtain a new general solution to the equations of motion in terms of time-independent oscillating modes. The constraint structure is adequate for quantization in the Gupta-Bleuler formalism. We construct the space of states as a one-parameter family of Hilbert spaces and give the Bargmann-Fock and Jordan-Schwinger representations of it. Also, we give a simple description of the physical subspace as an infinite product of ${\mathcal{D}}_{\frac{1}{2}}^{+}$ in the positive discreet series irreducible representations of the $SU(1,1)$ group. We construct the map generated by the Hamiltonian between states at two different values of time and show how it produces the entanglement of left and right excitations. Next, we derive the formula of the entanglement entropy of the reduced density matrix for the ground state acted upon by the Hamiltonian map. Finally, we determine the asymptotic form of the entanglement entropy of a single mode bi-oscillator in the limit of large values of time.

Discussion considerably enriched. References added. This version is consistent with the published article

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• A class of quantum many-body states that can be efficiently simulated
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• String spectra near some null cosmological singularities
• SU(1,1) symmetry of multimode squeezed states
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Cited by in corpus (8)

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• Non-locality effect on the entanglement entropy in deSitter
• Replica trick and string winding
• LREE of a Dynamical Unstable D$p$-brane
• LREE of a Dressed D$p$-brane with Transverse Motion in the Partially Compact Spacetime