Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1604.00037 · Physical review. B./Physical review. B

Detection of gapped phases of a one-dimensional spin chain with on-site and spatial symmetries

Abhishodh Prakash, Colin G. West, Tzu-Chieh Wei

Published Jul 27, 2016• 5 citations
arXivHTMLPDF

Abstract

We investigate the phase diagram of a quantum spin-1 chain whose Hamiltonian is invariant under a global on-site ${A}_{4}$, translation, and lattice inversion symmetries. We detect different gapped phases characterized by a symmetry protected topological (SPT) order and symmetry breaking using matrix product state order parameters. We observe a rich variety of phases of matter characterized by a combination of symmetry breaking and symmetry fractionalization and also the interplay between the on-site and spatial symmetries. Examples of continuous phase transitions directly between topologically nontrivial SPT phases are also observed.

DOI
10.1103/physrevb.94.045136

Related papers

Complete classification of one-dimensional gapped quantum phases in interacting spin systems

Xie Chen, Zheng‐Cheng Gu, Xiao-Gang Wen

Quantum phases with different orders exist with or without breaking the symmetry of the system. Recently, a classification of gapped quantum phases which do not break time reversal, parity, or on-site unitary symmetry has been given for 1D spin systems by X. Chen, Z.-C. Gu, and X.-G. Wen [Phys. Rev. B 83, 035107 (2011)]. It was found that such symmetry-protected topological (SPT) phases are labeled by the projective representations of the symmetry group which can be viewed as a symmetry fractionalization. In this paper, we extend the classification of 1D gapped phases by considering SPT phases with combined time reversal, parity, and/or on-site unitary symmetries and also the possibility of symmetry breaking. We clarify how symmetry fractionalizes with combined symmetries and also how symmetry fractionalization coexists with symmetry breaking. In this way, we obtain a complete classification of gapped quantum phases in 1D spin systems. We find that in general, symmetry fractionalization, symmetry breaking, and long-range entanglement (present in 2 or higher dimensions) represent three main mechanisms to generate a very rich set of gapped quantum phases. As an application of our classification, we study the possible SPT phases in 1D fermionic systems, which can be mapped to spin systems by Jordan-Wigner transformation.

arXiv:1103.3323•Physical Review B•Dec 20, 2011•421 citations

Gapped quantum phases for the S=1 spin chain with D2h symmetry

Zheng-Xin Liu, Min Liu, Xiao-Gang Wen

We study different quantum phases in integer spin systems with on-site ${D}_{2h}={D}_{2}\ensuremath{\bigotimes}{Z}_{2}$ and translation symmetry. We find four distinct nontrivial phases in $S=1$ spin chains despite the fact that they all have the same symmetry. All four phases have gapped bulk excitations, doubly degenerate end states, and the doubly degenerate entanglement spectrum. These nontrivial phases are examples of symmetry-protected topological (SPT) phases introduced by Gu and Wen [Phys. Rev. B 80, 155131 (2009)]. One of the SPT phases corresponds to the Haldane phase and the other three are new. These four SPT phases can be distinguished experimentally by their different responses of the end states to weak external magnetic fields. According to the Chen--Gu--Wen classification, the ${D}_{2h}$ symmetric spin chain can have a total of 64 SPT phases that do not break the symmetry. Here we constructed seven nontrivial phases from the seven classes of nontrivial projective representations of the ${D}_{2h}$ group. Four of these are found in $S=1$ spin chains and studied in this paper.

arXiv:1101.5680•Physical Review B•Aug 10, 2011•22 citations

Detecting and identifying two-dimensional symmetry-protected topological, symmetry-breaking, and intrinsic topological phases with modular matrices via tensor-network methods

Ching-Yu Huang, Tzu-Chieh Wei

Symmetry-protected topological (SPT) phases exhibit nontrivial order if symmetry is respected but are adiabatically connected to the trivial product phase if symmetry is not respected. However, unlike the symmetry-breaking phase, there is no local order parameter for SPT phases. Here we employ a tensor-network method to compute the topological invariants characterized by the simulated modular $S$ and $T$ matrices to study transitions in a few families of two-dimensional (2D) wave functions which are ${\mathbb{Z}}_{N}$ $(N=2\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}3)$ symmetric. We find that in addition to the topologically ordered phases, the modular matrices can be used to identify nontrivial SPT phases and detect transitions between different SPT phases as well as between symmetric and symmetry-breaking phases. Therefore modular matrices can be used to characterize various types of gapped phases in a unifying way.

arXiv:1512.07842•Physical review. B./Physical review. B•Apr 28, 2016•25 citations

Symmetry protected topological phases of spin chains

Kasper Duivenvoorden

Symmetry protected topological (SPT) phases are characterized by robust boundary features, which do not disappear unless passing through a phase transition. These boundary features can be quantified by a topological invariant which, in some cases, is related to a physical quantity, such as the spin conductivity for the quantum spin Hall insulators. In other cases, the boundary features give rise to new physics, such as the Majorana fermion. In all cases the boundary features can be analyzed with the help of an entanglement spectrum and their robustness make them promising candidates for storing quantum information. The topological invariant characterizing SPT phases is strictly only invariant under deformations which respect a certain \nsymmetry. For example, the boundary currents of the quantum spin Hall insulator are only robust against non-magnetic, i.e. time-reversal invariant, impurities. In this thesis we study the SPT phases of spin chains. \nAs a result of our work we find a topological invariant for SPT phases of spin chains which are protected by continuous symmetries. By means of a non-local order parameter we find a way to extract this invariant from the ground state wave function of the system. Using density-matrix-renormalization-group techniques we verify that this invariant is a tool to detect transitions between different topological phases. We find a non-local transformation that maps SPT phases to conventional phases characterized by a local order parameter. This transformation suggests an analogy between topological phases and conventional phases and thus give a deeper understanding of the role of topology in spin systems.

•Oct 31, 2013•0 citations

Towards a complete classification of 1D gapped quantum phases in interacting spin systems

Xie Chen, Zheng‐Cheng Gu, Xiao-Gang Wen

Quantum phases with different orders exist with or without breaking the symmetry of the system. Recently, a classification of gapped quantum phases which do not break time reversal, parity or onsite unitary symmetry has been given for 1D spin systems in [X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 83, 035107 (2011); arXiv:1008.3745]. It was found that, such symmetry protected topological (SPT) phases are labeled by the projective representations of the symmetry group which can be viewed as a symmetry fractionalization. In this paper, we extend the classification of 1D gapped phases by considering SPT phases with combined time reversal, parity, and/or on-site unitary symmetries and also considering the possibility of symmetry breaking. In this way, we obtain a complete classification of gapped quantum phases in 1D spin systems. We find that in general, symmetry fractionalization, symmetry breaking and long range entanglement(present in 2 or higher dimensions) represent three main mechanisms to generate a very rich set of gapped quantum phases. As an application of our classification, we study the possible SPT phases in 1D fermionic systems, which can be mapped to spin systems by Jordan-Wigner transformation.

•Mar 17, 2011•3 citations

Hidden symmetry-breaking picture of symmetry-protected topological order

Dominic V. Else, Stephen D. Bartlett, Andrew C. Doherty

We generalize the hidden symmetry-breaking picture of symmetry-protected topological (SPT) order developed by Kennedy and Tasaki in the context of the Haldane phase. Our generalization applies to a wide class of SPT phases in one-dimensional spin chains, protected by an on-site representation of a finite Abelian group. This generalization takes the form of a nonlocal unitary map that relates local symmetry-respecting Hamiltonians in an SPT phase to local Hamiltonians in a symmetry-broken phase. Using this unitary, we establish a relation between the two-point correlation functions that characterize fully symmetry-broken phases with the string-order correlation functions that characterize the SPT phases, therefore establishing the perspective in these systems that SPT phases are characterized by hidden symmetry breaking. Our generalization is also applied to systems with continuous symmetries, including $\mathrm{SO}(2k+1)$ and $\mathrm{SU}(k)$.

arXiv:1304.0783•Physical Review B•Aug 15, 2013•63 citations
We investigate the phase diagram of a quantum spin-1 chain whose Hamiltonian is invariant under a global on-site A4{A}_{4}A4​, translation, and lattice inversion symmetries. We detect different gapped phases characterized by a symmetry protected topological (SPT) order and symmetry breaking using matrix product state order parameters. We observe a rich variety of phases of matter characterized by a combination of symmetry breaking and symmetry fractionalization and also the interplay between the on-site and spatial symmetries. Examples of continuous phase transitions directly between topologically nontrivial SPT phases are also observed.
Quantum phases with different orders exist with or without breaking the symmetry of the system. Recently, a classification of gapped quantum phases which do not break time reversal, parity, or on-site unitary symmetry has been given for 1D spin systems by X. Chen, Z.-C. Gu, and X.-G. Wen [Phys. Rev. B 83, 035107 (2011)]. It was found that such symmetry-protected topological (SPT) phases are labeled by the projective representations of the symmetry group which can be viewed as a symmetry fractionalization. In this paper, we extend the classification of 1D gapped phases by considering SPT phases with combined time reversal, parity, and/or on-site unitary symmetries and also the possibility of symmetry breaking. We clarify how symmetry fractionalizes with combined symmetries and also how symmetry fractionalization coexists with symmetry breaking. In this way, we obtain a complete classification of gapped quantum phases in 1D spin systems. We find that in general, symmetry fractionalization, symmetry breaking, and long-range entanglement (present in 2 or higher dimensions) represent three main mechanisms to generate a very rich set of gapped quantum phases. As an application of our classification, we study the possible SPT phases in 1D fermionic systems, which can be mapped to spin systems by Jordan-Wigner transformation.
We study different quantum phases in integer spin systems with on-site ${D}_{2h}={D}_{2}\ensuremath{\bigotimes}{Z}_{2}$ and translation symmetry. We find four distinct nontrivial phases in S=1S=1S=1 spin chains despite the fact that they all have the same symmetry. All four phases have gapped bulk excitations, doubly degenerate end states, and the doubly degenerate entanglement spectrum. These nontrivial phases are examples of symmetry-protected topological (SPT) phases introduced by Gu and Wen [Phys. Rev. B 80, 155131 (2009)]. One of the SPT phases corresponds to the Haldane phase and the other three are new. These four SPT phases can be distinguished experimentally by their different responses of the end states to weak external magnetic fields. According to the Chen--Gu--Wen classification, the D2h{D}_{2h}D2h​ symmetric spin chain can have a total of 64 SPT phases that do not break the symmetry. Here we constructed seven nontrivial phases from the seven classes of nontrivial projective representations of the D2h{D}_{2h}D2h​ group. Four of these are found in S=1S=1S=1 spin chains and studied in this paper.
Symmetry-protected topological (SPT) phases exhibit nontrivial order if symmetry is respected but are adiabatically connected to the trivial product phase if symmetry is not respected. However, unlike the symmetry-breaking phase, there is no local order parameter for SPT phases. Here we employ a tensor-network method to compute the topological invariants characterized by the simulated modular SSS and TTT matrices to study transitions in a few families of two-dimensional (2D) wave functions which are ZN{\mathbb{Z}}_{N}ZN​ (N=2and3)(N=2\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}3)(N=2and3) symmetric. We find that in addition to the topologically ordered phases, the modular matrices can be used to identify nontrivial SPT phases and detect transitions between different SPT phases as well as between symmetric and symmetry-breaking phases. Therefore modular matrices can be used to characterize various types of gapped phases in a unifying way.
Symmetry protected topological (SPT) phases are characterized by robust boundary features, which do not disappear unless passing through a phase transition. These boundary features can be quantified by a topological invariant which, in some cases, is related to a physical quantity, such as the spin conductivity for the quantum spin Hall insulators. In other cases, the boundary features give rise to new physics, such as the Majorana fermion. In all cases the boundary features can be analyzed with the help of an entanglement spectrum and their robustness make them promising candidates for storing quantum information. The topological invariant characterizing SPT phases is strictly only invariant under deformations which respect a certain \nsymmetry. For example, the boundary currents of the quantum spin Hall insulator are only robust against non-magnetic, i.e. time-reversal invariant, impurities. In this thesis we study the SPT phases of spin chains. \nAs a result of our work we find a topological invariant for SPT phases of spin chains which are protected by continuous symmetries. By means of a non-local order parameter we find a way to extract this invariant from the ground state wave function of the system. Using density-matrix-renormalization-group techniques we verify that this invariant is a tool to detect transitions between different topological phases. We find a non-local transformation that maps SPT phases to conventional phases characterized by a local order parameter. This transformation suggests an analogy between topological phases and conventional phases and thus give a deeper understanding of the role of topology in spin systems.
Quantum phases with different orders exist with or without breaking the symmetry of the system. Recently, a classification of gapped quantum phases which do not break time reversal, parity or onsite unitary symmetry has been given for 1D spin systems in [X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 83, 035107 (2011); arXiv:1008.3745]. It was found that, such symmetry protected topological (SPT) phases are labeled by the projective representations of the symmetry group which can be viewed as a symmetry fractionalization. In this paper, we extend the classification of 1D gapped phases by considering SPT phases with combined time reversal, parity, and/or on-site unitary symmetries and also considering the possibility of symmetry breaking. In this way, we obtain a complete classification of gapped quantum phases in 1D spin systems. We find that in general, symmetry fractionalization, symmetry breaking and long range entanglement(present in 2 or higher dimensions) represent three main mechanisms to generate a very rich set of gapped quantum phases. As an application of our classification, we study the possible SPT phases in 1D fermionic systems, which can be mapped to spin systems by Jordan-Wigner transformation.
We generalize the hidden symmetry-breaking picture of symmetry-protected topological (SPT) order developed by Kennedy and Tasaki in the context of the Haldane phase. Our generalization applies to a wide class of SPT phases in one-dimensional spin chains, protected by an on-site representation of a finite Abelian group. This generalization takes the form of a nonlocal unitary map that relates local symmetry-respecting Hamiltonians in an SPT phase to local Hamiltonians in a symmetry-broken phase. Using this unitary, we establish a relation between the two-point correlation functions that characterize fully symmetry-broken phases with the string-order correlation functions that characterize the SPT phases, therefore establishing the perspective in these systems that SPT phases are characterized by hidden symmetry breaking. Our generalization is also applied to systems with continuous symmetries, including SO(2k+1)\mathrm{SO}(2k+1)SO(2k+1) and SU(k)\mathrm{SU}(k)SU(k).