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Chui, C. H. Otto; Pearce, Paul A.

doi:10.1023/a:1015113909363

Cited by 8 publications

(4 citation statements)

References 18 publications

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“…x z 1 1 . The spectra for these three spin-chain boundary conditions in the critical case has been found explicitly [32,35,36], but we will show in section 6 that the explicit expressions are not needed to relate the critical partition functions to those in conformal field theory.…”

Section: mentioning

confidence: 99%

Topological defects on the lattice: I. The Ising model

Aasen¹,

Mong²,

Fendley³

2016

J. Phys. A: Math. Theor.

174

222

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In this paper and its sequel, we construct topologically invariant defects in two-dimensional classical lattice models and quantum spin chains. We show how defect lines commute with the transfer matrix/Hamiltonian when they obey the defect commutation relations, cousins of the Yang-Baxter equation. These relations and their solutions can be extended to allow defect lines to branch and fuse, again with properties depending only on topology. In this part I, we focus on the simplest example, the Ising model. We define lattice spin-flip and duality defects and their branching, and prove they are topological. One useful consequence is a simple implementation of Kramers-Wannier duality on the torus and higher genus surfaces by using the fusion of duality defects. We use these topological defects to do simple calculations that yield exact properties of the conformal field theory describing the continuum limit. For example, the shift in momentum quantization with duality-twisted boundary conditions yields the conformal spin 1/16 of the chiral spin field. Even more strikingly, we derive the modular transformation matrices explicitly and exactly.

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Section: mentioning

confidence: 99%

Topological defects on the lattice: I. The Ising model

Aasen¹,

Mong²,

Fendley³

2016

J. Phys. A: Math. Theor.

174

222

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“…where the nonuniversal constant, f 0 , has the meaning of free energy per unit length. When applying to twodimensional statistical models with spatially isotropic couplings (see, e.g., [10,11]), the velocity is v = 1. We note that (7) is the seminal result obtained by Affleck [12] and Blöte, Cardy, and Nightingale [13] and ( 8) is the key result of the present work.…”

mentioning

confidence: 99%

Universal Entropy of Conformal Critical Theories on a Klein Bottle

2017

Phys. Rev. Lett.

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We show that rational conformal field theories in 1+1 dimensions on a Klein bottle, with length L and width β, satisfying L β, have a universal entropy. This universal entropy depends on the quantum dimensions of the primary fields and can be accurately extracted by taking a proper ratio between the Klein bottle and torus partition functions, enabling the characterization of conformal critical theories. The result is checked against exact calculations in quantum spin-1/2 XY and Ising chains.Introduction -The characterization of phases and phase transitions is an important task in condensed matter physics. The discovery of topological phases of matter [1], examples of which include integer and fractional quantum Hall states [2,3], as well as topological insulators and topological superconductors [4,5], greatly enriches our understanding of quantum phases. The topological systems exhibit intriguing behaviors, such as gapless edge states, robust ground-state degeneracy, and quasiparticles with fractional statistics. Because of these remarkable properties, they constitute an important candidate for quantum information processing devices.Much effort has been made for characterizing these socalled topological phases. In 2+1 dimensions, many topological phases have gapless edge states that are exponentially localized at the boundary. These edge modes are either unidirectional or bidirectional, generically have linear dispersion, and thus are described by chiral or nonchiral conformal field theory (CFT) in 1+1 dimensions. The best-understood examples are fractional quantum Hall states, for which the bulk properties are fully characterized by the edge chiral CFTs through a remarkable bulkedge correspondence [6]. The nonchiral CFTs can appear as the edge theory of (2+1)-dimensional symmetryprotected topological (SPT) phases, such as the Z 2 topological insulators [7] protected by time-reversal and charge conservation symmetries. Needless to say, identifying the edge CFT is an important step toward the full characterization of topological phases in 2+1 dimensions.In this Letter, we show that (1+1)-dimensional nonchiral rational CFTs, when placed on a Klein bottle (with length L and width β, satisfying L β), have a universal entropy S = ln g. This entropy depends on the quantum dimensions of the CFT primary fields and, therefore, provides a useful quantity which, at least partially, distinguishes different CFTs. This result is directly applicable to (1+1)-dimensional quantum chains and twodimensional classical statistical models, when their lowenergy effective theories are rational CFTs, and is potentially applicable for (2+1)-dimensional SPT phases with nonchiral gapless edge states. As a first step toward its applications in lattice models, we focus on (1+1)-

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“…Since only the diagonal terms of the partition function survive on the Klein bottle, all the non-diagonal terms cancel in (5.4). As we showed in [17], the non-diagonal terms have the characteristic that the zeros of the corresponding eigenvalue functions are not symmetric on the upper and lower half complex plane. Note that for the A even models the trace of F (2) D (r,a) is zero, since the action of Z 2 automorphism σ interchanges the sublattices of even and odd spins.…”

Section: Numerical Resultsmentioning

confidence: 89%

“…In particular, the lattice realizations [13,14] of conformal A-D-E unitary minimal models on a torus with defect lines [15,16] has been studied. The Ising model on the Klein bottle and Möbius strip has also been investigated [17,18]. In this paper, we study the critical sl(2) A-D-E minimal lattice models on the Klein bottle and on the Möbius strip with twisted boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Lattice realizations of the open descendants of twisted boundary conditions forsl(2) A–D–Emodels

Chui

Pearce

2005

J. Stat. Mech.

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The twisted boundary conditions and associated partition functions of the conformal sl(2) A-D-E models are studied on the Klein bottle and the Möbius strip. The A-D-E minimal lattice models give realization to the complete classification of the open descendants of the sl(2) minimal theories. We construct the transfer matrices of these lattice models that are consistent with non-orientable geometries. In particular, we show that in order to realize all the Klein bottle amplitudes of different crosscap states, not only the topological flip on the lattice but also the involution in the spin configuration space must be taken into account. This involution is the Z 2 symmetry of the Dynkin diagrams which corresponds to the simple current of the Ocneanu algebra.

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Cited by 8 publications

References 18 publications

Topological defects on the lattice: I. The Ising model

Topological defects on the lattice: I. The Ising model

Universal Entropy of Conformal Critical Theories on a Klein Bottle

Lattice realizations of the open descendants of twisted boundary conditions forsl(2) A–D–Emodels

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