Classical impurities and boundary Majorana zero modes in quantum chains

Abstract

We study the response of classical impurities in quantum Ising chains. The Z2 degeneracy they entail renders the existence of two decoupled Majorana modes at zero energy an exact property of a finite system at arbitrary values of its bulk parameters. We trace the evolution of these modes across the transition from the disordered phase to the ordered one and analyze the concomitant qualitative changes of local magnetic properties of an isolated impurity. In the disordered phase, the two ground states differ only close to the impurity, and they are related by the action of an explicitly constructed quasi-local operator. In this phase the local transverse spin susceptibility follows a Curie law. The critical response of a boundary impurity is logarithmically divergent and maps to the two-channel Kondo problem, while it saturates for critical bulk impurities, as well as in the ordered phase. The results for the Ising chain translate to the related problem of a resonant level coupled to a 1d p-wave superconductor or a Peierls chain, whereby the magnetic order is mapped to topological order. We find that the topological phase always exhibits a continuous impurity response to local fields as a result of the level repulsion of local levels from the boundary Majorana zero mode. In contrast, the disordered phase generically features a discontinuous magnetization or charging response. This difference constitutes a general thermodynamic fingerprint of topological order in phases with a bulk gap.

Universal scaling for the quantum Ising chain with a classical impurity

“…Both discrete modes are present in the intersection of R 1 and R 2 . On the solid green line (μ = 0 with h 1), mode 2 becomes an actual zero mode at any finite N ( 2 = 0) and the impurity becomes classical [52]. Finally, only quasicontinuous modes are present in the white region.…”

“…This is enough to ensure the twofold spectral degeneracy ofĤ μ , which is ultimately consistent with a nonzero ground-state expectation value σ z 1 playing the role of a characteristic parameter for such a phase. Remarkably, in this classical impurity limit, the degeneracy emerges at any finite N , without the need of the thermodynamic limit [52]. In Jordan-Wigner fermion coordinates, the twofold spectral degeneracy at μ = 0 occurs when the discrete mode η 2 becomes gapless: it corresponds to the zero-mode operator in Eq.…”

“…Accordingly, the Hamiltonian spectrum becomes twofold degenerate at any finite N and one may always construct a conserved operator commuting withĤ μ=0 , but anticommuting with the Jordan-Wigner total fermion parity operator (see Ref. [52] for a detailed discussion of this point). Taking the N → ∞ limit of Eq.…”

“…At variance, when μ → 0, the discrete mode 2 emerges with vanishing energy at any finite N . This corresponds to the impurity at n = 1 becoming classical [52]. Accordingly, the Hamiltonian spectrum becomes twofold degenerate at any finite N and one may always construct a conserved operator commuting withĤ μ=0 , but anticommuting with the Jordan-Wigner total fermion parity operator (see Ref.…”

“…In contrast, for μ = 0, we can explore the physics of a classical impurity (see Ref. [52]). As outlined there, a classical impurity gives rise to a twofold degeneracy for the spectrum of the whole system Hamiltonian.…”