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arXiv:1302.5517 · New Journal of Physics

Entanglement spectra and entanglement thermodynamics of Hofstadter bilayers

John Schliemann

Published May 9, 2013• 17 citations
arXivHTMLPDF

Abstract

We study Hofstadter bilayers, i.e. coupled hopping models on two-dimensional square lattices in a perpendicular magnetic field. Upon tracing out one of the layers, we find an explicit expression for the resulting entanglement spectrum in terms of the energy eigenvalues of the underlying monolayer system. For strongly coupled layers, the entanglement Hamiltonian is proportional to the energetic Hamiltonian of the monolayer system. The proportionality factor, however, cannot be interpreted as the inverse thermodynamic temperature, but represents a phenomenological temperature scale. We derive an explicit relationship between both temperature scales which is in close analogy to a standard result of classic thermodynamics. In the limit of vanishing temperature, thermodynamic quantities such as entropy and inner energy approach their ground-state values, but show a fractal structure as a function of magnetic flux.

DOI
10.1088/1367-2630/15/5/053017

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We study Hofstadter bilayers, i.e. coupled hopping models on two-dimensional square lattices in a perpendicular magnetic field. Upon tracing out one of the layers, we find an explicit expression for the resulting entanglement spectrum in terms of the energy eigenvalues of the underlying monolayer system. For strongly coupled layers, the entanglement Hamiltonian is proportional to the energetic Hamiltonian of the monolayer system. The proportionality factor, however, cannot be interpreted as the inverse thermodynamic temperature, but represents a phenomenological temperature scale. We derive an explicit relationship between both temperature scales which is in close analogy to a standard result of classic thermodynamics. In the limit of vanishing temperature, thermodynamic quantities such as entropy and inner energy approach their ground-state values, but show a fractal structure as a function of magnetic flux.
We study the entanglement spectra of bilayer quantum Hall systems at total filling factor $\ensuremath{\nu}=1$. In the interlayer-coherent phase at layer separations smaller than a critical value, the entanglement spectra show a striking similarity to the energy spectra of the corresponding monolayer systems around half-filling. The transition to the incoherent phase can be followed in terms of low-lying entanglement levels. Finally, we describe the connection between those two types of spectra in terms of an effective temperature leading to relations for the entanglement entropy which are in full analogy to canonical thermodynamics.
Abstract 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, on-site energy difference and uniform magnetic field. Employing single particle correlation functions, we present an explicit form for a layer–layer entanglement Hamiltonian whose spectrum is the entanglement spectrum. We demonstrate that in the absence of trigonal warping, at zero on-site energy difference exact correspondence is established between the entanglement spectrum and energy spectrum of a 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 the 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 the 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 as the Hofstadter butterfly. Our results also show that the on-site energy difference causes a transition from the Hofstadter butterfly to a tree-like picture.
We investigate further the relationship between the entanglement spectrum of a composite many-body system and the energy spectrum of a subsystem making use of concepts of canonical thermodynamics. In many important cases the entanglement Hamiltonian is, in the limit of strong coupling between subsystems, proportional to the energy Hamiltonian of the subsystem. The proportionality factor is an appropriately defined coupling parameter, suggesting to interpret the latter as a inverse temperature. We identify a condition on the entanglement Hamiltonian which rigorously guarantees this interpretation to hold and removes any ambiguity in the definition of the entanglement Hamiltonian regarding contributions proportional to the unit operator. Illustrations of our findings are provided by spin ladders of arbitrary spin length, and by bilayer quantum Hall systems at total filling factor ν = 2. Within mean-field description, the latter system realizes an entanglement spectrum of free fermions with just two levels of equal modulus where the analogies to canonical thermodynamics are particularly close.
We calculate exactly the von Neumann and topological entropies of the toric code as a function of system size and temperature. We do so for systems with infinite energy scale separation between magnetic and electric excitations, so that the magnetic closed loop structure is fully preserved while the electric loop structure is tampered with by thermally excited electric charges. We find that the entanglement entropy is a singular function of temperature and system size, and that the limit of zero temperature and the limit of infinite system size do not commute. The two orders of limit differ by a term that does not depend on the size of the boundary between the partitions of the system, but instead depends on the topology of the bipartition. From the entanglement entropy we obtain the topological entropy, which is shown to drop to half its zero-temperature value for any infinitesimal temperature in the thermodynamic limit, and remains constant as the temperature is further increased. Such discontinuous behavior is replaced by a smooth decreasing function in finite-size systems. If the separation of energy scales in the system is large but finite, we argue that our results hold at small enough temperature and finite system size, and a second drop in the topological entropy should occur as the temperature is raised so as to disrupt the magnetic loop structure by allowing the appearance of free magnetic charges. We discuss the scaling of these entropies as a function of system size, and how the quantum topological entropy is shaved off in this two-step process as a function of temperature and system size. We interpret our results as an indication that the underlying magnetic and electric closed loop structures contribute equally to the topological entropy (and therefore to the topological order) in the system. Since each loop structure per se is a classical object, we interpret the quantum topological order in our system as arising from the ability of the two structures to be superimposed and appear simultaneously.
We argue that the entanglement entropy for a very small subsystem obeys a property which is analogous to the first law of thermodynamics when we excite the system. In relativistic setups, its effective temperature is proportional to the inverse of the subsystem size. This provides a universal relationship between the energy and the amount of quantum information. We derive the results using holography and confirm them in two-dimensional field theories. We will also comment on an example with negative specific heat and suggest a connection between the second law of thermodynamics and the strong subadditivity of entanglement entropy.
Entropy is a quantity for counting physical degrees of freedom in a system. At a finite temperature, one can use thermal entropy to study thermodynamical properties. At zero temperature, entanglement entropy is expected to provide a suitable order parameter of a phase structure. Especially, the entanglement entropy exhibits an interesting codimension two area law in a strongly coupled conformal field theory. We compute thermal entropy in a non-relativistic model with an infinite fermion mass limit from an exact effective potential to obtain thermal entropy at an infinite strong coupling limit. The computational result provides vanishing thermal entropy at an infinite strong coupling limit with a finite lattice spacing. The non-trivial topological term can be included in the strongly coupled lattice system to obtain the non-trivial entropy and the topology can be marked from the entropy. We first compute the thermal entropy in a two dimensional lattice topological quantum field theory to study the lattice artifact and also argue that a theory possibly has translational invariance if a system does not have a volume law in the entanglement entropy. Finally, we show that a coefficient of a universal term of the entanglement entropy should not be affected by a choice of an entangling surface in two dimensional conformal field theory for one interval case. We also discuss a choice of the entangling surface in the entanglement entropy in two dimensional CPN−1\mathrm{CP}^{N-1}CPN−1 model at the large NNN limit.