Classification of gapped symmetric phases in one-dimensional spin systems
TL;DR: In this paper, the authors classify possible quantum phases for one-dimensional matrix product states, which represent well the class of 1D gapped ground states, and find that in the absence of any symmetry all states are equivalent to trivial product states.
Abstract: Quantum many-body systems divide into a variety of phases with very different physical properties. The questions of what kinds of phases exist and how to identify them seem hard, especially for strongly interacting systems. Here we make an attempt to answer these questions for gapped interacting quantum spin systems whose ground states are short-range correlated. Based on the local unitary equivalence relation between short-range-correlated states in the same phase, we classify possible quantum phases for one-dimensional (1D) matrix product states, which represent well the class of 1D gapped ground states. We find that in the absence of any symmetry all states are equivalent to trivial product states, which means that there is no topological order in 1D. However, if a certain symmetry is required, many phases exist with different symmetry-protected topological orders. The symmetric local unitary equivalence relation also allows us to obtain some simple results for quantum phases in higher dimensions when some symmetries are present.
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TL;DR: A review of quantum spin liquids can be found in this paper, where the authors discuss the nature of such phases and their properties based on paradigmatic models and general arguments, and introduce theoretical technology such as gauge theory and partons that are conveniently used in the study of spin liquids.
Abstract: Quantum spin liquids may be considered "quantum disordered" ground states of spin systems, in which zero point fluctuations are so strong that they prevent conventional magnetic long range order. More interestingly, quantum spin liquids are prototypical examples of ground states with massive many-body entanglement, of a degree sufficient to render these states distinct phases of matter. Their highly entangled nature imbues quantum spin liquids with unique physical aspects, such as non-local excitations, topological properties, and more. In this review, we discuss the nature of such phases and their properties based on paradigmatic models and general arguments, and introduce theoretical technology such as gauge theory and partons that are conveniently used in the study of quantum spin liquids. An overview is given of the different types of quantum spin liquids and the models and theories used to describe them. We also provide a guide to the current status of experiments to study quantum spin liquids, and to the diverse probes used therein.
1,339 citations
TL;DR: This review discusses the nature of such phases and their properties based on paradigmatic models and general arguments, and introduces theoretical technology such as gauge theory and partons, which are conveniently used in the study of quantum spin liquids.
Abstract: Quantum spin liquids may be considered 'quantum disordered' ground states of spin systems, in which zero-point fluctuations are so strong that they prevent conventional magnetic long-range order. More interestingly, quantum spin liquids are prototypical examples of ground states with massive many-body entanglement, which is of a degree sufficient to render these states distinct phases of matter. Their highly entangled nature imbues quantum spin liquids with unique physical aspects, such as non-local excitations, topological properties, and more. In this review, we discuss the nature of such phases and their properties based on paradigmatic models and general arguments, and introduce theoretical technology such as gauge theory and partons, which are conveniently used in the study of quantum spin liquids. An overview is given of the different types of quantum spin liquids and the models and theories used to describe them. We also provide a guide to the current status of experiments in relation to study quantum spin liquids, and to the diverse probes used therein.
1,288 citations
TL;DR: In this paper, a review of the physics of spin liquid states is presented, including spin-singlet states, which may be viewed as an extension of Fermi liquid states to Mott insulators, and they are usually classified in the category of SU(2), U(1), or Z2.
Abstract: This is an introductory review of the physics of quantum spin liquid states. Quantum magnetism is a rapidly evolving field, and recent developments reveal that the ground states and low-energy physics of frustrated spin systems may develop many exotic behaviors once we leave the regime of semiclassical approaches. The purpose of this article is to introduce these developments. The article begins by explaining how semiclassical approaches fail once quantum mechanics become important and then describe the alternative approaches for addressing the problem. Mainly spin-1/2 systems are discussed, and most of the time is spent in this article on one particular set of plausible spin liquid states in which spins are represented by fermions. These states are spin-singlet states and may be viewed as an extension of Fermi liquid states to Mott insulators, and they are usually classified in the category of so-called SU(2), U(1), or Z2 spin liquid states. A review is given of the basic theory regarding these states and the extensions of these states to include the effect of spin-orbit coupling and to higher spin (S>1/2) systems. Two other important approaches with strong influences on the understanding of spin liquid states are also introduced: (i) matrix product states and projected entangled pair states and (ii) the Kitaev honeycomb model. Experimental progress concerning spin liquid states in realistic materials, including anisotropic triangular-lattice systems [κ-(ET)2Cu2(CN)3 and EtMe3Sb[Pd(dmit)2]2], kagome-lattice system [ZnCu3(OH)6Cl2], and hyperkagome lattice system (Na4Ir3O8), is reviewed and compared against the corresponding theories.
1,108 citations
TL;DR: In this paper, it was shown that the boundary excitations of SPT phases can be described by a nonlocal Lagrangian term that generalizes the Wess-Zumino-Witten term for continuous nonlinear σ models.
Abstract: Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G. They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phases which is protected by SO(3) spin rotation symmetry. The topological insulator is another example of SPT phases which are protected by U(1) and time-reversal symmetries. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: Distinct d-dimensional bosonic SPT phases with on-site symmetry G (which may contain antiunitary time-reversal symmetry) can be labeled by the elements in H^(1+d)[G,UT(1)], the Borel (1+d)-group-cohomology classes of G over the G module UT(1). Our theory, which leads to explicit ground-state wave functions and commuting projector Hamiltonians, is based on a new type of topological term that generalizes the topological θ term in continuous nonlinear σ models to lattice nonlinear σ models. The boundary excitations of the nontrivial SPT phases are described by lattice nonlinear σ models with a nonlocal Lagrangian term that generalizes the Wess-Zumino-Witten term for continuous nonlinear σ models. As a result, the symmetry G must be realized as a non-on-site symmetry for the low-energy boundary excitations, and those boundary states must be gapless or degenerate. As an application of our result, we can use H^(1+d)[U(1)⋊ Z^(T)_(2),U_T(1)] to obtain interacting bosonic topological insulators (protected by time reversal Z2T and boson number conservation), which contain one nontrivial phase in one-dimensional (1D) or 2D and three in 3D. We also obtain interacting bosonic topological superconductors (protected by time-reversal symmetry only), in term of H^(1+d)[Z^(T)_(2),U_T(1)], which contain one nontrivial phase in odd spatial dimensions and none for even dimensions. Our result is much more general than the above two examples, since it is for any symmetry group. For example, we can use H1+d[U(1)×Z2T,UT(1)] to construct the SPT phases of integer spin systems with time-reversal and U(1) spin rotation symmetry, which contain three nontrivial SPT phases in 1D, none in 2D, and seven in 3D. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: (G_H,G_Ψ,H^(1+d)[G_Ψ,U_T(1)]), where G_H is the symmetry group of the Hamiltonian and G_Ψ the symmetry group of the ground states.
1,001 citations
TL;DR: Just as group theory allows us to construct 230 crystal structures in three-dimensional space, group cohomology theory is used to systematically construct different interacting bosonic SPT phases in any dimension and with any symmetry, leading to the discovery of bosonic topological insulators and superconductors.
Abstract: Symmetry-protected topological (SPT) phases are bulk-gapped quantum phases with symmetries, which have gapless or degenerate boundary states as long as the symmetries are not broken. The SPT phases in free fermion systems, such as topological insulators, can be classified; however, it is not known what SPT phases exist in general interacting systems. We present a systematic way to construct SPT phases in interacting bosonic systems. Just as group theory allows us to construct 230 crystal structures in three-dimensional space, we use group cohomology theory to systematically construct different interacting bosonic SPT phases in any dimension and with any symmetry, leading to the discovery of bosonic topological insulators and superconductors.
479 citations
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TL;DR: This special issue of Mathematical Structures in Computer Science contains several contributions related to the modern field of Quantum Information and Quantum Computing, with a focus on entanglement.
Abstract: This special issue of Mathematical Structures in Computer Science contains several contributions related to the modern field of Quantum Information and Quantum Computing.
The first two papers deal with entanglement. The paper by R. Mosseri and P. Ribeiro presents a detailed description of the two-and three-qubit geometry in Hilbert space, dealing with the geometry of fibrations and discrete geometry. The paper by J.-G.Luque et al. is more algebraic and considers invariants of pure k-qubit states and their application to entanglement measurement.
14,205 citations
TL;DR: In this paper, the authors proposed a method to solve the problem of "uniformity" in the literature.and.and, and, respectively, the authors' work.
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352 citations
02 Nov 2010
TL;DR: In this article, the authors discuss the geometry of phase transition transitions in two-dimensional (2D) systems, and present an algorithm for dissipation and transport near critical points.
Abstract: NEW DIRECTIONS AND NEW CONCEPTS IN QUANTUM PHASE TRANSITIONS Finite Temperature Dissipation and Transport Near Quantum Critical Points, Subir Sachdev Dissipation, Quantum Phase Transitions, and Measurement, Sudip Chakravarty Universal Dynamics Near Quantum Critical Points, Anatoli Polkovnikov and Vladimir Gritsev Fractionalization and Topological Order, Masaki Oshikawa Entanglement Renormalization: An Introduction, Guifre Vidal The Geometry of Quantum Phase Transitions, Gerardo Ortiz PROGRESS IN MODEL HAMILTONIANS AND IN SPECIFIC SYSTEMS Topological Order and Quantum Criticality, Claudio Castelnovo, Simon Trebst, and Matthias Troyer Quantum Criticality and the Kondo Lattice, Qimiao Si Spin-Boson Systems: Dissipation and Light Phenomena, Karyn Le Hur Topological Excitations in Superfluids with Internal Degrees of Freedom, Yuki Kawaguchi and Masahito Ueda Quantum Monte Carlo Studies of the Attractive Hubbard Hamiltonian, Richard T. Scalettar and George G. Batrouni Quantum Phase Transitions in Quasi-One-Dimensional Systems, Thierry Giamarchi Metastable Quantum Phase Transitions in a One-Dimensional Bose Gas, Lincoln D. Carr, Rina Kanamoto, and Masahito Ueda EXPERIMENTAL REALIZATIONS OF QUANTUM PHASES AND QUANTUM PHASE TRANSITIONS Quantum Phase Transitions in Quantum Dots, Ileana G. Rau, Sami Amasha, Yuval Oreg, and David Goldhaber-Gordon Quantum Phase Transitions in Two-Dimensional Electron Systems, Alexander Shashkin and Sergey Kravchenko Local Observables for QPTs in Strongly Correlated Systems, Eun-Ah Kim, Michael J. Lawler, and J.C. Davis Molecular Quasi-Triangular Lattice Antiferromagnets, Reizo Kato and Tetsuaki Itou Quantum Criticality and Superconductivity in Heavy Fermions, Philipp Gegenwart and Frank Steglich Ultracold Bosonic Atoms in Optical Lattices, Immanuel Bloch NUMERICAL SOLUTION METHODS FOR QUANTUM PHASE TRANSITIONS Worm Algorithm for Problems of Quantum and Classical Statistics, Nikolay Prokof 'ev and Boris Svistunov Cluster Monte Carlo Algorithms for Dissipative QPTs, Philipp Werner and Matthias Troyer Current Trends in Density Matrix Renormalization Group Methods, Ulrich Schollwock Simulations based on MPS and PEPS, Valentin Murg, Ignacio Cirac, and Frank Verstraete Continuous-Time Monte Carlo Methods, Philipp Werner and Andrew J. Millis QUANTUM PHASE TRANSITIONS ACROSS PHYSICS Quantum Phase Transitions in Dense QCD, Tetsuo Hatsuda and Kenji Maeda Quantum Phase Transitions in Coupled Atom-Cavity Systems, Andrew D. Greentree and Lloyd C. L. Hollenberg Quantum Phase Transitions in Nuclei, Francesco Iachello and Mark A. Caprio Quantum Critical Dynamics from Black Holes, Sean Hartnoll
263 citations