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arXiv:1502.01854 · Journal of Physics A Mathematical and Theoretical

The scaling of entanglement entropy in a honeycomb lattice on a torus

Wen‐Long You

Published Jun 5, 2014• 10 citations
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Abstract

The entanglement entropy of a noninteracting fermionic system confined to a two-dimensional honeycomb lattice on a torus is calculated. We find that the entanglement entropy can characterize Lifshitz phase transitions without a local order parameter. In the noncritical phase and critical phase with a nodal Fermi surface, the entanglement entropy satisfies an area law. The leading subarea term is a constant in the gapped phase rather than a logarithmic additive term in the gapless regime. The tuning of chemical potential allows for a nonzero Fermi surface, whose variation along a particular direction determines a logarithmic violation of the area law. We perform the scaling of entanglement entropy numerically and find agreement between the analytic and numerical results. Furthermore, we clearly show that an entanglement spectrum is equivalent to an edge spectrum.

DOI
10.1088/1751-8113/47/25/255301

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•Oct 1, 2012•0 citations
The entanglement entropy of a noninteracting fermionic system confined to a two-dimensional honeycomb lattice on a torus is calculated. We find that the entanglement entropy can characterize Lifshitz phase transitions without a local order parameter. In the noncritical phase and critical phase with a nodal Fermi surface, the entanglement entropy satisfies an area law. The leading subarea term is a constant in the gapped phase rather than a logarithmic additive term in the gapless regime. The tuning of chemical potential allows for a nonzero Fermi surface, whose variation along a particular direction determines a logarithmic violation of the area law. We perform the scaling of entanglement entropy numerically and find agreement between the analytic and numerical results. Furthermore, we clearly show that an entanglement spectrum is equivalent to an edge spectrum.
We investigate the scaling of the bipartite entanglement entropy across Lifshitz quantum phase transitions, where the topology of the Fermi surface changes without any changes in symmetry. We present both numerical and analytical results which show that Lifshitz transitions are characterized by a well-defined set of critical exponents for the entanglement entropy near the phase transition. In one dimension, we show that the entanglement entropy exhibits a length scale that diverges as the system approaches a Lifshitz critical point. In two dimensions, the leading and subleading coefficients of the scaling of entanglement entropy show distinct power-law singularities at critical points. The effect of weak interactions is investigated using the density matrix renormalization group algorithm.
We study the universal scaling behavior of the entanglement entropy of critical theories in 2 + 1 dimensions. We specially consider two fermionic scale-invariant models, free massless Dirac fermions and a model of fermions with quadratic band touching, and numerically study the two-cylinder entanglement entropy of the models on the torus. We find that in both cases the entanglement entropy satisfies the area law and has the subleading term which is a scaling function of the aspect ratios of the cylindrical regions. We test the scaling of entanglement in both the free fermion models using three possible scaling functions for the subleading term derived from (a) the quasi-1D conformal field theory, (b) the bosonic quantum Lifshitz model and (c) the holographic AdS/CFT correspondence. For the later case we construct an analytic scaling function using holography, appropriate for critical theories with a gravitational dual description. We find that the subleading term in the fermionic models is well described, for a range of aspect ratios, by the scaling form derived from the quantum Lifshitz model as well as that derived using the AdS/CFT correspondence (in this case only for the Dirac model). For the case where the fermionic models are placed on a square torus we find the fit to the different scaling forms is in agreement to surprisingly high precision.
We analyze in detail the effect of nontrivial band topology on the area-law behavior of the entanglement entropy in Kitaev's honeycomb model. By mapping the translationally invariant 2D spin model onto 1D fermionic subsystems, we identify those subsystems responsible for universal entanglement contributions in the gapped phases and those responsible for critical entanglement scaling in the gapless phases. For the gapped phases, we analytically show how the topological edge states contribute to the entanglement entropy and provide a universal lower bound for it. For the gapless semimetallic phases and topological phase transitions, the identification of the critical subsystems shows that they fall always into the Ising or the XY universality classes. As our study concerns the fermionic degrees of freedom in the honeycomb model, qualitatively similar results are expected to apply also to generic topological insulators and superconductors.
Quantum phases characterized by surfaces of gapless excitations are known to violate the otherwise ubiquitous boundary law of entanglement entropy in the form of a multiplicative log correction: $S\ensuremath{\sim}{L}^{d\ensuremath{-}1}logL$. Using variational Monte Carlo, we calculate the second R\'enyi entropy for a model wave function of the $\ensuremath{\nu}=1/2$ composite Fermi liquid (CFL) state defined on the two-dimensional triangular lattice. By carefully studying the scaling of the total R\'enyi entropy and, crucially, its contributions from the modulus and sign of the wave function on various finite-size geometries, we argue that the prefactor of the leading LlogLL\phantom{\rule{0.16em}{0ex}}logLLlogL term is equivalent to that in the analogous free fermion wave function. In contrast to the recent results of Shao et al. [Phys. Rev. Lett. 114, 206402 (2015)], we thus conclude that the ``Widom formula'' holds even in this non-Fermi liquid CFL state. More generally, our results further elucidate---and place on a more quantitative footing---the relationship between nontrivial wave function sign structure and $S\ensuremath{\sim}L\phantom{\rule{0.16em}{0ex}}logL$ entanglement scaling in such highly entangled gapless phases.
In Part I, the relationship between the topology of the Fermi surface and the entanglement entropy S is examined. Spinless fermionic systems on one and two dimensional lattices at fixed chemical potential are considered. The lattice is partitioned into sub-system of length L and environment, and the entanglement of the subsystem with the environment is calculated via the correlation matrix. S is plotted as a function of the next-nearest or next-next nearest neighbor hopping parameter, t. In 1 dimension, the entanglement entropy jumps at lifshitz transitions where the number of Fermi points changes. In 2 dimensions, a neck-collapsing transition is accompanied by a cusp in S, while the formation of electron or hole-like pockets coincides with a kink in the S as a function of the hopping parameter. The entanglement entropy as a function of subsystem length L is also examined. The leading order coefficient of the LlnL term in 2 dimensions was seen to agree well with the Widom conjecture. Of interest is the difference this coefficient and the coefficient of the term linear in L near the neck-collapsing point. The leading order term changes like |t-tc|1/2 whereas the first sub-leading term varies like |t-tc|1/3, where tc is the critical value of the hopping parameter at the transition. In Part II, we study the statistics of fractionalized excitations in a bosonic model which describes strongly interacting excitons in a N-band insulator. The elementary excitations of this system are strings, in a large N limit. A string is made of a series of bosons whose flavors are correlated such that the end points of a string carries a fractionalized flavor quantum number. When the tension of a string vanishes, the end points are deconfined. We determine the statistics of the fractionalized particles described by the end points of strings. We show that either bosons or Fermions can arise depending on the microscopic coupling constants. In the presence of the cubic interaction in the Hamiltonian as the only higher order interaction term, it was shown that bosons are emergent. In the presence of the quartic interaction with a positive coupling constant, it was revealed that the elementary excitations of the system possess Fermion statistics.