Deepak Dhar and Samarth Chandra- Exact entropy of dimer coverings on some lattices in three or more dimensions

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

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
http://find/http://goback/


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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



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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.



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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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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)


I t d ti


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Concluding Remarks

Outline

Introduction

Summary of Results




Concluding Remarks


Introduction


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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.


Introduction


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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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Concluding Remarks

Figure: A 3-dimensional lattice with corner-sharing triangles

C = 13 log2


Introduction


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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


Introduction


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Concluding Remarks

1

2

Figure: Unit cell of another 3-dimensional lattice obtained bydecorations. Here C = 1

10log 12.


Introduction


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Summary of ResultsSurvey of earlier results


Orientational correlationsConcluding Remarks

The Kasteleyn MethodCorrelation functionsHeight representation

Outline

Introduction

Summary of Results




Concluding Remarks


Introduction


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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.


Introduction


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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)


IntroductionS f R l


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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})


IntroductionS f R lt


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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.




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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



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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.




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Outline

Introduction

Summary of Results


Dimers on the kagome lattice

Generalization to higher dimensions


Concluding Remarks




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Figure: The kagome lattice




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ySurvey of earlier results



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.




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ySurvey of earlier results



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




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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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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.



S f li l


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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

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or, C =16 log[(z1 + z2)

2

z3].Deepak Dhar and Samarth Chandra Exact entropy of dimer coverings on some lattices in three or


S f li lt


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Concluding Remarks

Outline

Introduction

Summary of Results





Concluding Remarks



Survey of earlier results


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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.





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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.



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Concluding Remarks

Finally,We are not restricted to bivariate s.

1

2





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yDimers on the kagome lattice


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.





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yDimers on the kagome lattice


Concluding Remarks

Outline

Introduction

Summary of Results





Concluding Remarks





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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.





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Outline

Introduction

Summary of Results





Concluding Remarks



Survey of earlier resultsDi th k l tti


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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




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ReferenceDeepak Dhar and Samarth Chandra, Phys. Rev. Lett., 100,120602(2008). [ cond-mat/0711.0971]

Thank You.


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Deepak Dhar and Samarth Chandra- Exact entropy of dimer coverings on some lattices in three or more dimensions