Quantum dimer model: from triangular to square lattice and vice versa

Abstract

Quantum dimer models (QDM) were introduced to describe high-temperature supraconductors, and resulted from short-range resonating valence bond physics. Though connection to experimental results remains an open topic, QDM are nowadays widely studied for their theoretical interest (topological order, con- formal quantum critical points, . . .), particularly models on square and on tri- angular lattices. A simple geometric construction allows comparisons between these two models: adding a diagonal to the square lattice creates a lattice that is equivalent to the triangular one. A coefficient called fugacity applied to the diagonal then interpolates between the square and the triangular models.
Elementary excitations are known to be gapless in the square model: they are the “resonons” and are related to the dimer density. On the triangular lattice, in contrast, elementary excitations are strongly believed to be gapped and are called visons; they are non-local in terms of dimers. Mechanism of the gap apparition will be studied: what happens to resonons between the square and the triangular lattices, and how do visons behave for small fugacities?

Full text

Master Thesis in Physics
S´ebastien Perseguers
Quantum Dimer Model
from Triangular to Square Lattice and Vice Versa
October 2005 — February 2006
Institute : ITP
Chair : CPMSC
Supervisor : Prof. Dmitri Ivanov

Abstract
Quantum dimer models (QDM) were introduced to describe high-temperature
supraconductors, and resulted from short-range resonating valence bond physics.
Though connection to experimental results remains an open topic, QDM are
nowadays widely studied for their theoretical interest (topological order, con-
formal quantum critical points, ...), particularly models on square and on tri-
angular lattices. A simple geometric construction allows comparisons between
these two models: adding a diagonal to the square lattice creates a lattice that
is equivalent to the triangular one. A coefficient called fugacity applied to the
diagonal then interpolates between the square and the triangular models.
Elementary excitations are known to be gapless in the square model: they
are the “resonons” and are related to the dimer density. On the triangular
lattice, in contrast, elementary excitations are strongly believed to be gapped
and are called visons; they are non-local in terms of dimers. Mechanism of the
gap apparition will be studied: what happens to resonons between the square
and the triangular lattices, and how do visons behave for small fugacities?
i

Contents
1 Introduction 1
1.1 Conventions and Notations . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Coordinates and Unit Cells . . . . . . . . . . . . . . . . . 2
1.1.2 Plaquettes . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Hamiltonian and Ground State . . . . . . . . . . . . . . . . . . . 3
2 Preliminary Results 5
2.1 Expected Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Flippable Plaquettes . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 P(γ)for Small Fugacities . . . . . . . . . . . . . . . . . . 8
2.2.2 P(γ)for All Values of the Fugacity . . . . . . . . . . . . . 9
3 Resonons 11
3.1 Introduction.............................. 11
3.2 Conservation Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 Energy Matrix E........................... 12
3.3.1 Commutation Relations . . . . . . . . . . . . . . . . . . . 12
3.3.2 Explicit Calculation . . . . . . . . . . . . . . . . . . . . . 13
3.3.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.4 Normalization Matrix N....................... 15
3.4.1 Wick’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 15
3.4.2 Further Calculations . . . . . . . . . . . . . . . . . . . . . 16
3.5 Expected Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.5.1 Square Lattice . . . . . . . . . . . . . . . . . . . . . . . . 18
3.5.2 Gap and Fugacity . . . . . . . . . . . . . . . . . . . . . . 19
3.5.3 Triangular Lattice . . . . . . . . . . . . . . . . . . . . . . 19
4 Visons 21
4.1 Introduction.............................. 21
4.2 Energy Matrix E........................... 22
4.2.1 Explicit Calculation . . . . . . . . . . . . . . . . . . . . . 22
4.2.2 Triangular Lattice . . . . . . . . . . . . . . . . . . . . . . 23
4.2.3 Square Lattice and Small Fugacity . . . . . . . . . . . . . 24
4.3 Expected Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5 Conclusion 27
A Pfaffian Techniques 29
A.1 Kasteleyn’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 29
A.2 Grassmann Variables . . . . . . . . . . . . . . . . . . . . . . . . . 30
ii

CONTENTS iii
B Resonons: Eand N32
B.1 Energy Matrix E........................... 32
B.2 Normalization Matrix N....................... 32
C Visons: Eand N34
C.1 Energy Matrix E........................... 34
C.2 Normalization Matrix N....................... 34

Chapter 1
Introduction
Following Anderson’s ideas about resonating valence bonds in metals [1], quan-
tum dimer models (QDM) were introduced by Rokhsar and Kivelson for the
square lattice [2]; a number of generalizations then appeared, in particular the
triangular lattice model [3]. When all sites of the lattice are involved in exactly
one dimer (hard-core constraint), and for the special choice of coupling con-
stants v=t= 1 in the Hamiltonian H(Eq. 1.1), QDM exhibit a so called “RK
point”: at this point His non-negative and its ground state is constructed as
a sum over all states with equal amplitude. On the square lattice, elementary
excitations (resonons) are known to be gapless, while on the triangular one they
are believed to be gapped and non-local in terms of dimers [4]; they are called
visons.
H=X

v|ih | +| ih |−t| ih | +| ih |.(1.1)
Geometric Construction
A square lattice can be made from a triangular lattice by removing bonds in
one direction (Fig. 1.1). Setting a variable weight µ(0 ≤µ≤1) called fugacity
to these bonds allows one to perform a continuous deformation between the two
lattices: µ= 0 gives the square lattice while µ= 1 stands for the triangular one.
The main part of this diploma will be the study of resonons at all fugacities (in
particular the gap apparition with µ≪1). Visons will also be studied, but less
analyticaly than resonons (different kind of calculations involved).
Figure 1.1: From the triangular to the square lattice: we first skew the triangular
lattice and then apply a weight µto the diagonal.
1

CHAPTER 1. INTRODUCTION 2
τ1 2 3 4 5 6
Direction ˆx ˆx +ˆy ˆy −ˆx −ˆx −ˆy −ˆy
Table 1.1: The six directions of the bonds in the lattice.
x
y
(a) Ns,x ×Ns,y lattice
12
Y
X
(b) Nd,x ×Nd,y lattice
Figure 1.2: Choice of unit cells: simple (left), and doubled in the ˆ
xdirection
(right).
1.1 Conventions and Notations
Before writing explicit calculations, it is necessary to clearly define the various
notations. The first one relates to the six directions of the bonds in the lattice:
they are labelled by the greek letter τ(Tab. 1.1).
1.1.1 Coordinates and Unit Cells
We consider a Ns,x ×Ns,y lattice, where Ns,i refers to the number of sites in
the ˆıdirection; they are consequently Ns=Ns,xNs,y sites in this lattice. Co-
ordinates and corresponding wave vectors are written with small letters: rand k.
The unit cell needs to be doubled in two cases: when using Pfaffian tech-
niques (App. A), and when working with visons (gauge choice, Chap. 4). Our
convention is to double the unit cell in the ˆ
xdirection, and to write the vectors
of the direct or reciprocal lattice with capital letters: Rand K. Dimensions
of the lattice are then given by Nd,x and Nd,y, and the total number of cells is
Nd=Nd,xNd,y .
In order to locate a site on this lattice, one must give the cell coordinate Rin
which it lies as well as an index α= 1 or 2 that determines its relative position
in the cell (Fig. 1.2). The following relations are verified:
(rx, ry) = (2Rx+α−1, Ry)
(kx, ky) = (Kx/2, Ky)⇒k·r=K·(R+α−1
2ˆ
X),(1.2)
Ns,x = 2Nd,x
Ns,y =Nd,y ⇒Ns= 2Nd.(1.3)
1.1.2 Plaquettes
There are three types of plaquettes in the lattice (Fig. 1.3): square, horizontal
or vertical. Any plaquette can be labelled by the coordinate rof its bottom-left
corner and by an index γ=s,hor v. Because of the symmetry of the lattice it
is useful to group the horizontal and vertical plaquettes: they form the diagonal

CHAPTER 1. INTRODUCTION 3
(a) square (b) horizontal (c) vertical
Figure 1.3: The three types of plaquettes in the lattice: s,hand v. Plaquettes
hand vconstitute the diagonal plaquettes d.
F(s)
1g3
F(s)
3g1
F(h)
1g2
F(h)
2g1
F(v)
3g2
F(v)
2g3
Figure 1.4: The six operators which flip pairs of parallel dimers
plaquettes d.
The sum over all plaquettes which appears in the Hamiltonian can be written
as:
X
plaquettes ≡X
r,γ ≡X
R,α,γ
(1.4)
Operators Acting on Plaquettes
Let us choose a plaquette (r, γ). Operators that flip a pair of parallel dimers
from the direction τ1to the direction τ2in this plaquette are written F(γ)
τ1→τ2(r)
(Fig. 1.4). Operators that project on the subset of configurations containing
two parallel dimers in direction τare written P(γ)
τ(r).
1.2 Hamiltonian and Ground State
The Hilbert space Hconsists of all dimers configurations con the triangular
lattice. In order to interpolate between the square and the triangular lattice [5,
Chap. III], dimers along the diagonal should get a weight µin the ground state
|Gi. The Hamiltonian for the square plaquettes remains unchanged (Eq. 1.1),
but some slight modifications have to be done for the horizontal and vertical
plaquettes. Let c1and c2be two identical dimers configurations, except for an
horizontal plaquette where cihas a pair of parallel dimers in the direction τ=i
(the same reasoning can be done for the vertical plaquettes). The Hamiltonian
should satisfy the following properties:
i ) H|Gi= 0,

CHAPTER 1. INTRODUCTION 4
ii ) hc2|Gi=µhc1|Gi,
i.e. we want Hannihilating the state |c1i+µ|c2i. The Hamiltonian that
interpolates between the square and the triangular lattice is then:
H=H(s)+νH (h)+νH(v),(1.5)
where
H(s)=X
r
P(s)
1(r) + P(s)
3(r)−F(s)
1→3(r)−F(s)
3→1(r),
H(h)=X
r
µP (h)
1(r) + 1
µP(h)
2(r)−F(h)
1→2(r)−F(h)
2→1(r),
H(v)=X
r
µP (v)
3(r) + 1
µP(v)
2(r)−F(v)
3→2(r)−F(v)
2→3(r),
and νis a additionnal parameter introduced for some generalization. If D(c) is
the number of dimers along diagonals in the configuration c, the ground state
is:
|Gi=1
Z(µ)X
c
µD(c)/2|ci,(1.6)
and
Z(µ) = X
c
µD(c).(1.7)
Remark All formulae are correct for a fugacity µstrictly positive. In the limit
µ→0, one should use the original quantum dimer model on square lattice, or
set ν= 0 in Eq. 1.5. Values µ=ν= 1 stand for the model on triangular lattice.

Chapter 2
Preliminary Results
Some general results apply to both resonons and visons excitations: the non-
normalized expected energy is easily calculated by applying two commutators on
the Hamiltonian (Sect. 2.1), This calculation leads to the probability of flipping
a plaquette in the lattice (Sect. 2.2).
2.1 Expected Energy
Resonons and visons excitations can be approximated by variational wave func-
tions of the form:
|ψki= Ψk|Gi=X
r
eik·rΨ(r)|Gi,(2.1)
where Ψ(r) stands for the dimer density (resonons) or for the vison operators.
The state |ψkiis not an eigenvector of the Hamiltonian, but its expected energy
ǫ(k) measured from the ground state provides an upper bound on the lowest
excitations at momentum k. This energy is given by:
ǫ(k) = hHiψk=hΨ†
kHΨki
h؆
kΨki.(2.2)
A slight generalization of Eq. 2.2 is done by introducing a set of operators
{Ψi,1≤i≤n}. Lowest excitations are then upper bounded by:
ǫ= min eigenvalues EN−1,(2.3)
where Eij is the matrix element h؆
iHΨjiand Nij its normalization.
In the case of resonons, a method for computing the matrix Ewould be to
use Grassmann variables for each dimer that appears in the Hamiltonian and
in the dimer density operators, and then to express all correlations in terms
of Green’s functions according to Wick’s theorem. In practice, however, such
calculations are too laborious to be done. A much more efficient method, based
on the zero energy of the ground state |Gi, is the following one:
Let Aand Bbe any operators and |aiand |bitwo states defined as |ai=A|Gi
and |bi=B|Gi. The matrix element Eab equals:
5

CHAPTER 2. PRELIMINARY RESULTS 6
Eab =ha|H|bi=hG|A†HB|Gi
=hG|A†[H, B]|Gi+hG|A†BH|Gi
|{z }
=0
=hG|A†,[H, B]|Gi+hG|[H , B]A†|Gi
=hG|A†,[H, B]|Gi+hG|H BA†|Gi
|{z }
=0
− hG|BH A†|Gi,
hence
hA†HBi+hBHA†i=hA†,[H, B]i.(2.4)
For resonons or visons, the Aand Boperators can be chosen as: A= Ψ(a)
k
and B= Ψ(b)
k. Since Ψ†
k= Ψ−k, the second term in Eq. 2.4 becomes:
hBH A†i=hAHB†i
=hΨ(a)
kHΨ(b)
−ki=hΨ(a)
−kHΨ(b)
ki
=hA†HBi=Eab ,
thus
Eab =1
2hA†,[H, B]i.(2.5)
Because calculations used in this report are based on the variational method,
calculated energies will be higher than real values of the excitations. Therefore,
except special comments, all results should be thought as an upper bound.
2.2 Flippable Plaquettes
An important quantity that arises from the calculations of the resonons and
visons energy is the probability of flipping a plaquette, or more exactly the
expected value of the flips operators in the ground state:
P(γ)=1
NsX
rhF(γ)
τ1→τ2(r) + F(γ)
τ2→τ1(r)i,(2.6)
where τ1and τ2are the two possible directions for setting two pairs of dimers
in a plaquette of type γ. We will consider such probabilities only for very large
lattices, so that the result does not depend on the positions r:
P(γ)=hF(γ)
τ1→τ2+F(γ)
τ2→τ1i.(2.7)
Because of the symmetry of the lattice, the probabilities P(h)and P(v)are
the same for all values of the fugacity, and differ from P(s)except when the
fugacity is equal to one (isotropic triangular lattice). The probabilities P(γ)can
be computed with the use of Grassmann variables: we put such a variable at
each corner of the plaquette and then calculate their correlation by expanding
it in terms of Green’s functions (Fig. 2.1), according to Wick’s theorem.

CHAPTER 2. PRELIMINARY RESULTS 7
1 2
345
1 2
345
g1g2
g4
g3
Figure 2.1: Positions in the unit cell of the Grassmann variables ζi(left), and
the Green’s functions githat appear in Wick’s theorem (right).
Square Plaquettes
The correlation hζ1ζ2ζ3ζ4icounts the number of configurations cwith the four
sites excluded from the lattice, divided by the total number of configurations.
For each configuration c, one can construct two different dimers coverings: with
a pair of horizontal dimers (c1), or with two vertical dimers (c2). Therefore:
P(s)= 2 hζ1ζ2ζ3ζ4i
= 2 (hζ1ζ2ihζ3ζ4i − hζ1ζ3ihζ2ζ4i+hζ1ζ4ihζ2ζ3i)
= 4g2
1+ 2g2g3.(2.8)
For the square plaquettes, we can say that Eq. 2.7 represents the probability
of finding a flippable plaquette because c1and c2have the same amplitude in
the ground state:
P(s)=hF(s)
1→3+F(s)
3→1i=hP(s)
1+P(s)
3i.
Diagonal Plaquettes
By symmetry of the lattice, plaquettes hand vhave the same probability:
P(h)=P(v)=P(d). Calculations will be done for an horizontal plaquette.
Contrary to square plaquettes, the two dimers coverings that can be built with
a flippable plaquette hdo not have the same amplitude in |Gi: the configura-
tion with a pair of diagonal dimers is weighted by a factor µ. The correlation
hζ1ζ2ζ4ζ5igives the probability of finding a plaquette with two horizontal dimers.
Thus:
P(h)= 2µhζ1ζ2ζ4ζ5i
= 2µ(hζ1ζ2ihζ4ζ5i − hζ1ζ4ihζ2ζ5i+hζ1ζ5ihζ2ζ4i)
= 2µ(g2
1+g2
2−g1g4),(2.9)
and one notices that for µ6= 1:
P(h)=hF(h)
1→2+F(h)
2→1i 6=hP(h)
1+P(h)
2i= (1 + µ2)hζ1ζ2ζ4ζ5i.
Green’s Functions
The four Green’s functions needed to compute the probabilities P(s)and P(h)
are given by (App. A):

CHAPTER 2. PRELIMINARY RESULTS 8
g1=1
4π2Zdp A21(p)
=1
16π2Zdp 1−eipx+µ(e−ipy+ei(px+py))
sin2(py) + sin2(px
2) + µ2cos2(py+px
2),(2.10a)
g2=1
4π2Zdp eipyA21(p)
=1
16π2Zdp eipy−ei(px+py)+µ(1 + ei(px+2py))
sin2(py) + sin2(px
2) + µ2cos2(py+px
2),(2.10b)
g3=1
4π2Zdp eipyA12(p)
=1
16π2Zdp −eipy+e−i(px−py)−µ(e2ipy+e−ipx)
sin2(py) + sin2(px
2) + µ2cos2(py+px
2),(2.10c)
g4=1
4π2Zdp ei(px+py)A11(p)
=1
16π2Zdp eipx(1 −e2ipy)
sin2(py) + sin2(px
2) + µ2cos2(py+px
2).(2.10d)
All terms that are multiplied by a factor e±ipygive a zero contribution in
these integrals. In fact, let us write such a term by f(px, py). Since f(px, py+
π) = −f(px, py), it results that the integration over the entire Brillouin zone is
exactly zero. In addition, because the Green’s functions are real, the imaginary
parts vanish too. After some trigonometric simplifications one finds:
g1=1
2π2Zdp sin2(px
2)
∆(p),(2.11a)
g2=µ
2π2Zdp cos2(py+px
2)
∆(p),(2.11b)
g3=−µ
4π2Zdp cos(px) + cos(2py)
∆(p),(2.11c)
g4=1
2π2Zdp sin(py) sin(px+py)
∆(p),(2.11d)
where ∆(p) = 4 sin2(py) + sin2(px
2) + µ2cos2(py+px
2).
Remark Probabilities Pτof finding a dimer in the τdirection are directly
expressed in terms of the Green’s functions g1and g2:
P1=P4=hζ1ζ2i=g1,(2.12a)
P2=P5=µhζ1ζ4i=µ g2,(2.12b)
P3=P6=hζ1ζ3i=g1.(2.12c)
The Green’s function g2is multiplied by µbecause the corresponding configu-
ration gets a weight õfor the diagonal bond.
2.2.1 P(γ)for Small Fugacities
From now on we will consider gi=gi(µ) and P(γ)=P(γ)(µ). In order to
describe P(γ)for small fugacities, we first compute the lowest terms in the
Taylor’s series of the four Green’s function, and then combine them according
to Eq. 2.8 and Eq. 2.9.

CHAPTER 2. PRELIMINARY RESULTS 9
Green’s Function g1
For µ= 0, the Green’s function g1(Eq. 2.11a) becomes:
g1(0) = 1
8π2Zdp sin2(px
2)
sin2(py) + sin2(px
2).(2.13)
To calculate that integral, one easily checks that the result is symmetric over
the Brillouin zone (change of variables (px, py)7→ (2py, px/2), and use of the
periodicity of the functions to integrate from 0 to 2π). Hence:
g1(0) = 1
16π2 Zdp sin2(px
2)
sin2(py) + sin2(px
2)+Zdp sin2(py)
sin2(py) + sin2(px
2)!
=1
16π2Zdp =1
4.
That result is intuitive: dimers have exactly a probability of one fourth to point
in a given direction on the square lattice.
The next term in the Taylor’s series of g1is at least of order two since the
fugacity does not appear at the first order.
Other Green’s Functions
One immediately sees that g2and g3are of order O(µ) (Eq. 2.11b and Eq. 2.11c),
while a numerical integration yields (the last figure in brackets indicates the
precision): g4(0) = 0.0683099(1).Therefore:
g1(µ) = 0.25 + O(µ2),(2.14a)
g2(µ) = O(µ),(2.14b)
g3(µ) = O(µ),(2.14c)
g4(µ) = 0.0683099(1) + O(µ2).(2.14d)
Collecting the values of g1,g2,g3and g4, the probabilities P(s)and P(d)
(Eq. 2.8 and Eq. 2.9) are:
P(s)(µ) = 0.25 + O(µ2),(2.15a)
P(d)(µ) = 0.0908451(1) µ+O(µ3).(2.15b)
2.2.2 P(γ)for All Values of the Fugacity
Probabilities P(s)and P(d)are plotted in Fig. 2.2 for µbetween 0 and 1 (numeri-
cal integrations). However, it is worthful to do some analytical work for the value
µ= 1. As for the square lattice, we use the symmetry of the integrals over the
Brillouin zone to compute g1. After changes of variables (px, py)7→ (2py, px/2)
and (px, py)7→ (px+ 2py+π, −py), one finds:

CHAPTER 2. PRELIMINARY RESULTS 10
0.2 0.4 0.6 0.8 1
0.05
0.1
0.15
0.2
0.25
Figure 2.2: Probabilities P(s)(up) and P(d)(down) as functions of fugacity.
g1(1) = 1
3
1
8π2 Zdp sin2(px
2)
sin2(py) + sin2(px
2) + cos2(py+px
2)
+Zdp sin2(py)
sin2(py) + sin2(px
2) + cos2(py+px
2)
+Zdp cos2(py+px
2)
sin2(py) + sin2(px
2) + cos2(py+px
2)!
=1
24π2Zdp =1
6,(2.16)
result that we were expecting (it is the probability for a dimer to point in a
given direction on the isotropic triangular lattice). Looking at Eq. 2.16, one
immediately sees that g1(1) = g2(1). A numerical integration yields:
g3(1) = −g4(1) = −0.0531984(1),(2.17)
hence
P(s)(1) = P(d)(1) = 0.0933783(1).(2.18)
Notations
Probabilities of flipping a plaquette on the square or on the triangular lattice
are:
P=P(s)(0) = 0.25,(2.19a)
P△=P(γ)(1) = 0.0933783(1).(2.19b)

Chapter 3
Resonons
In this chapter, we will show that:
i ) resonons are gapless on the square lattice for k= (π, π ), and that near
this point the dispersion relation is ǫ(k) = π
8k2,
ii ) at k= (π, π), the gap is a continuous and monotonic function of µand ν,
which behaves as µ ν π
2−1for small fugacities and whose maximum is
0.7856(1) for µ=ν= 1,
iii ) on the triangular lattice, resonons reach their minimum (variational) en-
ergy 0.7726(1) for knear (2π
3,2π
3), while their maximum energy 2.3568(1)
is reached for ||k|| ≪ 1.
3.1 Introduction
Resonons were first introduced by Rokhsar and Kivelson [2] as the elementary
excitations of the quantum dimer model on the square lattice. They showed
that a gapless branch of excitations exists at (π, π) and that the dispersion is
proportional to k2. Other methods based on the bipartite nature of the lattice
were then used to describe the energy of long-wavelength mode: height rep-
resentation [6], mapping to the roughening problem [7] or to electrodynamics
through field theory [8, Chap. VI]. Here we cannot use such methods because
we are interested in the triangular lattice too, which is not bipartite.
The local operator Ψ(r) of the variational wave function Eq. 2.1 is nτ(r),
the dimer density operator on site rfor dimer pointing in the τdirection:
nτ(r)|ci=n†
τ(r)|ci=(|ciif a dimer points from rto r+τin c
0 otherwise.
The set of wave functions is given by:
|k, τ i=nk,τ |Gi=1
√NsX
r
eik·rnτ(r)|Gi,(3.1)
with τ∈ {1,2,3}. According to Eq. 2.3, we have to calculate the 3×3 hermitian
matrices E(energy) and N(normalization) to find the upper bound on the
resonons energy.
11

CHAPTER 3. RESONONS 12
3.2 Conservation Law
The hard-core constraint on the dimer coverings implies that exactly one dimer
is connected at each site:
6
X
τ=1
nτ(r) = 1 ∀r.(3.2)
For lattices with periodic boundary conditions, we also have:
X
r
eik·r= 0 ∀k6=0.(3.3)
Expressing Eq. 3.2 in terms of the three directions τ= 1,2,3 only and
multiplying it by Eq. 3.3 yields:
X
r
eik·rn1(r) + n1(r−ˆ
x) + n2(r) + n2(r−ˆ
x−ˆ
y) + n3(r) + n3(r−ˆ
y)= 0.
We then change the summation index so that all density operators are ap-
plied to the same site r:
X
r
eik·r1 + eikxn1(r) + 1 + ei(kx+ky)n2(r) + 1 + eikyn3(r)= 0,
which is by definition equivalent to:
1 + eikxnk,1+1 + ei(kx+ky)nk,2+1 + eikynk,3= 0.(3.4)
Therefore
ϕ0= (1 + eikx,1 + ei(kx+ky),1 + eiky) (3.5)
is always the eigenvector of Eand Nfor the eigenvalue 0. Since the matrix N
has to be inverted (Eq. 2.3), we must consider the subspace that is orthogonal
to ϕ0,i.e. we have to build 2 ×2 matrices.
3.3 Energy Matrix E
The elements of Eare given by (Eq. 2.5):
Eτ1τ2=1
2h[n−k,τ1,[H, nk,τ2]]i.(3.6)
3.3.1 Commutation Relations
Before computing explicitly the matrix elements Eij, it is worthful to calculate
some commutators between the dimer density operators nτand the plaquettes
operators P(γ)
τand F(γ)
τ1→τ2.
Projection Operators
It is obvious that the operators P(γ)
τalways commute with nτ. We simply have:
hnτ1(r), P (γ)
τ2(s)i= 0 ∀τ1, τ2, γ , r,s.(3.7)

CHAPTER 3. RESONONS 13
Flips Operators
Operators that flip a pair of parallel dimers in a given plaquette always commute
with the dimer density operator, except when this last dimer lies in the plaquette
being flipped. Looking at Fig. 1.4, one easily verifies the following commutators:
hnτ(r), F (s)
1→3(s))i=F(s)
1→3(s)δτ,3(δr,s+δr,s+x)−δτ,1(δr,s+δr,s+y)(3.8a)
hnτ(r), F (s)
3→1(s))i=F(s)
3→1(s)δτ,1(δr,s+δr,s+y)−δτ,3(δr,s+δr,s+x)(3.8b)
hnτ(r), F (h)
1→2(s))i=F(h)
1→2(s)δτ,2(δr,s+δr,s+x)−δτ,1(δr,s+δr,s+x+y)(3.8c)
hnτ(r), F (h)
2→1(s))i=F(h)
2→1(s)δτ,1(δr,s+δr,s+x+y)−δτ,2(δr,s+δr,s+x)(3.8d)
hnτ(r), F (v)
2→3(s))i=F(v)
2→3(s)δτ,3(δr,s+δr,s+x+y)−δτ,2(δr,s+δr,s+y)(3.8e)
hnτ(r), F (v)
3→2(s))i=F(v)
3→2(s)δτ,2(δr,s+δr,s+y)−δτ,3(δr,s+δr,s+x+y)(3.8f )
3.3.2 Explicit Calculation
Using the commutation relations Eq. 3.8, one can without difficulty compute all
matrix elements Eij (only six terms have to be evaluated since Eis hermitian).
We will show here the explicit calculation of the diagonal element E11:
E11 =1
2h[n−k,1,[H, nk,1]]i.
Let us compute the first commutator:
[H, nk,1] = 1
√NsX
r
eik·r[H, n1(r)]
=1
√NsX
r,s
eik·rn1(r), F (s)
1→3(s) + F(s)
3→1(s)+νn1(r), F (h)
1→2(s) + F(h)
2→1(s)
=1
√NsX
s
eik·s1 + eikyF(s)
3→1(s)−F(s)
1→3(s)
+ν1 + ei(kx+ky)F(h)
2→1(s)−F(h)
1→2(s).
The second commutator becomes:
[n−k,1,[H, nk,1]] = 1
√NsX
r
e−ik·r[n1(r),[H, nk,1]]
=1
NsX
r,s
eik·(s−r)1 + eikyn1(r), F (s)
3→1(s)−F(s)
1→3(s)
+ν1 + ei(kx+ky)n1(r), F (h)
2→1(s)−F(h)
1→2(s)
=2
NsX
s1 + cos(ky)F(s)
1→3(s) + F(s)
3→1(s)
+ν1 + cos(kx+ky)F(h)
1→2(s) + F(h)
2→1(s).

CHAPTER 3. RESONONS 14
−π 0 π
−π
0
π
(a) Square Lattice
−π 0 π
−π
0
π
(b) Triangular Lattice
Figure 3.1: The contour map of ε(k) for the square and the triangular lattices.
Extrema are plotted by the bullet points.
We finally have to take the average in the ground state and to divide the result
by two. From the definition of probabilities P(γ)(Eq. 2.6), one finds:
E11 =P(s)1 + cos(ky)+νP(h)1 + cos(kx+ky).(3.9)
The whole matrix Eis given in App. B.1.
3.3.3 Properties
The determinant of Eis zero for all values of µ,νand k, and one easily checks
that ϕ0(Eq. 3.5) is the eigenvector for the eigenvalue 0:
E

1 + eikx
1 + ei(kx+ky)
1 + eiky
=

0
0
0
∀µ, ν, k.
Though we have not calculated the normalization matrix yet, it is instructive
to study the smallest (non-zero) eigenvalue εof Eas a function of k. In fact,
symmetries of the excitations in the Brillouin zone are already contained in this
matrix (the normalization matrix Nthen does not break the general shape of
these symmetries). Some level sets of ε(k) for the square and the triangular
lattices are shown in Fig. 3.1.
Square Lattice
Setting µ=ν= 0, we have:
ε(k) = P2 + cos(kx) + cos(ky).(3.10)
Since ε(π, π) = 0, gapless excitations may exist on the square lattice (we still
need to show that the normalization of such wave functions is not zero).
Triangular Lattice
The eigenvalue εis (µ=ν= 1):
ε(k) = P△3 + cos(kx) + cos(ky) + cos(kx+ky),(3.11)

CHAPTER 3. RESONONS 15
so resonons are always gapped in the triangular lattice; the minimum is reached
at k= (2π
3,2π
3).
The correspondence with the hexagonal Brillouin zone, which is more natural
on the triangular lattice, is easily done by looking at Fig. 3.1(b).
3.4 Normalization Matrix N
Elements of the normalization matrix are given by:
Nτ1τ2(k) = hk, τ1|k, τ2i
=1
NsX
r1,r2
eik·(r2−r1)hn†
τ1(r1)nτ2(r2)i.(3.12)
The dimer-dimer correlation function appearing in Eq. 3.12 can be evaluated
with Pfaffian techniques (App. A); we now have to use the unit cell that is
doubled in the ˆ
xdirection. From Eq. 1.2, elements Nij become:
Nτ1τ2(K) = 1
NsX
R1,α1
R2,α2
eiK·(R2−R1+1
2(α2−α1)ˆ
X)hn†
τ1(R1, α1)nτ2(R2, α2)i.(3.13)
3.4.1 Wick’s Theorem
To calculate the dimer-dimer correlations in Eq. 3.13, we put one Grassmann
variable ζat each extremity of the dimers and use Wick’s theorem. Positions
in the lattice of the four Grassmann variables are:
ζ1↔(R1, α1)ζ′
1↔(R1+dR1, α′
1) (3.14a)
ζ2↔(R2, α2)ζ′
2↔(R2+dR2, α′
2) (3.14b)
where dRiand α′
idepend on τiand αi. The correlation is then expressed as:
hn†
τ1(R1, α1)nτ2(R2, α2)i=σ1σ2hζ1ζ′
1ζ2ζ′
2i,(3.15)
with σi=±1, a factor whose value is determined by the arrows orientation
between ζiand ζ′
iin the directed graph (Fig. A.1).
Remark Since the configurations ciare weighted by a coefficient µD(ci)/2in
the ground state (Eq. 1.6), it follows that the right part of Eq. 3.15 should also
get a weight depending on µwhen τ1= 2 or τ2= 2:
|σi|=(1 if τi∈ {1,3}
µotherwise.
Applying Wick’s theorem for the dimer-dimer correlation yields:
hζ1ζ′
1ζ2ζ′
2i=hζ1ζ′
1ihζ2ζ′
2i − hζ1ζ2ihζ′
1ζ′
2i+hζ1ζ′
2ihζ′
1ζ2i,(3.16)

CHAPTER 3. RESONONS 16
result that is always correct, except for a special positioning of the two dimers:
when they match each other perfectly. In fact, let ζ1=ζ2and ζ′
1=ζ′
2(and so
τ1=τ2). Wick’s theorem gives:
hζ1ζ′
1ζ1ζ′
1i=hζ1ζ′
1ihζ1ζ′
1i − hζ1ζ1ihζ′
1ζ′
1i
|{z }
=0
+hζ1ζ′
1i hζ′
1ζ1i
|{z}
=−hζ1ζ′
1i
= 0,
but the correct correlation value is:
hn2
τ(r)i=hnτ(r)i 6= 0 (in general) (3.17)
Since Wick’s theorem gives zero for this particular positioning, we still can
let the sum run over all dimers positions in Eq. 3.13, and then add the dimer
density (Eq. 3.17):
Nτ1τ2(K) = Wτ1τ2(K) + δτ1,τ2
1
NsX
rhnτ1(r)i,(3.18)
where
Wτ1τ2(K) = 1
NsX
R1,α1
R2,α2
eiK·(R2−R1+1
2(α2−α1)ˆ
X)σ1σ2hζ1ζ′
1ζ2ζ′
2i.(3.19)
3.4.2 Further Calculations
Using the Green’s functions (Eq. A.13) and the definitions of the Grassmann
variables positions (Eq. 3.14), we can rewrite Eq. 3.19 as:
Wτ1τ2(K) = 1
NsN2
dX
R1,R2
α1,α2X
P1,P2
eiK·(R2−R1+1
2(α2−α1)ˆ
X)σ1σ2
Aα′
1α1(P1)Aα′
2α2(P2)ei(P1·dR1+P2·dR2)
−Aα2α1(P1)Aα′
2α′
1(P2)ei(P1+P2)·(R2−R1)eiP2·(dR2−dR1)
+Aα′
2α1(P1)Aα2α′
1(P2)ei(P1+P2)·(R2−R1)ei(P1·dR2−P2·dR1).(3.20)
One notices that this last expression does not depend explicitly on the posi-
tions R1and R2, but only on their difference. The first simplification consists
in introducing a new variable R=R2−R1and in changing the sum over the
positions in the following way (fis a generic function):
X
R1,R2
f(R2−R1) = NdX
R
f(R).
Then, since the vectors K,P1and P2are defined on the reciprocal lattice
we can use the following equalities:
X
R
eiK·R=Ndδ(K) and X
R
ei(K+P1+P2)·R=Ndδ(K+P1+P2).
The first delta function gives a non zero contribution to Wonly for the particular
(and uninteresting) case K= 0. From now, we consider that Kis not zero. The
second delta function allows us to sum over P=P2and to set P1=−K−P.
These considerations yield:

CHAPTER 3. RESONONS 17
z z
z z
21
12
(a) τ= 1
z
z
z
z
1
2
2
1
(b) τ= 2
z
z
z
z
1
1
2
2
(c) τ= 3
Figure 3.2: Grassmann variables positions for the three directions τ. The figure
1 or 2 stands for α, while the arrow gives the sign of σ.
σdR α′
τ= 1 1 (α−1)ˆ
x¯α
τ= 2 −µ eiπα (α−1)ˆ
x+ˆ
y¯α
τ= 3 −eiπ α ˆ
yα
Table 3.1: Position of the Grassmann variable ζ′according to the index αof ζ
and to the direction τ. The symbol ¯αstands for the complementary site of αin
the unit cell: ¯α= 3 −α.
Wτ1τ2(K) = 1
NsX
α1,α2X
P
ei1
2Kx(α2−α1)σ1σ2
−Aα2α1(−K−P)Aα′
2α′
1(P)eiP·(dR2−dR1)
+Aα′
2α1(−K−P)Aα2α′
1(P)e−iP·(dR1+dR2)e−iK·dR2.(3.21)
Because we are interested in very large lattices, we can take the thermody-
namic limit: X
P≈Nd
4π2ZdP.
Finally, using the property Aij(−P) = −Aj i(P) and the equality Ns= 2Nd,
Eq. 3.21 becomes:
Wτ1τ2(K) = 1
8π2ZdP X
α1,α2
ei1
2Kx(α2−α1)+P·(dR2−dR1)σ1σ2
Aα′
2α′
1(P)Aα1α2(K+P)−Aα2α′
1(P)Aα1α′
2(K+P)e−i(K+2P)·dR2.(3.22)
We now need to describe the relative position of ζ′with regard to that of ζ.
The possible dimer configurations are shown in Fig. 3.2, and the corresponding
values of σi,dRiand α′
iare given in Tab. 3.1. After some simplifications, one
finds all elements of N(App. B.2).

CHAPTER 3. RESONONS 18
Conservation Law
As for the matrix E, one can check that ϕ0is the eigenvector of Nfor the
eigenvalue 0:
N

1 + eikx
1 + ei(kx+ky)
1 + eiky
=

0
0
0
∀µ, k.
3.5 Expected Energy
From the calculations of Eand N, we now are able to find ǫ, the lowest upper
bound to the energy, for all fugacities and all vectors of the Brillouin zone.
3.5.1 Square Lattice
In the square lattice (ν= 0, µ→0), there exists a gapless branch of excitations
at k= (π, π). At this particular point, the eigenvector ϕ0is (0,1,0) and the
matrices Eand Nare diagonal. So there are two “polarizations” for resonons,
corresponding to the dimer density along either the ˆ
xor ˆ
ydirection [2]. To
describe excitations near (π, π ), let us move the origin of the Brillouin zone:
k7→ k−(π, π).
Energy Matrix
The matrix Eis identically zero for k=0. At the smallest order in k, diagonal
elements become:
E11 =k2
y
8,E33 =k2
x
8.(3.23)
Normalization Matrix
All elements Nij are zero, except N11 and N33. A numerical integration yields:
lim
µ→0N11(µ) = lim
µ→0N33(µ) = 0.3183099(1) ≈1
π,(3.24)
where 1/π is thought to be the exact value of the normalization. From Eq. 3.23
and Eq. 3.24, it follows that the dispersion relation is quadratic in k:
ǫ(k) = π
8k2+O(k3).(3.25)
Remark N11 and N33 are not continuous in µat 0 for k= (π, π). For µ= 0,
in fact, N11 (same discussion for N33) exhibits two poles at (0,0) and (0, π) in
its integral:
N11(0) = 1
4+1
4π2Zdpcos2px
2cos(2py)−cos(px)
2−cos px−cos 2py2.
Expanding this function near the poles, one finds that the difference between
N11(0 < µ ≪1) and N11 (0) is exactly 1/2π. Hence:
N11(0) = 1
2π6=1
π= lim
µ→0N11(µ).

CHAPTER 3. RESONONS 19
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
Figure 3.3: Resonon gap as a function of µ, with ν= 1 and k= (π, π). This
function is linear for small fugacities: ǫ(µ)≈0.5708 µ.
3.5.2 Gap and Fugacity
We have shown that resonons are gapless on the square lattice, but we know
that any excitation is gapped on the triangular lattice: how does the gap appear
when a small fugacity is given to the diagonal bonds?
Let k= (π, π). Since φ0= (0,1,0) is the eigenvector for the eigenvalue 0,
we need to consider the submatrices:
E′=E11 E13
E31 E33and N′=N11 N13
N31 N33,
where (App. B.1)
E′= 2 νP(d)1 0
0 1.(3.26)
The coefficient νhas a trivial effect on ǫ: it simply multiplies it. The graph
of ǫ(µ) for ν= 1 is shown in Fig. 3.3 (numerical integrations); its maximum is
reached on the triangular lattice (µ=ν= 1): ǫmax(µ) = 0.7856(1).
Gap for Small Fugacities
From Eq. 2.15b and Eq. 3.26, we know that for µ≪1 the elements E′
ii are given
by:
E′
11 =E′
33 = 0.1816901(1) µ ν,
and that the diagonal elements N′
ii approach the constant 1/π (Eq. 3.24), while
N′
13 and N′
31 are at least of order O(µ). Hence:
ǫ(µ) = 0.5707963(1) µ ν +O(µ2).(3.27)
3.5.3 Triangular Lattice
Let µ=ν= 1 (isotropic triangular lattice). We first need to define the 2 ×2
submatrices E′and N′:
E′=MS−1E S MTand N′=M S−1N S MT,

CHAPTER 3. RESONONS 20
0
π
2
π0
π
2
π
0.77
2.36
π
32π
3π
1
1.5
2
2.36
0.77
Figure 3.4: Resonon gap in the triangular lattice for k∈[0, π]2(up) and section
along ˆx +ˆy (down).
with
M=1 0 0
0 1 0and S=ϕ1ϕ2ϕ0,
where ϕ0,ϕ1and ϕ2are the eigenvectors of E(or of Nsince both matrices have
the same eigenvector ϕ0for the eigenvalue 0).
Numerical results for the resonon gap ǫ(k) are shown in Fig. 3.4. Its maxi-
mum 2.3568(1) is reached for ||k|| ≪ 1, while the three points (π, π), (0, π) and
(π, 0) are local minima and have the same energy 0.7856(1). From the discus-
sion of Sect. 3.3.3, we know that the minimum of ǫ(k) should lie near (2π
3,2π
3)
in the direction ˆx +ˆy (symmetry of the excitation):
ǫmin = 0.7726(1) for k= 1.0181(1) 2π
3,2π
3.(3.28)

Chapter 4
Visons
The main results of this chapter are:
i ) there does not exist any local conservation law for visons,
ii ) vison excitations are gapped for all values of the fugacity,
iii ) on the triangular lattice, the variational method provides an upper bound
of 0.158(1) to their energy.
4.1 Introduction
For the triangular lattice, analytic studies show an exponential decay of ground
state correlation functions [3, 9] and so suggest a gapped spectrum. Numerical
simulations [4] agree with this fact, and the lowest-energy excitations are known
as “visons” (a concept that comes from general discussions of dimers liquids in
two dimensions and which is known as a Z2-vortex operator).
A vison operator VΓis defined for any contour Γ whose end points lie either
inside a plaquette or at the lattice boundary:
VΓ|ci= (−1)NΓ(c)|ci,(4.1)
where NΓ(c) is the number of dimers intersecting Γ in c. One immediately sees
that V2
Γ=I. From now on we will consider only horizontal paths Γ that start
from a point of the dual lattice and that end at the right boundary of the lattice.
The product of two visons defined on such paths can be computed in the fol-
lowing way [4]: we first chose a reference dimer configuration (“gauge choice”),
and then we draw a path Γ′that joins the two start points of the visons and
that does not intersect dimers of the reference configuration. The product of
the two visons is then given by VΓ′.
The convention for the reference dimers configuration is shown in Fig. 4.1;
the unit cell has to be doubled in the ˆx direction. They are four distinct vison
operators Vβ(R), β ∈ {1,2,3,4}, per cell Rof the lattice. The variational wave
functions are:
|K, βi=VK,β |Gi=1
√NdX
R
eiK·RVβ(R)|Gi.(4.2)
21

CHAPTER 4. VISONS 22
Y
X
13
24
(a) Unit cell
R
G
(b) Vison path Γ
Figure 4.1: (a) Unit cell used for the visons, and the four start points of the
visons paths Γ. The reference dimer configuration that fixes the gauge is drawn
in orange. (b) The contour Γ (dashed line) of the operator V1(R).
Contrary to resonons, there does not exist any analytical formulae for the
visons normalization (visons are non-local in terms of dimers, so Pfaffian tech-
niques do not apply.)
4.2 Energy Matrix E
Elements of the 4 ×4 energy matrix are given by (Eq. 2.5):
Eβ1β2=1
2h[V−K,β1,[H, VK,β2]]i.(4.3)
Before computing these elements, we first establish some useful commutators
between vison operators and plaquettes operators P(γ)
τ(S, α) and F(γ)
τ1→τ2(S, α):
i ) Vβ(R) always commutes with the projection operators,
ii ) Vβ(R) commutes with the flips operators except when its start point lies
in the plaquette being flipped. For example, we have the following com-
mutator:
[V1(R), F (s)
1→3(S, α)] = 2 δR,Sδα,1V1(S)F(s)
1→3(S,1).
4.2.1 Explicit Calculation
Let us calculate E12. The first commutator (Eq. 4.3) becomes:
[H, VK,2] = 1
√NdX
R
eiK·R[H, V2(R)]
=2
√NdX
S
eiK·SV2(S)F(s)
1→3(S,1) + F(s)
3→1(S,1)
+ν V2(S)F(h)
1→2(S,1) + F(h)
2→1(S,1)
+ν eiKyV2(S+ˆ
Y)F(v)
2→3(S,1) + F(v)
3→2(S,1).
We then have to calculate:
[V−K,1,[H, VK,2]] = 1
√NdX
R
e−iK·R[V1(R),[H, VK,2]] .
Commutators of the type
[V1, V2F] = [V1, V2]F+V2[V1, F ] = V2[V1, F ]

CHAPTER 4. VISONS 23
R1R2R3
Figure 4.2: Products of two visons lying in a flippable plaquette. For the speci-
fied gauge, these products are: V1(R1)V2(R1+ˆ
Y) = 1, V3(R2)V4(R2+ˆ
Y) = −1
and V2(R3)V3(R3) = 1.
appear while developing the last equation. Hence:
[V−K,1,[H, VK,2]] = 2
NdX
R,S
eiK·(S−R)V2(S)V1(R), F (s)
1→3(S,1) + F(s)
3→1(S,1)
+ν V2(S)V1(R), F (h)
1→2(S,1) + F(h)
2→1(S,1)
+ν eiKyV2(S+ˆ
Y)V1(R), F (v)
2→3(S,1) + F(v)
3→2(S,1)
=4
NdX
SV1(S)V2(S)F(s)
1→3(S,1) + F(s)
3→1(S,1)
+ν eiKyV1(S)V2(S+ˆ
Y)F(v)
2→3(S,1) + F(v)
3→2(S,1).
We always can draw a path that does not intersect with any dimers between
the start points of V1(S) and V2(S), and those of V1(S) and V2(S+ˆ
Y); in fact,
the corresponding plaquettes should be flippable and do not contain any dimers
from the reference covering (see Fig. 4.2.1). Hence:
E12 =2
NdX
ShF(s)
1→3(S,1) + F(s)
3→1(S,1) + ν eiKyF(v)
2→3(S,1) + F(v)
3→2(S,1)i
= 2 P(s)+ 2 ν eiKyP(v).
From this example, we can make some rules to compute all matrix elements:
i ) start points β1and β2that are not related by a plaquette do not contribute
to the energy,
ii ) there are three different contributions to the diagonal terms, one for each
type of plaquettes (γ=s, h or v); their value is proportional to P(γ),
iii ) since the start points β1and β2must lie in the same flippable plaque-
tte, the product Vβ1Vβ2equals +1 (except for the vertical plaquette that
crosses the reference dimers configuration),
iv ) the phase is not trivial only for the start points that are related by a
plaquette lying on two different cells.
One finally finds the whole energy matrix (App. C.1).
4.2.2 Triangular Lattice
Setting µ=ν= 1, the matrix Ebecomes:

CHAPTER 4. VISONS 24
0 π 2π
0
π
2
π
A
B1
B2
C1
C2
C3
Figure 4.3: Some level sets of the non-normalized energy ε(K) on the triangular
lattice. The high-symmetry points A,Band Care plotted by the bullet points.
E= 2 P△



3 1 + eiKy0e−iKx
1 + e−iKy3 1 0
0 1 3 1 −eiKy
eiKx0 1 −e−iKy3



,(4.4)
and its determinant is:
det(E) = 32 P4
△15 −cos(Kx)−cos(2Ky) + cos(Kx+ 2Ky).(4.5)
Since the determinant is always greater than zero, it follows that there exists
no local conservation law for the visons.
The smallest eigenvalue εof the energy matrix is:
ε= 2 P△3−r3 + q23 + cos(Kx) + cos(2Ky)−cos(Kx+ 2Ky).(4.6)
Some level sets of ε(K) are shown in Fig. 4.3. Though this energy is not
normalized, symmetries in the Brillouin zone are almost the same as those of
the real excitations [4]. High-symmetry points are:
i ) the high-energy point (6-fold symmetry): A=π, π
2,
ii ) the low-energy points (3-fold symmetry): B1=π
3,π
6and B2=5π
3,5π
6,
iii ) the saddle-points: C1= (0,0), C2=0,π
2and C3= (π, 0).
4.2.3 Square Lattice and Small Fugacity
The matrix Edoes not depend on Kon the square lattice:
E=1
2



1 1 0 0
1 1 0 0
0 0 1 1
0 0 1 1



.(4.7)

CHAPTER 4. VISONS 25
Eigenvectors for the eigenvalue 0 are ϕ0= (1,−1,0,0) and ϕ′
0= (0,0,1,−1),
while ϕ1= (1,1,0,0) and ϕ′
1= (0,0,1,1) are those for the eigenvalue 1. The
vectors ϕ0and ϕ′
0tell us that for all K:
VK,1−VK,2= 0 and VK,3−VK,4= 0,
X
R
eiK·RV1(R)−V2(R)= 0 and X
R
eiK·RV3(R)−V4(R)= 0.
Thus
V1(R) = V2(R) and V3(R) = V4(R).(4.8)
These equalities are trivial for a lattice without diagonal bonds; it results that
the normalization of the corresponding wave function is zero too.
Small Fugacities
Let µ≪1, thus (Eq. 2.15): P(s)= 0.25 + O(µ2) and P(d)=c µ +O(µ3), where
c= 0.090845(1). The smallest eigenvalue of Eis:
ε=c ν 4−q4 + 2cos(Kx) + cos(2Ky)µ+O(µ2),(4.9)
so that it is linear for all K.
4.3 Expected Energy
Though we are not able to write any analytical formulae for the normalization
matrix, we can give results for the expected energy using numerical simulations
or general considerations.
Triangular Lattice
Normalization matrices for the high-symmetry points A,Band Chave been
computed by D. Ivanov using the classical Monte Carlo method (numerical
values are given in App. C.2). The expected energies are:
ǫ(A) = 1.217(1)
ǫ(B1) = 0.158(1), ǫ(B2) = 0.158(1)
ǫ(C1) = 0.217(1), ǫ(C2) = 0.218(1) and ǫ(C3) = 0.216(1)
These variational results are compared in Tab. 4.1 to those found with Monte
Carlo simulations [4].
Method ǫ(A)ǫ(B)ǫ(C)
Variational 1.217(1) 0.158(1) 0.217(1)
Monte Carlo >0.3 0.089(1) 0.114(1)
Table 4.1: Vison gap ǫfor the high-symmetry points A,Band Cusing varia-
tional method or Monte Carlo simulations.

CHAPTER 4. VISONS 26
Small Fugacities
We know that the wave function corresponding to the smallest eigenvalue of E
vanishes for µ≪1 (Eq. 4.8). Hence, the first term of its Taylor’s series should
be at least of order one. Since the energy εis linear in µ(Eq. 4.9), it yields that
its normalized energy is strictly positive, and that visons are always gapped.

Chapter 5
Conclusion
The first part of this report shows a possible way to interpolate the QDM
between square and triangular lattices: we are slightly generalising the usual
Hamiltonian to take into consideration the fugacity applied on the diagonal
bonds. In particular, the choice of the configurations amplitude in the ground
state (which depends on the number of diagonal dimers) requires a “weighted”
version of Wick’s theorem when computing correlations through Grassmann
variables. With these considerations one can naturally define the probability
of flipping plaquettes, which in general differs from the probability of finding
flippable plaquettes.
Though the variational method provides only an upper bound on the energy,
it allows one to find all important results:
i) Resonons are gapless at k= (π, π) on the square lattice, with a quadratic
dispersion relation. A gap appears as soon as a non-zero fugacity is applied
on the diagonal: it is a continuous and monotonic function of µ. The hard-
core constraint on the dimers coverings implies that the wave functions
are lineary dependent (local conservation law).
ii) On the triangular lattice, visons have a much smaller energy than resonons;
this confirms the fact that elementary excitations should be non-local in
terms of dimers. The very nature of visons excitations is completely dis-
tinct from that of resonons: visons are gapped for all fugacities and do
not obey any local conservation law.
General symmetries in the Brillouin zone of the non-normalized energy, how-
ever, looks very similar for both resonons and visons. Further work on this topic
seems highly interesting and would perhaps yield to some analytical formulae
for the visons normalization, at least to a better understanding of the QDM.
Acknowledgments
I thank Dmitri Ivanov for having proposed me such an interesting topic, for
the helpful discussions and comments and for his Monte Carlo simulations that
complete my results.
General thanks also to all people that helped me in one way or another during
not only the diploma but my four years of studies at EPFL.
27

Bibliography
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Res. Bull, vol. 8, 1973.
[2] D. Rokhsar and S. Kivelson, “Superconductivity and the quantum hard-
core dimer gas,” Phys. Rev. Letters, vol. 61, no. 20, 1988.
[3] R. Moessner and S. Sondhi, “Resonating valence bond phase in the trian-
gular lattice,” Phys. Rev. Lett., vol. 86, no. 9, 2001.
[4] D. Ivanov, “Vison gap in the rokhsar-kivelson dimer model on the triangular
lattice,” ArXiv/cond-mat, no. 0403383, 2004.
[5] E. Ardonne, P. Fendley, and E. Fradkin, “Topological order and conformal
quantum critical points,” ArXiv/cond-mat, no. 0311466, 2004.
[6] C. Henley, “Relaxation time for a dimer covering with height representa-
tion,” ArXiv/cond-mat, no. 9607222, 1997.
[7] L. Levitov, “Equivalence of the dimer resonating-valence-bond problem to
the quantum roughening problem,” Phys. Rev. Letters, vol. 64, no. 1, 1990.
[8] E. Fradkin, Field Theories of Condensed Matter Systems, vol. 82 of Fron-
tiers In Physics. Addison-Wesley Publishing Company, 1991.
[9] A. Ioselevich, D. Ivanov, and M. Feigelman, “Ground-state properties of
the rokhsar-kivelson dimer model on the triangular lattice,” Phys. Rev. B,
vol. 66, no. 174405, 2002.
[10] M. Fisher and J. Stephenson, “Statistical mechanics of dimers on a plane
lattice,” Phys. Rev., vol. 132, no. 1411, 1963.
[11] P. Fendley, R. Moessner, and S. Sondhi, “Classical dimers on the triangular
lattice,” ArXiv/cond-mat, no. 0206159, 2003.
[12] P. Kasteleyn, “Statistics of dimers on a lattice,” Physica, vol. 27, no. 1209,
1961.
[13] P. D. Francesco, P. Mathieu, and D. S´en´echal, Conformal Field Theory.
Springer, 1997.
[14] S. Samuel, “The use of anticommuting variable integrals in statistical me-
chanics,” J. Math. Phys., vol. 21, no. 2806, 1980.
[15] F. Wegner, “Expectation values, wick’s theorem and normal ordering,”
2000.
28

Appendix A
Pfaffian Techniques
Some important quantities for the quantum dimer models, such as partition
functions or dimers correlations, can be calculated with the use of Pfaffian tech-
niques [10] and Grassmannian variables [11]. In this appendix only results that
are important for this report are provided, but they are not proved here.
A.1 Kasteleyn’s Theorem
Kasteleyn’s theorem [12] is based on a special disposition of arrows on the links
of a planar graph: the product of their orientation around any even-length closed
path should be -1. Such a disposition is given in Fig. A.1; one notices that the
unit cell of the lattice must be doubled to obey the sign rule. We define an
antisymmetric matrix M, where Mij = 1 if an arrow points from the site ito
the site j,Mij =−1 if the arrow points from jto i, and Mij = 0 otherwise.
The matrix elements Mij that connect two sites on a diagonal link are then
multiplied by the fugacity µ. Kasteleyn’s theorem states that the partition
function Zis given by the Pfaffian of M:
Z= Pf[M].(A.1)
A possible definition of the Pfaffian is the following one [13, Chap. II]: let
Mbe a n×nantisymmetric matrix, then:
Pf[M] = X
π∈Sn
(−1)πMπ1π2Mπ3π4. . . Mπn−1πn,(A.2)
with two constraints on the permutations π:
π1< π2, π3< π4, . . .
π1< π3< π5< . . .
12
y
x
Figure A.1: Arrows orientation defining the matrix M.
29

APPENDIX A. PFAFFIAN TECHNIQUES 30
The main property of the Pfaffian is that its square is the determinant of
the matrix:
Pf2(M) = Det(M) (A.3)
Finally, one can easily verify the equivalence of Eq. 1.7 and Eq. A.2.
A.2 Grassmann Variables
Pfaffian techniques have been reformulated in the language of fermionic path
integrals, or equivalently in terms of anticommuting variables [14]. Introducing
a Grassmann variable ζion each site iof the lattice and defining the action
S=Pi<j Mij ζiζj, a basic result of Grassmannian integrals yields:
Z=ZY
i
dζiexp−X
i<j
Mij ζiζj=Z[Dζ] exp(−S).(A.4)
Correlations
Dimers correlations may be expressed in terms of Grassmann variables: for m
dimers connecting pairwise sites 1 and 2, 3 and 4, . . . , 2m−1 and 2m, one places
Grassmann variables ζiat these sites, which is equivalent to exclude them from
the lattice [9]. Writing dij a dimer that connect sites iand j, the result is (up
to a sign):
hd1,2. . . d2m−1,2mi=hζ1ζ2. . . ζ2mi(A.5)
=1
ZZ[Dζ]ζ1ζ2. . . ζ2mexp(−S).(A.6)
Wick’s theorem for anticommuting variables is applicable, and using it expresses
all correlators in terms of Green’s functions hζiζji:
hζ1ζ2. . . ζ2ni=
2n
X
k=2
(−1)khζ1ζkih
2n
Y
l=2,l6=k
ζli,(A.7)
hζiζji= (M−1)ji =−(M−1)ij .(A.8)
Proves of these two last equations can be found in [15], and the Green’s
functions are easily calculated in Fourier space [11]. We define the Fourier
transform of M(R), the 2 ×2 matrix that connects the references sites 1 and 2
to the other sites (Fig. A.1) as:
˜
M(K) = X
R
eiK·RM(R) (A.9)
=2isin(Ky)g(K)
−g∗(K)−2isin(Ky),(A.10)
where g(K) = 1 −e−iKx+µeiKy+µe−i(Kx+Ky).
The determinant ∆(K) and the inverse A(K) of the matrix ˜
M(K) are:
∆(K) = 4 sin2(Ky) + sin2(Kx
2) + µ2cos2(Ky+Kx
2),(A.11)
A(K) = ˜
M−1(K) = 1
∆(K)−2isin(Ky)−g(K)
g∗(K) 2isin(Ky).(A.12)

APPENDIX A. PFAFFIAN TECHNIQUES 31
The correlation hζ1ζ2ibetween two sites (R1, α1) and (R2, α2) is finally given
by:
hζ1ζ2i=1
NdX
K
e−iK·(R1−R2)Aα2α1(K),(A.13)
where the sum runs over all vectors of the reciprocal lattice (periodic boundary
conditions):
K∈ 2π
Nd,x
nx,2π
Nd,y
ny0≤nx< Nd,x,0≤ny< Nd,y .
When the lattice is large enough (Nd,x ≫1 and Nd,y ≫1) the usual ther-
modynamic limit is valid and Eq. A.13 becomes:
hζ1ζ2i=1
4π2ZZ
[0,2π]2
dK e−iK·(R1−R2)Aα2α1(K).(A.14)
Remark In order to simplify notations, the double integral in Eq. A.14 is
simply written with the symbol R.

Appendix B
Resonons: Eand N
B.1 Energy Matrix E
The 3 ×3 matrix Efor the resonons is:
E(k) = P(s)

1 + cos(ky) 0 −1
21 + eikx1 + e−iky
000
−1
21 + e−ikx1 + eiky0 1 + cos(kx)

+νP(h)

1 + cos(kx+ky)−1
21 + eikx1 + e−i(kx+ky)0
−1
21 + e−ikx1 + ei(kx+ky)1 + cos(kx) 0
0 0 0

+νP(v)

0 0 0
0 1 + cos(ky)−1
21 + ei(kx+ky)1 + e−iky
0−1
21 + e−i(kx+ky)1 + eiky1 + cos(kx+ky)
.
B.2 Normalization Matrix N
For very large lattices (thermodynamic limit), elements of the hermitian matrix
Nare given by:
Nij (k) = Wij (k) + δi,j hnii,(B.1)
where
Wij (k) = 1
π2Zdp Fij (k,p)
∆(px, py) ∆(2kx+px, ky+py),(B.2)
with
∆(x, y) = 4 sin2(y) + sin2(x
2) + µ2cos2(y+x
2),(B.3)
and
F11(k,p) = sin2kx+px
2−cos(kx) + cos(ky) + cos(kx+px)−
cos(ky+ 2py) + µ2cos(kx+ky) + µ2cos(kx+px+ky+ 2py),
F22(k,p) = µ2cos2kx+px+ky
2+pycos(kx) + cos(ky)−cos(kx+px)−
cos(ky+ 2py)−µ2cos(kx+ky)−µ2cos(kx+px+ky+ 2py),
32

APPENDIX B. RESONONS: EAND N33
F33(k,p) = sin2ky
2+pycos(kx)−cos(ky)−cos(kx+px)+
cos(ky+ 2py) + µ2cos(kx+ky) + µ2cos(kx+px+ky+ 2py),
F12(k,p) = µ2ei1
2(kx+px+2py)sinkx+px
2−sin(kx−py)+
sin(kx+ky+py)−sin(kx+px+py)−sin(kx+ky+px+py),
F13(k,p) = ei1
2(π−ky−px)sinky
2+py−cos kx+px
2−py−
cos ky−px
2+py+ cos kx+px
2+py+ cos ky+px
2+py,
F23(k,p) = µ2e−i1
2(ky+px+2py)sinky
2+py−sinky−px
2+
sinkx+ky+px
2−2 coskx
2sinkx+ 2ky+px+ 4py
2.
Dimer densities hniiare given by:
hn1i=hn3i=1
2π2Zdp sin2(px
2)
∆(p),(B.4a)
hn2i=µ2
2π2Zdp cos2(py+px
2)
∆(p).(B.4b)

Appendix C
Visons: Eand N
C.1 Energy Matrix E
The 4 ×4 matrix Efor the visons is:
E(K) = 2 P(s)



1 1 0 0
1 1 0 0
0 0 1 1
0 0 1 1



+ 2 νP(d)



2eiKy0e−iKx
e−iKy2 1 0
0 1 2 −eiKy
eiKx0−e−iKy2



.
C.2 Normalization Matrix N
Normalization matrices for the high-symmetry points have been computed with
the classical Monte Carlo method [4]. These points are: the high-energy point
A=π, π
2, the low-energy points B1=π
3,π
6and B2=5π
3,5π
6, and the
saddle-points C1= (0,0), C2=0,π
2and C3= (π, 0).
N(A) = 



0.4210 0.1757 + 0.1754 i0.0004 −0.0001 i−0.1757 −0.0003 i
0.1757 −0.1754 i0.4214 0.1759 −0.0001 i−0.0004 −0.0002 i
0.0004 + 0.0001 i0.1759 + 0.0001 i0.4217 0.1755 −0.1756 i
−0.1757 + 0.0003 i−0.0004 + 0.0002 i0.1755 + 0.1756 i0.4218




N(B1) = 



2.6217 2.4186 + 0.64949 i1.2011 + 0.3211 i0.6486 −1.1228 i
2.4186 −0.6495 i2.6217 1.2965 −0.0015 i0.3214 −1.2015 i
1.2011 −0.3211 i1.2965 + 0.0015 i0.8641 0.1741 −0.6482 i
0.6486 + 1.1228 i0.3214 + 1.2015 i0.1741 + 0.6482 i0.8639




N(B2) = 



0.8616 0.1728 + 0.6459 i0.3214 + 1.1971 i0.6466 + 1.1188 i
0.1728 −0.6459 i0.8611 1.2924 −0.0015 i1.1962 −0.3221 i
0.3214 −1.1971 i1.2924 + 0.0015 i2.6149 2.4116 −0.6461 i
0.6466 −1.1188 i1.1962 + 0.3221 i2.4116 + 0.6461 i2.6135




N(C1) = 



2.0612 1.9439 0.7603 0.8218
1.9439 2.0603 0.8221 0.7595
0.7603 0.8221 0.5418 0.3001
0.8218 0.7595 0.3001 0.5409




34

APPENDIX C. VISONS: EAND N35
N(C2) = 



1.2941 0.8178 + 0.8167 i0.7557 + 0.7554 i1.1155 −0.0001 i
0.8178 −0.8167 i1.2951 1.1162 + 0.0009 i0.7562 −0.7551 i
0.7557 −0.7554 i1.1162 −0.0009 i1.295 0.8176 −0.8175 i
1.1155 + 0.0001 i0.7562 + 0.7551 i0.8176 + 0.8175 i1.2943




N(C3) = 



2.0681 1.9506 0.7634 −0.8238
1.9506 2.0684 0.8258 −0.7622
0.7634 0.8258 0.5430 −0.3012
−0.8238 −0.7622 −0.3012 0.5415




Because of symmetry arguments, some elements of the normalization matri-
ces should be equal. For example, we should have:
N11(C1) = N22 (C1) = N11(C3) = N22 (C3).
Comparing such values determines their absolute precision: it is less than 0.001.