Level crossings induced by a longitudinal coupling in the transverse field Ising chain

Abstract

We study the effect of antiferromagnetic longitudinal coupling on the one-dimensional transverse field Ising model with nearest-neighbour couplings. In the topological phase where, in the thermodynamic limit, the ground state is twofold degenerate, we show that, for a finite system of N sites, the longitudinal coupling induces N level crossings between the two lowest lying states as a function of the field. We also provide strong arguments suggesting that these N level crossings all appear simultaneously as soon as the longitudinal coupling is switched on. This conclusion is based on perturbation theory, and a mapping of the problem onto the open Kitaev chain, for which we write down the complete solution in terms of Majorana fermions.The topological properties of matter are currently attracting a considerable attention [1,2]. One of the hallmarks of a topologically non trivial phase is the presence of surface states. In one dimension, the first example was the spin-1 chain that was shown a long time ago to have a gapped phase [3] with two quasi-degenerate low-lying states (a singlet and a triplet) on open chains [4]. These low-lying states are due to the emergent spin-1/2 degrees of freedom at the edges of the chains which combine to make a singlet ground state with an almost degenerate low-lying triplet for an even number of sites, and a triplet ground state with an almost degenerate low-lying singlet when the number of sites is odd. In that system, the emergent degrees of freedom are magnetic since they carry a spin 1/2, and they can be detected by standard probes sensitive to local magnetisation such as NMR [5].In fermionic systems, a topological phase is present if the model includes a pairing term (as in the mean-field treatment of a p-wave superconductor), and the emergent degrees of freedom are two Majorana fermions localised at the opposite edges of the chain [6]. Their detection is much less easy than that of magnetic edge states, and it relies on indirect consequences such as their impact on the local tunneling density of states [7,8], or the presence of two quasi-degenerate low-lying states in open systems. In that respect, it has been suggested to look for situations where the low-lying states cross as a function of an external parameter, for instance the chemical potential, to prove that there are indeed two low-lying states [9].In a recent experiment with chains of Cobalt atoms evaporated onto a Cu 2 N/Cu(100) substrate [10], the presence of level crossings as a function of the external magnetic field has been revealed by scanning tunneling microscopy, which exhibits a specific signature whenever the ground state is degenerate. The relevant effective model for that system is a spin-1/2 XY model in an inplane magnetic field. The exact diagonalisation of finite XY chains has indeed revealed the presence of quasidegeneracy between the two lowest energy states, that are well separated from the rest of the spectrum, and a series of level crossings between them as a function of the magnetic field [11]. ...

Perturbative approach to tunneling and quantum interferences in spin clusters

“…On a 1D chain of spins with open boundary conditions, this model has an interesting limit in which it is exactly solvable. Taking J y = h x = 0, one can map the model with a Jordan-Wigner transformation [26] onto the Kitaev chain model [22], a free fermion model which one can solve exactly [21,27] to obtain field values in which not only the ground state is degenerate, but all eigenstates are:…”

Infinite coherence time of edge spins in finite-length chains

“…Furthermore, we show that the edge spins remain coherent for any initial state for the integrable case of transverse field because all states have level crossings at the same value of the field, while the coherence time is increasingly large for lower temperatures in the case of longitudinal field, which is non-integrable.In recent experiments on chains of cobalt adatoms[1], level crossings of the two lowest energy states have been observed as a function of the external magnetic field Γ. An analysis of the effective spin model of that system, the spin-1/2 XY chain with in-plane magnetic field, has revealed the presence of N level crossings as a function of the magnetic field Γ between the two lowest energy states [2,3]. In Ref.…”

“…In recent experiments on chains of cobalt adatoms[1], level crossings of the two lowest energy states have been observed as a function of the external magnetic field Γ. An analysis of the effective spin model of that system, the spin-1/2 XY chain with in-plane magnetic field, has revealed the presence of N level crossings as a function of the magnetic field Γ between the two lowest energy states [2,3]. In Ref.…”

Exact zero modes in frustrated Haldane chains

“…The presence of two quasidegenerate low-lying states can be most easily detected if these two states cross as a function of an external parameter, such as the chemical potential in fermionic chains 8 or an external magnetic field in spin chains. Such level crossings have been recently detected in chains of Co adatoms 9 , and their interpretation in terms of localized Majorana fermions worked out in details 10,11 . At each level crossing, there is an exact zero mode, i.e.…”

“…More generally, we expect the effect to be present whenever incommensurate correlations develop in a one-dimensional topological phase with edge states. In fact, the level crossings that have been recently discussed in the transverse field Ising model with an additional longitudinal coupling, or equivalently in the Kitaev chain when the pairing amplitude is smaller than the hopping term, have been explained in terms of oscillations of the Majorana edge state wave functions 11 , in a parameter range where incommensurate spin-spin correlations are indeed present 39 . Let us note to be complete that these zero modes do not appear to be strong zero modes in the terminology of Paul Fendley 40 .…”