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arXiv:2301.09168 · Physical Review A

Conditional global entanglement in a Kosterlitz-Thouless quantum phase transition

Elahe Samimi, Mohammad Hossein Zarei, Afshin Montakhab

Published May 17, 2023• 3 citations
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Abstract

Entanglement is known as an important indicator for characterizing different types of quantum phase transitions (QPTs); however, it faces some challenges in the Kosterlitz-Thouless (KT) phase transitions due to an essential singularity which cannot be identified in finite derivatives of the ground-state energy. In this paper, we consider global entanglement (GE) in a KT phase transition and show that while it does not indicate any clear signature of the phase transition, the conditional version of GE is a good indicator with strong signatures of the KT transition. In particular, we study a deformed version of the ${Z}_{d}$ Kitaev model which has an intermediate KT phase which separates a ${Z}_{d}$ topological phase from a magnetized phase at two different KT transition points. Using a mapping to the classical $d$-state clock model, we consider GE and the generalized GE and show that they do not provide a reliable indicator of transition points. However, their difference, called conditional global entanglement ($Q$), shows a peak at the first KT transition point. Additionally, we show that it can characterize various phases of the model as it behaves substantially differently in each phase. We therefore conclude that $Q$ is a useful measure that can characterize various phases of KT QPTs as well as their related critical points.

DOI
10.1103/physreva.107.052412

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Entanglement is known as an important indicator for characterizing different types of quantum phase transitions (QPTs); however, it faces some challenges in the Kosterlitz-Thouless (KT) phase transitions due to an essential singularity which cannot be identified in finite derivatives of the ground-state energy. In this paper, we consider global entanglement (GE) in a KT phase transition and show that while it does not indicate any clear signature of the phase transition, the conditional version of GE is a good indicator with strong signatures of the KT transition. In particular, we study a deformed version of the Zd{Z}_{d}Zd​ Kitaev model which has an intermediate KT phase which separates a Zd{Z}_{d}Zd​ topological phase from a magnetized phase at two different KT transition points. Using a mapping to the classical ddd-state clock model, we consider GE and the generalized GE and show that they do not provide a reliable indicator of transition points. However, their difference, called conditional global entanglement (QQQ), shows a peak at the first KT transition point. Additionally, we show that it can characterize various phases of the model as it behaves substantially differently in each phase. We therefore conclude that QQQ is a useful measure that can characterize various phases of KT QPTs as well as their related critical points.
A useful approach to characterize and identify quantum phase transitions lies in the concept of multipartite entanglement. In this paper, we consider well-known measures of multipartite (global) entanglement, i.e., average linear entropy of one-qubit and two-qubit reduced density matrices, in order to study topological quantum phase transition (TQPT) in the Kitaev Toric code Hamiltonian with a nonlinear perturbation. We provide an exact mapping from aforementioned measures in the above model to internal energy and energy-energy correlations in the classical Ising model. Accordingly, we find that the global entanglement shows a continuous and sharp transition from a maximum value in the topological phase to zero in the magnetized phase in a sense that its first-order derivative diverges at the transition point. In this regard, we conclude that not only can the global entanglement serve as a reasonable tool to probe quantum criticality at TQPTs, but it also can reveal the highly entangled nature of topological phases. Furthermore, we also introduce a conditional version of global entanglement which becomes maximum at the critical point. Therefore, regarding a general expectation that multipartite entanglement reaches maximum value at the critical point of quantum many-body systems, our result proposes that the conditional global entanglement can be a good measure of multipartite entanglement in TQPTs.
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We consider a spin ladder model which is known to have matrix product states as exact ground states with spin liquid characteristics. The model has two critical-point transitions at the parameter values u=0u=0u=0 and $\ensuremath{\infty}$. We study the variation of entanglement and fidelity measures in the ground states as a function of uuu and specially look for signatures of quantum phase transitions at u=0u=0u=0 and $\ensuremath{\infty}$. The two different entanglement measures used are S(i)S(i)S(i) (the single-site von Neumann entropy) and S(i,j)S(i,j)S(i,j) (the two-body entanglement). At the quantum critical point (QCP) $u=\ensuremath{\infty}$, the entanglement measure EEE [=S(i),S(i,j)][=S(i),S(i,j)][=S(i),S(i,j)] vanishes but remains nonzero at the other QCP u=0u=0u=0. The first and second derivatives of EEE with respect to the parameter uuu and the entanglement length associated with S(i,j)S(i,j)S(i,j) are further calculated to identify special features, if any, near the QCPs. We further determine the GS fidelity FFF and a quantity ln∣D∣\text{ln}|D|ln∣D∣ related to the second derivative of FFF and show that these quantities calculated for finite-sized systems are good indicators of QPTs occurring in the infinite system.
In this work, we investigate quantum phase transition (QPT) in a generic family of spin chains using the ground-state energy, the energy gap and the geometric measure of entanglement (GE). In many of prior works, GE per site was used. Here, we also consider GE per block with each block size being two. This can be regarded as a coarse grain of GE per site. We introduce a useful parameterization for the family of spin chains that includes the XY models with n-site interaction, the GHZ-cluster model and a cluster antiferromagnetic model, the last of which exhibits QPT between a symmetry-protected topological (SPT) phase and a symmetry-breaking antiferromagnetic phase. As the models are exactly solvable, their ground-state wavefunctions can be obtained, and thus, their GE can be studied. It turns out that the overlap of the ground states with translationally invariant product states can be exactly calculated, and hence, the GE can be obtained via further parameter optimization. The QPTs exhibited in these models are detected by the energy gap and singular behavior of geometric entanglement. In particular, the XzY model exhibits transitions from the nontrivial SPT phase to a trivial paramagnetic phase. Moreover, the halfway XY model exhibits a first-order transition across the Barouch–McCoy circle, on which it was only a crossover in the standard XY model.
We derive a general relation between the nonanalyticities of the ground state energy and those of a subclass of the multipartite generalized global entanglement (GGE) measure defined by de Oliveira et al. [Phys. Rev. A 73, 010305(R) (2006)] for many-particle systems. We show that GGE signals both a critical point location and the order of a quantum phase transition (QPT). We also show that GGE allows us to study the relation between multipartite entanglement and QPTs, suggesting that multipartite but not bipartite entanglement is favored at the critical point. Finally, using GGE we were able, at a second-order QPT, to define a diverging entanglement length (EL) in terms of the usual correlation length. We exemplify this with the XY spin-1/2 chain and show that the EL is half the correlation length.
From the perspective of entanglement in low-lying excited states, a profound analysis was carried out regarding the quantum phase transitions within three models that fall outside the Landau-Ginzburg-Wilson (LGW) paradigm. In the context of the deconfined quantum critical point (DQCP) in a one-dimensional quantum spin chain, our findings demonstrate a tight correlation between the reconstruction of low-lying excitation spectra and the DQCP. The precise location of the critical point and its continuous nature can be signaled by the singular behaviors of the entanglement of the first-excited state. Moreover, in comparison with two types of Berezinskii-Kosterlitz-Thouless (BKT) phase transitions, the entanglement presents three different singularity characteristics. These characteristics not only unveil the essence of diverse symmetry types on either side of the DQCP but also expose the disparate causes underlying the formation of the two BKT type phase transitions.