On Next-Nearest-Neighbor Interaction in Linear Chain. II

Majumdar, Chanchal K.; Ghosh, D.

doi:10.1063/1.1664979

Cited by 475 publications

(257 citation statements)

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“…This is the case with MajumdarGhosh [34,35] model [20]. Here we will see a similar phenomena, where the matrix product state is the sum of many ground states each of which breaks the translation symmetry of the original Hamiltonian.…”

Section: Introductionsupporting

confidence: 53%

A New Family of Matrix Product States with Dzyaloshinski-Moriya Interactions

Asoudeh

2011

Int J Theor Phys

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We define a new family of matrix product states which are exact ground states of spin 1/2 Hamiltonians on one dimensional lattices. This class of Hamiltonians contain both Heisenberg and Dzyaloshinskii-Moriya interactions but at specified and not arbitrary couplings. We also compute in closed forms the one and two-point functions and the explicit form of the ground state. The degeneracy structure of the ground state is also discussed.

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Section: Introductionsupporting

confidence: 53%

A New Family of Matrix Product States with Dzyaloshinski-Moriya Interactions

Asoudeh

2011

Int J Theor Phys

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“…7,8) These studies have revealed that the system undergoes a phase transition from the spin-liquid phase to the dimer phase at j = j c (∆) with increasing j.…”

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confidence: 99%

Ground State Phase Diagram of Frustrated S = 1 XXZ chains: Chiral Ordered Phases

et al. 2000

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The ground-state phase diagram of frustrated S = 1 XXZ spin chains with the competing nearest-and next-nearest-neighbor antiferromagnetic couplings is studied using the infinitesystem density-matrix renormalization-group method. We find six different phases, namely, the Haldane, gapped chiral, gapless chiral, double Haldane, Néel, and double Néel (uudd) phases. The gapped and gapless chiral phases are characterized by the spontaneous breaking of parity, in which the long-range order parameter is a chirality, κ l =S , whereas the spin correlation decays either exponentially or algebraically. These chiral ordered phases appear in a broad region in the phase diagram for ∆ < 0.95, where ∆ is an exchange-anisotropy parameter. The critical properties of phase transitions are also studied.KEYWORDS: frustration, spin-1 chain, ground-state phase diagram, density-matrix renormalization-group method, chiral ordering §1. Introduction Frustrated antiferromagnetic quantum spin systems have attracted considerable attention over the decades since they exhibit a wealth of fascinating phenomena in their ground states and lowlying excitations. In general, frustration supresses antiferromagnetic correlations and the tendency towards the Néel order. Classical systems, for example, often show a helical ordered state in their ground states in the presence of strong frustration. In quantum systems, the interplay of frustration and quantum fluctuations plays an important role which causes exotic phenomena, e.g., a spin-liquid state and a novel type of spontaneous symmetry breaking.In this paper, we study a one-dimensional anisotropic spin system with the antiferromagnetic nearest-neighbor coupling J 1 and the frustrating next-nearest-neighbor coupling J 2 . The model is * E-mail address: hikihara@phys03.phys.sci.kobe-u.ac. where S l is a spin-S spin operator at site l and ∆ ≥ 0 represents an exchange anisotropy. Hereafter, we put j ≡ J 2 /J 1 (j ≥ 0).In the classical limit, S → ∞, the system exhibits a magnetic long-range order (LRO) characterized by a wavenumber q. The order parameter is defined bywhere L is the total number of spins. While the magnetic LRO is of the Néel-type (q = π) when j is smaller than a critical value, j ≤ 1/4, it becomes of helical-type for j > 1/4 with q = cos −1 (−1/4j).It should be noticed that both the time-reversal and parity symmetries are broken in this helical ordered state. When the system has an XY -like anisotropy (∆ < 1), the helical ordered state possesses a two-fold discrete degeneracy according as the helix is either right-or left-handed, in addition to a continuous degeneracy associated with the original U (1) symmetry of the XY spin.This discrete degeneracy is characterized by mutually opposite signs of the total chirality definedNote that this chirality is distinct from the scalar chirality of the Heisenberg spin often discussed in the literature 2) defined by χ l = S l−1 · S l × S l+1 : The chirality O κ changes its sign under the parity operation but is invariant under the time-reversal op...

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“…In the gapped regime, the Lieb-Schultz-Mattis theorem [25] guarantees that the ground state is degenerate. Another possible generalization is the inclusion of local, but not "nearest neighbor," couplings (so-called "Majumdar-Ghosh" couplings [26]) of the form These couplings can dramatically change the behavior of the chain in the thermodynamic limit. For example, the AF spin-1 2 chain becomes gapped when J 2 = 1 2 J 1 and all other J k vanish [26].…”

Section: The Heisenberg Spin-1 2 Chain and Its Generalizationsmentioning

confidence: 99%

“…Then, any K appearing in the fusion of I and J satisfies S AK * /S 0K * = S A0 /S 00 . 26 Proof. We have…”

Section: A Fusion Theoremmentioning

confidence: 99%

Anyonic Chains, Topological Defects, and Conformal Field Theory

Buican

Gromov

2017

Commun. Math. Phys.

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Abstract:Motivated by the three-dimensional topological field theory/two-dimensional conformal field theory (CFT) correspondence, we study a broad class of one-dimensional quantum mechanical models, known as anyonic chains, which can give rise to an enormously rich (and largely unexplored) space of two-dimensional critical theories in the thermodynamic limit. One remarkable feature of these systems is the appearance of non-local microscopic "topological symmetries" that descend to topological defects of the resulting CFTs. We derive various model-independent properties of these theories and of this topological symmetry/topological defect correspondence. For example, by studying precursors of certain twist and defect fields directly in the anyonic chains, we argue that (under mild assumptions) the two-dimensional CFTs correspond to particular modular invariants with respect to their maximal chiral algebras and that the topological defects descending from topological symmetries commute with these maximal chiral algebras. Using this map, we apply properties of defect Hilbert spaces to show how topological symmetries give a handle on the set of allowed relevant deformations of these theories. Throughout, we give a unified perspective that treats the constraints from discrete symmetries on the same footing as the constraints from topological ones.

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On Next-Nearest-Neighbor Interaction in Linear Chain. II

Cited by 475 publications

References 7 publications

A New Family of Matrix Product States with Dzyaloshinski-Moriya Interactions

A New Family of Matrix Product States with Dzyaloshinski-Moriya Interactions

Ground State Phase Diagram of Frustrated S = 1 XXZ chains: Chiral Ordered Phases

Anyonic Chains, Topological Defects, and Conformal Field Theory

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