Entanglement and Disordered-Enhanced Topological Phase in the Kitaev Chain

Abstract

In recent years, tools from quantum information theory have become indispensable in characterizing many-body systems. In this work, we employ measures of entanglement to study the interplay between disorder and the topological phase in 1D systems of the Kitaev type, which can host Majorana end modes at their edges. We find that the entanglement entropy may actually increase as a result of disorder, and identify the origin of this behavior in the appearance of an infinite-disorder critical point. We also employ the entanglement spectrum to accurately determine the phase diagram of the system, and find that disorder may enhance the topological phase, and lead to the appearance of Majorana zero modes in systems whose clean version is trivial.Entanglement is the quintessential characteristic of the quantum world. Recently, much attention has been devoted to manifestations of entanglement in quantum many-body systems [1,2]. Motivations include the prospect of using many-body ground and excited states as a resource for quantum communication and computation; the amount of entanglement controls the applicability of a powerful tensor-network-based numerical method; entanglement is an indicator of quantum correlations which is independent of the details of the system, allowing, in particular, one to identify quantum phase transitions even without knowing the relevant order parameter. In this work, we will concentrate on the latter property, in the context of disordered 1D quantum many-body systems.Majorana zero modes [3][4][5][6] and their more complicated relatives [7] have been studied extensively recently, both due to their predicted exotic non-abelian braiding statistics, which gives rise to the prospect of topologically-protected quantum computation [8], and to many concrete proposals for their realization in the laboratory. Leading candidates are semiconductor quantum wires with strong spin orbit interaction, which are rendered effectively spinless by the application of an appropriate magnetic field and gate voltage, and driven into a topological phase by proximity-coupling to a superconductor, leading to the formation of Majorana end modes [9,10]. Indications for these modes have recently been measured experimentally [11][12][13][14][15]. These Majorana zero modes show up not only in the presence of a real edge, but also in the entanglement spectrum, that is, in the spectrum of the reduced density matrix of a subsystem [16].Disorder naturally occurs in all these systems, and may hamper their topological characteristics, especially the Majorana edge modes of 1D systems . In this work, we will study the interplay of disorder and the topological phase from the entanglement point of view. After introducing the model and our method for calculating the entanglement spectrum and entropy in Section 2, we will

Symmetry resolved entanglement: exact results in 1D and beyond

“…This definition of k implies that q = exp −π I(k ) I(k) (I was defined in (40)) [49], and thus it agrees with the definition of k previously presented in (38). We also write k n (q) ≡ k (q n ) and k n (q) ≡ k (q n ), and rely on the following identities from [49] that hold for every 0 < q < 1:…”

“…Both S (−) n and S (−) illustrate a striking property of the phase in which the system is found for h < 2: since S (−) n = S (−) = 0, we obtain that for h < 2, the system satisfies S (even) n = S (odd) n and S (even) = S (odd) . This property stems from the fact that we can write the RDM as ρ A = exp (−H A ) where the entanglement Hamiltonian H A is quadratic [38,48], and treat H A as the Hamiltonian of an effective system of a 1D open fermionic chain with L sites. H A is expected to have the same modes at the virtual edges of the subsystem as the original system (the Kitaev chain) would host at a physical edge [29].…”

“…To resolve this ambiguity we turn to the large h limit of the above expression. The definition of k in (38) implies that as h → ∞, k → 0 and therefore k → 1 and q → 0. Furthermore, one can show that as q → 0, k ∼ 4q…”

Robust multipartite entanglement in dirty topological wires

“…Generally speaking, the characterization of quantum phases and quantum phase transitions by entanglement-based approaches is an intriguing problem, at the frontier between quantum information [35,36] and many-body physics [37][38][39][40][41]. The literature has mostly focused on bipartite entanglement, with witnesses such as the Von Neumann entropy [28,29,[42][43][44], the entanglement spectrum [32,[45][46][47][48][49] and pairwise entanglement [50,51], also in the presence of disorder [52][53][54]. Instead, multipartite entanglement (ME) has been much less studied [55,56], while it captures a more complex entanglement structure than than identified by bipartite and pairwise entanglement.…”

Pattern-dependent proximity effect and Majorana edge mode in one-dimensional quasicrystals

“…Introduction-Over a decade, the topological Majorana edge mode, theoretically predicted by the Kitaev chain model, is one of the most studied physical phenomena in condensed matter physics [1][2][3][4][5][6][7][8]. The topological robustness of the Majorana edge mode has been attracted broad research interest including information theory and quantum computing, and has been discussed in experimental realization with general aspects [9][10][11][12][13][14][15][16][17][18][19][20][21]. Along with it, the stability of the Majorana edge modes has been studied in the presence of spatially varying hopping amplitudes or inhomogeneous potentials, for instance.…”