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Universal order parameters and quantum phase transitions: a finite-size approach.

Shi QQ, Zhou HQ, Batchelor MT - Sci Rep (2015)

Bottom Line: We propose a method to construct universal order parameters for quantum phase transitions in many-body lattice systems.The method exploits the H-orthogonality of a few near-degenerate lowest states of the Hamiltonian describing a given finite-size system, which makes it possible to perform finite-size scaling and take full advantage of currently available numerical algorithms.An explicit connection is established between the fidelity per site between two H-orthogonal states and the energy gap between the ground state and low-lying excited states in the finite-size system.

View Article: PubMed Central - PubMed

Affiliation: 1] College of Materials Science and Engineering, Chongqing University, Chongqing 400044, The People's Republic of China [2] Centre for Modern Physics, Chongqing University, Chongqing 400044, The People's Republic of China.

ABSTRACT
We propose a method to construct universal order parameters for quantum phase transitions in many-body lattice systems. The method exploits the H-orthogonality of a few near-degenerate lowest states of the Hamiltonian describing a given finite-size system, which makes it possible to perform finite-size scaling and take full advantage of currently available numerical algorithms. An explicit connection is established between the fidelity per site between two H-orthogonal states and the energy gap between the ground state and low-lying excited states in the finite-size system. The physical information encoded in this gap arising from finite-size fluctuations clarifies the origin of the universal order parameter. We demonstrate the procedure for the one-dimensional quantum formulation of the q-state Potts model, for q = 2, 3, 4 and 5, as prototypical examples, using finite-size data obtained from the density matrix renormalization group algorithm.

No MeSH data available.


Related in: MedlinePlus

The effective relation between the correlation length ξ∞ and the UOP.In each case we calculate the correlation length ξ∞(λ) and UOP  for control parameter λ < 1 then fit ξ∞(λ) and ln d∞(λ) to the relation ξ∞ = −a/ln d∞, with . A simple linear fit gives the values (a) a = −0.503, (b) a = −0.490, (c) a = −0.491 and (d) a = −0.506.
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f2: The effective relation between the correlation length ξ∞ and the UOP.In each case we calculate the correlation length ξ∞(λ) and UOP for control parameter λ < 1 then fit ξ∞(λ) and ln d∞(λ) to the relation ξ∞ = −a/ln d∞, with . A simple linear fit gives the values (a) a = −0.503, (b) a = −0.490, (c) a = −0.491 and (d) a = −0.506.

Mentions: Fig. 2 shows this expected relation between d∞(λ) and ξ∞(λ) for different values of λ. Here, the data are mainly obtained using the iTEBD algorithm for infinite-size systems. The results are consistent with the relation (14) holding throughout the SSB phase λ < 1. At the critical point λ = 1, the correlation length ξ and energy gap δL scale as ξ ~ 1/δL. With scale invariance at criticality, ξ ~ L, and thus δL ~ 1/L. Then with the expected relation between the fidelity per site of the H-orthogonality states and finite size L at criticality is ln dL ~ − ln L/L. The results presented in Fig. 3 indicate that this relation is more precisely


Universal order parameters and quantum phase transitions: a finite-size approach.

Shi QQ, Zhou HQ, Batchelor MT - Sci Rep (2015)

The effective relation between the correlation length ξ∞ and the UOP.In each case we calculate the correlation length ξ∞(λ) and UOP  for control parameter λ < 1 then fit ξ∞(λ) and ln d∞(λ) to the relation ξ∞ = −a/ln d∞, with . A simple linear fit gives the values (a) a = −0.503, (b) a = −0.490, (c) a = −0.491 and (d) a = −0.506.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4286789&req=5

f2: The effective relation between the correlation length ξ∞ and the UOP.In each case we calculate the correlation length ξ∞(λ) and UOP for control parameter λ < 1 then fit ξ∞(λ) and ln d∞(λ) to the relation ξ∞ = −a/ln d∞, with . A simple linear fit gives the values (a) a = −0.503, (b) a = −0.490, (c) a = −0.491 and (d) a = −0.506.
Mentions: Fig. 2 shows this expected relation between d∞(λ) and ξ∞(λ) for different values of λ. Here, the data are mainly obtained using the iTEBD algorithm for infinite-size systems. The results are consistent with the relation (14) holding throughout the SSB phase λ < 1. At the critical point λ = 1, the correlation length ξ and energy gap δL scale as ξ ~ 1/δL. With scale invariance at criticality, ξ ~ L, and thus δL ~ 1/L. Then with the expected relation between the fidelity per site of the H-orthogonality states and finite size L at criticality is ln dL ~ − ln L/L. The results presented in Fig. 3 indicate that this relation is more precisely

Bottom Line: We propose a method to construct universal order parameters for quantum phase transitions in many-body lattice systems.The method exploits the H-orthogonality of a few near-degenerate lowest states of the Hamiltonian describing a given finite-size system, which makes it possible to perform finite-size scaling and take full advantage of currently available numerical algorithms.An explicit connection is established between the fidelity per site between two H-orthogonal states and the energy gap between the ground state and low-lying excited states in the finite-size system.

View Article: PubMed Central - PubMed

Affiliation: 1] College of Materials Science and Engineering, Chongqing University, Chongqing 400044, The People's Republic of China [2] Centre for Modern Physics, Chongqing University, Chongqing 400044, The People's Republic of China.

ABSTRACT
We propose a method to construct universal order parameters for quantum phase transitions in many-body lattice systems. The method exploits the H-orthogonality of a few near-degenerate lowest states of the Hamiltonian describing a given finite-size system, which makes it possible to perform finite-size scaling and take full advantage of currently available numerical algorithms. An explicit connection is established between the fidelity per site between two H-orthogonal states and the energy gap between the ground state and low-lying excited states in the finite-size system. The physical information encoded in this gap arising from finite-size fluctuations clarifies the origin of the universal order parameter. We demonstrate the procedure for the one-dimensional quantum formulation of the q-state Potts model, for q = 2, 3, 4 and 5, as prototypical examples, using finite-size data obtained from the density matrix renormalization group algorithm.

No MeSH data available.


Related in: MedlinePlus