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arXiv:2302.09630 · AVS Quantum Science

Investigating global and topological orders of states by local measurement and classical communication: Study on SPT phase diagrams by quantum energy teleportation

Kazuki Ikeda

Published Sep 1, 2023• 18 citations
arXivHTMLPDF

Abstract

Distinguishing non-local orders, including global and topological orders of states through solely local operations and classical communications (LOCC), is a highly non-trivial and challenging task since the topology of states is determined by the global characteristics of the many-body system, such as the system's symmetry and the topological space it is based on. Here, we report that we reproduced the phase diagram of Ising model and symmetry protected topological phases using the quantum energy teleportation protocol, which foresees non-trivial energy transfer between remote observers using the entanglement nature of the ground state and LOCC. The model we use includes the Haldane model, the AKLT model, and the Kitaev model. Therefore, our method paves a new general experimental framework to determine and quantify phase transitions in various condensed matter physics and statistical mechanics.

DOI
10.1116/5.0164999

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Distinguishing non-local orders, including global and topological orders of states through solely local operations and classical communications (LOCC), is a highly non-trivial and challenging task since the topology of states is determined by the global characteristics of the many-body system, such as the system's symmetry and the topological space it is based on. Here, we report that we reproduced the phase diagram of Ising model and symmetry protected topological phases using the quantum energy teleportation protocol, which foresees non-trivial energy transfer between remote observers using the entanglement nature of the ground state and LOCC. The model we use includes the Haldane model, the AKLT model, and the Kitaev model. Therefore, our method paves a new general experimental framework to determine and quantify phase transitions in various condensed matter physics and statistical mechanics.
We consider one-dimensional Hamiltonian systems whose ground states display symmetry-protected topological order. We show that ground states within the topological phase cannot be connected with each other through local operations and classical communication between a bipartition of the system. Our claim is demonstrated by analyzing the entanglement spectrum and R\'enyi entropies of different physical systems that provide examples for symmetry-protected topological phases. Specifically, we consider the spin-1/21/21/2 cluster-Ising model and a class of spin-1 models undergoing quantum phase transitions to the Haldane phase. Our results provide a probe for symmetry-protected topological order. Since the picture holds even at the system's local scale, our analysis can serve as a local experimental test for topological order.
Abstract We study the entanglement structure of symmetry-protected topological (SPT) phases from an operational point of view by considering entanglement distillation in the presence of symmetries. We demonstrate that non-trivial SPT phases in one-dimension necessarily contain some entanglement which is inaccessible if the symmetry is enforced. More precisely, we consider the setting of local operations and classical communication (LOCC) where the local operations commute with a global onsite symmetry group G, which we call G-LOCC, and we define the inaccessible entanglement E inacc as the entanglement that cannot be used for distillation under G-LOCC. We derive a tight bound on E inacc which demonstrates a direct relation between inaccessible entanglement and the SPT phase, namely log ( D ω 2 ) ⩽ E i n a c c ⩽ log ( | G | ) , where D ω is the topologically protected edge mode degeneracy of the SPT phase ω with symmetry G. For particular phases such as the Haldane phase, D ω = | G | so the bound becomes an equality. We numerically investigate the distribution of states throughout the bound, and show that typically the region near the upper bound is highly populated, and also determine the nature of those states lying on the upper and lower bounds. We then discuss the relation of E inacc to string order parameters, and also the extent to which it can be used to distinguish different SPT phases of matter.
Symmetry-protected topological (SPT) phases exhibit nontrivial short-ranged entanglement protected by symmetry and cannot be adiabatically connected to trivial product states while preserving the symmetry. In contrast, intrinsic topological phases do not need ordinary symmetry to stabilize them and their ground states exhibit long-range entanglement. It is known that for a given symmetry group GGG, the 2D SPT phase protected by GGG is dual to the 2D topological phase exemplified by the twisted quantum double model ${D}^{\ensuremath{\omega}}(G)$ via gauging the global symmetry GGG. Recently it was realized that such a general gauging map can be implemented by some local unitaries and local measurements when GGG is a finite, solvable group. Here, we review the general approach to gauging a $G\text{\ensuremath{-}}\mathrm{SPT}$ starting from a fixed-point ground-state wave function and applying a NNN-step gauging procedure. We provide an in-depth analysis of the intermediate states emerging during the NNN-step gauging and provide tools to measure and identify the emerging symmetry-enriched topological order (SET) of these states. We construct the generic lattice parent Hamiltonians for these intermediate states and show that they form an entangled superposition of a twisted quantum double (TQD) with an SPT-ordered state. Notably, we show that they can be connected to the TQD through a finite-depth, local quantum circuit which does not respect the global symmetry of the SET order. We introduce the so-called symmetry branch line operators and show that they can be used to extract the symmetry fractionalization classes (SFC) and symmetry defectification classes (SDC) of the SET phases with the input data GGG and $[\ensuremath{\omega}]\ensuremath{\in}{H}^{3}(G,U(1))$ of the pregauged SPT-ordered state. We illustrate the procedure of preparing and characterizing the emerging SET-ordered states for some Abelian and non-Abelian examples such as dihedral groups Dn{D}_{n}Dn​ and the quaternion group Q8{Q}_{8}Q8​.
Symmetry-protected topological (SPT) phases are short-range entangled phases of matter with a non-local order parameter which are preserved under a local symmetry group. Here, by using unsupervised learning algorithm, namely the diffusion maps, we demonstrate that can differentiate between symmetry broken phases and topologically ordered phases, and between non-trivial topological phases in different classes. In particular, we show that the phase transitions associated with these phases can be detected in different bosonic and fermionic models in one dimension. This includes the interacting SSH model, the AKLT model and its variants, and weakly interacting fermionic models. Our approach serves as an inexpensive computational method for detecting topological phases transitions associated with SPT systems which can be also applied to experimental data obtained from quantum simulators.
Abstract Classifying and understanding phases of matter is an important task in condensed matter physics. The class of ‘conventional’ symmetry-broken phases is well understood in terms of Landau’s theory. A paradigmatic example is the Z2-symmetric Ising model with symmetric (paramagnetic) and symmetry-broken (ferromagnet) phases, which can be distinguished by measuring the magnetization as a local order parameter. Topological phases of matter, however, are less understood and no complete classification is known. These lecture notes discuss schemes that provide an understanding and characterization of certain topological phases of matter in models with local Hamiltonians. The main focus is on symmetry-protected topological (SPT) phases in one-dimensional bosonic systems. On the basis of the entanglement properties of such systems, a motivation is given for the matrix product state (MPS) representation of ground states. Using the MPS framework, it is demonstrated how SPT phases can be classified using projective representations of the symmetries.
Topological properties are associated with nonlocal entanglement and global properties. On the other hand, in some topological systems, it has been shown that quasilocal operators detect topological transitions. We show in this work that in the case of noninteracting long-range symmetry-protected topological models, energy bonds signal topological transitions. Interestingly, we also find that they display some signatures at the Berezinskii-Kosterlitz-Thouless transition that occurs in a spin chain with first and second neighbor interactions, if we consider the first excited state instead of the ground state of the system. Moreover, we show that the ground-state topological transition in the spin Kitaev model in a honeycomb lattice, which displays topological long-range order, is also detected by the energy bond correlator. Despite the model being interacting, in the ground state it reduces to a noninteracting model. Even for the spin-liquid phase of the two-dimensional Heisenberg model with first and second neighbor interactions, where the system has true long-range entanglement and topological order, the local bonds do signal the topological phase transition.