Critical phase boundaries of static and periodically kicked long-range Kitaev chain

Bhattacharya, Utso; Maity, Somnath; Dutta, Amit; Sen, Diptiman

doi:10.1088/1361-648x/ab03b9

Cited by 20 publications

(17 citation statements)

References 59 publications

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“…In the case of a constant chemical potential f (i) = 1, the model has well known limits: for α, ξ 1, it is topologically equivalent to the short-range Kitaev model with nearest-neighbor pairing terms [42]. In contrast, for α 1 and ξ 1 the model can host a long-range topological phase with massive Dirac modes (MDM) characterized by a half integer topological invariant [43][44][45][46]. For α 1 and ξ 1, we recover a one-dimensional chain of spinless fermions with long-range hopping.…”

Section: The Modelmentioning

confidence: 99%

Topological properties of the long-range Kitaev chain with Aubry-André-Harper modulation

Fraxanet

Bhattacharya

Graß

et al. 2021

Phys. Rev. Research

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Section: The Modelmentioning

confidence: 99%

Topological properties of the long-range Kitaev chain with Aubry-André-Harper modulation

Fraxanet

Bhattacharya

Graß

et al. 2021

Phys. Rev. Research

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“…39 and references therein). Similarly, the emergent topological features of the long range chain under periodic drives has also been investigated 40 .…”

Section: B Topological Invariants For Periodic Drivingmentioning

confidence: 99%

Driven quantum many-body systems and out-of-equilibrium topology

Bandyopadhyay,

Bhattacharjee,

Sen

2021

Preprint

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In this review we present some of the work done in India in the area of driven and out-of-equilibrium systems with topological phases. After presenting some well-known examples of topological systems in one and two dimensions, we discuss the effects of periodic driving in some of them. We discuss the unitary as well as the non-unitary dynamical preparation of topologically non-trivial states in one and two dimensional systems. We then discuss the effects of Majorana end modes on transport through a Kitaev chain and a junction of three Kitaev chains. Transport through the surface states of a three-dimensional topological insulator is discussed. The effects of hybridization between the top and bottom surfaces and the application of electromagnetic radiation on a strip-like region on the top surface are described. Two unusual topological systems are mentioned briefly, namely, a spin system on a kagome lattice and a Josephson junction of three superconducting wires. We have also included a pedagogical discussion on topology and topological invariants in the appendices, where the connection between topological properties and the intrinsic geometry of quantum states is also elucidated. Contents V. Generically driven two-dimensionalChern topological systems 24 A. The unitary no-go theorem 24 B. The non-unitary no-go theorem 26 C. Dissipative preparation of mixed states 28 1. Dissipators having a finite range 28 2. Macroscopic electric polarization and mixed state Chern number 29 3. Long-range dissipators and Floquet Chern insulators 32 D. Going around the unitary no-go theorem: bilayer Chern systems 34 VI. Majorana modes in one-dimensional systems and transport 35 VII. Transport on surfaces of topological insulators 37 VIII. Other topological systems 41 IX. Summary and future directions 42 A. A pedagogical introduction to the geometry and topology of quantum states43 1. Adiabatic transformations 43 a. Topology of adiabatically connected states44 2. The underlying geometry of quantum states 45 3. Topological invariants 48 a. Chern invariants 48 b. Bott invariant 50 B. A perturbative expansion of the steady state Majorana correlations 51 References 52

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“…Before getting into the details of each method, we would like to stop and briefly analyze the long-range case α 1 [22][23][24][25][26]. On the one hand, the long-range character of the Hamiltonian affects the edge states of the system: the MMs hybridize and become massive and form MDMs.…”

Section: Winding Numbermentioning

confidence: 99%

“…The short-range Kitaev chain is a version of the inte-grable p-wave superconducting chain of fermions. The primary focus of the present work is a long-range Kitaev chain, in which the superconducting pairing term decays with distance as a power-law [22][23][24][25][26]. While the topological phases of the short-range Kitaev chain are characterized by Z topological invariants, the present model posseses a new unconventional topological phase characterized by half-integer Z/2 winding numbers, even though the system belongs to the same BDI class and respects the same discrete symmetries of the short-range model.…”

Section: Introductionmentioning

confidence: 99%

“…One must note that the quantum critical behavior of systems with long-range interactions/pairings can be very different from that of the short-range ones, due to the non-local nature of the interactions. Therefore, over the recent years, much effort has been devoted to understanding the static [27,28] and dynamic properties [29] of long-range systems through quantum quenches [30][31][32], periodic drives [26], through establishment of the Lieb-Robinson bounds [33], and through studies of the growth of entanglement entropy [34], Bell inequalities [35], and thermalization [36,37] (for a detailed review, see [38]).…”

Section: Introductionmentioning

confidence: 99%

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Topological properties of the long-range Kitaev chain with Aubry-André-Harper modulation

Fraxanet,

Bhattacharya,

Grass

et al. 2020

Preprint

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We present a detailed study of the topological properties of the Kitaev chain with long-range pairing terms and in the presence of an Aubry-André-Harper on-site potential. Specifically, we consider algebraically decaying superconducting pairing amplitudes; the exponent of this decay is found to determine a critical pairing strength, below which the chain remains topologically trivial. Above the critical pairing, topological edge modes are observed in the central gap. For sufficiently fast decay of the pairing, these modes are identified as Majorana zero-modes. However, if the pairing term decays slowly, the modes become massive Dirac modes. Interestingly, these massive modes still exhibit a true level crossing at zero energy, which points towards an initimate relation to Majorana physics. We also observe a clear lack of bulk-boundary correspondence in the long-range system, where bulk topological invariants remain constant, while dramatic changes appear in the behavior at the edge of the system. In addition to the central gap around zero energy, the Aubry-André-Harper potential also leads to other energy gaps at non-zero energy. As for the analogous short-range model, the edge modes in these gaps can be characterized through a 2D Chern invariant. However, in contrast to the short-range model, this topological invariant does not correspond to the number of edge mode crossings anymore. This provides another example for the weakening of the bulk-boundary correspondence occurring in this model. Finally, we discuss possible realizations of the model with ultracold atoms and condensed matter systems.

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Critical phase boundaries of static and periodically kicked long-range Kitaev chain

Cited by 20 publications

References 59 publications

Topological properties of the long-range Kitaev chain with Aubry-André-Harper modulation

Topological properties of the long-range Kitaev chain with Aubry-André-Harper modulation

Driven quantum many-body systems and out-of-equilibrium topology

Topological properties of the long-range Kitaev chain with Aubry-André-Harper modulation

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