Cardy states, defect lines and chiral operators of coset CFTs on the lattice

Lootens, Laurens; Vanhove, Robijn; Verstraete, Frank

doi:10.48550/arxiv.1907.02520

Cited by 7 publications

(10 citation statements)

References 40 publications

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“…The local Hamiltonian commuting with the corresponding MPO is precisely the cluster state Hamiltonian with critical magnetic field (Bridgeman and Williamson, 2017), which is equivalent to the XY model (Lahtinen and Ardonne, 2015). Similarly, if one starts from the Ising fusion rules, one obtains the critical Ising model in transverse magnetic field, and for the Fibonacci fusion rules, the critical "golden chain" Hamiltonian emerges (Lootens et al, 2019;Vanhove et al, 2018).…”

Section: Critical Spin Systems: Mpo Symmetriesmentioning

confidence: 99%

“…A systematic study of these MPO symmetries allows to represent all fields, including the chiral ones, in terms of the so-called tube algebra, which is an MPO algebra representing the Drinfeld Center of the input category. Many of the intriguing properties of conformal field theories can as such be transmuted to the lattice, including notions like orbifolding and the coset construction (Lootens et al, 2019).…”

Section: Critical Spin Systems: Mpo Symmetriesmentioning

confidence: 99%

See 1 more Smart Citation

Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems

Cirac,

Perez-Garcia,

Schuch

et al. 2020

Preprint

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111

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The theory of entanglement provides a fundamentally new language for describing interactions and correlations in many body systems. Its vocabulary consists of qubits and entangled pairs, and the syntax is provided by tensor networks. We review how matrix product states and projected entangled pair states describe many-body wavefunctions in terms of local tensors. These tensors express how the entanglement is routed, act as a novel type of non-local order parameter, and we describe how their symmetries are reflections of the global entanglement patterns in the full system. We will discuss how tensor networks enable the construction of real-space renormalization group flows and fixed points, and examine the entanglement structure of states exhibiting topological quantum order. Finally, we provide a summary of the mathematical results of matrix product states and projected entangled pair states, highlighting the fundamental theorem of matrix product vectors and its applications.

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Section: Critical Spin Systems: Mpo Symmetriesmentioning

confidence: 99%

Section: Critical Spin Systems: Mpo Symmetriesmentioning

confidence: 99%

Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems

Cirac,

Perez-Garcia,

Schuch

et al. 2020

Preprint

Self Cite

111

View full text Add to dashboard Cite

show abstract

“…In the case that these models are critical, their low-energy physics is described by a conformal field theory, in which these non-local symmetries are identified with the topological defects [33][34][35][36][37][38]. These quantum spin Hamiltonians can then be understood as the gapless edge theories of a (2+1)d system with topological order [21,36,[39][40][41][42], realizing a lattice version of the well known holographic relation between topological and conformal field theories [43][44][45][46][47].…”

Section: Sec I | Introductionmentioning

confidence: 99%

Dualities in one-dimensional quantum lattice models: symmetric Hamiltonians and matrix product operator intertwiners

Lootens¹,

Delcamp²,

Ortíz³

et al. 2021

Preprint

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We present a systematic approach for generating duality transformations in quantum lattice models. Within our formalism, dualities are completely characterized by equivalent but distinct realizations of a given (possibly non-abelian and non-invertible) symmetry. These different realizations are encoded into fusion categories, and dualities are methodically generated by considering all Morita equivalent categories. The full set of symmetric operators can then be constructed from the categorical data. We construct explicit intertwiners, in the form of matrix product operators, that convert local symmetric operators of one realization into local symmetric operators of its dual. Concurrently, it maps local operators that transform non-trivially into non-local ones. This guarantees that the structure constants of the algebra of all symmetric operators are equal in both dual realizations. Families of dual Hamiltonians, possibly with long range interactions, are then designed by taking linear combinations of the corresponding symmetric operators. We illustrate this approach by establishing matrix product operator intertwiners for well-known dualities such as Kramers-Wannier and Jordan-Wigner, consider theories with two copies of the Ising category symmetry, and present an example with quantum group symmetries. Finally, we comment on generalizations to higher dimensions of this categorical approach to dualities.

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“…There are several reasons for this. On the one hand, this is part of the more general question of how one can approximate field theories using discrete systems, with the ultimate goal of carrying out more efficient (quantum) simulations [3][4][5]. On the other hand, the possibility of observing, on the lattice, properties "analogous" to those of CFTs opens the door to the introduction of new discrete mathematical tools with a smorgasbord of exciting potential applications [6,7].…”

Section: Introductionmentioning

confidence: 99%

“…Of course, the co-standard module would be formally obtained by reversing the arrows, which corresponds formally to propagating "towards the past", or acting with the transpose of the transfer matrix to build partition and correlation functions. It is not clear what this means physically 5. In the present paper we consistently use calligraphic fonts for the lattice analogs of some key quantities: the Hamiltonian H, the momentum P-with their corresponding densities h j and p j -, the Virasoro generators Ln, Ln and the stress-energy tensor T, T. We denote the corresponding continuum quantities by Roman fonts: H, P and Ln, Ln, as well as T , T .…”

mentioning

confidence: 99%

The action of the Virasoro algebra in quantum spin chains. Part I. The non-rational case

Grans-Samuelsson

Jacobsen

Saleur

2021

J. High Energ. Phys.

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We investigate the action of discretized Virasoro generators, built out of generators of the lattice Temperley-Lieb algebra (“Koo-Saleur generators” [1]), in the critical XXZ quantum spin chain. We explore the structure of the continuum-limit Virasoro modules at generic central charge for the XXZ vertex model, paralleling [2] for the loop model. We find again indecomposable modules, but this time not logarithmic ones. The limit of the Temperley-Lieb modules Wj,1 for j ≠ 0 contains pairs of “conjugate states” with conformal weights (hr,s, hr,−s) and (hr,−s, hr,s) that give rise to dual structures: Verma or co-Verma modules. The limit of $$ {W}_{0,{\mathfrak{q}}^{\pm 2}} $$ W 0 , q ± 2 contains diagonal fields (hr,1, hr,1) and gives rise to either only Verma or only co-Verma modules, depending on the sign of the exponent in $$ {\mathfrak{q}}^{\pm 2} $$ q ± 2 . In order to obtain matrix elements of Koo-Saleur generators at large system size N we use Bethe ansatz and Quantum Inverse Scattering methods, computing the form factors for relevant combinations of three neighbouring spin operators. Relations between form factors ensure that the above duality exists already at the lattice level. We also study in which sense Koo-Saleur generators converge to Virasoro generators. We consider convergence in the weak sense, investigating whether the commutator of limits is the same as the limit of the commutator? We find that it coincides only up to the central term. As a side result we compute the ground-state expectation value of two neighbouring Temperley-Lieb generators in the XXZ spin chain.

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Cardy states, defect lines and chiral operators of coset CFTs on the lattice

Cited by 7 publications

References 40 publications

Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems

Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems

Dualities in one-dimensional quantum lattice models: symmetric Hamiltonians and matrix product operator intertwiners

The action of the Virasoro algebra in quantum spin chains. Part I. The non-rational case

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