von Neumann entropy spectra and entangled excitations in spin-orbital models

Abstract

We consider the low-energy excitations of one-dimensional spin-orbital models which consist of spin waves, orbital waves, and joint spin-orbital excitations. Among the latter we identify strongly entangled spin-orbital bound states which appear as peaks in the von Neumann entropy (vNE) spectral function introduced in this work. The strong entanglement of bound states is manifested by a universal logarithmic scaling of the vNE with system size, while the vNE of other spin-orbital excitations saturates. We suggest that spin-orbital entanglement can be experimentally explored by the measurement of the dynamical spin-orbital correlations using resonant inelastic x-ray scattering, where strong spin-orbit coupling associated with the core hole plays a role. Introduction.-The spin-orbital interplay is one of the important topics in the theory of strongly correlated electrons [1]. In many cases, the intertwined spin-orbital interaction is decoupled by mean-field approximation, and the spin and orbital dynamics are independent from each other. Thus a spinonly Heisenberg model can be derived by averaging over the orbital state, which successfully explains magnetism and optical excitations in some materials, for instance in LaMnO 3 [2]. But in others, especially in t 2g systems [3], the orbital degeneracy plays an indispensable role in understanding the lowenergy properties in the Mott insulators of transition metal oxides (TMOs), such as LaTiO 3 [4], LaVO 3 and YVO 3 [5], and also in recently discussed RbO 2 [6]. The well known cases are also strong spin-orbit coupling which leads to locally entangled states [7], and entanglement on the superexchange bonds in K 3 Cu 2 F 7 [8]. For such models, the mean-field-type approximation and the decoupling of composite spin-orbital correlations fail and generate uncontrolled errors, even when the orbitals are polarized [9]. The strong spin-orbital fluctuations on the exchange bonds will induce the violation of the Goodenough-Kanamori rules [10]. Furthermore, the flavors may form exotic composite spin-orbital excitations.Model and system.-A paradigmatic model derived for a TMO in Mott-insulating limit is the one-dimensional (1D) spin-orbital Hamiltonian, which reads

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

“…(2.1) has the following parameters: (i) x and y which determine the amplitudes of orbital and spin ferroexchange interactions, −Jx and −Jy, respectively, and (ii) ∆ which interpolates between the Heisenberg (∆ = 1) and Ising (∆ = 0) limit for orbital interactions. When ∆ = 1, the spin and orbital interactions are on equal footing and the symmetry of the Hamiltonian (2.1) is enhanced to SU(2)⊗SU(2) -this model describes a generic competition between FM and AF spin, and between FO and AO bond correlations [43].…”

“…To investigate SOE we use here as two subsystems A and B the spin and orbital degrees of freedom in the entire chain. Standard spin-orbital phases may have entanglement in only one sector and here we concentrate on joint SOE [43]. This choice is distinct from the one conventionally made when the system is separated into two spatially complementary parts [44], for instance in frustrated spin chains [45] or in the periodic 1D Anderson model [46].…”

“…Here we report the complete phase diagram of the anisotropic SU(2)⊗XXZ spin-orbital model, with two phases of similar nature which gain energy from singlet correlations leading to dimerization, either in spin or in orbital sector. These phases were overlooked before in the fully symmetric case, i.e., in the phase diagram of the isotropic SU(2)⊗SU(2) model [43].…”

“…For a spin-orbital coupled system, the division of spin and orbital operators retains the real-space symmetries, which is beneficial to the calculation of mutual entanglement. In two-particle states, the SOE is determined by the inter-component coherence length [43], as though the state has sufficient decay of correlations [45].…”

“…Though much attention was devoted to the ground state in the past [47], it has been noticed only recently that the entanglement entropy of low-energy excitations may provide even more valuable insights [43,48,49] which are of crucial importance to understand the origin of quantum phase transitions in spin-orbital systems [50]. The well known area law of the bipartite entanglement entropy restricts the Hilbert space accessible to a ground state of gapped systems [51,52], while the area law is violated by a leading logarithmic correction in critical systems, whose prefactor is determined by the number of chiral modes and precisely given by Widom conjecture [53].…”