Topological Order in an Entangled SU(2)⊗XYSpin-Orbital Ring

Abstract

We present rigorous topological order which emerges in a one-dimensional spin-orbital model due to the ring topology. Although an exact solution of a spin-orbital ring with SU(2) spin and XY orbital interactions separates spins from orbitals by means of a unitary transformation, the spins are not independent when the ring is closed, but form two half-rings carrying opposite pseudomomenta. We show that an inverse transformation back to the physical degrees of freedom entangles the spin half-rings with the orbitals once again. This surprising correlation arises on changing the topology from an open to a closed chain, which reduces the degeneracy of the ground-state manifold, leaving in it only the states in which pseudomomenta compensate each other. Spin-orbital physics is one of the foundation stones in the theory of frustrated magnetism [1][2][3][4][5]. When degenerate 3d orbitals in a transition-metal oxide are partly filled, electrons localize due to large on-site Coulomb interaction and superexchange between magnetic ions includes both spin and orbital degrees of freedom that are strongly interrelated [6]. The orbital degeneracy leads in many cases to a dramatic increase of quantum fluctuation [7], which may trigger exotic order [8] or may stabilize a spin-liquid [9, 10] when different states compete near a quantum critical point. While spin-orbital separation is possible in one-dimensional (1D) systems [11], as observed recently in Sr 2 CuO 3 [12], spins and orbitals are usually entangled strongly, as in the archetypal KugelKhomskii model [13]. In the S = 1/2 SU(2)⊗SU(2) chain [14,15], both ground state [16] and excited states [17] are entangled, similar to the S = 1 SU(2)⊗SU(2) chain which plays a prominent role in perovskite vanadates [3][4][5]18]. Only in exceptional cases can such 1D models be solved exactly, for example at the SU(4) point [19] or for a valence-bond state [20] of alternating spin and orbital singlets [21], but even in these situations the spins and orbitals cannot be separated from each other.In real materials the symmetry between spin and orbital interactions is absent. Orbital interactions generically have lower symmetry than spin ones [22], being usually Ising-or XY-like [23]. The XY case is quantum and in general the orbitals cannot be separated from the spins [24]. In this context the 1D spin-orbital SU(2)⊗XY model introduced by Kumar [25] is surprising -by a change of basis, the S = 1/2 spins decouple from the orbitals in an open chain. The orbital interactions remain formally unchanged but the spin ones are gauged away. The spins then appear free and the ground state has large degeneracy (2 L for chain length L) [25].Frustrated spin systems are at the forefront of modern condensed matter theory and experiment [26][27][28], in large part for the investigation of topology in manybody physics. A particular manifestation is the topological spin liquid (TSL) [29], a category including resonating valence-bond (RVB) states [30] and states hosting excitations with non-Abelian fractional ...

Entanglement driven phase transitions in spin–orbital models

“…It includes two phases with dimer correlations [45] which arise near the SU(4) point. Some of its ground states could be even determined exactly at selected x y ( , , ) Δ points [46][47][48][49][50]. Here we are interested in the complementary and less explored model with negative (ferromagnetic) coupling (J = 1), possibly realized in multi-well optical lattices [51], which has been studied so far only for SU(2) orbital interaction ( 1 Δ = ) [30].…”

Quantum entanglement in the one-dimensional spin-orbitalSU(2)⊗XXZmodel

“…The latter two show a tendency towards dimerised spin or orbital correlations which occur here in the proximity of the SU(4) point. For some special choice of parameters the model can be solved exactly: (i) when ∆ = 1 and x = y = 3/4, the exact ground state is doubly degenerate with the spins and the orbitals forming singlets on alternate bonds, while (ii) when ∆ = 0, x = 3/4 and y = 1/2, the non-Haldane spin-liquid ground state can be analytically obtained [64,65], and (iii) several integrable cases were presented for interactions with special symmetries [66,67], or (iv) with XY orbital interactions (∆ = ∞) [20].…”

“…In the noninteracting case, we have the imaginary and real parts of Eq. (5.6), 20) where θ(x) is Heaviside step function whose value is zero for negative argument and 1 for nonnegative argument. The frequency dependence of the imaginary part exhibits square-root singularities in Eq.…”

“…The strong Coulomb interactions and the relativistic spinorbit interaction entangle locally the spin and orbital degrees of freedom [13] which display an amazing variety of fundamentally new and fascinating phenomena, ranging from topologically nontrivial states [14], relativistic Mott-insulating behavior in 5d [15,16] and 4d [17,18] transition-metal oxides and entanglement on superexchange bonds in spin-orbital models [6,19]. Other more recent developments include entangled spin-orbital excitations [20,21], doped spin-orbital systems [22], skyrmion lattices in the chiral metal MnSi [23], multiferroics, spinHall effects [24], Majorana and Weyl fermions [25], topological surface states [26], Kondo systems [27], exotic spin textures in disordered systems, to name just a few.…”

Exotic Spin Order due to Orbital Fluctuations

“…Studies of such models require more sophisticated approaches than the singlesite mean field (MF) approximation or linear spin-wave expansion. Exact solutions are possible only for some one-dimensional spin-orbital models [35][36][37][38] -they also highlight the importance of quantum effects beyond simple classical approaches.…”