Matrix product density operators: Renormalization fixed points and boundary theories

Cirac, J. I.; Pérez-Garcı́a, David; Schuch, Norbert; Verstraete, Frank

doi:10.1016/j.aop.2016.12.030

Cited by 71 publications

(160 citation statements)

References 40 publications

(87 reference statements)

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“…This has two consequences [26,27] with some left-invertible X s . Second, there can in addition be off-diagonal blocks D s,t coupling blocks s and t, which however-up to reordering of blocks-must be upper triangular, i.e., D s,t ≡ 0 if s > t. This implies that any product K , K → ∞.…”

Section: Structure Ofẽ C Cmentioning

confidence: 99%

“…We now distinguish two cases: If g / ∈ H, then U ⊗N g |ρ c0 = |ρ c with c = c 0 , and since different representations of an injective MPS are related by a local gauge transformation [26,27], it holds that . and thus , and since λ max (E c c0 ) < 1, (ρ|S g;ᾱ − g;α |ρ) → 0 as → ∞.…”

Section: Behavior Of String Order Parameters Imentioning

confidence: 99%

“…Specifically, given an MPS with periodic boundary conditions, it can be brought into a standard form (possibly involving blocking of sites) where it can be understood as a superposition of distinct injective MPS |ρ c ) (possibly with size-dependent amplitudes) [26,27]. While the |ρ c ) are not necessarily fixed points of the transfer operator themselves, such as in the case of an antiferromagnet where the transfer operator acts by permuting the |ρ c ), they will be fixed points of the transfer operator obtained after suitable blocking.…”

Section: B the Assumption: Matrix Product Fixed Pointsmentioning

confidence: 99%

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Entanglement phases as holographic duals of anyon condensates

et al. 2017

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Anyon condensation forms a mechanism which allows to relate different topological phases. We study anyon condensation in the framework of Projected Entangled Pair States (PEPS) where topological order is characterized through local symmetries of the entanglement. We show that anyon condensation is in one-to-one correspondence to the behavior of the virtual entanglement state at the boundary (i.e., the entanglement spectrum) under those symmetries, which encompasses both symmetry breaking and symmetry protected (SPT) order, and we use this to characterize all anyon condensations for abelian double models through the structure of their entanglement spectrum. We illustrate our findings with the Z4 double model, which can give rise to both Toric Code and Doubled Semion order through condensation, distinguished by the SPT structure of their entanglement. Using the ability of our framework to directly measure order parameters for condensation and deconfinement, we numerically study the phase diagram of the model, including direct phase transitions between the Doubled Semion and the Toric Code phase which are not described by anyon condensation.

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Section: Structure Ofẽ C Cmentioning

confidence: 99%

Section: Behavior Of String Order Parameters Imentioning

confidence: 99%

Section: B the Assumption: Matrix Product Fixed Pointsmentioning

confidence: 99%

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Entanglement phases as holographic duals of anyon condensates

et al. 2017

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“…Thus the local matrices is shown in Eq. (46). Therefore, the transfer matrices for the whole ground state subspace of symmetry breaking phases are given by…”

Section: B Degeneracy Of Transfer Matrix Spectrummentioning

confidence: 99%

“…In this appendix, we will list the definitions of these concepts, which are summarized from the literature 7,45,46 .…”

Section: Appendix: Some Useful Conceptsmentioning

confidence: 99%

Classifying parafermionic gapped phases using matrix product states

Zhang

2018

Phys. Rev. B

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In the Fock representation, we construct matrix product states (MPS) for one-dimensional gapped phases for Zp parafermions. From the analysis of irreducibility of MPS, we classify all possible gapped phases of Zp parafermions without extra symmetry other than Zp charge symmetry, including topological phases, spontaneous symmetry breaking phases and a trivial phase. For all phases, we find the irreducible forms of local matrices of MPS, which span different kinds of graded algebras. The topological phases are characterized by the non-trivial simple Zp graded algebras with the characteristic graded centers, yielding the degeneracies of the full transfer matrix spectra uniquely. But the spontaneous symmetry breaking phases correspond to the trivial semisimple Z p/n graded algebras, which can be further reduced to the trivial simple Z p/n graded algebras, where n is the divisor of p. So the present results provide the complete classification of the parafermionic gapped phases and deepen our understanding of topological phases in one dimension.

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Tensor Network Contractions

Ran

Tirrito

Peng

et al. 2020

Lecture Notes in Physics

105

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PrefaceTensor network (TN), a young mathematical tool of high vitality and great potential, has been undergoing extremely rapid developments in the last two decades, gaining tremendous success in condensed matter physics, atomic physics, quantum information science, statistical physics, and so on. In this lecture notes, we focus on the contraction algorithms of TN as well as some of the applications to the simulations of quantum many-body systems. Starting from basic concepts and definitions, we first explain the relations between TN and physical problems, including the TN representations of classical partition functions, quantum many-body states (by matrix product state, tree TN, and projected entangled pair state), time evolution simulations, etc. These problems, which are challenging to solve, can be transformed to TN contraction problems. We present then several paradigm algorithms based on the ideas of the numerical renormalization group and/or boundary states, including density matrix renormalization group, time-evolving block decimation, coarsegraining/corner tensor renormalization group, and several distinguished variational algorithms. Finally, we revisit the TN approaches from the perspective of multilinear algebra (also known as tensor algebra or tensor decompositions) and quantum simulation. Despite the apparent differences in the ideas and strategies of different TN algorithms, we aim at revealing the underlying relations and resemblances in order to present a systematic picture to understand the TN contraction approaches. v AcknowledgementsWe are indebted to

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Matrix product density operators: Renormalization fixed points and boundary theories

Cited by 71 publications

References 40 publications

Entanglement phases as holographic duals of anyon condensates

Entanglement phases as holographic duals of anyon condensates

Classifying parafermionic gapped phases using matrix product states

Tensor Network Contractions

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