Edge insulating topological phases in a two-dimensional superconductor with long-range pairing

Abstract

We study the zero-temperature phase diagram of a two dimensional square lattice loaded by spinless fermions, with nearest-neighbor hopping and algebraically decaying pairing. We find that for sufficiently long-range pairing, new phases occur, not continuously connected with any short-range phase and not belonging to the standard families of topological insulators/superconductors. These phases are signaled by the violation of the area law for the Von Neumann entropy, by semi-integer Chern numbers, and by edge modes with nonzero mass. The latter feature results in the absence of single-fermion edge conductivity, present instead in the short-range limit. The definition of a bulk-topology and the presence of a bulk-boundary correspondence is suggested also for the long-range phases. Recent experimental proposals and advances open the possibility to probe the described long-range effects in near-future realistic set-ups. PACS numbers:Introduction -In recent years, the topological phases of matter have become a central focus for physical investigation. A groundbreaking result is the classification of the symmetry-protected topologically inequivalent classes for non interacting fermionic systems (topological insulators/superconductors) [1][2][3][4][5][6]. This theoretical breakthrough has been probed on particular solid-state compounds [7][8][9][10].Notwithstanding the presence of a nonvanishing bulk energy gap, the most relevant feature displayed by nontrivial topological insulators/superconductors is a conductivity localized on the edges, due to massless edge mode with a dynamics well distinguished from the bulk excitations. Moreover, continuous transitions with a vanishing mass gap generally divide phases with different topology (even if also first order transitions seem possible if perturbative interactions between fermions are added [11]). Regarding the entanglement content, topological insulators/superconductors display exponentially saturating entanglement and correlations, giving rise to the area-law for the Von Neumann entropy between the two elements of a real-space bipartition.These features characterize quantum systems governed by Hamiltonians with short-range (SR) terms. However, in recent years, long-range (LR) classical and quantum systems [12], obtained renewed attention. Independent theoretical works have concluded that LR systems can exhibit many interesting and unusual properties, essentially due to the violation of locality (see e.g. [13][14][15][16][17][18][19][20]). In particular, one-dimensional LR models can host new phases, manifesting striking properties, not present in the SR limit. [15][16][17][21][22][23][24][25][26][27][28][29][30].In spite of these relevant achievements, full comprehension and classification phases emerging from Hamiltonians with LR terms are still not available.In [29] a partial solution to these problems has been discussed, exploiting one-dimensional topological superconductive chains as playgrounds, but inferring nontrivial properties also for LR topological insulators...

Long-range topological insulators and weakened bulk-boundary correspondence

“…The analysis of the conditions for the emergence of massive edge states seems to be a promising approach; (ii) the corresponding investigation of the nature of the edge states in LR fermionic systems with dimensionality bigger than one, in order to probe the weakened bulk-boundary correspondence (also following the logic in [80,81]). Interestingly from the experimental and technological points of view, this issue also concerns the possible absence of edge conductivity, present instead for SR topological insulators and superconductors(first examples have been recently given in [87,88]); (iii) the generalization to interacting LR models, also exploiting entanglement indicators (for instance, in the SR limit the ES is proved to be effective also in the presence of explicit interactions [62]); (iv) the study of the effects of stronger deviations from the area-law for the von Neumann entropy, for instance assuming a (almost) volume-law scaling, as in the set of systems investigated in [41]; (v) the understanding of the role of disorder on the singular dynamics in interacting LR systems, as for the LR Ising model. There effects of many-body localization [82] are expected to play a relevant role; (vi) the identification of a general scheme for the experimental detection of the LR phases, for instance based on direct measurements of the entanglement or of some topological invariants, e.g.…”

Topological quantum phase transitions and criticality in a longer-range Kitaev chain

“…And there are two different arguments about the reduction of the long-range model to the short-range model. Some works suggest that the reduction happens when α > 1 [25,55,56] and some studies obtain a short-range limit for α > 1.5 [27]. Solving these issues through the study of universality class of critical exponents and CFT can give the better understanding.…”

“…On the other hand, long-range topological models are the more generalized version of novel phases of matter [23]. This includes realization of new phases like edge insulating topological phases [24,25] with fractional topological invariants [26] and quasiparticles like Majorana zero modes (MZM), massive majorana modes [27]. In this work, we carry out a theoretical study of a topological longerrange as well as truly long-range model.…”

Multipartite-entanglement tomography of a quantum simulator

“…The interesting physics associated to LR interaction concerns also fermionic lattice systems, characterized by nontrivial topological invariants [36-38, 69, 72-77]. Notably, for these systems, BE is known to characterize only partially the LR regimes, not being able to distinguish in general the different LR phases [38,74], while ME appears to be more indicative [50][51][52].…”