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arXiv:0910.2488 · Physical Review B

Visualizing elusive phase transitions with geometric entanglement

Román Orús, Tzu-Chieh Wei

Published Oct 13, 2010• 70 citations
arXivHTMLPDF

Abstract

We show that by examining the global geometric entanglement it is possible to identify ``elusive'' or hard to detect quantum phase transitions. We analyze several one-dimensional quantum spin chains and demonstrate the existence of nonanalyticities in the geometric entanglement, in particular, across a Kosterlitz-Thouless transition and across a transition for a gapped deformed Affleck-Kennedy-Lieb-Tasaki chain. The observed nonanalyticities can be understood and classified in connection to the nature of the transitions, and are in sharp contrast to the analytic behavior of all the two-body reduced density operators and their derived entanglement measures.

DOI
10.1103/physrevb.82.155120

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We show that by examining the global geometric entanglement it is possible to identify ``elusive'' or hard to detect quantum phase transitions. We analyze several one-dimensional quantum spin chains and demonstrate the existence of nonanalyticities in the geometric entanglement, in particular, across a Kosterlitz-Thouless transition and across a transition for a gapped deformed Affleck-Kennedy-Lieb-Tasaki chain. The observed nonanalyticities can be understood and classified in connection to the nature of the transitions, and are in sharp contrast to the analytic behavior of all the two-body reduced density operators and their derived entanglement measures.
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.
In this paper we provide the analytical derivation of the global geometric entanglement per block for the valence-bond solid ground state of the spin-1 Affleck-Kennedy-Lieb-Tasaki chain. In particular, we show that this quantity saturates exponentially fast to a constant when the sizes of the blocks are sufficiently large. Our result provides an example of an analytical calculation of the geometric entanglement for a gapped quantum many-body system in one dimension and far away from a quantum critical point.
Geometric entanglement (GE), as a measure of multipartite entanglement, has been investigated as a universal tool to detect phase transitions in quantum many-body lattice models. In this paper we outline a systematic method to compute GE for two-dimensional (2D) quantum many-body lattice models based on the translational invariant structure of infinite projected entangled pair state (iPEPS) representations. By employing this method, the qqq-state quantum Potts model on the square lattice with $q\ensuremath{\in}{2,3,4,5}$ is investigated as a prototypical example. Further, we have explored three 2D Heisenberg models: the antiferromagnetic spin-1/2 XXXXXXXXX and anisotropic XYXXYXXYX models in an external magnetic field, and the antiferromagnetic spin-1 XXZXXZXXZ model. We find that continuous GE does not guarantee a continuous phase transition across a phase transition point. We observe and thus classify three different types of continuous GE across a phase transition point: (i) GE is continuous with maximum value at the transition point and the phase transition is continuous, (ii) GE is continuous with maximum value at the transition point but the phase transition is discontinuous, and (iii) GE is continuous with nonmaximum value at the transition point and the phase transition is continuous. For the models under consideration, we find that the second and the third types are related to a point of dual symmetry and a fully polarized phase, respectively.
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.
The entanglement in quantum XY spin chains of arbitrary length is investigated via the geometric measure of entanglement. The emergence of entanglement is explained intuitively from the perspective of perturbations. The model is solved exactly and the energy spectrum is determined and analyzed in particular for the lowest two levels for both finite and infinite systems. The overlaps for these two levels are calculated analytically for arbitrary number of spins. The entanglement is hence obtained by maximizing over a single parameter. The corresponding ground-state entanglement surface is then determined over the entire phase diagram, and its behavior can be used to delineate the boundaries in the phase diagram. For example, the field-derivative of the entanglement becomes singular along the critical line. The form of the divergence is derived analytically and it turns out to be dictated by the universality class controlling the quantum phase transition. The behavior of the entanglement near criticality can be understood via a scaling hypothesis, analogous to that for free energies. The entanglement density vanishes along the so-called disorder line in the phase diagram, the ground space is doubly degenerate and spanned by two product states. The entanglement for the superposition of the lowest two states is also calculated. The exact value of the entanglement depends on the specific form of superposition. However, in the thermodynamic limit the entanglement density turns out to be independent of the superposition. This proves that the entanglement density is insensitive to whether the ground state is chosen to be the spontaneously Z2Z_2Z2​ symmetry broken one or not. The finite-size scaling of entanglement at critical points is also investigated from two different view points. First, the maximum in the field-derivative of the entanglement density is computed and fitted to a logarithmic dependence of the system size, thereby deducing the correlation length exponent for the Ising class using only the behavior of entanglement. Second, the entanglement density itself is shown to possess a correction term inversely proportional to the system size, with the coefficient being universal (but with different values for the ground state and the first excited state, respectively).
We investigate the quantum phase transition of the Su-Schrieffer-Heeger (SSH) model by inspecting the two-site entanglements in the ground state. It is shown that the topological phase transition of the SSH model is signified by a nonanalyticity of local entanglement, which becomes discontinuous for finite even system sizes, and that this nonanalyticity has a topological origin. Such a peculiar singularity has a universal nature in one-dimensional topological phase transitions of noninteracting fermions. We make this clearer by pointing out that an analogous quantity in the Kitaev chain exhibiting the identical nonanalyticity is the local electron density. As a byproduct, we show that there exists a different type of phase transition, whereby the pattern of the two-site entanglements undergoes a sudden change. This transition is characterised solely by quantum information theory and does not accompany the closure of the spectral gap. We analyse the scaling behaviours of the entanglement in the vicinities of the transition points.