Entanglement measures for quasi-two-dimensional fractional quantum Hall states

Biddle, John; Peterson, Michael R.; Sarma, S. Das

doi:10.1103/physrevb.84.125141

Cited by 15 publications

(13 citation statements)

References 87 publications

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“…In the spherical geometry, we divide the system in two pieces A and B [71][72][73][74][75] and obtain the reduced density matrix for one-half by tracing out the degrees of freedom of the remaining half. The eigenvalues ρ n of the density matrix are interpreted as energies ρ n ≡ e −ξ n =2 [76].…”

Section: Entanglement Spectrummentioning

confidence: 99%

“…This phase-transition point can also be identified by computing the bipartite entanglement entropy, which is the von Neumann entropy of the reduced density matrix [71][72][73][74][75], discussed in Sec. V. Figure 13 shows that the resulting entanglement entropy displays two nearby peaks as a function of κ (only a single peak for w=l 0 > 1.5).…”

Section: Entanglement Properties and Phase Diagrammentioning

confidence: 99%

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Phase Diagram of theν=5/2Fractional Quantum Hall Effect: Effects of Landau-Level Mixing and Nonzero Width

et al. 2015

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Interesting non-Abelian states, e.g., the Moore-Read Pfaffian and the anti-Pfaffian, offer candidate descriptions of the ν ¼ 5=2 fractional quantum Hall state. But, the significant controversy surrounding the nature of the ν ¼ 5=2 state has been hampered by the fact that the competition between these and other states is affected by small parameter changes. To study the phase diagram of the ν ¼ 5=2 state, we numerically diagonalize a comprehensive effective Hamiltonian describing the fractional quantum Hall effect of electrons under realistic conditions in GaAs semiconductors. The effective Hamiltonian takes Landau-level mixing into account to lowest order perturbatively in κ, the ratio of the Coulomb energy scale to the cyclotron gap. We also incorporate the nonzero width w of the quantum-well and subband mixing. We find the ground state in both the torus and spherical geometries as a function of κ and w. To sort out the nontrivial competition between candidate ground states, we analyze the following four criteria: its overlap with trial wave functions, the magnitude of energy gaps, the sign of the expectation value of an order parameter for particle-hole symmetry breaking, and the entanglement spectrum. We conclude that the ground state is in the universality class of the Moore-Read Pfaffian state, rather than the anti-Pfaffian, for κ < κ c ðwÞ, where κ c ðwÞ is a w-dependent critical value 0.6 ≲ κ c ðwÞ ≲ 1. We observe that both Landau-level mixing and nonzero width suppress the excitation gap, but Landau-level mixing has a larger effect in this regard. Our findings have important implications for the identification of non-Abelian fractional quantum Hall states.

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Section: Entanglement Spectrummentioning

confidence: 99%

Section: Entanglement Properties and Phase Diagrammentioning

confidence: 99%

Phase Diagram of theν=5/2Fractional Quantum Hall Effect: Effects of Landau-Level Mixing and Nonzero Width

et al. 2015

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“…This would exclude the spin-unpolarized 331 state. At the same time, several papers 56,68,76,77 lend support to a spin-polarized ground state. Another limitation comes from the incomplete understanding of the Landau-level mixing effects [78][79][80][81] .…”

Section: Discussionmentioning

confidence: 99%

“…At the same time, edge reconstruction 54 or bulk transport 55 may affect the interpretation of the experiments on upstream modes. On the other hand, various proposed 5/2 states are rather close in energy 56 and it was suggested that more than one 5/2 state might be present at different conditions or in different samples 57 . It is also possible that the true 5/2 state is not one of the simplest states considered above.…”

Section: Discussionmentioning

confidence: 99%

Influence of device geometry on tunneling in theν=52quantum Hall liquid

Yang

Feldman

2013

Phys. Rev. B

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Two recent experiments [I. P. Radu et al., Science 320, 899 (2008) and X. Lin et al., Phys. Rev. B 85, 165321 (2012)] measured the temperature and voltage dependence of the quasiparticle tunneling through a quantum point contact in the ν = 5/2 quantum Hall liquid. The results led to conflicting conclusions about the nature of the quantum Hall state. In this paper, we show that the conflict can be resolved by recognizing different geometries of the devices in the experiments. We argue that in some of those geometries there is significant unscreened electrostatic interaction between the segments of the quantum Hall edge on the opposite sides of the point contact. Coulomb interaction affects the tunneling current. We compare experimental results with theoretical predictions for the Pfaffian, SU (2)2, 331 and K = 8 states and their particle-hole conjugates. After Coulomb corrections are taken into account, measurements in all geometries agree with the spin-polarized and spin-unpolarized Halperin 331 states.

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“…We mention that entanglement is nowadays a central concept in many branches of quantum physics. In condensed matter physics, entanglement entropy and spectrum have proven to be powerful tools characterizing topological phases [4,[61][62][63][64], quantum phases transitions [65][66][67], and many-body localization [68][69][70], etc. A number of theoretical proposals have been introduced to measure entanglement entropy [71][72][73][74] and spectrum [75] in many-body systems.…”

Section: Neural-network Representation and Quantum Entanglement: mentioning

confidence: 99%

Quantum Entanglement in Neural Network States

2017

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Machine learning, one of today's most rapidly growing interdisciplinary fields, promises an unprecedented perspective for solving intricate quantum many-body problems. Understanding the physical aspects of the representative artificial neural-network states is recently becoming highly desirable in the applications of machine learning techniques to quantum many-body physics. Here, we study the quantum entanglement properties of neural-network states, with a focus on the restricted-Boltzmann-machine (RBM) architecture. We prove that the entanglement of all short-range RBM states satisfies an area law for arbitrary dimensions and bipartition geometry. For long-range RBM states we show by using an exact construction that such states could exhibit volume-law entanglement, implying a notable capability of RBM in representing efficiently quantum states with massive entanglement. We further examine generic RBM states with random weight parameters. We find that their averaged entanglement entropy obeys volume-law scaling and meantime strongly deviates from the Page-entropy of the completely random pure states. We show that their entanglement spectrum has no universal part associated with random matrix theory and bears a Poisson-type level statistics. Using reinforcement learning, we demonstrate that RBM is capable of finding the ground state (with power-law entanglement) of a model Hamiltonian with long-range interaction. In addition, we show, through a concrete example of the one-dimensional symmetry-protected topological cluster states, that the RBM representation may also be used as a tool to analytically compute the entanglement spectrum. Our results uncover the unparalleled power of artificial neural networks in representing quantum many-body states, which paves a novel way to bridge computer science based machine learning techniques to outstanding quantum condensed matter physics problems.Comment: 17 pages, 8 figures. Version published in Phys. Rev.

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Entanglement measures for quasi-two-dimensional fractional quantum Hall states

Cited by 15 publications

References 87 publications

Phase Diagram of theν=5/2Fractional Quantum Hall Effect: Effects of Landau-Level Mixing and Nonzero Width

Phase Diagram of theν=5/2Fractional Quantum Hall Effect: Effects of Landau-Level Mixing and Nonzero Width

Influence of device geometry on tunneling in theν=52quantum Hall liquid

Quantum Entanglement in Neural Network States

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