Irreducible many-body correlations in topologically ordered systems

Abstract

Topologically ordered systems exhibit large-scale correlation in their ground states, which may be characterized by quantities such as topological entanglement entropy. We propose that the concept of irreducible many-body correlation (IMC), the correlation that cannot be implied by all local correlations, may also be used as a signature of topological order. In a topologically ordered system, we demonstrate that for a part of the system with holes, the reduced density matrix exhibits IMCs which become reducible when the holes are removed. The appearance of these IMCs then represents a key feature of topological phase. We analyze the many-body correlation structures in the ground state of the toric code model in external magnetic fields, and show that the topological phase transition is signaled by the IMCs.

Discontinuity of maximum entropy inference and quantum phase transitions

“…Usually, the state ρ will be chosen to be a reduced state of the ground state of the Hamiltonian. It is known that the quantum conditional mutual information is an upper bound of C ABC [13,16]. Namely, we have…”

“…In other words, the corresponding ABC * ρ of the ground state is always a quantum Markovian state (there are reasons to believe this, see e.g. [8,16]). Assuming this conjecture, I A C B ( : ) | is indeed a good quantity to signal the discontinuity and phase transition in the thermodynamic limit.…”

“…For each of these cuttings, a nontrivial I A C B ( : ) | detects different orders of the system. For figure 11(a), I A C B ( : ) | detects both symmetrybreaking order and SPT order and topological phase transitions [16]. Figure 11(b) is nothing but the choices of A B C , , to define the topological entanglement entropy by Levin and Wen [13], which detects topological order.…”

Signatures of quantum phase transitions from the boundary of the numerical range

“…In addition to the entropy of entanglement, a plethora of other entropic quantities is used to study critical phenomena. Examples are conditional mutual information and irreducible many-body correlation [17,50]. In some cases [39], irreducible many-body correlation is closely related to topological entanglement entropy known from the classification of quantum phases [47,41,34].…”

Maximum-Entropy Inference and Inverse Continuity of the Numerical Range

“…They have been discussed as a signature of a quantum phase transition [8]. They are passed [46,32] from the inference map to a correlation quantity called irreducible correlation [23,49] which is connected to the topological entanglement entropy used to characterize topological order [25,18].…”