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arXiv:0812.3445 · Physical Review B

Edge states of a spin- 12 two-leg ladder with four-spin ring exchange

Mitsuhiro Arikawa, Shou Tanaya, Isao Maruyama, Yasuhiro Hatsugai

Published May 11, 2009• 21 citations
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Abstract

A topological insulator and its spin analog as a gapped spin liquid have characteristic low-energy excitations (edge states) within the gap when the systems have boundaries. This is the bulk-edge correspondence, which implies that the edge states themselves characterize the gapped bulk spin liquid. Based on the general principle, we analyzed the vector chirality and rung-singlet phases of the spin-$\frac{1}{2}$ ladder with ring exchange by using the edge states and the entanglement entropy.

DOI
10.1103/physrevb.79.205107

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A topological insulator and its spin analog as a gapped spin liquid have characteristic low-energy excitations (edge states) within the gap when the systems have boundaries. This is the bulk-edge correspondence, which implies that the edge states themselves characterize the gapped bulk spin liquid. Based on the general principle, we analyzed the vector chirality and rung-singlet phases of the spin-12\frac{1}{2}21​ ladder with ring exchange by using the edge states and the entanglement entropy.
A spin-12\frac{1}{2}21​ two-leg ladder with four-spin ring exchange is studied by quantized Berry phases, used as local-order parameters. Reflecting local objects, nontrivial $(\ensuremath{\pi})$ Berry phase is founded on a rung for the rung-singlet phase and on a plaquette for the vector-chiral phase. Since the quantized Berry phase is topologically invariant for gapped systems with the time-reversal symmetry, topologically identical models can be obtained by the adiabatic modification. The rung-singlet phase is adiabatically connected to a decoupled rung-singlet model and the vector-chiral phase is connected to a decoupled vector-chiral model. Decoupled models reveal that the local objects are a local singlet and a plaquette singlet, respectively.
We suggest a class of two-dimensional lattice spin Hamiltonians describing non-Abelian SU(2) chiral spin liquids---spin analogs of fractional non-Abelian quantum Hall states---with gapped bulk and gapless chiral edge excitations described by the SU(2)n{}_{n}n​ Wess-Zumino-Novikov-Witten conformal field theory. The models are constructed from an array of generalized spin-n/2n/2n/2 ladders with multi-spin-exchange interactions which are coupled by isolated spins. Such models allow a controllable analytic treatment starting from the one-dimensional limit and are characterized by a bulk gap and non-Abelian SU(2)n{}_{n}n​ gapless edge excitations.
We investigate the properties of antiferromagnetic spin-SSS ladders with the help of local Berry phases defined by imposing a twist on one or a few local bonds. In gapped systems with time-reversal symmetry, these Berry phases are quantized, hence able in principle to characterize different phases. In the case of a fully frustrated ladder where the total spin on a rung is a conserved quantity that changes abruptly upon increasing the rung coupling, we show that two Berry phases are relevant to detect such phase transitions: the rung Berry phase defined by imposing a twist on one rung coupling, and the twist Berry phase defined by twisting the boundary conditions along the legs. In the case of nonfrustrated ladders, we have followed the fate of both Berry phases when interpolating between standard ladders and dimerized spin chains, with the surprising conclusion that, at least far enough from dimerized chains, they define different domains in parameter space. A careful investigation of the spin gap and of edge states shows that a change of twist Berry phase is associated with a quantum phase transition at which the bulk gap closes, and at which, with appropriate boundary conditions, edge states appear or disappear, while a change of rung Berry phase is not necessarily associated with a quantum phase transition. The difference is particularly acute for regular ladders, in which the twist Berry phase does not change at all upon increasing the rung coupling from zero to infinity while the rung Berry phase changes 2S2S2S times. By analogy with the fully frustrated ladder, these changes are interpreted as crossovers between domains in which the rungs are in different states of total spin from 0 in the strong rung limit to 2S2S2S in the weak rung limit. This interpretation is further supported by the observation that these crossovers turn into real phase transitions as a function of rung coupling if one rung is strongly ferromagnetic, or equivalently if one rung is replaced by a spin 2S2S2S impurity.
We show that spin-S ladders undergo, for antiferromagnetic rung coupling, a series of 4S topological phase transitions when the ratio of the leg to rung coupling changes from strongly antiferromagnetic to strongly ferromagnetic. These phase transitions are characterized by a change of Berry phase on the rungs, and, with appropriate boundary conditions, by the appearance and disappearance of edge states without the bulk gap ever closing. The various phases can be distinguished by the number of valence bond singlets on the rungs, which decreases from 2S for strong rung coupling to 0 for dominant leg coupling. In particular, the isotropic spin-3/2 ladder with equal leg and rung coupling is explicitly shown to be in the valence bond solid phase with one singlet per bond.
We construct a two-dimensional (2D) lattice model that is argued to realize a gapped chiral spin liquid with (Ising) non-Abelian topological order. The building blocks are spin-12\frac{1}{2}21​ two-leg ladders with SU(2)\text{SU}(2)SU(2)-symmetric spin-spin interactions. The two-leg ladders are then arranged on rows and coupled through SU(2)\text{SU}(2)SU(2)-symmetric interactions between consecutive ladders. The intraladder interactions are tuned so as to realize c=1/2c=1/2c=1/2 Ising conformal field theory, a fact that we establish numerically via density matrix renormalization group studies. Time-reversal breaking interladder interactions are tuned so as to open a bulk gap in the 2D lattice system. This 2D system supports gapless chiral edge modes with Ising non-Abelian excitations but no charge excitations, in contrast to the Pfaffian non-Abelian fractional quantum Hall state.
A spin-1/2 frustrated two-leg ladder with four-spin exchange interaction is studied by quantized Berry phases. We found that the Berry phase successfully characterizes the Haldane phase in addition to the rung-singlet phase, and the dominant vector-chirality phase. The Hamiltonian of the Haldane phase is topologically identical to the S=1 antiferromagnetic Heisenberg chain. Decoupled models connected to the dominant vector-chirality phase revealed that the local object identified by the non-trivial (πππ) Berry phase is the direct product of two diagonal singlets.