Valence bond and von Neumann entanglement entropy in Heisenberg ladders.

Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada.
Physical Review Letters (Impact Factor: 7.73). 09/2009; 103(11):117203. DOI: 10.1103/PhysRevLett.103.117203
Source: arXiv

ABSTRACT We present a direct comparison of the recently proposed valence bond entanglement entropy and the von Neumann entanglement entropy on spin-1/2 Heisenberg systems using quantum Monte Carlo and density-matrix renormalization group simulations. For one-dimensional chains we show that the valence bond entropy can be either less or greater than the von Neumann entropy; hence, it cannot provide a bound on the latter. On ladder geometries, simulations with up to seven legs are sufficient to indicate that the von Neumann entropy in two dimensions obeys an area law, even though the valence bond entanglement entropy has a multiplicative logarithmic correction.

  • [Show abstract] [Hide abstract]
    ABSTRACT: We study the von Neumann block entropy in the Kondo necklace model for different anisotropies η in the XY interaction between conduction spins using the density matrix renormalization group method. It was found that the block entropy presents a maximum for each η considered, and, comparing it with the results of the quantum criticality of the model based on the behavior of the energy gap, we observe that the maximum block entropy occurs at the quantum critical point between an antiferromagnetic and a Kondo singlet state, so this measure of entanglement is useful for giving information about where a quantum phase transition occurs in this model. We observe that the block entropy also presents a maximum at the quantum critical points that are obtained when an anisotropy Δ is included in the Kondo exchange between localized and conduction spins; when Δ diminishes for a fixed value of η, the critical point increases, favoring the antiferromagnetic phase.
    Physical Review A 06/2010; 81(6). · 3.04 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We present an exact expression for the entanglement entropy generated at a quantum point contact between noninteracting electronic leads in terms of the full counting statistics of charge fluctuations, which we illustrate with examples from both equilibrium and nonequilibrium transport. The formula is also applicable to ground-state entanglement entropy in systems described by noninteracting fermions in any dimension, which in one dimension include the critical spin-1/2 XX and Ising models where conformal field theory predictions for the entanglement entropy are reproduced from the full counting statistics. These results may play an important role in experimental measurements of entanglement entropy in mesoscopic structures and cold atoms in optical lattices.
    Physical review. B, Condensed matter 04/2011; 83. · 3.66 Impact Factor
  • [Show abstract] [Hide abstract]
    ABSTRACT: The set of valence-bond states --- states in which localized spin-1/2 particles are correlated in singlet pairs said to be connected by valence bonds --- provides a useful basis for visualizing singlet ground states of quantum spin systems. For example, the ground state of the uniform one-dimensional nearest-neighbor spin-1/2 antiferromagnetic (AFM) Heisenberg model (the prototypical spin-liquid state) can be viewed as a strongly fluctuating liquid of valence bonds with a power-law length distribution. This intuitive picture directly reflects the long-range spin correlations in this state, as well as the existence of gapless excitations created by breaking long bonds. Valence-bond states also play a key role in describing the physics of random spin-1/2 AFM Heisenberg chains. For these systems, it was shown by Fisher, using a real space renormalization group analysis, that on long-length scales the ground state is described by a single valence-bond state known as a random singlet state. This single valence-bond state should be viewed as a caricature of the true ground state, which will certainly exhibit bond fluctuations on short-length scales. In valence-bond Monte Carlo (VBMC) simulations valence-bond states are used to stochastically sample singlet ground states of quantum spin systems. One of the appealing features of VBMC is that if one imagines viewing the sampled valence-bond states over many Monte Carlo time steps the resulting ``movie" would correspond closely to the intuitive resonating valence bond picture described above. For random Heisenberg chains (and related models) VBMC should therefore provide a useful method for directly studying the phenomenon of random singlet formation on long-length scales, while at the same time capturing the short-range fluctuations which will always be present. In this dissertation I present results of VBMC studies for a class of models which include the uniform and random spin-1/2 AFM Heisenberg chains, as well as models describing chains of interacting non-Abelian quasiparticles --- exotic quasiparticles conjectured to exist in certain fractional quantum Hall states. In addition to numerically computing and analyzing the so-called valence-bond entanglement scaling in these models, I introduce a new quantity which I refer to as the valence-bond fluctuation (the central new result and the main contribution of this dissertation). It is shown that this quantity, which is easy to compute in valence-bond Monte Carlo, provides a direct signature of random singlet phase formation by essentially allowing one to directly ``see" the ``locking" of the ground state into a particular valence-bond state on long-length scales. A detailed scaling analysis of this new quantity is then used to extract the dependence of the fluctuation length scale on disorder strength. Where possible, the results are compared to previous numerical and analytic work on the relevant models.


1 Download
Available from