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arXiv:2111.03506 · Physical review. B./Physical review. B

Detecting topological phase transitions through entanglement between disconnected partitions in a Kitaev chain with long-range interactions

Saikat Mondal, Souvik Kumar Bandyopadhyay, Sourav Bhattacharjee, Amit Dutta

Published Feb 4, 2022• 18 citations
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Abstract

We explore the behavior of the disconnected entanglement entropy (DEE) across the topological phases of a long-range interacting Kitaev chain where the long-range interactions decay as a power law with an exponent $\ensuremath{\alpha}$. We show that while the DEE may not remain invariant deep within the topologically nontrivial phase when $\ensuremath{\alpha}<1$, it nevertheless shows a quantized discontinuous jump at the quantum critical point and can act as a strong marker for the detection of topological phase transition. We also study the time evolution of the DEE after a sudden quench of the chemical potential within the same phase. In the short-range limit of a finite chain, the DEE is expected to remain constant up to a critical time after the quench, which diverges in the thermodynamic limit. However, no such critical time is found to exist when the long-range interactions dominate (i.e., $\ensuremath{\alpha}<1$).

DOI
10.1103/physrevb.105.085106

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We explore the behavior of the disconnected entanglement entropy (DEE) across the topological phases of a long-range interacting Kitaev chain where the long-range interactions decay as a power law with an exponent \ensuremathα\ensuremath{\alpha}\ensuremathα. We show that while the DEE may not remain invariant deep within the topologically nontrivial phase when \ensuremathα<1\ensuremath{\alpha}<1\ensuremathα<1, it nevertheless shows a quantized discontinuous jump at the quantum critical point and can act as a strong marker for the detection of topological phase transition. We also study the time evolution of the DEE after a sudden quench of the chemical potential within the same phase. In the short-range limit of a finite chain, the DEE is expected to remain constant up to a critical time after the quench, which diverges in the thermodynamic limit. However, no such critical time is found to exist when the long-range interactions dominate (i.e., \ensuremathα<1\ensuremath{\alpha}<1\ensuremathα<1).
We explore the behavior of the disconnected entanglement entropy (DEE) across the topological phases of a long range interacting Kitaev chain where the long range interactions decay as a power law with an exponent α\alphaα. We show that while the DEE may not remain invariant deep within the topologically non-trivial phase when α<1\alpha<1α<1, it still shows a quantized discontinuous jump at the quantum critical point and can act as a strong marker for the detection of topological phase transition for the long range interacting systems. We also study the time evolution of the disconnected entanglement entropy after a sudden quench of chemical potential within same phase. In the short range limit of a finite chain, the DEE is expected to remain constant upto a critical time which diverges in the thermodynamic limit. However, we find that no such critical time exists in the presence of long range interactions (α<1\alpha<1α<1).
Long-range interactions exhibit surprising features which have been less explored so far. Here, studying a one-dimensional fermionic chain with long-range hopping and pairing, we discuss some general features associated to the presence of long-range entanglement. In particular, after determining the algebraic decays of the correlation functions, we prove that a long-range quantum mutual information exists if the exponent of the decay is not larger than one. Moreover, we show that the time evolution triggered by a quantum quench between short-range and long-range regions, can be characterized by dynamical quantum phase transitions without crossing any phase boundary. We show, also, that the adiabatic dynamics is dictated by the divergence of a topological length scale at the quantum critical point, clarifying the violation of the Kibble-Zurek mechanism for long-range systems.
We study the topological properties of a monitored Rashba chain along quantum-jump trajectories, investigating the persistence of the initial topological value of the disconnected entanglement entropy (DEE). We find that the DEE persists in its topological value for a time linear in the system size, even if the dissipation acts on the boundary and affects the topological Majorana modes. The reason for this phenomenon lies in the absence of particle conservation and in the degeneracy of the topological manifold, allowing the monitoring to let the system switch between different topological states -- alternatively creating and annihilating a Majorana mode -- while producing a poisoning of finite-energy ballistically propagating quasiparticles that eventually destroy the topological entanglement structure.
We propose and analyze a generalization of the Kitaev chain for fermions with long-range p-wave pairing, which decays with distance as a power law with exponent α. Using the integrability of the model, we demonstrate the existence of two types of gapped regimes, where correlation functions decay exponentially at short range and algebraically at long range (α > 1) or purely algebraically (α < 1). Most interestingly, along the critical lines, long-range pairing is found to break conformal symmetry for sufficiently small α. This is accompanied by a violation of the area law for the entanglement entropy in large parts of the phase diagram in the presence of a gap and can be detected via the dynamics of entanglement following a quench. Some of these features may be relevant for current experiments with cold atomic ions.
A bstract Thanks to their prominent collective character, long-range interactions promote information spreading and generate forms of entanglement scaling, which cannot be observed in traditional systems with local interactions. In this work, we study the asymptotic behavior of the entanglement entropy for Kitaev chains with long-range hopping and pairing couplings decaying with a power law of the distance. We provide a fully-fledged analytical and numerical characterization of the asymptotic growth of the ground state entanglement in the large subsystem size limit, finding that the truly non-local nature of the model leads to an extremely rich phenomenology. Most significantly, in the strong long-range regime, we discovered that the system ground state may have a logarithmic, fractal, or volume-law entanglement scaling, depending on the value of the chemical potential and on the strength of the power law decay.
We study the disconnected entanglement entropy, S^\mathrm{D} S D , of the Su-Schrieffer-Heeger model. S^\mathrm{D} S D is a combination of both connected and disconnected bipartite entanglement entropies that removes all area and volume law contributions and is thus only sensitive to the non-local entanglement stored within the ground state manifold. Using analytical and numerical computations, we show that S^\mathrm{D} S D behaves like a topological invariant, i.e., it is quantized to either 0 0 or 2\log(2) 2 log ( 2 ) in the topologically trivial and non-trivial phases, respectively. These results also hold in the presence of symmetry-preserving disorder. At the second-order phase transition separating the two phases, S^\mathrm{D} S D displays a finite-size scaling behavior akin to those of conventional order parameters, that allows us to compute entanglement critical exponents. To corroborate the topological origin of the quantized values of S^\mathrm{D} S D , we show how the latter remain quantized after applying unitary time evolution in the form of a quantum quench, a characteristic feature of topological invariants associated with particle-hole symmetry.