Tensor networks as conformal transformations

Abstract

Tensor networks are often used to accurately represent ground states of quantum spin chains. Two popular choices of such tensor network representations can be seen to implement linear maps that correspond, respectively, to euclidean time evolution and to global scale transformations. In this paper, by exploiting the local structure of the tensor networks, we explain how to also implement local or non-uniform versions of both euclidean time evolution and scale transformations. We demonstrate our proposal with a critical quantum spin chain on a finite circle, where the low energy physics is described by a conformal field theory (CFT), and where non-uniform euclidean time evolution and local scale transformations are conformal transformations acting on the Hilbert space of the CFT. We numerically show, for the critical quantum Ising chain, that the proposed tensor networks indeed transform the low energy states of the periodic spin chain in the same way as the corresponding conformal transformations do in the CFT.

arXiv comment4 pages, 5 figures, plus appendices

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Cited by in corpus (11)

• Quantum Computation as Gravity
• Time Evolution of Complexity: A Critique of Three Methods
• Developments in the Tensor Network -- from Statistical Mechanics to Quantum Entanglement
• Holographic Quantum Circuits from Splitting/Joining Local Quenches
• Entanglement branes and factorization in conformal field theory
• Path Integral Complexity and Kasner singularities
• Quantum circuit approximations and entanglement renormalization for the Dirac field in 1+1 dimensions
• Boundary theories of critical matchgate tensor networks
• Multi-boundary generalization of thermofield double states and their realization in critical quantum spin chains
• Entanglement distillation toward minimal bond cut surface in tensor networks
• The Quantum Gravity Disk: Discrete Current Algebra