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Register a new userGeneralization of Blochs theorem for arbitrary boundary conditions: Interfaces and topological surface band structure
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We describe a method for exactly diagonalizing clean $D$-dimensional lattice systems of independent fermions subject to arbitrary boundary conditions in one direction, as well as systems composed of two bulks meeting at a planar interface. Our method builds on the generalized Bloch theorem [A. Alase et al., Phys. Rev. B 96, 195133 (2017)] and the fact that the bulk-boundary separation of the Schrodinger equation is compatible with a partial Fourier transform operation. Bulk equations may display unusual features because they are relative eigenvalue problems for non-Hermitian, bulk-projected Hamiltonians. Nonetheless, they admit a rich symmetry analysis that can simplify considerably the structure of energy eigenstates, often allowing a solution in fully analytical form. We illustrate our extension of the generalized Bloch theorem to multicomponent systems by determining the exact Andreev bound states for a simple SNS junction. We then analyze the Creutz ladder model, by way of a conceptual bridge from one to higher dimensions. Upon introducing a new Gaussian duality transformation that maps the Creutz ladder to a system of two Majorana chains, we show how the model provides a first example of a short-range chiral topological insulator hosting topological zero modes with a power-law profile. Additional applications include the complete analytical diagonalization of graphene ribbons with both zigzag-bearded and armchair boundary conditions, and the analytical determination of the edge modes in a chiral $p+ip$ two-dimensional topological superconductor. Lastly, we revisit the phenomenon of Majorana flat bands and anomalous bulk-boundary correspondence in a two-band gapless $s$-wave topological superconductor. We analyze the equilibrium Josephson response of the system, showing how the presence of Majorana flat bands implies a substantial enhancement in the $4pi$-periodic supercurrent.
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We present a generalization of Blochs theorem to finite-range lattice systems of independent fermions, in which translation symmetry is broken only by arbitrary boundary conditions, by providing exact, analytic expressions for all energy eigenvalues and eigenstates. By transforming the single-particle Hamiltonian into a corner-modified banded block-Toeplitz matrix, a key step is a bipartition of the lattice, which splits the eigenvalue problem into a system of bulk and boundary equations. The eigensystem inherits most of its solutions from an auxiliary, infinite translation-invariant Hamiltonian that allows for non-unitary representations of translation symmetry. A reformulation of the boundary equation in terms of a boundary matrix ensures compatibility with the boundary conditions, and determines the allowed energy eigenstates. We show how the boundary matrix captures the interplay between bulk and boundary properties, leading to efficient indicators of bulk-boundary correspondence. Remarkable consequences of our generalized Bloch theorem are the engineering of Hamiltonians that host perfectly localized, robust zero-energy edge modes, and the predicted emergence, e.g. in Kitaevs chain, of localized excitations whose amplitudes decay exponentially with a power-law prefactor. We further show how the theorem yields diagonalization algorithms for the class of Hamiltonians under consideration, and use the proposed bulk-boundary indicator to characterize the topological response of a multi-band time-reversal invariant s-wave superconductor under twisted boundary conditions, showing how a fractional Josephson effect can occur without a fermionic parity switch. Finally, we establish connections to the transfer matrix method and demonstrate, using the paradigmatic Kitaevs chain example, that a non-diagonalizable transfer matrix signals the presence of solutions with a power-law prefactor.
We consider an open isotropic Heisenberg quantum spin chain, coupled at the ends to boundary reservoirs polarized in different directions, which sets up a twisting gradient across the chain. Using a matrix product ansatz, we calculate the exact magnetization profiles and magnetization currents in the nonequilibrium steady steady state of a chain with N sites. The magnetization profiles are harmonic functions with a frequency proportional to the twisting angle {theta}. The currents of the magnetization components lying in the twisting plane and in the orthogonal direction behave qualitatively differently: In-plane steady state currents scale as 1/N^2 for fixed and sufficiently large boundary coupling, and vanish as the coupling increases, while the transversal current increases with the coupling and saturates to 2{theta}/N.
We extend the exact periodic Bethe Ansatz solution for one-dimensional bosons and fermions with delta-interaction and arbitrary internal degrees of freedom to the case of hard wall boundary conditions. We give an analysis of the ground state properties of fermionic systems with two internal degrees of freedom, including expansions of the ground state energy in the weak and strong coupling limits in the repulsive and attractive regimes.
We present a detailed study of squared local roughness (SLRDs) and local extremal height distributions (LEHDs), calculated in windows of lateral size $l$, for interfaces in several universality classes, in substrate dimensions $d_s = 1$ and $d_s = 2$. We show that their cumulants follow a Family-Vicsek type scaling, and, at early times, when $xi ll l$ ($xi$ is the correlation length), the rescaled SLRDs are given by log-normal distributions, with their $n$th cumulant scaling as $(xi/l)^{(n-1)d_s}$. This give rise to an interesting temporal scaling for such cumulants $leftlangle w_n rightrangle_c sim t^{gamma_n}$, with $gamma_n = 2 n beta + {(n-1)d_s}/{z} = left[ 2 n + {(n-1)d_s}/{alpha} right] beta$. This scaling is analytically proved for the Edwards-Wilkinson (EW) and Random Deposition interfaces, and numerically confirmed for other classes. In general, it is featured by small corrections and, thus, it yields exponents $gamma_n$s (and, consequently, $alpha$, $beta$ and $z$) in nice agreement with their respective universality class. Thus, it is an useful framework for numerical and experimental investigations, where it is, usually, hard to estimate the dynamic $z$ and mainly the (global) roughness $alpha$ exponents. The stationary (for $xi gg l$) SLRDs and LEHDs of Kardar-Parisi-Zhang (KPZ) class are also investigated and, for some models, strong finite-size corrections are found. However, we demonstrate that good evidences of their universality can be obtained through successive extrapolations of their cumulant ratios for long times and large $l$s. We also show that SLRDs and LEHDs are the same for flat and curved KPZ interfaces.
Time reversal invariance (TRI) of particles systems has many consequences, among~which the celebrated Onsager reciprocal relations, a milestone in Statistical Mechanics dating back to 1931. Because for a long time it was believed that (TRI) dos not hold in presence of a magnetic field, a modification of such relations was proposed by Casimir in 1945. Only in the last decade, the~strict traditional notion of reversibility that led to Casimirs work has been questioned. It was then found that other symmetries can be used, which allow the Onsager reciprocal relations to hold without modification. In this paper we advance this investigation for classical Hamiltonian systems, substantially increasing the number of symmetries that yield TRI in presence of a magnetic field. We~first deduce the most general form of a generalized time reversal operation on the phase space of such a system; secondly, we express sufficient conditions on the magnetic field which ensure TRI. Finally, we examine common examples from statistical mechanics and molecular dynamics. Our main result is that TRI holds in a much wider generality than previously believed, partially explaining why no experimental violation of Onsager relations has so far been reported.
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