Entanglement thermodynamics

We analytically evaluate the entanglement spectra of the superconductivity states in graphene, primarily focusing on the s-wave and chiral d x 2 −y 2 + idxy superconductivity states. We demonstrate that the topology of the entanglement Hamiltonian can differ from that of the subsystem Hamiltonian. In particular, the topological properties of the entanglement Hamiltonian of the chiral d x 2 −y 2 + idxy superconductivity state obtained by tracing out one spin direction clearly differ from those of the time-reversal invariant Hamiltonian of noninteracting fermions on the honeycomb lattice.

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“…The canonical entanglement Hamiltonian at half-filling is independent of the inverse temperature [37] …”

Section: Entanglement Spectramentioning

confidence: 99%

“…Particular focus has been placed on various spin ladder systems [26][27][28][29][30][31][32][33][34] and on bilayer systems [35][36][37], where a proportionality between the entanglement and subsystem Hamiltonians is realized by the strong coupling limit. However, this relationship is not valid in general, as indicated in Ref.…”

Section: Introductionmentioning

confidence: 99%

Entanglement spectra of superconductivity ground states on the honeycomb lattice

Predin

Schliemann

2017

Eur. Phys. J. B

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“…A particular situation arises if the edge comprises the entire remaining subsystem, as is the case for spin ladders [12][13][14][15][16][17][18][19][20][21] and various bilayer systems [22][23][24]. A typical observation in such scenarios is, in the regime of strongly coupled subsystems, a proportionality between the energy Hamiltonian of the remaining subsystem and the appropriately defined entanglement Hamiltonian.…”

Section: Introductionmentioning

confidence: 99%

“…A typical observation in such scenarios is, in the regime of strongly coupled subsystems, a proportionality between the energy Hamiltonian of the remaining subsystem and the appropriately defined entanglement Hamiltonian. We note that the entanglement Hamiltonian entering the reduced density matrix (1) is only determined up to multiples of the unit operator, which has consequences regarding thermodynamic relations between the entanglement entropy and the subsystem energy [22][23][24].…”

Section: Introductionmentioning

confidence: 99%

Trigonal warping in bilayer graphene: Energy versus entanglement spectrum

2016

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We present a mainly analytical study of the entanglement spectrum of Bernal-stacked graphene bilayers in the presence of trigonal warping in the energy spectrum. Upon tracing out one layer, the entanglement spectrum shows qualitative geometric differences to the energy spectrum of a graphene monolayer. However, topological quantities such as Berry-phase-type contributions to Chern numbers agree. The latter analysis involves not only the eigenvalues of the entanglement Hamiltonian but also its eigenvectors. We also discuss the entanglement spectra resulting from tracing out other sublattices. As a technical basis of our analysis, we provide closed analytical expressions for the full eigensystem of bilayer graphene in the entire Brillouin zone with a trigonally warped spectrum.

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“…14 , the Kitaev model 15 , one dimensional quantum spin systems [16][17][18][19][20][21][22][23][24][25] , Hofstadter poblem [26][27][28] , interacting fermions on honeycomb lattice 29 and bosonic critical system in three dimensions 30 . As a result of these studies, the ES depends on the chosen basis to partition the many body Hilbert space.…”

Section: Introductionmentioning

confidence: 99%

Entanglement spectrum of fermionic bilayer honeycomb lattice: Hofstadter butterfly

Moradi¹,

Abouie²

2016

J. Stat. Mech.

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We perform an analytical study of the energy and entanglement spectrum of non-interacting fermionic bilayer honeycomb lattices in the presence of trigonal warping in the energy spectrum, onsite energy difference and uniform magnetic field. Employing single particle correlation functions, we present an explicit form for layer-layer entanglement Hamiltonian whose spectrum is entanglement spectrum. We demonstrate that in the absence of trigonal warping, at zero on-site energy difference exact correspondence is established between entanglement spectrum and energy spectrum of monolayer which means that the entanglement spectrum perfectly reflects the edge state properties of the bilayer. We also show that trigonal warping breaks down such a perfect correspondence, however, in Γ-K direction in hexagonal Brillouin zone, their behaviors are remarkably the same for particular relevances of hopping parameters. In the presence of an on-site energy difference the symmetry of entanglement spectrum is broken with opening an indirect entanglement gap. We also study the effects of a perpendicular magnetic field on both energy and the entanglement spectrum of the bilayer in the presence of trigonal warping and on-site energy difference. We demonstrate that the entanglement spectrum versus magnetic flux has a self similar fractal structure, known Hofstadter butterfly. Our results also show that the on-site energy difference causes a transition from the Hofstadter butterfly to a tree-like picture.

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Entanglement thermodynamics

Cited by 18 publications

References 39 publications

Entanglement spectra of superconductivity ground states on the honeycomb lattice

Entanglement spectra of superconductivity ground states on the honeycomb lattice

Trigonal warping in bilayer graphene: Energy versus entanglement spectrum

Entanglement spectrum of fermionic bilayer honeycomb lattice: Hofstadter butterfly

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