5,745 research outputs found
Singular Behaviour of the Potts Model in the Thermodynamic Limit
The self-duality transformation is applied to the Fisher zeroes near the
critical point in the thermodynamic limit in the q>4 state Potts model in two
dimensions. A requirement that the locus of the duals of the zeroes be
identical to the dual of the locus of zeroes (i) recovers the ratio of specific
heat to internal energy discontinuity at criticality and the relationships
between the discontinuities of higher cumulants and (ii) identifies duality
with complex conjugation. Conjecturing that all zeroes governing ferromagnetic
critical behaviour satisfy the latter requirement, the full locus of Fisher
zeroes is shown to be a circle. This locus, together with the density of zeroes
is shown to be sufficient to recover the singular form of all thermodynamic
functions in the thermodynamic limit.Comment: Contribution to Lattice 97, LaTeX, 3 pages, 0 figure
On the algebraic approach to solvable lattice models
We develop an algebraic approach to solvable lattice models based on a chain
of algebras obeyed by the models. In each subalgebra we use a unit, giving a
chain of ideals. Thus, we divide the models into distinct sectors which do not
mix. This method gives the usual Bethe anzats results in cases it is known, but
generalizes it to non integrable models. We exemplify the method on the
Temperley--Lieb and Fuss--Catalan algebras. For the Fuss--Catalan algebra we
show that the ground state energy is zero and there is a mass gap of one for
, and that for we seem to get an RCFT as the scaling
limit.Comment: 14 pages, one table. Minor typos correcte
Determinant of the Potts model transfer matrix and the critical point
By using a decomposition of the transfer matrix of the -state Potts Model
on a three dimensional simple cubic lattice its
determinant is calculated exactly. By using the calculated determinants a
formula is conjectured which approximates the critical temperature for a
d-dimensional hypercubic lattice.Comment: 8 page
Fisher Zeroes and Singular Behaviour of the Two Dimensional Potts Model in the Thermodynamic Limit
The duality transformation is applied to the Fisher zeroes near the
ferromagnetic critical point in the q>4 state two dimensional Potts model. A
requirement that the locus of the duals of the zeroes be identical to the dual
of the locus of zeroes in the thermodynamic limit (i) recovers the ratio of
specific heat to internal energy discontinuity at criticality and the
relationships between the discontinuities of higher cumulants and (ii)
identifies duality with complex conjugation. Conjecturing that all zeroes
governing ferromagnetic singular behaviour satisfy the latter requirement gives
the full locus of such Fisher zeroes to be a circle. This locus, together with
the density of zeroes is then shown to be sufficient to recover the singular
form of the thermodynamic functions in the thermodynamic limit.Comment: 10 pages, 0 figures, LaTeX. Paper expanded and 2 references added
clarifying duality relationships between discontinuities in higher cumulant
The antiferromagnetic transition for the square-lattice Potts model
We solve the antiferromagnetic transition for the Q-state Potts model
(defined geometrically for Q generic) on the square lattice. The solution is
based on a detailed analysis of the Bethe ansatz equations (which involve
staggered source terms) and on extensive numerical diagonalization of transfer
matrices. It involves subtle distinctions between the loop/cluster version of
the model, and the associated RSOS and (twisted) vertex models. The latter's
continuum limit involves two bosons, one which is compact and twisted, and the
other which is not, with a total central charge c=2-6/t, for
sqrt(Q)=2cos(pi/t). The non-compact boson contributes a continuum component to
the spectrum of critical exponents. For Q generic, these properties are shared
by the Potts model. For Q a Beraha number [Q = 4 cos^2(pi/n) with n integer]
the two-boson theory is truncated and becomes essentially Z\_{n-2}
parafermions. Moreover, the vertex model, and, for Q generic, the Potts model,
exhibit a first-order critical point on the transition line, i.e., the critical
point is also the locus of level crossings where the derivatives of the free
energy are discontinuous. In that sense, the thermal exponent of the Potts
model is generically nu=1/2. Things are profoundly different for Q a Beraha
number, where the transition is second order, with nu=(t-2)/2 determined by the
psi\_1 parafermion. As one enters the adjacant Berker-Kadanoff phase, the model
flows, for t odd, to a minimal model of CFT with c=1-6/t(t-1), while for t even
it becomes massive. This provides a physical realization of a flow conjectured
by Fateev and Zamolodchikov in the context of Z\_N integrable perturbations.
Finally, we argue that the antiferromagnetic transition occurs as well on other
two-dimensional lattices
Chromatic Polynomials for Strip Graphs and their Asymptotic Limits
We calculate the chromatic polynomials for -vertex strip graphs of the
form , where and are various subgraphs on the
left and right ends of the strip, whose bulk is comprised of -fold
repetitions of a subgraph . The strips have free boundary conditions in the
longitudinal direction and free or periodic boundary conditions in the
transverse direction. This extends our earlier calculations for strip graphs of
the form . We use a generating function method. From
these results we compute the asymptotic limiting function ; for this has physical significance as
the ground state degeneracy per site (exponent of the ground state entropy) of
the -state Potts antiferromagnet on the given strip. In the complex
plane, is an analytic function except on a certain continuous locus . In contrast to the strip graphs, where
(i) is independent of , and (ii) consists of arcs and possible line segments
that do not enclose any regions in the plane, we find that for some
strip graphs, (i) does depend on and
, and (ii) can enclose regions in the plane. Our study elucidates the
effects of different end subgraphs and and of boundary conditions on
the infinite-length limit of the strip graphs.Comment: 33 pages, Latex, 7 encapsulated postscript figures, Physica A, in
press, with some typos fixe
Dynamics of parental work hours, job insecurity, and child wellbeing during middle childhood in Australian dual-income families
This study examines the relationship between parental employment characteristics and child well-being during middle childhood in Australian dual-earner families. Parental employment provides important resources for childrenâs wellbeing, but may also be associated with variations in parental time availability, parental stress levels and wellbeing, differences in parenting styles and variations in household dynamics. Further, there may be gender differences in how mothersâ and fathersâ employment characteristics relate to child wellbeing, as well as variations by age. Our study contributes to existing research by 1) examining longitudinal data that enables us to examine changes in the association between parental work hours, job insecurity and child wellbeing, within and across parent-child relationships; 2) focusing on dual-employed households to examine the effects of mothersâ and fathersâ employment characteristics on girlsâ and boysâ wellbeing; and 3) testing possible mediators in the relationship between parental employment characteristics and child well-being. Drawing on 3 waves of data from two cohorts of the Longitudinal Study of Australian Children (N = 3,216), from 2004 to 2012, we find that mothers who work long hours on average over the study period have children with poorer socio emotional development, while fathers with increasing work hours have children with poorer socio-emotional development. Mothersâ job security is associated with better child development comparing both across mothers and within mothers over time. We find little evidence that these associations are mediated by parenting style or work-family balance, suggesting further research is needed to understand the mechanisms linking parental employment with childrenâs outcomes
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