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arXiv:2203.08254 · SciPost Physics Core

The resistance of quantum entanglement to temperature in the Kugel-Khomskii model

V. E. Valiulin, Andrey V. Mikheyenkov, N. M. Chtchelkatchev, Климент Ильич Кугель

Published Apr 4, 2023• 3 citations
arXivHTMLPDF

Abstract

The Kugel–Khomskii model with entangled spin and orbital degrees of freedom is a good testing ground for many important features in quantum information processing, such as robust gaps in the entanglement spectra. Here, we demonstrate that the entanglement can be also robust under effect of temperature within a wide range of parameters. It is shown, in particular, that the temperature dependence of entanglement often exhibits a nonmonotonic behavior. Namely, there turn out to be ranges of the model parameters, where entanglement is absent at zero temperature, but then, with an increase in temperature, it appears, passes through a maximum, and again vanishes.

DOI
10.21468/scipostphyscore.6.2.025

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The Kugel–Khomskii model with entangled spin and orbital degrees of freedom is a good testing ground for many important features in quantum information processing, such as robust gaps in the entanglement spectra. Here, we demonstrate that the entanglement can be also robust under effect of temperature within a wide range of parameters. It is shown, in particular, that the temperature dependence of entanglement often exhibits a nonmonotonic behavior. Namely, there turn out to be ranges of the model parameters, where entanglement is absent at zero temperature, but then, with an increase in temperature, it appears, passes through a maximum, and again vanishes.
We study the quantum entanglement of the spin and orbital degrees of freedom in the one-dimensional Kugel-Khomskii model, which includes both gapless and gapped phases, using analytical techniques and exact diagonalization with up to 16 sites. We compute the entanglement entropy and the entanglement spectra using a variety of partitions or ``cuts'' of the Hilbert space, including two distinct real-space cuts and a momentum-space cut. Our results show that the Kugel-Khomski model possesses a number of new features not previously encountered in studies of the entanglement spectra. Notably, we find robust gaps in the entanglement spectra for both gapped and gapless phases with the orbital partition, and show these are not connected to each other. The counting of the low-lying entanglement eigenvalues shows that the ``virtual edge'' picture, which equates the low-energy Hamiltonian of a virtual edge, here one gapless leg of a two-leg ladder, to the ``low-energy'' entanglement Hamiltonian, breaks down for this model, even though the equivalence has been shown to hold for a similar cut in a large class of closely related models. In addition, we show that a momentum space cut in the gapless phase leads to qualitative differences in the entanglement spectrum when compared with the same cut in the gapless spin-1/2 Heisenberg spin chain. We emphasize the new information content in the entanglement spectra compared to the entanglement entropy, and using quantum entanglement, we present a refined phase diagram of the model. Using analytical arguments, exploiting various symmetries of the model, and applying arguments of adiabatic continuity from two exactly solvable points of the model, we are also able to prove several results regarding the structure of the low-lying entanglement eigenvalues.
Typically two particles (spins) could be maximally entangled at zero temperature, and for a certain temperature the phenomenon of entanglement vanishes at the threshold temperature. For the Heisenberg coupled model or even the Ising model with a transverse magnetic field, one can observe some rise of entanglement even for a disentangled region at zero temperature. So we can understand this emergence of entanglement at finite temperature as being due to the mixing of some maximally entangled states with some other untangled states. Here, we present a simple one-dimensional Ising model with alternating Ising and Heisenberg spins in an arbitrarily oriented magnetic field, which can be mapped onto the classical Ising model with a magnetic field. This model does not show any evidence of entanglement at zero temperature, but surprisingly at finite temperature rise a pairwise thermal entanglement between two untangled spins at zero temperature when an arbitrarily oriented magnetic field is applied. This effect is a purely magnetic field, and the temperature dependence, as soon as the temperature increases, causes a small increase in concurrence, achieving its maximum at around 0.1. Even for long-range entanglement, a weak concurrence still survives. There are also some real materials that could serve as candidates that would exhibit this effect, such as Dy(NO3{}_{3}3​)(DMSO)2{}_{2}2​Cu(opba)(DMSO)2{}_{2}2​ [DMSO = dimethyl sulfoxide; opba = ooo-phenylenebis(oxamoto)] [J. Stre\ifmmode \check{c}\else \v{c}\fi{}ka, M. Hagiwara, Y. Han, T. Kida, Z. Honda, and M. Ikeda, Condens. Matter Phys. 15, 43002 (2012)].
An isolated spin system that is in internal thermodynamic equilibrium and that has an upper limit to its allowed energy states can possess a negative temperature. We calculate the thermodynamic characteristics and the concurrence in this system over the entire range of positive and negative temperatures. Our calculation was performed for different real structures, which can be used in experiments. It is found that the temperature dependence of the concurrence is substantially asymmetrical similarly to other thermodynamic characteristics. At a negative temperature the maximum concurrence and the absolute temperature of the entanglement appearance are significantly larger than those at a positive temperature. The concurrence can be characterized by two dimensionless parameters: the ratio between the Zeeman and dipolar energies and the ratio of the thermal and dipolar energies. It was shown that for all considered structures the dimensionless temperatures of the transition between entanglement and separability of the first and second spins are independent of spin structure and the number of spins.
In this paper, we consider entanglement in a system out of equilibrium, adopting the viewpoint given by the formalism of superstatistics. Such an approach yields a good effective description for a system in a slowly fluctuating environment within a weak interaction between the system and the environment. For this purpose, we introduce an alternative version of the formalism within a quantum mechanical picture and use it to study entanglement in the Heisenberg XY model, subject to temperature fluctuations. We consider both isotropic and anisotropic cases and explore the effect of different temperature fluctuations (χ^{2}, log-normal, and F distributions). Our results suggest that particular fluctuations may enhance entanglement and prevent it from vanishing at higher temperatures than those predicted for the same system at thermal equilibrium.
We investigate the quantum phases of a frustrated antiferromagnetic Heisenberg spin-1/2 model Hamiltonian on a kagom\'e-strip chain (KSC), a one-dimensional analog of the kagom\'e lattice, and construct its phase diagram in an extended exchange-parameter space. The isolated unit cell of this lattice comprises of five spin-1/2 particles, giving rise to several types of magnetic ground states in a unit cell: a spin-3/23/23/2 state as well as spin-1/21/21/2 states with and without additional degeneracies. We explore the ground-state properties of the fully connected thermodynamic system using exact diagonalization and density matrix renormalization group methods, identifying several distinct quantum phases. All but one of the phases exhibit gapless spin excitations. The exception is a dimerized spin-gapped phase that covers a large part of the phase diagram and includes the uniformly exchange coupled system. We argue that this phase can be understood by perturbing around the limit of decoupled unit cells where each unit cell has six degenerate ground states. We use degenerate perturbation theory to obtain an effective Hamiltonian, a Kugel-Khomskii model with an anisotropic spin-one orbital degree of freedom, which helps explain the origin of dimerization.
We investigate the quantum phases of a frustrated antiferromagnetic Heisenberg spin-1/2 model Hamiltonian on a Kagome-strip chain (KSC), a one-dimensional analogue of the Kagome lattice, and construct its phase diagram in an extended exchange parameter space. The isolated unit cell of this lattice comprises of five spin-1/2 particles, giving rise to several types of magnetic ground states in a unit cell: a spin-3/23/23/2 state as well as spin-1/21/21/2 states with and without additional degeneracies. We explore the ground state properties of the fully connected thermodynamic system using exact diagonalization and density matrix renormalization group methods, identifying several distinct quantum phases. All but one of the phases exhibit gapless spin excitations. The exception is a dimerized spin-gapped phase that covers a large part of the phase diagram and includes the uniformly exchange coupled system. We argue that this phase can be understood by perturbing around the limit of decoupled unit cells where each unit cell has six degenerate ground states. We use degenerate perturbation theory to obtain an effective Hamiltonian, a Kugel-Khomskii model with an anisotropic spin-one orbital degree of freedom, which helps explain the origin of dimerization.