Deepak Dhar and Samarth Chandra- Exact entropy of dimer coverings on some lattices in three or more...
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IntroductionSummary of Results
Survey of earlier resultsDimers on the kagome lattice
Generalization to higher dimensionsOrientational correlations
Concluding Remarks
Exact entropy of dimer coverings on some lattices
in three or more dimensions
Deepak Dhar and Samarth Chandra
Tata Institute of Fundamental Research
Mumbai, INDIA
E S I , Vienna, May 18-31, 2008
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
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8/3/2019 Deepak Dhar and Samarth Chandra- Exact entropy of dimer coverings on some lattices in three or more dimensions
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IntroductionSummary of Results
Survey of earlier resultsDimers on the kagome lattice
Generalization to higher dimensionsOrientational correlations
Concluding Remarks
Outline
Introduction
Summary of Results
Survey of earlier resultsThe Kasteleyn MethodCorrelation functionsHeight representation
Dimers on the kagome latticeGeneralization to higher dimensions
Orientational correlations
Concluding Remarks
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
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8/3/2019 Deepak Dhar and Samarth Chandra- Exact entropy of dimer coverings on some lattices in three or more dimensions
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IntroductionSummary of Results
Survey of earlier resultsDimers on the kagome lattice
Generalization to higher dimensionsOrientational correlations
Concluding Remarks
How many ways can we cover an m n chessboard with 1 2 tiles?
Problem can be generalized to other lattices, higher dimensions.
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
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IntroductionSummary of Results
Survey of earlier resultsDimers on the kagome lattice
Generalization to higher dimensionsOrientational correlations
Concluding Remarks
A geometrical model of phase transitionsHard core models e.g. Percolation, Hard spheres in a box
Nonspherical molecules (needles, discs , stars, bananas ..) can
show a variety of ordered structures. e.g. in liquid crystals, manyphase-transitions
Iso.liq Nem.liq SmecA SmecB Solid
Simplest nonspherical shape is dimers.Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
I d i
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IntroductionSummary of Results
Survey of earlier resultsDimers on the kagome lattice
Generalization to higher dimensionsOrientational correlations
Concluding Remarks
No. of different covers exp(CN)
[ We can set up bounds 2N/4
2N/2
]C is called entropy of dimer coverings per site.
Assuming all dimer coverings are equally likely, we can studydimer-dimer orientation correlation functions
Gxx(R),Gxy(R),Gyy(R)
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
I t d ti
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IntroductionSummary of Results
Survey of earlier resultsDimers on the kagome lattice
Generalization to higher dimensionsOrientational correlations
Concluding Remarks
Outline
Introduction
Summary of Results
Survey of earlier resultsThe Kasteleyn MethodCorrelation functionsHeight representation
Dimers on the kagome latticeGeneralization to higher dimensions
Orientational correlations
Concluding Remarks
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
Introduction
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IntroductionSummary of Results
Survey of earlier resultsDimers on the kagome lattice
Generalization to higher dimensionsOrientational correlations
Concluding Remarks
In a classic paper in 1961, Kasteleyn showed how the entropyof dimer coverings can be determined for arbitrarytwo-dimensional planar graphs.
For three dimensional lattices, few exact results are known.
We show that one can determine C exactly for a class oflattices with dimension 2. Using elementary arguments.
Correlation functions can be calculated exactly. Short ranged.
Some examples of lattices for which we can determine theentropy are given here.
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
Introduction
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IntroductionSummary of Results
Survey of earlier resultsDimers on the kagome lattice
Generalization to higher dimensionsOrientational correlations
Concluding Remarks
Figure: The kagome lattice
C=
1
3 log2Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
Introduction
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IntroductionSummary of Results
Survey of earlier resultsDimers on the kagome lattice
Generalization to higher dimensionsOrientational correlations
Concluding Remarks
Figure: A 3-dimensional lattice with corner-sharing triangles
C = 13 log2
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
Introduction
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IntroductionSummary of Results
Survey of earlier resultsDimers on the kagome lattice
Generalization to higher dimensionsOrientational correlations
Concluding Remarks
Figure: The unit cell of the Na4Ir3O8 lattice. There are 12 Iridium atomsper unit cell (shown in red or blue). Sodium and oxygen atoms are notshown. Atoms belonging to other unit cells are shown in dark red orblack. Here also C = 1
3
log2
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
Introduction
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IntroductionSummary of Results
Survey of earlier resultsDimers on the kagome lattice
Generalization to higher dimensionsOrientational correlations
Concluding Remarks
1
2
Figure: Unit cell of another 3-dimensional lattice obtained bydecorations. Here C = 1
10log 12.
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
Introduction
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Summary of ResultsSurvey of earlier results
Dimers on the kagome latticeGeneralization to higher dimensions
Orientational correlationsConcluding Remarks
The Kasteleyn MethodCorrelation functionsHeight representation
Outline
Introduction
Summary of Results
Survey of earlier resultsThe Kasteleyn MethodCorrelation functionsHeight representation
Dimers on the kagome latticeGeneralization to higher dimensions
Orientational correlations
Concluding Remarks
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
Introduction
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Summary of ResultsSurvey of earlier results
Dimers on the kagome latticeGeneralization to higher dimensions
Orientational correlationsConcluding Remarks
The Kasteleyn MethodCorrelation functionsHeight representation
Summary of Kasteleyns method Arbitrary planar graph Assign arrows to each bond so that each elementary loop has
odd number of counterclockwise bonds.
Figure: Assigning arrows to the planar graph.
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
Introduction
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Summary of ResultsSurvey of earlier results
Dimers on the kagome latticeGeneralization to higher dimensions
Orientational correlationsConcluding Remarks
The Kasteleyn MethodCorrelation functionsHeight representation
Define N N matrix A so that Ai,j = +1(1), if there is aforward (reverse) bond between i and j, 0 otherwise.
Number of confs=
Det(A)
For the M N square lattice one gets
=M
m=1
Nn=1
[cos(m
M + 1) + i cos(
n
N + 1)] (1)
For translationally invariant graphs, the entropy per site has
the general form
C =
20
d
2
20
d
2log DetM(, ) (2)
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
IntroductionS f R l
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Summary of ResultsSurvey of earlier results
Dimers on the kagome latticeGeneralization to higher dimensions
Orientational correlationsConcluding Remarks
The Kasteleyn MethodCorrelation functionsHeight representation
Relation to other models
The similarity to Ising model partition function is not
accidental. The Ising model problem in 2d can be rephrasedas a dimer model with some weights on a suitable graph.
Relation to free-fermion models.
Quantum dimer models : antiferromagnetic spin systemsresonating valence bond approximation
=
dimerpairings({pairing})
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
IntroductionS f R lt
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Summary of ResultsSurvey of earlier results
Dimers on the kagome latticeGeneralization to higher dimensions
Orientational correlationsConcluding Remarks
The Kasteleyn MethodCorrelation functionsHeight representation
Correlation functions Heilmann and Lieb proved that for all densities = 1, all
correlation functions decay exponentially.
On square lattice, for full packing, Gxx(R) R1/2
on triangular lattice, exponentially damped even at closepacking.
In higher dimensions, on bipartite lattices, Huse et al have
argued that one gets a power-law decay of correlations in alldimensions.
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
IntroductionSummary of Results
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Summary of ResultsSurvey of earlier results
Dimers on the kagome latticeGeneralization to higher dimensions
Orientational correlationsConcluding Remarks
The Kasteleyn MethodCorrelation functionsHeight representation
Height RepresentationFor planar lattices, dimer coverings can be put in one-to-onecorrespondence with a configuration of heights on the dual graph.This is illustrated here.Choose edge labels periodically as shown.
Assign a height h = 0 to some chosen site (origin).
1 1 1
1 1 1
1 1 1
2 2 2
2 2 2
2 22
1 1 1
1
1
1
1
1
1
2
2
2 2
2
2 2
2
2
2 2 2
2
2
2
2
22
2
2
2 2 2
2
1
1
11
1
1
1
1
1
1
1
1
1
1
1
0 00 01 1 11
1 1 1 12 22
1 12
30
Figure: The edges are labelled 1 or 2 as shown. Height assignment for a
dimer configuration.Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
IntroductionSummary of Results
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Summary of ResultsSurvey of earlier results
Dimers on the kagome latticeGeneralization to higher dimensions
Orientational correlationsConcluding Remarks
The Kasteleyn MethodCorrelation functionsHeight representation
If an edge belongs to boundary of a dimer, then moving along it inthe +x or +y changes height by +1( 1) if it is label 1 (2).
In the continuum limit, the height field has gaussian correlations.
h(r)h(r) K log |r r| (3)
This result is very useful in discussing the continuum limit of this
theory. Conformal invariance.
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
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Summary of ResultsSurvey of earlier results
Dimers on the kagome latticeGeneralization to higher dimensions
Orientational correlationsConcluding Remarks
Outline
Introduction
Summary of Results
Survey of earlier resultsThe Kasteleyn MethodCorrelation functionsHeight representation
Dimers on the kagome lattice
Generalization to higher dimensions
Orientational correlations
Concluding Remarks
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
IntroductionSummary of Results
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Summary of ResultsSurvey of earlier results
Dimers on the kagome latticeGeneralization to higher dimensions
Orientational correlationsConcluding Remarks
Figure: The kagome lattice
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
IntroductionSummary of Results
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ySurvey of earlier results
Dimers on the kagome latticeGeneralization to higher dimensions
Orientational correlationsConcluding Remarks
The kagome LatticeFor the kagome lattice, entropy per site is expressed in terms of a6 6 matrix M(, ).
C =
2
0
d2
2
0
d2
Det[M(, )] (4)
But on evaluating the determinant, we get the remarkably simpleresult
Det[M(, )] = 4Wang and Wu (2007) tried to find an alternative 8-vertexformulation, which is not much simpler.
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
IntroductionSummary of Results
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ySurvey of earlier results
Dimers on the kagome latticeGeneralization to higher dimensions
Orientational correlationsConcluding Remarks
We tried a different approach. This is somewhat similar to earlierwork by Elser [1993].At each red site of the kagome lattice, for a covering C, define a
variablei = +1,if the other site of dimer covering i is above i.
= 1,if below.Select an arbitrary set{i}.How many dimer configurations consistent with this?
If for a site j, j = 1, we can delete the two bonds below jSimilarly, ifj = 1
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
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Survey of earlier resultsDimers on the kagome lattice
Generalization to higher dimensionsOrientational correlations
Concluding Remarks
1
3
2 4
1
1
1
Figure: On deleting two edges from each red site, the kagome lattice
breaks into disconnected chains. The values of the variables at some ofthe red sites are indicated by numbers in parantheses.
The lattice breaks up into mutually disconnected chains.Assume free boundary conditions on the right edge: Vertices at
the right edge may be covered by dimer, or left uncovered.Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
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Survey of earlier resultsDimers on the kagome lattice
Generalization to higher dimensionsOrientational correlations
Concluding Remarks
With this choice of boundary conditionsEach chain can be covered by dimers in exactly one way.
There N/3 red sites, and 2N/3 choices of {}. Hence the number of dimer configurations = 2N/3, and
C =1
3log2.
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
IntroductionSummary of Results
S f li l
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Survey of earlier resultsDimers on the kagome lattice
Generalization to higher dimensionsOrientational correlations
Concluding Remarks
1 1 1 1 1 1
2 2 2 2 2 2
3
3
3
3
3
3
333333
Figure: Three different weights of bonds for the the kagome lattice.
If we attach a weight z1 ( z2) with a bond coming to a red vertexfrom above(below), and z3 to horizontal vertices, we get
= (z1 + z2)N/3z
N/63
or, C =16 log[(z1 + z2)
2
z3].Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
IntroductionSummary of Results
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Survey of earlier resultsDimers on the kagome lattice
Generalization to higher dimensionsOrientational correlations
Concluding Remarks
Outline
Introduction
Summary of Results
Survey of earlier resultsThe Kasteleyn MethodCorrelation functionsHeight representation
Dimers on the kagome lattice
Generalization to higher dimensions
Orientational correlations
Concluding Remarks
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
IntroductionSummary of Results
Survey of earlier results
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Survey of earlier resultsDimers on the kagome lattice
Generalization to higher dimensionsOrientational correlations
Concluding Remarks
The argument can easily be extended to higher dimensionallattices.
Consider a 3-dimensional simple cubic lattice, with bonds onlyin the z-direction. This is a collection of 1-dimensional chains.
We make this into a three dimensional lattice, by adding links
between the vertical lines.
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
IntroductionSummary of Results
Survey of earlier results
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Survey of earlier resultsDimers on the kagome lattice
Generalization to higher dimensionsOrientational correlations
Concluding Remarks
We define the added sites as red sites, and assign a variable
to each red site. Then as before, there is exacly one dimer configuration,
consistent with a given {}.
Hence we get C = 13
log 2.
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
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IntroductionSummary of Results
Survey of earlier results
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Survey of earlier resultsDimers on the kagome lattice
Generalization to higher dimensionsOrientational correlations
Concluding Remarks
Finally,We are not restricted to bivariate s.
1
2
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
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yDimers on the kagome lattice
Generalization to higher dimensionsOrientational correlations
Concluding Remarks
Select a p1 p2 p3 unit cell.
A red site is added to body-center of a cube to three or fourneighboring vertical lines by two edges each, formingcorner-sharing triangles.
Here 1 takes 4 values, and 2 takes 3 values.
Repeat periodically in space
In this case, C = 110
log 12.
One can also have different weights to different edges.
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
IntroductionSummary of Results
Survey of earlier results
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yDimers on the kagome lattice
Generalization to higher dimensionsOrientational correlations
Concluding Remarks
Outline
Introduction
Summary of Results
Survey of earlier resultsThe Kasteleyn MethodCorrelation functionsHeight representation
Dimers on the kagome lattice
Generalization to higher dimensions
Orientational correlations
Concluding Remarks
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
IntroductionSummary of Results
Survey of earlier results
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Dimers on the kagome latticeGeneralization to higher dimensions
Orientational correlationsConcluding Remarks
Calculating orientational correlations
Consider the kagome lattice, for simplicity.
The variables {i} at different sites are independent randomvariables.
Orientations of dimers at sites in different horizontal rows areuncorrelated.
By symmetry, on the same row, sites are uncorrelated ifdistance > 2.
Orientations of dimers at sites i and j are uncorrelated, ifdistance dij > 2.
Similar result holds in higher dimensions.
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
IntroductionSummary of Results
Survey of earlier results
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Dimers on the kagome latticeGeneralization to higher dimensions
Orientational correlationsConcluding Remarks
Outline
Introduction
Summary of Results
Survey of earlier resultsThe Kasteleyn MethodCorrelation functionsHeight representation
Dimers on the kagome lattice
Generalization to higher dimensions
Orientational correlations
Concluding Remarks
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
IntroductionSummary of Results
Survey of earlier resultsDi th k l tti
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Dimers on the kagome latticeGeneralization to higher dimensions
Orientational correlationsConcluding Remarks
The {} variables provide a set of independent, local variablesfor a complete description of the dimer configuration.However, configurations of other dimers are fully fixed by {}.This is a reflection of the strong correlations in the system.Changing one will affect the dimer configuration very far
away.Orientations at other sites are given by variables that arecomplicated nonlocal functions of these.
Our approach is qualitatively different from the usualtreatment of integrable d-dimensional models in statistical
physics.In the standard transfer matrix approach, a d -dimensionalsystem is studied by thinking of it as a set (d 1)-dimensionallayers, which then have some conservation laws.Here we break a d -dimensional system directly into 1-dchains.Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
IntroductionSummary of Results
Survey of earlier resultsDimers on the kagome lattice
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Dimers on the kagome latticeGeneralization to higher dimensions
Orientational correlationsConcluding Remarks
ReferenceDeepak Dhar and Samarth Chandra, Phys. Rev. Lett., 100,120602(2008). [ cond-mat/0711.0971]
Thank You.
Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or
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