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Peschel, Ingo; Kaulke, M.; Legeza, Örs

doi:10.1002/(sici)1521-3889(199902)8:2<153::aid-andp153>3.3.co;2-e

Cited by 81 publications

(162 citation statements)

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“…The usefulness of this lemma relies in the fact that the spectrum ͕ i ͖ is typically decaying very fast for noncritical systems, and hence the upper bound decays typically very fast with increasing D. Let us illustrate this on the hand of, e.g., the XXZ Hamiltonian on a chain For ⌬Ͻ−1, the spectrum of the halve-infinite chain can be calculated exactly. 32 This is a very nice illustration for the superpolynomial accuracy of the infinite DMRG method for the XXZ model. The other integrable models will have a very similar behavior, and it is expected that non-integrable models share the same features.…”

Section: ͑7͒mentioning

confidence: 86%

“…This lemma is very interesting in the light of the fact that in the case of critical systems, arguable the hardest ones to simulate, 31 the Renyi-entropy of a contiguous block of L spins scales as [32][33][34][35][36] …”

Section: Lemma 2 Given a Density Operator If 0ͻmentioning

confidence: 99%

“…The fact that the eigenvalues of L decay fast has previously been identified as a indication for the validity of the DMRG approach. 32 The truncation error, 2,25 which has been used in the DMRG community as a check for convergence, is essentially given by ⑀͑D͒ − ⑀͑2D͒ and therefore indeed gives a good idea of the error in a simulation. Let us investigate how the computational effort to simulate such critical systems scales as a function of the length N =2L of the chain.…”

Section: Lemma 2 Given a Density Operator If 0ͻmentioning

confidence: 99%

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Matrix product states represent ground states faithfully

2006

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We quantify how well matrix product states approximate exact ground states of one-dimensional quantum spin systems as a function of the number of spins and the entropy of blocks of spins. We also investigate the convex set of local reduced density operators of translational invariant systems. The results give a theoretical justification for the high accuracy of renormalization group algorithms and justifies their use even in the case of critical systems.

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Section: ͑7͒mentioning

confidence: 86%

Section: Lemma 2 Given a Density Operator If 0ͻmentioning

confidence: 99%

Section: Lemma 2 Given a Density Operator If 0ͻmentioning

confidence: 99%

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Matrix product states represent ground states faithfully

2006

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“…On the other hand, we are also able to prove continuous majorization relations as a function of the parameters defining the model. On top we also provide explicit analytical examples for models with a boundary based on previous work of Peschel, Kaulke and Legeza [9]. This paper is structured as follows: in sec.II we remember the concepts of global, monotonous and fine-grained entanglement loss, as defined in [5].…”

Section: Introductionmentioning

confidence: 99%

Erratum: Entanglement and majorization in(1+1)-dimensional quantum systems [Phys. Rev. A71, 052327 (2005)]

Orús¹

2006

Phys. Rev. A

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Motivated by the idea of entanglement loss along Renormalization Group flows, analytical majorization relations are proven for the ground state of (1 + 1)-dimensional conformal field theories. For any of these theories, majorization is proven to hold in the spectrum of the reduced density matrices in a bipartite system when changing the size L of one of the subsystems. Continuous majorization along uniparametric flows is also proven as long as part of the conformal structure is preserved under the deformation and some monotonicity conditions hold as well. As particular examples of our derivations, we study the cases of the XX, Heisenberg and XY quantum spin chains. Our results provide in a rigorous way explicit proves for all the majorization conjectures raised by Latorre, Lütken, Rico, Vidal and Kitaev in previous papers on quantum spin chains.

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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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Cited by 81 publications

References 0 publications

Matrix product states represent ground states faithfully

Matrix product states represent ground states faithfully

Erratum: Entanglement and majorization in(1+1)-dimensional quantum systems [Phys. Rev. A71, 052327 (2005)]

Renormalization group and quantum information

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