Corner Transfer Matrix Algorithm for Classical    Renormalization Group

Nishino, Tomotoshi; Okunishi, Kouichi

doi:10.1143/jpsj.66.3040

Cited by 173 publications

(199 citation statements)

References 55 publications

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“…In order to construct an MPS representation of the ground state |Ψ XY , we exploit the 1D-quantum to 2D-classical mapping (commonly used e.g. in the context of Quantum Monte Carlo [12], Corner Transfer Matrix DMRG [13], or Bethe Ansatz at finite temperature [14], to name just a few), and above all the original observation by Suzuki [15] that H XY commutes and -more importantly -shares the ground state with an operator V given by…”

Section: The Ground State Of the Xy Model And Its Exact Mps Reprementioning

confidence: 99%

Truncating an exact matrix product state for the XY model: Transfer matrix and its renormalization

et al. 2015

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We discuss how to analytically obtain an -essentially infinite -Matrix Product State (MPS) representation of the ground state of the XY model. On the one hand this allows to illustrate how the Ornstein-Zernike form of the correlation function emerges in the exact case using standard MPS language. On the other hand we study the consequences of truncating the bond dimension of the exact MPS, which is also part of many tensor network algorithms, and analyze how the truncated MPS transfer matrix is representing the dominant part of the exact quantum transfer matrix. In the gapped phase we observe that the correlation length obtained from a truncated MPS approaches the exact value following a power law in effective bond dimension. In the gapless phase we find a good match between a state obtained numerically from standard MPS techniques with finite bond dimension, and a state obtained by effective finite imaginary time evolution in our framework. This provides a direct hint for a geometric interpretation of Finite Entanglement Scaling at the critical point in this case. Finally, by analyzing the spectra of transfer matrices, we support the interpretation put forward by [V. Zauner at. al., New J. Phys. 17, 053002 (2015)] that the MPS transfer matrix emerges from the quantum transfer matrix though the application of Wilson's Numerical Renormalisation Group along the imaginary-time direction.

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Section: The Ground State Of the Xy Model And Its Exact Mps Reprementioning

confidence: 99%

Truncating an exact matrix product state for the XY model: Transfer matrix and its renormalization

et al. 2015

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“…Finally, tensor networks can also be used to encode and manipulate the partition function of 2D classical lattice systems. [7][8][9] The computational cost of a simulation using tensor network algorithms is roughly proportional to the size of the lattice. However, when the system is invariant under translations, this cost can be made independent of the system's size.…”

Section: Introductionmentioning

confidence: 99%

Infinite time-evolving block decimation algorithm beyond unitary evolution

2008

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The infinite time-evolving block decimation algorithm ͓G. Vidal, Phys. Rev. Lett. 98, 070201 ͑2007͔͒ allows to simulate unitary evolution and to compute the ground state of one-dimensional ͑1D͒ quantum lattice systems in the thermodynamic limit. Here we extend the algorithm to tackle a much broader class of problems, namely, the simulation of arbitrary one-dimensional evolution operators that can be expressed as a ͑translationally invariant͒ tensor network. Relatedly, we also address the problem of finding the dominant eigenvalue and eigenvector of a one-dimensional transfer matrix that can be expressed in the same way. New applications include the simulation, in the thermodynamic limit, of open ͑i.e., master equation͒ dynamics and thermal states in 1D quantum systems, as well as calculations with partition functions in two-dimensional ͑2D͒ classical systems, on which we elaborate. The present extension of the algorithm also plays a prominent role in the infinite projected entangled-pair states approach to infinite 2D quantum lattice systems.

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“…This connection was realized by Nishino and Okunishi [22] and developed by Peschel and collaborators [23].…”

Section: Corner Transfer Matrix and Density Matrixmentioning

confidence: 88%

Renormalization group and quantum information

Gaite¹

2006

J. Phys. A: Math. Gen.

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The renormalization group is a tool that allows one to obtain a reduced description of systems with many degrees of freedom while preserving the relevant features. In the case of quantum systems, in particular, one-dimensional systems defined on a chain, an optimal formulation is given by White's "density matrix renormalization group". This formulation can be shown to rely on concepts of the developing theory of quantum information. Furthermore, White's algorithm can be connected with a peculiar type of quantization, namely, angular quantization. This type of quantization arose in connection with quantum gravity problems, in particular, the Unruh effect in the problem of black-hole entropy and Hawking radiation. This connection highlights the importance of quantum system boundaries, regarding the concentration of quantum states on them, and helps us to understand the optimal nature of White's algorithm.

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Corner Transfer Matrix Algorithm for Classical Renormalization Group

Cited by 173 publications

References 55 publications

Truncating an exact matrix product state for the XY model: Transfer matrix and its renormalization

Truncating an exact matrix product state for the XY model: Transfer matrix and its renormalization

Infinite time-evolving block decimation algorithm beyond unitary evolution

Renormalization group and quantum information

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