Criticality of the Anisotropic Quantum Heisenberg Model on a Self-Dual Hierarchical Lattice

Caride, A. O.; Tsallis, Constantino; Zanette, S. I.

doi:10.1103/physrevlett.51.145

Cited by 61 publications

(14 citation statements)

References 12 publications

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“…We have examined the Hubbard model using the small cluster diagonalization [9,10] and quantum Monte Carlo [11,12]. The Heisenberg model is studied within a real-space renormalization group framework [13,14,15].…”

Section: Introductionmentioning

confidence: 99%

Correlated electron systems on the Apollonian network

Souza

Herrmann

2007

Phys. Rev. B

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Strongly correlated electrons on an Apollonian network are studied using the Hubbard model. Ground-state and thermodynamic properties, including specific heat, magnetic susceptibility, spinspin correlation function, double occupancy and one-electron transfer, are evaluated applying direct diagonalization and quantum Monte Carlo. The results support several types of magnetic behavior. In the strong-coupling limit, the quantum anisotropic spin 1 2Heisenberg model is used and the phase diagram is discussed using the renormalization group method. For ferromagnetic coupling, we always observe the existence of long-range order. For antiferromagnetic coupling, we find a paramagnetic phase for all finite temperatures.

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Section: Introductionmentioning

confidence: 99%

Correlated electron systems on the Apollonian network

Souza

Herrmann

2007

Phys. Rev. B

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“…Many other systems (e.g., related to those mentioned in [102][103][104][105]) are awaiting for approaches along the above and similar lines. They would be very welcome.…”

Section: Further Applications and Final Wordsmentioning

confidence: 99%

Approach of Complexity in Nature: Entropic Nonuniqueness

Tsallis

2016

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Boltzmann introduced in the 1870s a logarithmic measure for the connection between the thermodynamical entropy and the probabilities of the microscopic configurations of the system. His celebrated entropic functional for classical systems was then extended by Gibbs to the entire phase space of a many-body system and by von Neumann in order to cover quantum systems, as well. Finally, it was used by Shannon within the theory of information. The simplest expression of this functional corresponds to a discrete set of W microscopic possibilities and is given by S BG = −k ∑ W i=1 p i ln p i (k is a positive universal constant; BG stands for Boltzmann-Gibbs). This relation enables the construction of BGstatistical mechanics, which, together with the Maxwell equations and classical, quantum and relativistic mechanics, constitutes one of the pillars of contemporary physics. The BG theory has provided uncountable important applications in physics, chemistry, computational sciences, economics, biology, networks and others. As argued in the textbooks, its application in physical systems is legitimate whenever the hypothesis of ergodicity is satisfied, i.e., when ensemble and time averages coincide. However, what can we do when ergodicity and similar simple hypotheses are violated, which indeed happens in very many natural, artificial and social complex systems. The possibility of generalizing BG statistical mechanics through a family of non-additive entropies was advanced in 1988, namely S q = k, which recovers the additive S BG entropy in the q → 1 limit. The index q is to be determined from mechanical first principles, corresponding to complexity universality classes. Along three decades, this idea intensively evolved world-wide (see the Bibliography in http://tsallis.cat.cbpf.br/biblio.htm) and led to a plethora of predictions, verifications and applications in physical systems and elsewhere. As expected, whenever a paradigm shift is explored, some controversy naturally emerged, as well, in the community. The present status of the general picture is here described, starting from its dynamical and thermodynamical foundations and ending with its most recent physical applications.

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“…We can obtain the recurrence relation between the original parameters (K, ∆) and the new parameters (K ′ , ∆ ′ ) by solving the trace Tr 3 with the method developed in Refs [29,31].…”

Section: Model and Methodsmentioning

confidence: 99%

“…This decimation RSRG method [27,29,38] has proved to be successful in spin chain and especially the fractal 3 lattices. For the spin chain, this decimation procedure is illustrated in Fig.…”

Section: Model and Methodsmentioning

confidence: 99%

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Thermal entanglement between non-nearest-neighbor spins on fractal lattices

Wang

Kong

2013

Phys. Rev. A

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We investigate thermal entanglement between two non-nearest-neighbor sites in ferromagnetic Heisenberg chain and on fractal lattices by means of the decimation renormalization-group (RG) method. It is found that the entanglement decreases with increasing temperature and it disappears beyond a critical value T c . Thermal entanglement at a certain temperature first increases with the increase of the anisotropy parameter ∆ and then decreases sharply to zero when ∆ is close to the isotropic point. We also show how the entanglement evolves as the size of the system L becomes large via the RG method. As L increases, for the spin chain and Koch curve the entanglement between two terminal spins is fragile and vanishes when L ≥ 17, but for two kinds of diamond-type hierarchical (DH) lattices the entanglement is rather robust and can exist even when L becomes very large. Our result indicates that the special fractal structure can affect the change of entanglement with system size.

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Criticality of the Anisotropic Quantum Heisenberg Model on a Self-Dual Hierarchical Lattice

Cited by 61 publications

References 12 publications

Correlated electron systems on the Apollonian network

Correlated electron systems on the Apollonian network

Approach of Complexity in Nature: Entropic Nonuniqueness

Thermal entanglement between non-nearest-neighbor spins on fractal lattices

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