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arXiv:1601.05486 · Physical Review Letters

Exact Solution of Quadratic Fermionic Hamiltonians for Arbitrary Boundary Conditions

Abhijeet Alase, Emilio Cobanera, Gerardo Ortíz, Lorenza Viola

Published Aug 11, 2016• 54 citations
arXivHTMLPDF

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 Hamiltonian-dependent 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.

DOI
10.1103/physrevlett.117.076804

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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 Hamiltonian-dependent 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.
A dynamical equivalent of the bulk-boundary correspondence has been suggested to occur in one and two dimensional topological models following sudden quenches. Depending on the topological invariant of the time evolving and initial phases involved large boundary contributions to a dynamical free energy occur. Moreover they occur periodically between the critical times at which this dynamical free energy becomes non-analytic, \emph{i.e.}~at dynamical quantum phase transitions. At these critical times the eigenvalue spectrum of the Loschmidt matrix which underlies the dynamical free energy closes its gap. The boundary contributions are understood to be due to zero-modes or in-gap bands of this matrix, forming a close analogy with equilibrium topological models and their edge modes. The exact cause of this phenomena and its generality remain unknown. In this article we test the dynamical bulk-boundary correspondence for a complicated two dimensional topological superconductor with a rich phase diagram, allowing quenches between many different Chern numbers. We show that there is no straightforward correspondence between the equilibrium phases quenched between and the dynamical bulk boundary correspondence. Furthermore the correspondence can depend on the orientation of the edges, suggesting a possible weak topological variant.
Majorana zero modes are well studied in the gapped phases of topological systems. We investigate Majorana zero modes at the topological quantum criticality in one dimensional topological superconducting model with longer range interaction. We identify stable localized Majorana zero modes appearing at criticality under certain conditions. Topological invariant number for these non-trivial criticalities is obtained from zeros of a complex function associated with the Hamiltonian. Behavior of parametric curve at criticalities validate the invariant obtained and account for the appearance of Majorana zero modes at criticality. Trivial and non-trivial topological nature of criticality due to the presence of multicritical point cause an unusual topological transition along the critical line. We observe and investigate this unique transition in terms of eigenvalue spectrum. Appearance of MZMs at criticality demands integer value of topological invariant number in order to validate the concept of bulk-boundary correspondence. Hence we propose a scheme to separate the invariant number into fractional and integer contribution to establish bulk-boundary correspondence at criticality.
In this paper, we study five-dimensional Dirac fermions of which the extra-dimension is compactified on quantum graphs. We find that there is a nontrivial correspondence between matrices specifying boundary conditions at the vertex of the quantum graphs and zero-dimensional Hamiltonians in gapped free-fermion systems. Based on the correspondence, we provide a complete topological classification of the boundary conditions in terms of noninteracting fermionic topological phases. The ten symmetry classes of topological phases are fully identified in the language of five-dimensional Dirac fermions, and topological numbers of the boundary conditions are given. In analogy with the bulk-boundary correspondence in noninteracting fermionic topological phases, the boundary condition topological numbers predict four-dimensional massless fermions localized at the vertex of the quantum graphs and thus govern the low energy physics in four dimensions.
Abstract In this paper, we study the localized states of a generic quadratic fermionic chain with finite-range couplings and an inhomogeneity in the hopping (defect) that breaks translational invariance. When the hopping of the defect vanishes, which represents an open chain, we obtain a simple bulk-edge correspondence: the zero-energy modes localized at the ends of the chain are related to the roots of a polynomial determined by the couplings of the Hamiltonian of the bulk. From this result, we define an index that characterizes the different topological phases of the system and can be easily computed by counting the roots of the polynomial. As the defect is turned on and varied adiabatically, the zero-energy modes may cross the energy gap and connect the valence and conduction bands. We analyze the robustness of the connection between bands against perturbations of the Hamiltonian. The pumping of states from one band to the other allows the creation of particle–hole pairs in the bulk. An important ingredient for our analysis is the transformation of the Hamiltonian under the standard discrete symmetries, C , P , T , as well as a fourth one, peculiar to our system, that is related to the existence of a gap and localized states.
Certain one-dimensional Fermi systems have an energy gap in the bulk spectrum while boundary states are described by one Majorana operator per boundary point. A finite system of length L possesses two ground states with an energy difference proportional to exp(- L / l 0 ) and different fermionic parities. Such systems can be used as qubits since they are intrinsically immune to decoherence. The property of a system to have boundary Majorana fermions is expressed as a condition on the bulk electron spectrum. The condition is satisfied in the presence of an arbitrary small energy gap induced by proximity of a three-dimensional p-wave superconductor, provided that the normal spectrum has an odd number of Fermi points in each half of the Brillouin zone (each spin component counts separately).
Certain one-dimensional Fermi systems have an energy gap in the bulk spectrum while boundary states are described by one Majorana operator per boundary point. A finite system of length L possesses two ground states with an energy difference proportional to exp(−L/l0) and different fermionic parities. Such systems can be used as qubits since they are intrinsically immune to decoherence. The property of a system to have boundary Majorana fermions is expressed as a condition on the bulk electron spectrum. The condition is satisfied in the presence of an arbitrary small energy gap induced by proximity of a 3-dimensional p-wave superconductor, provided that the normal spectrum has an odd number of Fermi points in each half of the Brillouin zone (each spin component counts separately).