Exact Solution of Quadratic Fermionic Hamiltonians for Arbitrary Boundary Conditions

Abstract

We present a procedure for exactly diagonalizing finite-range quadratic fermionic Hamiltonians with arbitrary boundary conditions in one of D dimensions, and periodic in the remaining D − 1. The key is a Hamiltoniandependent separation of the bulk from the boundary. By combining information from the two, we identify a matrix function that fully characterizes the solutions, and may be used to construct an efficiently computable indicator of bulk-boundary correspondence. As an illustration, we show how our approach correctly describes the zero-energy Majorana modes of a time-reversal-invariant s-wave two-band superconductor in a Josephson ring configuration, and predicts that a fractional 4π-periodic Josephson effect can only be observed in phases hosting an odd number of Majorana pairs per boundary.Developing a quantitative understanding of the physical properties of fermionic systems in the presence of non-trivial boundaries has widespread significance from both a fundamental and applied perspective. Not only has the behavior of fermions at a boundary informed leading materialcharacterization techniques like angle-resolved photoemission spectroscopy [1] and the revolution in metrology brought about by the integer quantum Hall effect [2]; nowadays, surface states of topological insulators and Majorana boundary modes of topological superconductors [3,4] play a central role in state-of-the-art proposals ranging from coherent spintronics [5,6] to topological quantum computation [7,8].All of the above phenomena are linked by a common theme: topologically non-trivial band structures [4]. Band structure theory, including the topological classification of mean-field fermionic systems [9], rests on a manifestation of crystal translational symmetry, the Bloch theorem. Since translational symmetry is broken by the presence of a boundary, it is remarkable that there exists a connection between the topological nature of the bulk and the boundary physics -the bulkboundary (BB) correspondence [4,10]. This principle states that a topologically non-trivial bulk mandates the emergence of fermionic states localized on the boundary, when boundary conditions (BCs) are changed from periodic to open, and that such states are distinguished by their robustness against symmetry-preserving local perturbations. While this heuristics has been numerically validated in a variety of cases, and rigorous results exist for discrete-time systems described by one-dimensional quantum walks [11], no general analytic insight is available as yet. Allowing for arbitrary BCs is necessary for any theory of BB correspondence to capture the robustness of the emerging localized modes to different perturbations [12]. Further motivation stems from studies of quantum quenches [13,14], where robustness against changes of the BCs has been argued to control the (quasi)local symmetries that characterize the stationary properties in the bulk. Tackling these issues calls for a procedure to determine energy eigenstates of lattice Hamiltonians with arbitrary BCs, compa...

Exact zero modes in twisted Kitaev chains

“…Then, we obtain the condition for the presence of Majorana zero modes in finite chains, as well as explicitly determine the spatial profile of the zero modes. We note that the emergence of Majorana zero modes (protected level crossing) in the fractional Josephson effect has been well-established since the seminal work of Kitaev 2,45,51,[58][59][60] . However, we emphasize that the boundary couplings studied in this paper are more general than those in previous work, as their amplitudes and phases can be arbitrary.…”

“…The usual periodic boundary condition (PBC) and the anti-periodic boundary condition (APBC) are included in the TBC as limiting cases. In practice, such a boundary condition can be realized by magnetic fluxes and Josephson junctions 6,45,[49][50][51][52][53][54][55][56][57][58][59][60][61] . Our work is motivated by the observation that the fermionic parities in the ground states of the Kitaev chain with the PBC and the APBC have opposite signs in the topological phase, hence indicating a level crossing when one continuously changes the parameters so as to connect the PBC with the APBC.…”

Entangled end states with fractionalized spin projection in a time-reversal-invariant topological superconducting wire

“…23 In this work we study these low-energy end states of a finite chain. By using a method presented recently by Alase et al 28,29 we analytically calculate the zero-energy eigenstates of an infinite chain of the model introduced by Zhang et al in Ref. 12 in the particle-hole symmetric configuration of the normal system (which means that the chemical potential µ = 0).…”

Generalization of Bloch's theorem for arbitrary boundary conditions: Interfaces and topological surface band structure

“…In this section, we revisit this phenomenon and show that the indicator of bulk-boundary correspondence we introduced in Ref. [3] captures it precisely. Furthermore, in the phase hosting a MFB, we demonstrate by combining our Bloch ansatz with numerical root evaluation, that the characteristic length of the MFB wavefunctions diverges as we approach the critical values of momentum, similarly to what was observed in graphene [Eq.…”