Conformal Fields and Operator Product Expansion in Critical Quantum Spin Chains

Zou, Yijian; Milsted, Ashley; Vidal, Guifré

doi:10.1103/physrevlett.124.040604

Cited by 42 publications

(30 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We compare the time-dependent CFT states (A.11) with the spin chain states (A.13). As the CFT states e −iHt P(0, x)|0 and e −iHt Q(0, x)|0 (A.11) are not regularized and normalized, we will not keep track of the overall normalization of the operators (the interested reader can find some recent progress on more careful identification of the CFT and spin chain local operators in [72,73]). We find that in (A.13) the modes with negative momenta correspond to the holomorphic state |ψ and its derivatives,…”

Section: A2 Operator Correspondence In the Critical Ising Spin Chainmentioning

confidence: 99%

Subsystem distance after a local operator quench

Zhang

Calabrese

2020

J. High Energ. Phys.

View full text Add to dashboard Cite

We investigate the time evolution of the subsystem trace distance and Schatten distances after local operator quenches in two-dimensional conformal field theory (CFT) and in one-dimensional quantum spin chains. We focus on the case of a subsystem being an interval embedded in the infinite line. The initial state is prepared by inserting a local operator in the ground state of the theory. We only consider the cases in which the inserted local operator is a primary field or a sum of several primaries. While a nonchiral primary operator can excite both left-moving and rightmoving quasiparticles, a holomorphic primary operator only excites a right-moving quasiparticle and an anti-holomorphic primary operator only excites a left-moving one. The reduced density matrix (RDM) of an interval hosting a quasiparticle is orthogonal to the RDM of the interval without any quasiparticles. Moreover, the RDMs of two intervals hosting quasiparticles at different positions are also orthogonal to each other. We calculate numerically the entanglement entropy, Rényi entropy, trace distance, and Schatten distances in time-dependent states excited by different local operators in the critical Ising and XX spin chains. These results match the CFT predictions in the proper limit.

show abstract

Section: A2 Operator Correspondence In the Critical Ising Spin Chainmentioning

confidence: 99%

Subsystem distance after a local operator quench

Zhang

Calabrese

2020

J. High Energ. Phys.

View full text Add to dashboard Cite

show abstract

“…They also lead to a scheme to numerically extract conformal data such as operator product expansion coefficients OPE of a CFT by computing wavefunction overlaps (as opposed to related techniques in Refs. [9,10], which require computing a lattice version of the CFT primary operators). Finally, for illustrative purposes, we have numerically constructed the states for the Ising CFT using novel free fermion techniques and computed entanglement quantities.…”

Section: Discussionmentioning

confidence: 99%

“…A second merit of our proposal is that it allows us to numerically extract the operator product expansion (OPE) coefficients and multi-point correlation functions of a CFT from a critical spin chain that realizes the latter, by computing overlaps between wavefunctions on the lattice. In recent years, related proposals were made that required however computing lattice versions of the CFT primary operators [9,10]. Here, no such lattice scaling operators are needed: the OPE coefficients follows solely from the computation of wavefunction overlaps.…”

mentioning

confidence: 99%

Multi-boundary generalization of thermofield double states and their realization in critical quantum spin chains

Zou,

Vidal

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We propose a multi-boundary generalization of thermofield double states (TFD) of a twodimensional conformal field theory (CFT) and show, through a conformal map to the complex plane, that they are closely related to multi-point correlation functions. We then also describe how to approximately realize these multi-boundary TFD states numerically on the lattice, starting from a critical quantum spin chain Hamiltonian. In addition, finite size corrections on the lattice are seen to be significantly reduced by the use of smoothers -numerically optimized unitary transformations that locally re-arrange the quantum spin degrees of freedom. One merit of the spin chain realization is that it allows us to probe the properties of the proposed multi-boundary TFD states through numerical experiments, including the characterization of their entanglement structure. As an illustration, we explicitly construct generalized TFD states with three and four boundaries for the Ising CFT and compute entanglement quantities using novel free fermion techniques. We find ranges of parameters where their multipartite entanglement is significant or negligible.

show abstract

“…A great deal of our understanding of the fields with degenerate weights in the Potts model comes from indirect arguments, such as the solution of the bootstrap equations for correlation functions and the presence of an underlying "interchiral" algebra, responsible for relations between some of the conformal-block amplitudes [3]. The purpose of this paper is to explore this issue much more directly using the lattice regularization of Vir ⊗ Vir first introduced in [2], and explored in further detail, in particular, in a companion paper on XXZ spin chains [20] (see also [21][22][23] for other applications).…”

Section: Jhep10(2020)109mentioning

confidence: 99%

The action of the Virasoro algebra in the two-dimensional Potts and loop models at generic Q

Grans-Samuelsson

Liu

et al. 2020

J. High Energ. Phys.

View full text Add to dashboard Cite

The spectrum of conformal weights for the CFT describing the two-dimensional critical Q-state Potts model (or its close cousin, the dense loop model) has been known for more than 30 years [1]. However, the exact nature of the corresponding Vir ⊗ $$ \overline{\mathrm{Vir}} $$ Vir ¯ representations has remained unknown up to now. Here, we solve the problem for generic values of Q. This is achieved by a mixture of different techniques: a careful study of “Koo-Saleur generators” [2], combined with measurements of four-point amplitudes, on the numerical side, and OPEs and the four-point amplitudes recently determined using the “interchiral conformal bootstrap” in [3] on the analytical side. We find that null-descendants of diagonal fields having weights (hr,1, hr,1) (with r ∈ ℕ*) are truly zero, so these fields come with simple Vir ⊗ $$ \overline{\mathrm{Vir}} $$ Vir ¯ (“Kac”) modules. Meanwhile, fields with weights (hr,s, hr,−s) and (hr,−s, hr,s) (with r, s ∈ ℕ*) come in indecomposable but not fully reducible representations mixing four simple Vir ⊗ $$ \overline{\mathrm{Vir}} $$ Vir ¯ modules with a familiar “diamond” shape. The “top” and “bottom” fields in these diamonds have weights (hr,−s, hr,−s), and form a two-dimensional Jordan cell for L0 and $$ {\overline{L}}_0 $$ L ¯ 0 . This establishes, among other things, that the Potts-model CFT is logarithmic for Q generic. Unlike the case of non-generic (root of unity) values of Q, these indecomposable structures are not present in finite size, but we can nevertheless show from the numerical study of the lattice model how the rank-two Jordan cells build up in the infinite-size limit.

show abstract

Conformal Fields and Operator Product Expansion in Critical Quantum Spin Chains

Cited by 42 publications

References 35 publications

Subsystem distance after a local operator quench

Subsystem distance after a local operator quench

Multi-boundary generalization of thermofield double states and their realization in critical quantum spin chains

The action of the Virasoro algebra in the two-dimensional Potts and loop models at generic Q

Contact Info

Product

Resources

About