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Quantum Phase Characterization via Entanglement Scaling in Fermionic Quantum Wires

Alexander Carlo Cameron

Published Oct 8, 2019• 0 citations
Publisher

Abstract

Quantum entanglement is studied in the context of its use as a probe of quantum phases within fermionic non-interacting quantum wires. In particular, we consider the uniform lattice, as well as the Su-Schreiffer-Heeger (SSH) model, and use the bipartite entropy of entanglement to characterize their quantum phases. In 1D, it has been shown analytically that the von Neumann entropy defined for a subsystem $\mathcal{H}_A \subset \mathcal{H}_A\otimes\mathcal{H}_B$ takes distinct forms dependent on the whether or not the state is thermodynamically gapped. It is also known that entanglement can be used to identify topological phases of a quantum system, such as the symmetry protected phase found in the SSH model. These results are confirmed numerically, and some common techniques for the calculation of bipartite entanglement are compared according to their aptitude for simulating larger and more complicated systems.

DOI
10.11575/prism/37205

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Quantum entanglement is studied in the context of its use as a probe of quantum phases within fermionic non-interacting quantum wires. In particular, we consider the uniform lattice, as well as the Su-Schreiffer-Heeger (SSH) model, and use the bipartite entropy of entanglement to characterize their quantum phases. In 1D, it has been shown analytically that the von Neumann entropy defined for a subsystem HA⊂HA⊗HB\mathcal{H}_A \subset \mathcal{H}_A\otimes\mathcal{H}_BHA​⊂HA​⊗HB​ takes distinct forms dependent on the whether or not the state is thermodynamically gapped. It is also known that entanglement can be used to identify topological phases of a quantum system, such as the symmetry protected phase found in the SSH model. These results are confirmed numerically, and some common techniques for the calculation of bipartite entanglement are compared according to their aptitude for simulating larger and more complicated systems.
Entanglement entropy (EE) provides a powerful probe of quantum phases, yet its role in identifying topological phase transitions in disordered systems remains underexplored. We introduce an exact EE-based framework that captures topological phase transitions even in the presence of disorder. Specifically, for a class of Su-Schrieffer-Heeger (SSH) model variants, we show that the difference in EE between half-filled and near-half-filled ground states,ΔSA, vanishes in the topological phase but remains finite in the trivial phase, a direct consequence of edge-state localization. This behavior persists even in the presence of quasiperiodic or binary disorder. By analyzing domain-wall configurations in the SSH chain, we further show how subsystem tuning allows one to distinguish genuine topological zero-energy eigenstates from trivial localized states. Exact phase boundaries, derived from Lyapunov exponents via transfer matrices, agree closely with numerical results fromΔSAand the topological invariant𝓠, with instances whereΔSAoutperforms𝓠. Our results highlight EE as a robust diagnostic tool and a potential bridge between quantum information and condensed matter approaches to topological matter.
Abstract Among many types of quantum entanglement properties, the entanglement spectrum provides more abundant information than other observables. Exact diagonalization and density matrix renormalization group methods could handle the system in one-dimension properly, while in a higher dimension, it exceeds the capacity of the algorithms. To expand the ability of existing numerical methods, we take a different approach via quantum Monte Carlo algorithm. By exploiting the particle number and spin conservation, we realize an efficient algorithm to solve the entanglement spectrum in the interacting fermionic system. Taking the two-dimensional interacting Su–Schrieffer–Heeger (SSH) model as an example, we verify the existence of topological phase transition under different types of many-body interactions. The calculated particle number distribution and wavefunction of the entanglement Hamiltonian indicate that the two belong to distinct types of topological phase transitions.
We present a detailed study of the fidelity, the entanglement entropy, and the entanglement spectrum, for a dimerized chain of spinless fermions---a simplified Su-Schrieffer-Heeger (SSH) model---with open boundary conditions which is a well-known example for a model supporting a symmetry protected topological (SPT) phase. In the non-interacting case the Hamiltonian matrix is tridiagonal and the eigenvalues and -vectors can be given explicitly as a function of a single parameter which is known analytically for odd chain lengths and can be determined numerically in the even length case. From a scaling analysis of these data for essentially semi-infinite chains we obtain the fidelity susceptibility and show that it contains a boundary contribution which is different in the topologically ordered than in the topologically trivial phase. For the entanglement spectrum and entropy we confirm predictions from massive field theory for a block in the middle of an infinite chain but also consider blocks containing the edge of the chain. For the latter case we show that in the SPT phase additional entanglement---as compared to the trivial phase---is present which is localized at the boundary. Finally, we extend our study to the dimerized chain with a nearest-neighbour interaction using exact diagonalization, Arnoldi, and density-matrix renormalization group methods and show that a phase transition into a topologically trivial charge-density wave phase occurs.
We present a detailed study of the fidelity, the entanglement entropy, and the entanglement spectrum, for a dimerized chain of spinless fermions---a simplified Su-Schrieffer-Heeger (SSH) model---with open boundary conditions which is a well-known example for a model supporting a symmetry protected topological (SPT) phase. In the non-interacting case the Hamiltonian matrix is tridiagonal and the eigenvalues and -vectors can be given explicitly as a function of a single parameter which is known analytically for odd chain lengths and can be determined numerically in the even length case. From a scaling analysis of these data for essentially semi-infinite chains we obtain the fidelity susceptibility and show that it contains a boundary contribution which is different in the topologically ordered than in the topologically trivial phase. For the entanglement spectrum and entropy we confirm predictions from massive field theory for a block in the middle of an infinite chain but also consider blocks containing the edge of the chain. For the latter case we show that in the SPT phase additional entanglement---as compared to the trivial phase---is present which is localized at the boundary. Finally, we extend our study to the dimerized chain with a nearest-neighbour interaction using exact diagonalization, Arnoldi, and density-matrix renormalization group methods and show that a phase transition into a topologically trivial charge-density wave phase occurs.
Abstract Entanglement Hamiltonians provide the most comprehensive characterisation of entanglement in extended quantum systems. A key result in unitary quantum field theories is the Bisognano-Wichmann theorem, which establishes the locality of the entanglement Hamiltonian. In this work, our focus is on the non-Hermitian Su-Schrieffer-Heeger (SSH) chain. We study the entanglement Hamiltonian both in a gapped phase and at criticality. In the gapped phase we find that the lattice entanglement Hamiltonian is compatible with a lattice Bisognano-Wichmann result, with an entanglement temperature linear in the lattice index. At the critical point, we identify a new imaginary chemical potential term absent in unitary models. This operator is responsible for the negative entanglement entropy observed in the non-Hermitian SSH chain at criticality.