[Show abstract][Hide abstract] ABSTRACT: We develop a method to calculate the bipartite entanglement entropy of
quantum lattice models in the thermodynamic limit, using a Numerical
Linked Cluster Expansion (NLCE) involving only rectangular clusters. The
NLCE is based on exact diagonalization of all N x M rectangular clusters
at the interface between entangled subsystems A and B. We show that the
method can be used to obtain the Renyi entanglement entropy of the
two-dimensional transverse field Ising model, for arbitrary real Renyi
index. Furthermore, extrapolating these results as a function of the
order of the calculation, one can obtain subleading universal pieces of
the entanglement entropy at a quantum critical point. These results are
compared with series expansions, quantum Monte Carlo simulations and
field theories, where available, and they demonstrate the utility of the
NLCE in obtaining accurate results for the universal properties of this
critical point for von Neumann and non-integer Renyi entropies.
[Show abstract][Hide abstract] ABSTRACT: Spin ice materials, such as Dy2Ti2O7 and Ho2Ti2O7, have been the subject
of much interest for over the past fifteen years. Their low temperature
strongly correlated state can be mapped onto the proton disordered state
of common water ice and, consequently, spin ices display the same low
temperature residual Pauling entropy as water ice. Interestingly, it was
found in a previous study [X. Ke {\it et. al.} Phys. Rev. Lett. {\bf
99}, 137203 (2007)] that, upon dilution of the magnetic rare-earth ions
(Dy^{3+} and Ho^{3+}) by non-magnetic Yttrium (Y^{3+}) ions, the
residual entropy depends {\it non-monotonically} on the concentration of
Y^{3+} ions. In the present work, we report results from Monte Carlo
simulations of site-diluted microscopic dipolar spin ice models (DSIM)
that account quantitatively for the experimental specific heat
measurements, and thus also for the residual entropy, as a function of
dilution, for both Dy2Ti2O7 and Ho2Ti2O7. The main features of the
dilution physics displayed by the magnetic specific heat data are
quantitatively captured by the diluted DSIM up to, and including, 85% of
the magnetic ions diluted (x=1.7). The previously reported departures in
the residual entropy between Dy2Ti2O7 versus Ho2Ti2O7, as well as with a
site-dilution variant of Pauling's approximation, are thus rationalized
through the site-diluted DSIM. For 90% (x=1.8) and 95% (x=1.9) of the
magnetic ions diluted, we find a significant discrepancy between the
experimental and Monte Carlo specific heat results. We discuss some
possible reasons for this disagreement.
Physical Review B 03/2013; 90(21). DOI:10.1103/PhysRevB.90.214433 · 3.74 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: By extending the calculation of the Renyi entropy from quantum models
[Phys. Rev. B 82, 100409(R) (2010)] to classical modes, we introduce a
general procedure to calculate the Renyi mutual information in Monte
Carlo simulations. Examining an array of quantum and classical models we
show that the mutual information is able to detect general finite
temperature phase transitions from different universality classes
without knowledge of the specific order parameter or any special
thermodynamic estimators. We demonstrate this technique on a standard
symmetry breaking phase transition, the classical Ising model and
anisotropic Heisenberg model, and a vortex-unbinding transition without
a local order parameter, the classical and quantum XY model, and present
the details necessary to implement this procedure on other models
[arXiv:1210.2403].
[Show abstract][Hide abstract] ABSTRACT: We implement a Wang-Landau sampling technique in quantum Monte Carlo (QMC) simulations for the purpose of calculating the Rényi entanglement entropies and associated mutual information. The algorithm converges an estimate for an analog to the density of states for stochastic series expansion QMC, allowing a direct calculation of Rényi entropies without explicit thermodynamic integration. We benchmark results for the mutual information on two-dimensional (2D) isotropic and anisotropic Heisenberg models, a 2D transverse field Ising model, and a three-dimensional Heisenberg model, confirming a critical scaling of the mutual information in cases with a finite-temperature transition. We discuss the benefits and limitations of broad sampling techniques compared to standard importance sampling methods.
Physical Review E 01/2013; 87(1-1):013306. DOI:10.1103/PhysRevE.87.013306 · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This Chapter outlines the fundamental construction of the Stochastic Series Expansion, a highly efficient and easily implementable quantum Monte Carlo method for quantum lattice models. Originally devised as a finite-temperature simu-lation based on a Taylor expansion of the partition function, the method has recently been recast in the formalism of a zero-temperature projector method, where a large power of the Hamiltonian is applied to a trial wavefunction to project out the ground-state. Although these two methods appear formally quite different, their implemen-tation via non-local loop or cluster algorithms reveals their underlying fundamental similarity. Here, we briefly review the finite-and zero-temperature formalisms, and discuss concrete manifestations of the algorithm for the spin 1/2 Heisenberg and transverse field Ising models.
[Show abstract][Hide abstract] ABSTRACT: By developing a method to represent the Renyi entropies via a replica-trick
on classical statistical mechanical systems, we introduce a procedure to
calculate the Renyi Mutual Information in any Monte Carlo simulation. Through
simulations on several classical models, we demonstrate that the Renyi Mutual
Information can detect finite-temperature critical points, and even identify
their universality class, without knowledge of an order parameter or other
thermodynamic estimators. Remarkably, in addition to critical points mediated
by symmetry breaking, the Renyi Mutual Information is able to detect
topological vortex-unbinding transitions, as we explicitly demonstrate on
simulations of the XY model.
[Show abstract][Hide abstract] ABSTRACT: We study resonating-valence-bond (RVB) states on the square lattice of spins
and of dimers, as well as SU(N)-invariant states that interpolate between the
two. These states are ground states of gapless models, although the
SU(2)-invariant spin RVB state is also believed to be a gapped liquid in its
spinful sector. We show that the gapless behavior in spin and dimer RVB states
is qualitatively similar by studying the R\'enyi entropy for splitting a torus
into two cylinders, We compute this exactly for dimers, showing it behaves
similarly to the familiar one-dimensional log term, although not identically.
We extend the exact computation to an effective theory believed to interpolate
among these states. By numerical calculations for the SU(2) RVB state and its
SU(N)-invariant generalizations, we provide further support for this belief. We
also show how the entanglement entropy behaves qualitatively differently for
different values of the R\'enyi index $n$, with large values of $n$ proving a
more sensitive probe here, by virtue of exhibiting a striking even/odd effect.
New Journal of Physics 07/2012; 15(1). DOI:10.1088/1367-2630/15/1/015004 · 3.56 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: At low temperatures, a spin ice enters a Coulomb phase - a state with
algebraic correlations and topologically constrained spin configurations. In
Ho2Ti2O7, we have observed experimentally that this process is accompanied by a
non-standard temperature evolution of the wave vector dependent magnetic
susceptibility, as measured by neutron scattering. Analytical and numerical
approaches reveal signatures of a crossover between two Curie laws, one
characterizing the high temperature paramagnetic regime, and the other the low
temperature topologically constrained regime, which we call the spin liquid
Curie law. The theory is shown to be in excellent agreement with neutron
scattering experiments. On a more general footing, i) the existence of two
Curie laws appears to be a general property of the emergent gauge field for a
classical spin liquid, and ii) sheds light on the experimental difficulty of
measuring a precise Curie-Weiss temperature in frustrated materials; iii) the
mapping between gauge and spin degrees of freedom means that the susceptibility
at finite wave vector can be used as a local probe of fluctuations among
topological sectors.
Physical Review X 04/2012; 3(1). DOI:10.1103/PhysRevX.3.011014 · 9.04 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We discuss designer Hamiltonians---lattice models tailored to be free from
sign problems ("de-signed") when simulated with quantum Monte Carlo methods but
which still host complex many-body states and quantum phase transitions of
interest in condensed matter physics. We focus on quantum spin systems in which
competing interactions lead to non-magnetic ground states. These states and the
associated quantum phase transitions can be studied in great detail, enabling
direct access to universal properties and connections with low-energy effective
quantum field theories. As specific examples, we discuss the transition from a
Neel antiferromagnet to either a uniform quantum paramagnet or a spontaneously
symmetry-broken valence-bond solid in SU(2) and SU(N) invariant spin models. We
also discuss anisotropic (XXZ) systems harboring topological Z2 spin liquids
and the XY* transition. We briefly review recent progress on quantum Monte
Carlo algorithms, including ground state projection in the valence-bond basis
and direct computation of the Renyi variants of the entanglement entropy.
[Show abstract][Hide abstract] ABSTRACT: We study the bipartite entanglement entropy of the two-dimensional (2D)
transverse-field Ising model in the thermodynamic limit using series expansion
methods. Expansions are developed for the Renyi entropy around both the
small-field and large-field limits, allowing the separate calculation of the
entanglement associated with lines and corners at the boundary between
sub-systems. Series extrapolations are used to extract subleading power laws
and logarithmic singularities as the quantum critical point is approached. In
1D, we find excellent agreement with exact results as well as quantum Monte
Carlo simulations. In 2D, we find compelling evidence that the entanglement at
a corner is significantly different from a free boson field theory. These
results demonstrate the power of the series expansion method for calculating
entanglement entropy in interacting systems, a fact that will be particularly
useful in future searches for exotic quantum criticality in models with and
without the sign problem.
[Show abstract][Hide abstract] ABSTRACT: We introduce a generalized loop move (GLM) update for Monte Carlo simulations of frustrated Ising models on two-dimensional lattices with bond-sharing plaquettes. The GLM updates are designed to enhance Monte Carlo sampling efficiency when the system's low-energy states consist of an extensive number of degenerate or near-degenerate spin configurations, separated by large energy barriers to single spin flips. Through implementation on several frustrated Ising models, we demonstrate the effectiveness of the GLM updates in cases where both degenerate and near-degenerate sets of configurations are favored at low temperatures. The GLM update's potential to be straightforwardly extended to different lattices and spin interactions allows it to be readily adopted on many other frustrated Ising models of physical relevance.
Physical Review E 03/2012; 85(3-2):036704. DOI:10.1103/PhysRevE.85.036704 · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Spin ice materials Dy2Ti2O7 and
Ho2Ti2O7 have been the subject of
ongoing interest for over ten years. The cooperative magnetic ground
state can be mapped onto the proton disordered ground state in water
ice, and its residual entropy follows the same Pauling's estimate.
Interestingly it was found in a previous study that, upon dilution of
the magnetic rare earth ions Dy^3+ and Ho^3+ by non-magnetic substitutes
Y^3+, the residual entropy depends non-monotonically on the dilution
level. In this work we investigate through Monte Carlo simulations
microscopic models to account quantitatively for the calorimetric
experimental measurements, and thus also the residual entropies as a
function of dilution. Features of the dilution physics in the specific
heat are captured quantitatively by the microscopic models and the
interplay between dilution and frustration is understood on the basis of
a Bethe lattice calculation. The effect of the dipolar interactions
between magnetic spins are exposed numerically for various dilution
concentrations. Our work explains the previous discrepancy of the
residual entropy between different species of rare earth ions and the
generalized Pauling's estimate.
[Show abstract][Hide abstract] ABSTRACT: Entanglement entropy is a quantity that is desirable to examine at
quantum critical points in condensed matter systems, because it is
expected that sub-leading scaling terms should contain universal
coefficients. In dimensions higher than one, these universal
coefficients (that are sub-leading to the area law) may possibly be used
to identify the universality class of the quantum critical point, much
like the central charge in 1D systems. The recent development of zero
temperature projector methods for the transverse field Ising model in
combination with replica methods for stochastic series expansion quantum
Monte Carlo (QMC) allows us to examine this idea, using measurements of
Renyi entanglement entropies. We compare zero- and finite- temperature
QMC results with series expansion, and discuss the scaling of the Renyi
entropies at the 2D critical point in the transverse field Ising model.
[Show abstract][Hide abstract] ABSTRACT: We use a Loop-Ratio Valence Bond quantum Monte Carlo algorithm to study
the scaling of the bipartite Renyi entanglement entropy in the 2D
Heisenberg ground state. We uncover the surprising result that
finite-size scaling supports a logarithmic correction to the entropic
area law even with the absence of corners in the entangled region. In
addition, examining the scaling within a single system, we observe an
aspect-ratio dependent scaling term resembling the ``conformal
distance'' term that appears in one-dimensional systems with conformal
symmetry.
[Show abstract][Hide abstract] ABSTRACT: The resonating valence bond (RVB) state on a two-dimensional lattice is
a superposition of all permutations of singlet spin pairs. This
wavefunction was first proposed by Anderson as a simple spin liquid
ground state, showing no long range order at T=0. Using a loop-algorithm
Monte Carlo method that samples all nearest-neighbor singlet pairs, we
examine the entanglement entropy of the nearest neighbor SU(2) RVB
wavefunction on the square lattice. In addition to the area law, we show
that the entanglement entropy splits into two branches, due to the
different topological sectors of the RVB wavefunction. These branches
individually scale with a logarithmic dependence on the size of the
entangled region, the functional form of which appears to be similar to
the conformal distance observed in scaling at conformal critical points
in 1D. We comment on the implication for the search for topological
order, and on generalizations of this wavefunction, including models
involving SU(N) spins.
[Show abstract][Hide abstract] ABSTRACT: Motivated by recent experiments on the organic materials
κ-(ET)2Cu2(CN)3 and
EtMe3Sb[Pd(dmit)2]2, we numerically
investigate the Mott metal-insulator transition in a system of
interacting, itinerant electrons at half-filling on the two-leg
triangular strip (i.e., zigzag chain). Previous work [1] has revealed
that an exotic ``spin Bose-metal'' (SBM) phase with three gapless modes
is stabilized on the zigzag strip in a pure spin model of Heisenberg
exchange supplemented with four-site cyclic ring exchange, a model
appropriate for describing weak Mott insulators near the Mott
transition. Indeed, a physically appealing picture of the realized SBM
phase is to view it as a particular Mott insulating instability out of a
two-band metal of interacting electrons. Guided by this idea, we perform
large-scale DMRG calculations across the Mott transition in various
Hubbard-type models (e.g., with on-site repulsion, longer-ranged
repulsion, and/or explicit spin exchange terms). We focus on the
successes and failures of describing the insulating phase near the
transition within the SBM framework. Finally, the implications of our
findings to the full 2D triangular lattice will be discussed.[4pt] [1]
D. N. Sheng et al., PRB 79, 250112 (2009).
[Show abstract][Hide abstract] ABSTRACT: Ground states of certain materials can support exotic excitations with a charge equal to a fraction of the fundamental electron charge. The condensation of these fractionalized particles has been predicted to drive unusual quantum phase transitions. Through numerical and theoretical analysis of a physical model of interacting lattice bosons, we establish the existence of such an exotic critical point, called XY*. We measure a highly nonclassical critical exponent η = 1.493 and construct a universal scaling function of winding number distributions that directly demonstrates the distinct topological sectors of an emergent Z(2) gauge field. The universal quantities used to establish this exotic transition can be used to detect other fractionalized quantum critical points in future model and material systems.
[Show abstract][Hide abstract] ABSTRACT: We numerically determine subleading scaling terms in the ground-state
entanglement entropy of several two-dimensional (2D) gapless systems, including
a Heisenberg model with N\'eel order, a free Dirac fermion in the {\pi}-flux
phase, and the nearest-neighbor resonating-valence-bond wavefunction. For these
models, we show that the entanglement entropy between cylindrical regions of
length x and L - x, extending around a torus of length L, depends upon the
dimensionless ratio x/L. This can be well-approximated on finite-size lattices
by a function ln(sin({\pi}x/L)), akin to the familiar chord-length dependence
in one dimension. We provide evidence, however, that the precise form of this
bulk-dependent contribution is a more general function in the 2D thermodynamic
limit.
[Show abstract][Hide abstract] ABSTRACT: We study the ground state phase diagram of a two-dimensional kagome lattice
spin-1/2 XY model (J) with a four-site ring exchange interaction (K) using
quantum Monte Carlo simulations. We find that the superfluid phase, existing in
the regime of small ring exchange, undergoes a direct transition to a Z_2
quantum spin liquid phase at (K/J)_c ~ 22, which is related to the phase
proposed by Balents, Girvin and Fisher [Phys. Rev. B, 65 224412 (2002)]. The
quantum phase transition between the superfluid and the spin liquid phase has
exponents z and \nu falling in the 3D XY universality class, making it a
candidate for an exotic XY* quantum critical point, mediated by the
condensation of bosonic spinons.
[Show abstract][Hide abstract] ABSTRACT: We compute the bipartite entanglement properties of the spin-half
square-lattice Heisenberg model by a variety of numerical techniques that
include valence bond quantum Monte Carlo (QMC), stochastic series expansion
QMC, high temperature series expansions and zero temperature coupling constant
expansions around the Ising limit. We find that the area law is always
satisfied, but in addition to the entanglement entropy per unit boundary
length, there are other terms that depend logarithmically on the subregion
size, arising from broken symmetry in the bulk and from the existence of
corners at the boundary. We find that the numerical results are anomalous in
several ways. First, the bulk term arising from broken symmetry deviates from
an exact calculation that can be done for a mean-field Neel state. Second, the
corner logs do not agree with the known results for non-interacting Boson
modes. And, third, even the finite temperature mutual information shows an
anomalous behavior as T goes to zero, suggesting that T->0 and L->infinity
limits do not commute. These calculations show that entanglement entropy
demonstrates a very rich behavior in d>1, which deserves further attention.