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:1208.0697 · Physical Review B

Discriminating string order parameter for topological phases of gapped SU(N) spin chains

Kasper Duivenvoorden, Thomas Quella

Published Dec 26, 2012• 30 citations
arXivHTMLPDF

Abstract

One-dimensional gapped spin chains with symmetry $PSU(N)=SU(N)/{\mathbb{Z}}_{N}$ are known to possess $N$ different topological phases. In this paper, we introduce a nonlocal string order parameter which characterizes each of these $N$ phases unambiguously. Numerics confirm that our order parameter allows one to extract a quantized topological invariant from a given nondegenerate gapped ground state wave function. Discontinuous jumps in the discrete topological order that arise when varying physical couplings in the Hamiltonian may be used to detect quantum phase transitions between different topological phases.

DOI
10.1103/physrevb.86.235142

Related papers

Detection of symmetry-protected topological phases in one dimension

Frank Pollmann, Ari M. Turner

A topological phase is a phase of matter which cannot be characterized by a local order parameter. It has been shown that gapped symmetric phases in one-dimensional (1D) systems can be completely characterized using tools related to projective representations of the symmetry groups. We explain two ways to detect these symmetry protected topological phases in 1D. First, we give a numerical approach for directly extracting the projective representations from a matrix-product state representation. Second, we derive nonlocal order parameters for time-reversal and inversion symmetry, and discuss a generalized string order for local symmetries for which the regular string-order parameter cannot be applied. We furthermore point out that the nonlocal order parameter for these topological phases is actually related to topological surfaces.

arXiv:1204.0704•Physical Review B•Sep 24, 2012•336 citations

Quantum phase transitions and topological orders in spin chains and ladders

Toplal Pandey

Dimerized antiferromagnetic spin-1/2 chains and ladders demonstrate quantum critical phase transition, the existence or absence of which is dependent on the dimerization and the dimerization pattern of the chain and the ladder, respectively. The gapped phases can not be distinguished by the conventional Landau long-range order parameters. However, they possess non-local topological string order parameters which can be used to classify different phases. We utilize the self-consistent free fermionic approximation and some standard results for exactly solved models to analytically calculate the string order parameters of dimerized spin chains. As a complement parameter the gapped phases possess the topological number, called the winding number and they are characterized by different integer values of the winding number. In order to calculate the string order parameters and winding numbers in dimerized spin chains and two-leg ladders we use analytical methods such as the Jordan-Wigner transformation, mean-field approximation, duality transformations, and some standard results available for the exactly 1D solve models. It is shown that the winding number provides the complementary framework to the string order parameter to characterize the topological gapped phases.

•Mar 17, 2014•0 citations

Asymptotic Correlations in Gapped and Critical Topological Phases of 1D Quantum Systems

N. G. Jones, Ruben Verresen

Topological phases protected by symmetry can occur in gapped and—surprisingly—in critical systems. We consider non-interacting fermions in one dimension with spinless time-reversal symmetry. It is known that the phases are classified by a topological invariant $$\omega $$ ω and a central charge c. We investigate the correlations of string operators, giving insight into the interplay between topology and criticality. In the gapped phases, these non-local string order parameters allow us to extract $$\omega $$ ω . Remarkably, ratios of correlation lengths are universal. In the critical phases, the scaling dimensions of these operators serve as an order parameter, encoding $$\omega $$ ω and c. We derive exact asymptotics of these correlation functions using Toeplitz determinant theory. We include physical discussion, e.g., relating lattice operators to the conformal field theory. Moreover, we discuss the dual spin chains. Using the aforementioned universality, the topological invariant of the spin chain can be obtained from correlations of local observables.

arXiv:1805.06904•Journal of Statistical Physics•Apr 8, 2019•43 citations

Landau Framework for Topological Order: Quantum Chains and Ladders

Gennady Y. Chitov, Toplal Pandey

We analyzed quantum phase transitions in the antiferromagnetic dimerized spin-1/2 XY chain and two-leg ladders. Their gapped phases possess distinct nonlocal string order parameters (SOPs) and the hidden symmetry breaking in each phase is identified. After mapping of the original spin Hamiltonians onto the tight-binding models with Dirac or Majorana fermions and the mean-field approximation (for ladders), the essential physics of nonlocal orders can be grasped analytically. By utilizing duality transformations the calculation of SOPs is mapped onto the local order problem in some dual representation and done without further approximations. The phases with hidden orders can also be distinguished by topological winding numbers. The latter provide a complimentary description. Our main result is the framework to treat nonlocal orders and hidden symmetries which unifies the key elements of the Landau paradigm with the new concept of topological order. This unified framework can be straightforwardly applied for spin chain and ladders, topological insulators and superconductors. Applications of this theory to other systems are under way.

•Mar 15, 2016•0 citations

Spatial reflection and associated string order in quantum spin chains

Li-Xiang Cen

We investigate spatial reflection and associated nonlocal order in spin-chain quantum systems. The proposed string order parameters, e.g., reflected via operations of the spatial reflection or combinations of it with spin reflection, are able to characterize a variety of physical systems and allow us to gain insights to the statistical mechanism underlying phenomena such as the Haldane gap and quantum phase transitions. Besides revealing further the potential application of the generalized parity symmetry in numerical algorithm, we build an explicit scheme to determine the symmetry and the related string order for matrix-product states so that one can construct ansatz models with presumed properties.

arXiv:0904.2886•Physical Review B•Oct 20, 2009•7 citations

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

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

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.

arXiv:1604.00037•Physical review. B./Physical review. B•Jul 27, 2016•5 citations
One-dimensional gapped spin chains with symmetry PSU(N)=SU(N)/ZNPSU(N)=SU(N)/{\mathbb{Z}}_{N}PSU(N)=SU(N)/ZN​ are known to possess NNN different topological phases. In this paper, we introduce a nonlocal string order parameter which characterizes each of these NNN phases unambiguously. Numerics confirm that our order parameter allows one to extract a quantized topological invariant from a given nondegenerate gapped ground state wave function. Discontinuous jumps in the discrete topological order that arise when varying physical couplings in the Hamiltonian may be used to detect quantum phase transitions between different topological phases.
A topological phase is a phase of matter which cannot be characterized by a local order parameter. It has been shown that gapped symmetric phases in one-dimensional (1D) systems can be completely characterized using tools related to projective representations of the symmetry groups. We explain two ways to detect these symmetry protected topological phases in 1D. First, we give a numerical approach for directly extracting the projective representations from a matrix-product state representation. Second, we derive nonlocal order parameters for time-reversal and inversion symmetry, and discuss a generalized string order for local symmetries for which the regular string-order parameter cannot be applied. We furthermore point out that the nonlocal order parameter for these topological phases is actually related to topological surfaces.
Dimerized antiferromagnetic spin-1/2 chains and ladders demonstrate quantum critical phase transition, the existence or absence of which is dependent on the dimerization and the dimerization pattern of the chain and the ladder, respectively. The gapped phases can not be distinguished by the conventional Landau long-range order parameters. However, they possess non-local topological string order parameters which can be used to classify different phases. We utilize the self-consistent free fermionic approximation and some standard results for exactly solved models to analytically calculate the string order parameters of dimerized spin chains. As a complement parameter the gapped phases possess the topological number, called the winding number and they are characterized by different integer values of the winding number. In order to calculate the string order parameters and winding numbers in dimerized spin chains and two-leg ladders we use analytical methods such as the Jordan-Wigner transformation, mean-field approximation, duality transformations, and some standard results available for the exactly 1D solve models. It is shown that the winding number provides the complementary framework to the string order parameter to characterize the topological gapped phases.
Topological phases protected by symmetry can occur in gapped and—surprisingly—in critical systems. We consider non-interacting fermions in one dimension with spinless time-reversal symmetry. It is known that the phases are classified by a topological invariant ω\omega ω ω and a central charge c. We investigate the correlations of string operators, giving insight into the interplay between topology and criticality. In the gapped phases, these non-local string order parameters allow us to extract ω\omega ω ω . Remarkably, ratios of correlation lengths are universal. In the critical phases, the scaling dimensions of these operators serve as an order parameter, encoding ω\omega ω ω and c. We derive exact asymptotics of these correlation functions using Toeplitz determinant theory. We include physical discussion, e.g., relating lattice operators to the conformal field theory. Moreover, we discuss the dual spin chains. Using the aforementioned universality, the topological invariant of the spin chain can be obtained from correlations of local observables.
We analyzed quantum phase transitions in the antiferromagnetic dimerized spin-1/2 XY chain and two-leg ladders. Their gapped phases possess distinct nonlocal string order parameters (SOPs) and the hidden symmetry breaking in each phase is identified. After mapping of the original spin Hamiltonians onto the tight-binding models with Dirac or Majorana fermions and the mean-field approximation (for ladders), the essential physics of nonlocal orders can be grasped analytically. By utilizing duality transformations the calculation of SOPs is mapped onto the local order problem in some dual representation and done without further approximations. The phases with hidden orders can also be distinguished by topological winding numbers. The latter provide a complimentary description. Our main result is the framework to treat nonlocal orders and hidden symmetries which unifies the key elements of the Landau paradigm with the new concept of topological order. This unified framework can be straightforwardly applied for spin chain and ladders, topological insulators and superconductors. Applications of this theory to other systems are under way.
We investigate spatial reflection and associated nonlocal order in spin-chain quantum systems. The proposed string order parameters, e.g., reflected via operations of the spatial reflection or combinations of it with spin reflection, are able to characterize a variety of physical systems and allow us to gain insights to the statistical mechanism underlying phenomena such as the Haldane gap and quantum phase transitions. Besides revealing further the potential application of the generalized parity symmetry in numerical algorithm, we build an explicit scheme to determine the symmetry and the related string order for matrix-product states so that one can construct ansatz models with presumed properties.
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.