Classical impurities and boundary Majorana zero modes in quantum chains

Abstract

We study the response of classical impurities in quantum Ising chains. The ${\mathbb{Z}}_2$ 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.

Introduction

In recent years there was a substantial boost in the search for signatures of Majorana zero modes (MZM) that may emerge as localized quasiparticles in various condensed matter realizations because of their potential for quantum computation  [1], [2]. MZMs occurring at boundaries or defects (domain walls and vortices) in low-dimensional topological superconductors are of particular importance because of their non-Abelian anyonic statistics that uncovers new prospects for storage and manipulation of quantum information  [3], [4].

In his seminal paper Kitaev  [5] proposed a one-dimensional model of a spinless $p$-wave superconductor (1DPS)

$$H=-\mu \sum _{n=1}^N(a_n^{†}{a}_n-1/2)+\frac{1}{2}\sum _{n=1}^{N-1}(t{a}_n^{†}{a}_{n+1}+\Delta a_n^{†}{a}_{n+1}^{†}+\text{h.c.})$$

(the usual negative sign of the hopping term can be obtained by the transformation $a_n\to {(-1)}^n{a}_n$, which changes the signs of $t$ and $\Delta$). This model has a topologically non-trivial massive phase that supports localized Majorana modes at the ends of the chain. For a macroscopically large system these boundary modes can be regarded as unpaired, in which case they represent a non-local realization of a doubly degenerate fermionic zero-energy state. The spatial separation of the two MZMs ensures the immunity of the topologically degenerate ground state of the 1DPS against weak local perturbations (as long as quasi-particle poisoning can be neglected, and thus fermion parity is conserved), making such a system potentially useful for the needs of quantum computation. Thus it is of great theoretical interest and practical importance to identify the physical properties of the edge of such a 1D system that can serve as evidence for the existence of boundary MZMs.

It has soon been realized that principal features of the Kitaev 1D model  [5] can be reproduced experimentally using a quantum wire with a strong spin–orbit coupling in the presence of an external magnetic field and the proximity effect with a conventional s-wave superconducting substrate  [6], [7]. Much theoretical and experimental effort is currently going into finding unambiguous signatures of MZMs in various set-ups. Important steps forward in this direction include tunneling spectroscopy experiments  [8], [9], whose findings, in particular, the zero-bias conductance peak observed in one-dimensional semiconductor–superconductor contacts, were consistent with theoretical predictions (see Ref.  [1] for a recent review).

Closely related to the 1DPS model is the quantum Ising chain (QIC), described by the Hamiltonian:

$$H=-J\sum _{n=1}^{N-1}{\sigma }_n^x{\sigma }_{n+1}^x-h\sum _{n=1}^N{\sigma }_n^z\text{.}$$

Here ${\sigma }_n^{\alpha }$ are Pauli matrices, $J>0$ is the exchange interaction and $h$ is a transverse magnetic field which endows the spins with quantum dynamics. The model possesses a ${\mathbb{Z}}_2$-symmetry associated with the global transformation $P_S{\sigma }_n^x{P}_S^{-1}=-{\sigma }_n^x$, where $P_S={\prod }_{j=1}^N{\sigma }_j^z$, $[H,P_S]=0$. This is an exactly solvable quantum 1D model which, by virtue of the transfer matrix formalism, is related to the classical 2D Ising model  [10], [11], [12]. The Jordan–Wigner (JW) transformation maps the many-body problem (2) onto a quadratic model of spinless fermions, the latter actually being a particular realization of the 1DPS (1) with a fine-tuned pairing amplitude $\Delta =\pm t$. Close to criticality, in the field-theoretical limit, the QIC represents a ($1+1$)-dimensional theory of a massive Majorana fermion  [12]. The topological phase of the 1DPS corresponds to the ordered phase of the QIC, which (in the thermodynamic limit) is characterized by spontaneously broken symmetry and a two-fold degeneracy of the ground state (up to an exponentially small splitting). The ordered phase ($J>h$) is separated from the topologically trivial, disordered phase ($J<h$) by a quantum critical point ($J=h$). As follows from the Kramers–Wannier duality  [11], [12], the two massive phases of the QIC (2) have identical bulk spectrum; however, they differ in the boundary conditions at the edges of a finite chain, reflecting their topological distinction.

This difference is clearly seen from the Kitaev–Majorana (KM) representation of the QIC  [5], [13]: the $N$-site spin chain (2) is equivalent to a $2N$-site tight-binding model of real (Majorana) fermions with nearest-neighbor couplings, as will be briefly reviewed below:

$$H=i\sum _{j=1}^N(h{c}_{2j-1}{c}_{2j}+J{c}_{2j}{c}_{2j+1})\text{,}{c}_i^{†}=c_i\text{,}\{c_i,c_j\}=2{\delta }_{ij}\text{.}$$

In the representative limits $h/J\to 0$ and $J/h\to 0$ a greatly simplified qualitative picture emerges. For $h\to 0$ one finds two decoupled boundary MZMs, $c_1$ and $c_{2N}$, in the otherwise dimerized chain, implying a two-fold degeneracy of the ground state, while for $J\to 0$ the KM lattice has a full dimer covering and the ground state is unique. At finite $h/J<1$ the exact degeneracy between the two boundary MZMs in the ordered phase is removed. In spin language, the level splitting is caused by quantum tunneling between the two classical Ising vacua caused by the propagation of a magnetization kink from one end of the chain to the other. For a macroscopically long chain, the tunneling amplitude is exponentially small, $t_{\mathrm{eff}}\sim Jexp(-L/\xi )$ ($L$ and $\xi$ being the length of the chain and the correlation length, respectively). However, within this accuracy the two boundary Majorana modes remain true zero modes, and their existence implies the two-fold degeneracy of the ground state in the ordered phase of the QIC.

It is worth mentioning that the usefulness of the Majorana fermions in the QIC hinges on the exact ${\mathbb{Z}}_2$ symmetry of the spin model (2). Local terms (in the ${\sigma }^x$ or ${\sigma }^y$) which break the Ising symmetry induce non-local couplings among the Majoranas and spoil the degeneracy of the edge modes. In contrast, the topological phase of the 1DPS is robust against any local perturbations in the fermionic basis. Nevertheless, within the symmetry protected sector, the QIC exhibits very similar physical properties as the 1DPS.

In this paper we aim at identifying clear physical differences associated with the presence or absence of Majorana edges modes in the topologically ordered or non-ordered phases, respectively. We focus on the effects of an impurity that interrupts an otherwise homogeneous 1d chain, or terminates it. We study its spectral weight and its response to locally applied external fields. Our main result is the finding summarized in Table 1: the local susceptibility of such an impurity can serve as a probe for Majorana edges modes in the adjacent bulk phase(s). In particular, we find that the non-topological phase is characterized by a discontinuity in the polarization response of the impurity to an external field, and a concomitant divergence of the susceptibility. In contrast, topological order and the associated MZMs quench such a divergence. This effect may serve as a thermodynamic, equilibrium tool in the search for direct traces of Majorana edge modes, which so far have been sought mostly in transport properties at zero bias.

For illustrative purposes and analytical convenience we focus on the QIC model in which the presence of impurities of a certain kind enforces all energy levels of the system to be two-fold degenerate even for a finite system. The impurities we have in mind represent lattice sites where the local transverse magnetic field vanishes. The spins residing at these sites are unable to flip and, therefore, are classical. The ${\mathbb{Z}}_2$ degeneracy of the ground state makes the existence of two decoupled Majorana modes at zero energy an exact property of a finite system at arbitrary values of the bulk parameters $J$ and $h$. The goal of this work is to trace the evolution of the associated MZMs across the transition from the disordered phase to the ordered one and describe the corresponding qualitative changes of measurable quantities—the spectral weight (density of states) of the impurity spin and the local magnetic susceptibility defined as the linear response to a small transverse magnetic field. For a 1D p-wave superconductor the equivalent quantities are the average occupancy of the impurity fermionic level and the local charge susceptibility.

The paper is organized as follows. In Section  2 we briefly overview the QIC in the KM representation, which in Section  3 is used to qualitatively describe the main features of a quantum Ising chain containing classical-spin impurities: the presence of a free local spin with a local Curie susceptibility in the disordered phase, its delocalization at the phase transition and the transformation of the spectral degeneracy from locally differing ground states to globally differing Ising symmetry-broken states.

In Section  4 we explicitly construct a triplet of conserved operators which obey the standard spin 1/2 algebra, and are quasi-local in the disordered phase. We relate their existence to the integrable character of the considered models, and compare with similar conserved operators in many-body localized systems.

In Section  5 we consider a single classical impurity in an Ising chain close to criticality. Taking the scaling limit, we establish the connection with massive versions of previously studied resonant-level models where the impurity couples to two channels of Majorana fermions, and we explain how to compute the observables of interest using the Green’s functions of the auxiliary Majorana fermions. Section  6 contains our central results for a boundary impurity. We evaluate the impurity spectral weight in both phases and calculate the temperature dependence of the transverse susceptibility of the impurity spin finding a rich behavior across the quantum critical window. At criticality, the QIC with a boundary impurity coincides with the Majorana resonant-level model discussed earlier by Emery and Kivelson  [14] in their studies of the two-channel Kondo problem. In this regime the impurity spin has a logarithmically divergent low-temperature susceptibility. This is intermediate between the Curie asymptotics of the disordered phase and the saturating susceptibility in the ordered phase. Section  8 establishes the connection with the one-dimensional p-wave superconductors to which the QIC maps under JW transformation. In particular, we find that the local compressibility of an impurity site (a quantum dot coupled to a superconducting wire) provides a thermodynamic signature of the presence or absence of topological order in the superconductor: The topological phase with its boundary Majorana zero mode forces the charge occupation of the quantum dot to be a smooth function of local potential acting on the dot. This is in contrast to the topologically trivial superconducting phase of the wire, in the presence of which the occupation of the dot generically undergoes discrete jumps as a function of applied gate voltage. Section  7 analyzes an impurity in the bulk and summarizes the salient features of the susceptibility, and how it differs from a boundary impurity. The symmetrically coupled impurity is shown to map to a semi-infinite Peierls chain coupled to a boundary impurity. The concluding Section  9 summarizes the main results, and discusses how generally topological order in 1 dimension may be detected by the absence of discontinuous response to local fields acting on impurities.