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arXiv:1512.03846 · Physical review. B./Physical review. B

Topological conformal defects with tensor networks

Markus Hauru, Glen Evenbly, Wen Wei Ho, Davide Gaiotto A, Guifré Vidal

Published Sep 9, 2016• 70 citations
arXivHTMLPDF

Abstract

The critical two-dimensional classical Ising model on the square lattice has two topological conformal defects: the ${\mathbb{Z}}_{2}$ symmetry defect ${D}_{\ensuremath{\epsilon}}$ and the Kramers-Wannier duality defect ${D}_{\ensuremath{\sigma}}$. These two defects implement antiperiodic boundary conditions and a more exotic form of twisted boundary conditions, respectively. On the torus, the partition function ${Z}_{D}$ of the critical Ising model in the presence of a topological conformal defect $D$ is expressed in terms of the scaling dimensions ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}}$ and conformal spins ${s}_{\ensuremath{\alpha}}$ of a distinct set of primary fields (and their descendants, or conformal towers) of the Ising conformal field theory. This characteristic conformal data ${{{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}},{s}_{\ensuremath{\alpha}}}}_{D}$ can be extracted from the eigenvalue spectrum of a transfer matrix ${M}_{D}$ for the partition function ${Z}_{D}$. In this paper, we investigate the use of tensor network techniques to both represent and coarse grain the partition functions ${Z}_{{D}_{\ensuremath{\epsilon}}}$ and ${Z}_{{D}_{\ensuremath{\sigma}}}$ of the critical Ising model with either a symmetry defect ${D}_{\ensuremath{\epsilon}}$ or a duality defect ${D}_{\ensuremath{\sigma}}$. We also explain how to coarse grain the corresponding transfer matrices ${M}_{{D}_{\ensuremath{\epsilon}}}$ and ${M}_{{D}_{\ensuremath{\sigma}}}$, from which we can extract accurate numerical estimates of ${{{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}},{s}_{\ensuremath{\alpha}}}}_{{D}_{\ensuremath{\epsilon}}}$ and ${{{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}},{s}_{\ensuremath{\alpha}}}}_{{D}_{\ensuremath{\sigma}}}$. Two key ingredients of our approach are (i) coarse graining of the defect $D$, which applies to any (i.e., not just topological) conformal defect and yields a set of associated scaling dimensions ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}}$, and (ii) construction and coarse graining of a generalized translation operator using a local unitary transformation that moves the defect, which only exist for topological conformal defects and yields the corresponding conformal spins ${s}_{\ensuremath{\alpha}}$.

DOI
10.1103/physrevb.94.115125

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The critical two-dimensional classical Ising model on the square lattice has two topological conformal defects: the Z2{\mathbb{Z}}_{2}Z2​ symmetry defect D\ensuremathϵ{D}_{\ensuremath{\epsilon}}D\ensuremathϵ​ and the Kramers-Wannier duality defect D\ensuremathσ{D}_{\ensuremath{\sigma}}D\ensuremathσ​. These two defects implement antiperiodic boundary conditions and a more exotic form of twisted boundary conditions, respectively. On the torus, the partition function ZD{Z}_{D}ZD​ of the critical Ising model in the presence of a topological conformal defect DDD is expressed in terms of the scaling dimensions \ensuremathΔ\ensuremathα{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}}\ensuremathΔ\ensuremathα​ and conformal spins s\ensuremathα{s}_{\ensuremath{\alpha}}s\ensuremathα​ of a distinct set of primary fields (and their descendants, or conformal towers) of the Ising conformal field theory. This characteristic conformal data \ensuremathΔ\ensuremathα,s\ensuremathαD{{{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}},{s}_{\ensuremath{\alpha}}}}_{D}\ensuremathΔ\ensuremathα​,s\ensuremathα​D​ can be extracted from the eigenvalue spectrum of a transfer matrix MD{M}_{D}MD​ for the partition function ZD{Z}_{D}ZD​. In this paper, we investigate the use of tensor network techniques to both represent and coarse grain the partition functions ZD\ensuremathϵ{Z}_{{D}_{\ensuremath{\epsilon}}}ZD\ensuremathϵ​​ and ZD\ensuremathσ{Z}_{{D}_{\ensuremath{\sigma}}}ZD\ensuremathσ​​ of the critical Ising model with either a symmetry defect D\ensuremathϵ{D}_{\ensuremath{\epsilon}}D\ensuremathϵ​ or a duality defect D\ensuremathσ{D}_{\ensuremath{\sigma}}D\ensuremathσ​. We also explain how to coarse grain the corresponding transfer matrices MD\ensuremathϵ{M}_{{D}_{\ensuremath{\epsilon}}}MD\ensuremathϵ​​ and MD\ensuremathσ{M}_{{D}_{\ensuremath{\sigma}}}MD\ensuremathσ​​, from which we can extract accurate numerical estimates of \ensuremathΔ\ensuremathα,s\ensuremathαD\ensuremathϵ{{{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}},{s}_{\ensuremath{\alpha}}}}_{{D}_{\ensuremath{\epsilon}}}\ensuremathΔ\ensuremathα​,s\ensuremathα​D\ensuremathϵ​​ and \ensuremathΔ\ensuremathα,s\ensuremathαD\ensuremathσ{{{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}},{s}_{\ensuremath{\alpha}}}}_{{D}_{\ensuremath{\sigma}}}\ensuremathΔ\ensuremathα​,s\ensuremathα​D\ensuremathσ​​. Two key ingredients of our approach are (i) coarse graining of the defect DDD, which applies to any (i.e., not just topological) conformal defect and yields a set of associated scaling dimensions \ensuremathΔ\ensuremathα{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}}\ensuremathΔ\ensuremathα​, and (ii) construction and coarse graining of a generalized translation operator using a local unitary transformation that moves the defect, which only exist for topological conformal defects and yields the corresponding conformal spins s\ensuremathα{s}_{\ensuremath{\alpha}}s\ensuremathα​.
We present a comprehensive study on extracting conformal field theory data using tensor network methods, especially, from the fixed-point tensor of the linearized tensor renormalization group (lTRG) for the classical two-dimensional Ising model near the critical temperature. Utilizing two different methods, we extract operator scaling dimensions and operator product expansion coefficients by introducing defects on the lattice and by employing the fixed-point tensor. We also explore the effects of pointlike defects in the lattice on the coarse-graining process. We find that there is a correspondence between coarse-grained defect tensors and conformal states obtained from the lTRG fixed-point equation. We also analyze the capabilities and limitations of our proposed coarse-graining scheme for tensor networks with pointlike defects, including graph-independent local truncation (GILT) and higher-order tensor renormalization group (HOTRG). Our results provide a better understanding of the capacity and limitations of the tensor renormalization group scheme in coarse-graining defect tensors, and we show that GILT+HOTRG can be used to give accurate two- and four-point functions under specific conditions. We also find that employing the minimal canonical form further improves the stability of the RG flow.
Abstract 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.
Tensor networks are a class of methods for studying many-body systems. They give a geometrical description of the internal structure of many-body states, operators, and partition functions, that can be used to implement efficient algorithms to simulate them numerically. In this thesis, after providing a very brief overview of the field, we present new tensor network methods for three different use cases. \n \nFirst, we present a new real-space renormalization group algorithm for tensor networks. Compared to existing methods, its advantages are its low computational cost, simplicity of implementation, and applicability to any network. We benchmark it on the 2D classical Ising model and find accuracy comparable with the best existing tensor network methods. Due to its simplicity and generalizability, we consider our algorithm to be an excellent candidate for implementation of real-space renormalization in higher dimensions, and discuss some of the details and the remaining challenges in 3D. \n \nSecond, we show how certain topological conformal defects of critical lattice theories can be represented as tensor networks, using the 2D classical Ising model as an example. Furthermore, we show how such tensor network descriptions, combined with a renormalization group algorithm, can be used to obtain accurate estimates of the universal properties of these defects. We also show how coarse-graining of defects can be applied to any conformal defect (i.e. not just topological ones), and yields a set of associated scaling dimensions. \n \nFinally, we leave behind the focus on renormalization group methods, and present a method for spatially resolving the overlap between two tensor network states, to obtain localized information about the similarities and differences between them. For a given region, the similarity of two states in this region can be quantified by the Uhlmann fidelity of their reduced density matrices, and we show how such fidelities can be efficiently computed in many cases when the two states are represented as tensor networks. We demonstrate the usefulness of evaluating such subsystem fidelities with three example applications: studying local quenches, comparing critical and non-critical states, and quantifying convergence in tensor network simulations.
Abstract We explore the topological defects of the critical three-state Potts spin system on the torus, Klein bottle and cylinder. A complete characterization is obtained by breaking down the Fuchs–Runkel–Schweigert construction of 2D rational CFT to the lattice setting. This is done by applying the strange correlator prescription to the recently obtained tensor network descriptions of string-net ground states in terms of bimodule categories (Lootens et al 2021 SciPost Phys. 10 053). The symmetries are represented by matrix product operators (MPO), as well as intertwiners between the diagonal tetracritical Ising model and the non-diagonal three-state Potts model. Our categorical construction lifts the global transfer matrix symmetries and intertwiners, previously obtained by solving Yang–Baxter equations, to MPO symmetries and intertwiners that can be locally deformed, fused and split. This enables the extraction of conformal characters from partition functions and yields a comprehensive picture of all boundary conditions.
The twofold twist defects in the D(Zk)D({\mathbb{Z}}_{k})D(Zk​) quantum double model (Abelian topological phase) carry non-Abelian fractional Majorana-like characteristics. We align these twist defects in a line and construct a one-dimensional Hamiltonian which only includes the pairwise interaction. For the defect chain with an even number of twist defects, it is equivalent to the Zk{\mathbb{Z}}_{k}Zk​ clock model with a periodic boundary condition (up to some phase factor for the boundary term), while for the odd number case, it maps to the Zk{\mathbb{Z}}_{k}Zk​ clock model with a duality twisted boundary condition. At the critical point, for both cases, the twist defect chain enjoys an additional translation symmetry, which corresponds to the Kramers-Wannier duality symmetry in the Zk{\mathbb{Z}}_{k}Zk​ clock model and can be generated by a series of braiding operators for twist defects. We further numerically investigate the low energy excitation spectrum for k=3,4,5k=3,4,5k=3,4,5, and 6. For the even-defect chain, the critical points are the same as the Zk{\mathbb{Z}}_{k}Zk​ clock conformal field theories (CFTs), while for the odd-defect chain, when $k\ensuremath{\ne}4$, the critical points correspond to orbifolding a Z2{\mathbb{Z}}_{2}Z2​ symmetry of CFTs of the even-defect chain. For k=4k=4k=4 case, we numerically observe some similarity to the Z4{\mathbb{Z}}_{4}Z4​ twist fields in the SU(2)1/D4SU{(2)}_{1}/{D}_{4}SU(2)1​/D4​ orbifold CFT.
The Ising model in two dimensions with special toroidal boundary conditions is analyzed. These boundary conditions, which we call duality-twisted boundary conditions, may be interpreted as inserting a specific defect line ("seam") in the system, along noncontractible circles of the cylinder, before closing it into a torus. We derive exact expressions for the eigenvalues of a transfer matrix for the critical ferromagnetic Ising model on the M×N square lattice wrapped on the torus with a specific defect line. As a result we have obtained analytically the partition function for the Ising model with such boundary conditions. In the case of infinitely long cylinders of circumference L with duality-twisted boundary conditions we obtain the asymptotic expansion of the free energy and the inverse correlation lengths. We find that the ratio of subdominant finite-size correction terms in the asymptotic expansion of the free energy and the inverse correlation lengths should be universal. We verify such universal behavior in the framework of a perturbating conformal approach by calculating the universal structure constant C_{n1n} for descendent states generated by the operator product expansion of the primary fields. For such states the calculations of an universal structure constants is a difficult task, since it involves knowledge of the four-point correlation function, which in general is not fixed by conformal invariance except for some particular cases, including the Ising model.