Amplitudeexponent relations for 3D O(n) spin models
Making use of the conformal invariance of systems of statistical mechanics at a
continuous phase transition, a rather complete understanding of twodimensional
critical phenomena is possible [1]. Among the most prominent
predictions of the resulting conformal field theory for critical systems in two
dimensions are finitesize scaling laws for semifinite geometries including
the exact scaling amplitudes, in particular the scaling of correlation lengths
on
cylinders of finite circumference L [2]. Although the bulk of exact
results from conformal field theory is not available for threedimensional
systems, a similar scaling relation has been numerically found probable for the Ising
model on the 3D geometry S^{1} x S^{1} x R
with a special choice of
boundary conditions [3].
Using highprecision Monte Carlo simulations and an elaborate set of data analysis
tools, we examine the finitesize scaling behavior of correlation lengths of the more
general O(n) spin models on this 3D geometry. We find that,
for the Ising, XY and
generalized Heisenberg models, a scaling relation of the type valid in 2D can be
retained, when using antiperiodic instead of periodic boundary conditions
[4,5]. For the case of periodic boundary conditions, a
generalized scaling law is conjectured [6]. A striking mismatch between
the n > oo extrapolation of the simulation results and analytical
calculations is traced back to a breakdown of the identification of this limit with
the spherical model [6]. For another, conformally flat 3D geometry of
the type S^{2} x R the relations valid in 2D can be extended to 3D by
analytical means [7]. Earlier attempts of a numerical reproduction of
this result failed due to problems with the discretization of the sphere S^{2}
[8]. We use slightly irregular lattices for this
discretization and
find the generalized scaling relation valid for this geometry to high precision
[9].
Dynamical quadrangulations and quantum gravity
Dynamical triangulations provide a discrete, nonperturbative approach to quantum
gravity alternative to the various further directions pursued, including string
theory, loop quantum gravity, noncommutative geometry or causal sets
[10]. Starting from a pathintegral approach, the
integral over metrics
is being performed as a discrete sum over simplicial complexes of a given topology
[11,12,13]. The Euclidean case in two dimensions can be
solved exactly as a combinatorial problem by virtue of the powerful methods of matrix
integrals and generating functions [14,15]. Additionally, matrix
models for the coupling of certain matter variables such as Ising, Potts or O(n)
models to the triangulations can be formulated and partially solved
[16]. For the general case, the dressing of the
weights of conformal
matter of central charge c < 1 coupled to the Euclidean
dynamical triangulations
model in two dimensions is conjectured by the KPZ/DDK formula from Liouville theory
[17,18,19], the predictions of which coincide with all
known exact results.
We consider the coupling of the sixvertex F model to
planar, "fat" \phi^4
Feynman diagrams corresponding to dynamical "quadrangulations". On regular
lattices, vertex models exhibit rich phase diagrams [20] and the F
model has a critical line of central charge c=1, i.e.,
directly at the "c=1 barrier'' where the KPZ/DDK description of the coupling of matter variables to the
dynamical triangulations model breaks down. A matrix model for this problem can be
formulated; its analytical treatment, however, does not reveal the detailed,
finitesize behavior of most matter related and some graph related properties
[21,22]. We thus analyze the coupled system with a set
of
Monte Carlo simulations of fluctuating graphs and coupled matter variables. While
numerical simulations of dynamical triangulations have been extensively
applied before, for the case of quadrangulations a set of update moves had to be
devised first, where a suitable generalization of the "Pachner moves"
[23] has to be augmented with additional "twolink
flips" along
selfenergy subgraphs to ensure ergodicity [24]. To alleviate the severe
critical slowing down of the dynamics resulting from the fractal structure of the
graphs, a nonlocal, clustertype update in the form of the "babyuniverse surgery"
method is generalized to the quartic case [25,26]. From the
regularlattice system as well as the matrix model treatment, the system is expected
to exhibit a KosterlitzThouless phase transition to an antiferroelectrically
ordered phase [20,21,22]. An order parameter for
this transition on a random lattice can only be defined indirectly through a duality
transformation of the model [26]. The scaling analysis is found to be
hampered by the combined effect of a comparable smallness of the effective linear
extent of the graphs resulting from their large fractal dimension and the logarithmic
corrections expected from a theory with central charge c=1. Nevertheless, a careful
analysis of the results guided by a newly performed detailed analysis of the scaling
behavior of the squarelattice model [27] confirms the predictions
of the KPZ/DDK formula [17,18,19] with respect to the
critical exponents related to the staggered polarization of the model. Concerning the
location of the transition point and the order of the transition, agreement can be
achieved with the matrix model results [21,22].
Additionally, various aspects of the backreaction of the matter variables unto
geometric properties of the graphs are investigated, including the coordination
number distribution, the string susceptibility exponent and the internal Hausdorff
dimension d_{h} of the coupled system [26]. With respect to the latter,
we find d_{h} = 4 as for the pure gravity model, in contradiction
with two different
analytical conjectures for the change of d_{h}
with the central charge c [28,29].
In an independent investigation, we consider the fractal properties of geometrical
and FortuinKasteleyn clusters of q = 2, 3 Potts models on
dynamical triangulations
and quadrangulations [30]. On regular lattices, there exists an intimate
relation between the critical properties of the pure Potts model and the tricritical
point of the (annealed) diluted Potts model and the geometrical and FortuinKasteleyn
clusters of the pure model [31]. Both sets of critical exponents are found
to be related by a duality transformation preserving the central charge. We propose
that this relation can be lifted to the level of fluctuating graphs by means of the
KPZ/DDK formula and perform simulations and a finitesize scaling analysis of the
Potts models on dynamical lattices to substantiate this
claim
[30].
Quenched disorder from random graphs
The Harris criterion judges the relevance of uncorrelated, quenched disorder for
altering the universal properties of systems of statistical mechanics close to a
continuous phase transition [32]. In this case, a change of universality
class is expected for models with a positive specific heat exponent α. For
the physically more realistic case of spatially correlated disorder degrees of
freedom, Harris' scaling argument can be generalized, yielding a shifted relevance
threshold oo < α_{c} < 1 known as Luck criterion
[33,34]. The value of α_{c}
depends on the quality and
strength of the spatial disorder correlations as expressed in a socalled
wandering exponent. We consider the effect
of a different, topologically defined
type of disorder on the universal behavior of coupled spin models, namely the result
of connectivity disorder produced by placing spin models on random
graphs. As it turns out, the HarrisLuck argument can be generalized to this
situation, leading to a criterion again involving a suitably defined wandering
exponent of the underlying random graph ensemble. Using a carefully tailored series
of finitesize scaling analyses, we precisely determine the wandering exponents of
the twodimensional ensembles of Poissonian VoronoïDelaunay random lattices as
well as the quantum gravity graphs of the dynamical triangulations model, thus
arriving at explicit predictions for the relevance threshold α_{c} for these
lattices [35].
According to these results, for the case of the Poissonian VoronoïDelaunay
random
graphs the Harris criterion should stay in effect, whereas
for the
dynamical triangulations the threshold is shifted to the remarkably small value
α_{c} = 2. The latter result is in perfect agreement with Monte
Carlo
simulations of the qstates Potts model [36] as well as an available exact
solution for the case of the percolation limit
q > 1 [37].
For the Poissonian VoronoïDelaunay
triangulations, while the Ising case q =
2 with α = 0 is marginal and a change of universal properties
cannot normally be
expected, the q = 3 Potts model,
exhibiting α = 1/3, should be shifted to a new
universality class. Following up on a first exploratory study of this problem for
small graphs [38], we perform highprecision clusterupdate Monte
Carlo
simulations for rather large lattices of up to triangles
to investigate
this model. Astonishingly, however, the (exactly known) critical exponents of the
squarelattice q = 3 Potts model are reproduced to high
precision by our randomgraph
simulations [39,40]. To clarify this situation, a generalized model
introducing a distance dependence of the interactions is currently under
investigation [38,41].
Development and analysis of new Monte Carlo techniques
The multihistogram method [42] is widely applied in Monte Carlo
simulations near phase transition points. It allows for the combination of data from
simulations at different points of the coupling parameter space for the evaluation of
thermal averages as continuous functions of the coupling parameters. Generalizing
the original scheme to the randomcluster representation of the qstate Potts model
and making use of an exactly known property of the density of states in this
formulation, we show how results for noninteger q values
can be found from Monte
Carlo simulations [43]. We analyze the reasons for the failure of a
similar previously proposed technique [44] and suggest a new method to
circumvent the supercritical slowing down of the Potts model with large q
values. In contrast to simulations in the usual energy/magnetization language, the
use of cluster estimators enables a consistent definition of magnetization and
magnetic susceptibility in the broken and unbroken phases even for finitesize
lattices.
Finitesize scaling techniques enable a precise determination of transition
temperatures and critical exponents. Often, different estimates for the same
quantities are available. It is common practice to obtain final estimates from plain
or varianceweighted means of the single estimates. A detailed analysis of the
crosscorrelations between these different estimates based on resampling techniques
such as the jackknife method [45], shows that this approach leads to a
systematic underestimation of statistical errors and additionally introduces a bias
[46]. Instead, the weighting should be done taking
the full covariance
matrix into account.
Zerotemperature properties of spinglass models with continuous symmetry
The problem of clarifying the phase structure and the static and dynamical properties
of spin glass models has attracted an enormous amount of analytical and numerical
work in the past decades [47,48]. Exploration of the
landscape of ground states provides valuable insights in the lowtemperature behavior
of these systems [49]. Due to their relative simplicity, Ising
spin
glass models such as the EdwardsAnderson (EA) Ising model have received much more
attention than the continuous symmetry variants. As a consequence, the properties of
the EA Ising model in two and three dimensions are now comparatively well understood,
whereas for the in many cases more realistic continuoussymmetry cases even the
fundamental question of the existence of a spinglass phase transition is still a
matter of recent debate [50].
Finding groundstates of spin glassy system is an inherently difficult problem known
to be NPhard in general [49]. Thus, for the treatment of reasonably
sized systems, approximation schemes have to be devised. For the analysis of
(classical) O(n) vector spin glasses in two dimensions, we
exploit the fact that
the corresponding Ising model is one of the few cases exactly solvable in polynomial
time through a mapping to a minimumweight perfect matching problem [51]
and apply the socalled "blossom algorithm" to Ising variables randomly embedded in
the vector spins. Inserting this procedure into a specially tailored genetic
algorithm using a clusterexchange crossover operation allows to find numerically
exact ground states for
XY systems of up to
28 x 28 spins
[52]. In contrast to previous claims and widespread
belief, we find the
groundstate of the +/ J case to be unique up to a global
O(n) transformation
for finite systems. Comparing the disorderaveraged fluctuations of the groundstate
energies between systems with periodic and antiperiodic boundary conditions allows
for an estimation of the spin stiffness, for which contradicting predictions have
been made from the replicasymmetry breaking and droplet pictures of the theory of
spin glasses [47,48]. Utilization of various novel sets of
boundary conditions and the socalled "aspectratio scaling" technique enable an
independent and reliable determination of the spin and chiral stiffness
exponents. The results, featuring different stiffness exponents for spin and chiral
degreesoffreedom, strongly support the occurrence of spinchirality decoupling in
this system, a much debated issue over which agreement could not be reached in the
scientific community for the past 25 years [52].
Considering the intrinsically quantum nature of the physical systems modeled by the
EdwardsAnderson spin glass, lowtemperature quantum excitations should be taken into
account for a proper description. This is done via a HolsteinPrimakoff spinwave
approximation, modeling quantum excitations around the (classical) ground states
found by the techniques outlined above. These calculations yield information about
quantum corrections to the magnetization and susceptibilities, about the
participation ratios and spinwave velocities etc. [53].
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