Abstract
We present a new method for calculating Renyi entanglement entropies for fermionic field-theories originating from microscopic Hamiltonians. The method builds on an operator identity which we discover for the first time. The identity leads to the representation of traces of operator products, and thus Renyi entropies of a subsystem, in terms of fermionic-displacement operators. This allows for a very transparent path-integral formulation, both in and out-of-equilibrium, having a simple boundary condition on the fermionic fields. The method is validated by reproducing well known expressions for entanglement entropy in terms of the correlation matrix for non-interacting fermions. We demonstrate the effectiveness of the method by explicitly formulating the field theory for Renyi entropy in a few zero and higher-dimensional large-$N$ interacting models based on the Sachdev-Ye-Kitaev (SYK) model, and for the Hubbard model within dynamical mean-field theory (DMFT) approximation. We use the formulation to compute Renyi entanglement entropy of interacting Fermi liquid (FL) and non-Fermi liquid (NFL) states in the large-$N$ models and compare successfully with the results obtained via exact diagonalization for finite $N$. We elucidate the connection between entanglement entropy and residual entropy of the NFL ground state in the SYK model and extract sharp signatures of quantum phase transition in the entanglement entropy across an NFL to FL transition. Furthermore, we employ the method to obtain nontrivial system-size scaling of entanglement in an interacting diffusive metal described by a chain of SYK dots.
Highlights
Quantum entanglement has emerged as a major tool to characterize quantum phases and phase transitions [1,2,3,4,5,6] and to distill the fundamental quantum mechanical nature of nontrivial many-body states, e.g., those with topological order that is otherwise hard to quantify [7,8]
In the other major part of the paper, we explicitly demonstrate the utility of the method by computing the second Rényi entropy (S(2)) for subsystems in several large-N model in thermal equilibrium: (i) zero-dimensional SYK model having infinite-range random four-fermion or two-body interaction with a non-Fermi liquid (NFL) ground state, (ii) SYK model with additional quadratic hopping between fermions having a Fermi liquid (FL) ground state, (iii) a generalized SYK model, the Banerjee-Altman (BA) model [30], having quantum phase transition (QPT) between SYK NFL and FL, and (iv) an extended system, a chain of SYK dots [31,32], describing an interacting diffusive metal
We obtain the following important results using our method for the large-N models: (1) We show that for the SYK model, in the N → ∞ limit, the zero-temperature residual entropy [27,33,34,35] of the SYK NFL contributes to the T = 0 subsystem Rényi entropy, making it difficult to recover the true quantum entanglement of the NFL ground state starting from a thermal ensemble
Summary
Quantum entanglement has emerged as a major tool to characterize quantum phases and phase transitions [1,2,3,4,5,6] and to distill the fundamental quantum mechanical nature of nontrivial many-body states, e.g., those with topological order that is otherwise hard to quantify [7,8]. A lot of progress has been made to obtain entanglement entropy, both numerically and analytically, for noninteracting bosonic and fermionic systems [3,4,17], and at critical points described by conformal field theories [1,2] The latter rely on field-theoretic techniques using replicas and path integrals, typically in imaginary time, with complicated boundary conditions on fields and associated Green’s function along the time direction [1,2,3]. The effect of the time-dependent self-energy can be implemented for interacting systems treated within standard perturbative and nonperturbative field-theoretic approximation and diagrammatic continuous-time Monte Carlo simulation [26] We elucidate this by deriving the subsystem second Rényi entropy within two well-known approaches to treat correlated fermions: (a) strongly interacting large-N fermionic models based on Sachdev-Ye-Kitaev model [27,28] and (b) dynamical mean-field theory (DMFT) [29]. Additional details of the derivations of the operator identities, path-integral formulations, and their analytical and numerical implementations in various models for computing Rényi entropies are given in the Appendixes
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