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Generalised route to effective field theories for quantum systems with local constraints
Attila Szabó,1Garry Goldstein,2Claudio Castelnovo,1and Alexei M. Tsvelik3
1TCM Group, Cavendish Laboratory, University of Cambridge,
J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
2Physics and Astronomy Department, Rutgers University, Piscataway, NJ 08854, USA
3Condensed Matter Physics and Materials Science Division,
Brookhaven National Laboratory, Upton, NY 11973-5000, USA
Some of the exciting phenomena uncovered in strongly correlated systems in recent years – for instance
quantum topological order, deconfined quantum criticality, and emergent gauge symmetries – appear in systems
in which the Hilbert space is effectively projected at low energies in a way that imposes local constraints on
the original degrees of freedom. Cases in point include spin liquids, valence bond systems, dimer models, and
vertex models. In this work, we use a slave boson description coupled to a large-Spath integral formulation to
devise a generalised route to obtain effective field theories for such systems. We demonstrate the validity and
capability of our approach by studying quantum dimer models and by comparing our results with the existing
literature. Field-theoretic approaches to date are limited to bipartite lattices, they depend on a gauge-symmetric
understanding of the constraint, and they lack generic quantitative predictive power for the coefficients of the
terms that appear in the Lagrangians of these systems. Our method overcomes all these shortcomings and we
show how the results up to quadratic order compare with the known height description of the square lattice
quantum dimer model, as well as with the numerical estimate of the speed of light of the photon excitations
on the diamond lattice. Finally, instanton considerations allow us to infer properties of the finite-temperature
behaviour in two dimensions.
I. INTRODUCTION
Low-energy descriptions of strongly correlated many-body
systems sometimes require the introduction of projected
Hilbert spaces where the degrees of freedom are subject to
local constraints. Notable examples include valence bond sys-
tems, quantum dimer models, and vertex models. The action
of the system Hamiltonian within the restricted Hilbert space
often gives rise to exotic and unexpected behaviour, from
emergent gauge symmetries and deconfined quantum critical-
ity to quantum topological order, which have been the subject
of much attention in recent years.
Field-theoretic descriptions of these systems have proven
to be a powerful tool to study their properties, in particular to
understand the nature of their correlations and critical points.
However, conventional routes to construct such field theories
often do not apply, and instead ad hoc methods have been de-
vised throughout the years. Such methods largely hinge on a
physical understanding of the constraint and how to best rep-
resent it in a (free) field theory language. While on the one
hand such approaches have provided great insight into the rel-
evant systems, they cannot easily be generalised. A system-
atic way to arrive at a field-theoretic description of quantum
systems with local constraints is currently lacking.
In this paper, we propose a generalised route to obtain field-
theoretic actions from microscopic Hamiltonians based on a
slave boson representation of the relevant degrees of freedom
and their constraints, combined with a large-Spath integral
formulation. We demonstrate the validity and capability of
our approach by deploying it to study quantum dimer mod-
els (QDMs), recovering known and obtaining new results on
bipartite lattices, and showing that it can be used straight-
forwardly on hitherto inaccessible non-bipartite lattices. Our
approach also paves the way to semiclassical simulations of
these systems, as discussed in Ref. 1by some of the authors.
QDMs were introduced to describe a magnetically disor-
dered (resonating valence bond) phase in high-temperature su-
perconducting materials [2]. They can also arise in Bose–Mott
insulators, electronic Mott insulators at fractional fillings [3],
and in mixed valence systems on frustrated lattices [4]. For a
review of these models, we refer the reader to Ref. 5.
When QDMs are defined on bipartite lattices, they are
amenable to a height mapping description in two dimensions
(2D), which generalises to quantum electrodynamics (QED)
in 3D [5–8]. The height mapping is built upon a gauge-
symmetric understanding of the constraint [9] and has great
predictive power for bipartite quantum dimer models, en-
abling one to answer detailed questions pertaining to their
long-wavelength properties. All its good features notwith-
standing, the height mapping has one essential drawback: a
quantitative derivation of its action from the microscopic lat-
tice Hamiltonian is currently not available. The state-of-the-
art derivation of the height mapping has been so far phe-
nomenological [5,10], and the coefficients (including their
signs) are a posteriori determined by theoretical considera-
tions (e.g., using the knowledge of the exact ground state at
fine-tuned critical points) combined with comparisons to nu-
merical results [5–8,10,11]. Furthermore, the approach can-
not be applied to nonbipartite lattices, which remain compar-
atively unexplored from a field-theoretic perspective.
Using our generalised route, we show that one can sys-
tematically derive the height Lagrangian for bipartite QDMs
in 2D and 3D, from the corresponding microscopic Hamil-
tonians. We compare our results against the existing litera-
ture, where we find good agreement considering the large-
Sand quadratic approximations that we employ. For exam-
ple, we derive the stiffness of the 2D square lattice QDM
at a well-known critical point called the Rokhsar–Kivelson
(RK) point [2], where the ground-state wave function of the
system is known exactly: Our result, 1/4, is comparable to
arXiv:1904.12868v2 [cond-mat.str-el] 5 Aug 2019
2
e2
e1
FIG. 1. Square lattice showing the choice of basis vectors e1,2.
the exact value, π/18 [5,12]. We use instanton consider-
ations to discuss the fate of 2D phases at finite tempera-
ture. We also obtain the speed of light to quadratic order in
the 3D QED long-wavelength theory of dimers on the cu-
bic lattice, c=√2J(J−V)S/3, and on the diamond lat-
tice, c=√J(J−V)/6S2. The latter is known numerically
from quantum Monte Carlo simulations to be c(S=1) '
√0.8J(J−V) [13,14]. We further show that our approach ap-
plies straightforwardly to the non-bipartite case of the QDM
on the triangular lattice, where we observe a curious analyti-
cal similarity with the formalism for the QDM on the 3D cubic
lattice – a similarity that we plan to explore further in future
work.
The paper is organised as follows. In Sec. II, we introduce
our approach by studying in detail the introductory and well-
known case of the QDM on the square lattice. In Sec. III, we
study the cubic lattice and obtain the dispersion of its photon
excitations in the gapless phase. We then briefly consider the
case of the nonbipartite triangular lattice in Sec. IV, which
is shown to be curiously similar in its analytical form to the
cubic lattice. Finally, in Secs. Vand VI, we consider for com-
pleteness the QDMs on the honeycomb and diamond lattices,
respectively. A comparison with the conventional height map-
ping and gauge theoretic formulations is presented in Sec. VII,
and we conclude in Sec. VIII.
II. SQUARE LATTICE
A. Bosonic representation and large S
We consider in the first instance the well-known case of the
RK Hamiltonian for the square lattice QDM, which allows us
to introduce our approach in the simplest setting. Generalisa-
tions to the honeycomb, triangular, cubic and diamond lattices
will be discussed in later sections.
The quantum dimer model can be mapped exactly onto a
slave boson model by considering a secondary Hilbert space
where we assign a bosonic mode br,η to each link (r,r+eη) of
the square lattice (eη=ˆx,ˆy) [15] (see Fig. 1). We associate
the number of dimers on a link with the occupation number
of the bosons on that link, thus embedding the dimer Hilbert
space in the larger Hilbert space of the bosons.
The constraint that each site of the lattice has one and ex-
actly one dimer attached to it can be expressed in the boson
language as
Πr≡X
l∈vr
b†
lbl−1=0.(2.1)
Here, for convenience of notation, `∈vrlabels the four
links r0, η that are attached to the vertex r. We note that the
constraint in Eq. (2.1) implies that the bosons are hard-core:
nr,η ≡b†
r,ηbr,η =0,1.
Any dimer Hamiltonian has a bosonic representation; in
particular, the RK Hamiltonian can be written as
HD=X
rn−J b†
r,1b†
r+e2,1br,2br+e1,2+(1 ↔2)
+V b†
r,1br,1b†
r+e2,1br+e2,1+(1 ↔2)o.(2.2)
To generalise this construction to a large-Sformulation, we
keep the same Hilbert space as before and replace Eq. (2.1)
with the constraint
Πr≡X
l∈vr
b†
lbl−S=0 (2.3)
without changing the Hamiltonian (2.2).
B. Path integral formulation and Gaussian approximation
In what follows, it is convenient to use the radial gauge
for the bosonic fields in the path integral formulation of the
model:
br,η =qρη(r+eη/2) exp hiΦη(r+eη/2)i(2.4)
≡rS
z+δρη(r+eη/2) exp hiΦη(r+eη/2)i,
where zis the lattice coordination number; for the square lat-
tice, z=4. For later convenience, we think of ρη(r+eη/2) and
Φη(r+eη/2) as functions defined on bond midpoints. The RK
Hamiltonian is then given by
3
HD=X
rn−2Jpρ1(r+e1/2)ρ2(r+e1+e2/2)ρ1(r+e2+e1/2)ρ2(r+e2/2)
×cos [Φ1(r+e1/2) −Φ2(r+e1+e2/2) + Φ1(r+e2+e1/2) −Φ2(r+e2/2)]
+Vρ1(r+e1/2)ρ1(r+e2+e1/2) +Vρ2(r+e2/2)ρ2(r+e1+e2/2)o,(2.5)
and the constraint in Eq. (2.3) can be written as
X
ηhδρη(r+eη/2) +δρη(r−eη/2)i=0.(2.6)
We now introduce the Fourier decomposition
δρη(r+eη/2) =1
√NX
k
δρη(k) exp h−ik(r+eη/2)i,(2.7)
Φη(r+eη/2) =1
√NX
k
Φη(k) exp h−ik(r+eη/2)i,(2.8)
where k(r+eη/2) ≡~
k·(~
r+~
eη/2) for brevity and Nis the
number of lattice sites. In these terms, the constraint can be
written as
X
η
cos(keη/2) δρη(k)=0.(2.9)
It will be useful in the following to introduce the shorthand
notation cη=cos(keη/2) and sη=sin(keη/2).
The constraint clearly imposes a relation between the two
field variables δρ1(k) and δρ2(k). The same conclusion can
be readily drawn about the fields Φµonce we notice that
the Hamiltonian depends only on the specific combination
of them that appears in the argument of the cosine term in
Eq. (2.5):
˜
φ(r)≡Φ1(r+e1/2) −Φ2(r+e1+e2/2)
+Φ1(r+e2+e1/2) −Φ2(r+e2/2),(2.10)
whose Fourier transform is
˜
φ(k)=e−ik(e1+e2)/22[c2Φ1(k)−c1Φ2(k)].(2.11)
Note that the cosine function depends only on powers of φ(r)2
and therefore phase factors in ˜
φ(k) are immaterial, and we de-
fine for convenience
φ(k)≡eik(e1+e2)/2˜
φ(k)
=2[c2Φ1(k)−c1Φ2(k)]≡ ZηΦη,(2.12)
where Zη=(2c2,−2c1). In real space, this amounts to in-
troducing a φ(r) living on the centres of the plaquettes rather
than a corner.
Notice that the constraints on δρηand on Φηare in fact two
sides of the same coin – indeed, conjugate variables come in
pairs, so their numbers have to be the same. In our case, one
can easily verify that imposing one of them implies the other.
This is a consequence of how the RK Hamiltonian is designed:
the plaquette-flipping term inherently preserves the number
of dimers at each vertex; and vice versa, if one imposes the
hard core dimer constraint, then any kinetic contribution in the
Hamiltonian is projected onto a combination of loop updates,
of which the plaquette-flipping term is an example.
To make further progress in the path integral formulation,
we shall expand the action to quadratic order in φ(r, τ)'2nπ,
n∈Z, and δρr,µ 1. Firstly, it is convenient to rewrite
cos(φ)=1−[1 −cos(φ)] and notice that the term in square
brackets contains only quadratic and higher-order contribu-
tions. Therefore, the square root in the second term of
√ρρρρ cos(φ)=√ρρρρ −√ρρρρ 1−cos(φ),(2.13)
needs to be expanded only to leading order in S:√ρρρρ '
S2/16. Upon expanding the first term, one obtains both linear
and quadratic terms in δρη(r+eη/2). However, one can readily
convince oneself that the linear terms vanish upon summing
over rbecause of the dimer constraint (2.6). The same is true
for the linear contributions due to the terms multiplying Vin
Eq. (2.5), leading to the following contributions to quadratic
order:
√ρρρρ 'S2
16 +1
4hδρ1(r+e1/2)δρ2(r+e1+e2/2) +δρ1(r+e1/2)δρ1(r+e2+e1/2) (2.14)
+δρ1(r+e1/2)δρ2(r+e2/2) +δρ2(r+e1+e2/2)δρ1(r+e2+e1/2)
+δρ2(r+e1+e2/2)δρ2(r+e2/2) +δρ1(r+e2+e1/2)δρ2(r+e2/2)i
−1
8hδρ1(r+e1/2)2+δρ1(r+e2+e1/2)2+δρ2(r+e1+e2/2)2+δρ2(r+e2/2)2i
ρρ +ρρ =S2
8+δρ1(r+e1/2)δρ1(r+e2+e1/2) +δρ2(r+e2/2)δρ2(r+e1+e2/2) .(2.15)
4
Writing the sum of these terms in Fourier space, we get (for
the quadratic contributions only):
−Jc1c2δρ1(k)δρ2(−k)+δρ2(k)δρ1(−k)
+h(2V−J)c2
2+(J−V)iδρ1(k)δρ1(−k)
+h(2V−J)c2
1+(J−V)iδρ2(k)δρ2(−k).(2.16)
The dynamics of the model is generated by the standard
bosonic Berry phase Pnbn∂τbn. In the radial representation
(2.4), this gives rise to the term Pniδρn∂τΦn, as well as total
derivative terms that do not contribute to the action. Alto-
gether, we obtain
S=ZdτX
k,µ
iδρµ(k, τ)∂τΦµ(−k, τ)
+ZdτX
r
JS 2
81−cos(φ(r, τ))
+ZdτX
k,µ,ν
Dµν(k)
2δρµ(k, τ)δρν(−k, τ),(2.17)
where
Dµν =2 J−V+(2V−J)c2
2−Jc1c2
−Jc1c2J−V+(2V−J)c2
1!.
To proceed further, we can either resolve the constraint ex-
plicitly, or keep it implicit. The former allows to relate di-
rectly with the customary height field representation for the
QDM on the square lattice; the latter is more concise and will
be useful to reduce the algebra and obtain analytic results for
three dimensional models in Secs. III and VI. For this reason,
we present them both in the following sections.
C. Implicit constraint
Let us consider a given cosine minimum at first, and assume
φ(r, τ)1. (We shall discuss the effect of instantons later in
Sec. II D 2.) The middle term in Eq. (2.17) then reduces to
X
r
JS 2
16 φ2(r, τ)=X
k
JS 2
16 ZµZνΦµ(k, τ)Φν(−k, τ).(2.18)
After integrating the first term in Eq. (2.17) by parts in τ, one
can integrate the fields Φµout and obtain an action only in
terms of the fields δρµ:
S=1
2ZdτX
khM−1µν ∂τδρµ(k, τ)∂τδρν(−k, τ)
+Dµν(k)δρµ(k, τ)δρν(−k, τ)i,(2.19)
where we define M ≡ J S 2ZZT/8 for convenience. One can
now readily obtain the dispersion by computing the eigenval-
ues of the matrix MD.
This approach gives us two modes while we know that the
physical system is constrained to only one real scalar field.
Fortunately, Eq. (2.9) states that PµZµδρµ(k)=0. Therefore,
0.0
0.5
1.0
1.5
2.0
ΓM X Γ
Photon frequency ω(k)/JS
Wave vector
V/J=1
0.5
0
FIG. 2. Photon dispersion relation of the large-SQDM on the square
lattice. The spectrum is gapless at the M=(π, π) point for all values
of V/J; near this point, the dispersion is quadratic at the RK point,
and linear away from it. Another minimum forms at the X=(π, 0)
points for lower V: this drives an ordering transition at V=0.
one of the two eigenvalues of MD corresponds to an unphys-
ical mode and vanishes, whereas the other (finite) eigenvalue
corresponds to the physical dispersion of the system,
ω2=JS 2h(2V−J)(c4
1+c4
2)+(J−V)(c2
1+c2
2)+2Jc2
1c2
2i.(2.20)
This dispersion is plotted for three values of V/Jin Fig. 2. It
is interesting to note that the dispersion vanishes at (π, π) and
symmetry related points for all values of Jand V. An instabil-
ity develops for V>Jwhen the dispersion becomes negative
near the (π, π) point (not shown). As we lower the value of
V, secondary minima appear at (π, 0) and related points in the
Brillouin zone, and they drive the system through an insta-
bility for V<0 which leads to (plaquette) dimer ordering at
these wave vectors.
D. Explicit constraint
The implicit approach in Sec. II C allows us to arrive at the
dispersion of the system with minimal algebra, but it does not
produce an action in terms of the physical degree of freedom
only. In order to achieve this, we need to resolve the constraint
explicitly. A convenient way to do so is to look for a conjugate
field h(k) such that the Berry phase in the path integral can be
written as ih(k)∂τφ(−k):
h(k)∂τφ(−k)=X
η
δρη(k)∂τΦη(−k).(2.21)
Substituting the expression for φ(k), Eq. (2.12), into the equa-
tion above, we obtain
δρ1(k)=2c2h(k), δρ2(k)=−2c1h(k).(2.22)
5
One can then straightforwardly verify that introducing the
field h(k) automatically resolves the constraint:
X
η
cηδρη=2(c1c2−c2c1)h(k)=0.(2.23)
Once again, this result should not come as a surprise. It is a re-
flection, at a field-theoretic level, of the fact that the plaquette
terms in the Hamiltonian respect the dimer constraint. There-
fore, if the field theory is built from plaquette kinetic terms
only, then the constraint is implied.
We are thus in the position to write the full large-Saction
for the system, including both the Berry phase and Hamil-
tonian contributions, in terms of the fields h(k) and φ(k) only.
Adding more complicated ring exchange type terms to the RK
Hamiltonian does not invalidate this conclusion, as the phase
in each ring exchange term may be written as a sum of phases
over single plaquettes. Substituting the expressions of δρη(k)
in terms of h(k), and ignoring trivial constants, we obtain the
action:
S=ZdτX
r(ih(r, τ)∂τφ(r, τ)+JS 2
81−cos(φ(r, τ)))
+ZdτX
k
D0(k)
2h(k, τ)h(−k, τ) (2.24)
where
D0(k)=8h(2V−J)(c4
1+c4
2)+(J−V)(c2
1+c2
2)+2Jc2
1c2
2i.(2.25)
1. Action without instantons
Ignoring for the time being the contribution to the action
due to instantons between different minima of the cosine term,
we can expand about one given minimum and integrate over
φto arrive at
S=1
2ZdτX
kh8
JS 2∂τh(k, τ)∂τh(−k, τ)
+D0(k)h(k, τ)h(−k, τ)i.(2.26)
One can easily see that the dispersion is indeed the same as in
Eq. (2.20).
Expanding D0about its minimum at (π, π),
D0[(π, π)+(kx,ky)] '2(J−V)(k2
x+k2
y)
+"7V
6−2J
3#(k4
x+k4
y)+Jk2
xk2
y,
we obtain the action
S=1
2Zdτd2r8
JS 2(∂τh)2+2(J−V)(∇h)2(2.27)
+7V−4J
6h∂4
x+∂4
yh+Jh ∂2
x∂2
yh+. . .
At the RK point, the (∇h)2term vanishes and the terms with
quartic derivatives add up to
J
2(∇2h)2,(2.28)
yielding the spectrum ω=k2/2mwith m=2/JS . We
note that the known value at the RK point for S=1 (and
J=V=1) in this normalisation is m=9/π (corresponding to
K=π/18 in Refs. 5and 12), which can be obtained exactly
from the ground-state wave function of the QDM, available
only at the RK point). This shows that, expanding to quadratic
order, our estimate is within 40% accuracy. We note that such
a discrepancy at quadratic order in a large-Sexpansion is con-
sistent for instance with similar results obtained in Ref. 16 for
quantum spin ice. Our results can be improved by going to
higher orders, and – more importantly with respect to earlier
work on field theories for quantum dimer models – their va-
lidity is not limited to the fine tuned RK point.
2. Instantons
We will now incorporate the instanton effects which, as we
shall demonstrate, always generate a mass for the photons for
V<J, as it generally happens in compact electrodynamics.
To this end, we are going to integrate out the field φ(r, τ)tak-
ing into account the fact that the action is periodic in it.
Firstly, we proceed by the standard Villain approach and
replace
1−cos φ→1
2hφ−2πX
j
qjθ(τ−τj)i2
,(2.29)
where the qj=q(rj, τ j) are integers representing instanton
events, and θ(τ) is the Heaviside step function. By integrating
over φand h, we obtain the following action:
S=(2π)2
2X
j,k
q(rj, τ j)Gqq(rj−rk;τj−τk)q(rk, τk)
Gqq(k, ω)=hω2/M+(ρ2k2+ρ4k4)i−1,(2.30)
where M=JS 2/8 and we introduced the symbolic terms
ρ2k2and ρ4k4to represent the quadratic and quartic deriva-
tive terms in the action: D0(k)'ρ2k2+ρ4k4+. . . The re-
sulting partition function is that of a Coulomb gas of charges
q=±1,±2, . . . , where the fugacity of charge qis given by
I=exp(−q2S0) and S0is the contribution to the action from
a single instanton with q=1:
S0=1
4πZdωd2k
ω2/M+(ρ2k2+ρ4k4)≈πS
8sJ
2ρ4
ln(ρ4/ρ2),
(2.31)
and hence
I=(ρ2/ρ4)q2πS/8,(2.32)
where we performed the calculations with the RK form of the
quartic term and substituted ρ4=J/2 for simplicity.
Since Iis a rapidly decaying function of the instanton
charge, we can restrict our consideration to the gas of charges
q=±1. Following Polyakov [17], we approximate the par-
tition function of the Coulomb gas (2.30) as the one of the
6
sine-Gordon model with action (2.27) augmented by the term
δS=−2µIZdτX
r
cos(2πh),(2.33)
where µdτis the preexponential part of the instanton measure
(see Appendix A); at the RK point, 2µ=JS 3/2√π/2. The
presence of this term makes the excitations massive:
ω2=c2k2+m2,m2=8π2MµI.(2.34)
As we see from (2.32), this mass vanishes at the RK point.
At finite temperatures, instantons interact logarithmically:
E=−2πT2
ρ2X
j<k
qjqkln |rj−rk|
r0!,(2.35)
where r2
0=ρ4/ρ2. The corresponding contribution to the free
energy density
δF∝q2IqZd2r
rdq,dq=2πq2T/ρ2,(2.36)
diverges at small T. The contribution of the static fluctuations
of his encoded in the free energy functional
F=Zd2rρ2
2(∇h)2+ρ4
2(∇2h)2−2µIcos(2πh).
(2.37)
The scaling dimension d1of the cosine term is given by (2.36).
The critical temperature above which the cosine term is irrel-
evant is determined by the condition d1=2:
Tc=ρ2/π. (2.38)
Above Tc, we have a critical phase; below it, the correlation
length of the hfield is finite:
ξ∼r0(µI/Tc)−1/(2−d1)(2.39)
This corresponds to melting the valence bond crystal. As it is
typical for 2D crystals, it melts via a Berezinskii–Kosterlitz–
Thouless transition.
We finally take a moment to comment on the difference be-
tween our result and the one obtained in Ref. 18, which in fact
addresses a somewhat different problem. In Ref. 18 the au-
thors considered equal time correlations at the RK point. The
remarkable property of this point is that the ground-state wave
function can be represented as a path integral with a Gaus-
sian action. It was argued that the periodicity of the height
field generates the irrelevant perturbation cos(2πh). Here, we
obtained a formally equivalent perturbation (2.33) also away
from the RK point; however, the prefactor of the cosine term
in our case vanishes precisely at the RK point.
E. Large-Sphase diagram
As S→ ∞, zero point fluctuations of any soft modes are
negligible, and ρand Φcan be treated as commuting, classical
S
S=1
V/J
S→ ∞
V/J
0−0.5
staggered
U(1)RVB
plaq.columnar
1
staggeredcolumnar
?
FIG. 3. Phase diagram of the square lattice quantum dimer model at
S=∞(this work) and S=1 [19–26].
variables. The ground state energy of the system in this limit
is therefore given by classical minimisation of the Hamilto-
nian (2.5). Since √ρρρρ ≥0 always, the Φin such an optimal
state satisfy cos(Φ1−Φ2+ Φ3−Φ4)=1 which is achieved,
for instance, by setting Φ≡0.
Finding the optimal values of ρin full generality is more
difficult. However, one can always compare the ground state
energies of phases suggested in the literature, or develop a
variational ansatz that captures several such phases. In the
case of the square lattice, we considered states in which ρis
constant within each set of bonds populated in the four colum-
nar ordered states. Such an ansatz can capture columnar and
plaquette ordered states as well as the RVB liquid phase.
Comparing the ground state energies of these phases yields
the S→ ∞ phase diagram shown in Fig. 3. As expected, the
ground state is staggered for V>Jand columnar at V→ −∞.
At intermediate V/J, we see a plaquette ordered phase as well
as an extended U(1) RVB liquid, with phase boundaries corre-
sponding to the instabilities shown in Fig. 2. The latter is un-
stable at finite Sdue to instanton effects, as discussed above.
The fate of the plaquette phase is, however, less clear: since
it is ordered, instantons are unlikely to substantially affect its
stability, and so the evolution of the columnar–plaquette phase
boundary must mostly depend on lattice effects. It may well
be possible that the plaquette order survives at S>1 and has
a proximity effect near the RK point even at S=1. This
could explain why numerical simulations of the square lattice
dimer model struggle to establish its true ground state in this
regime [19–26].
III. CUBIC LATTICE
The calculations for other lattices are straightforward gen-
eralisations of the square lattice case, with minimal but infor-
mative modifications. We begin with the cubic lattice, where
we have the three lattice vectors eη=ˆx,ˆy,ˆz. The correspond-
7
ing RK Hamiltonian can be written as:
HD=X
rn−J b†
r,1b†
r+e2,1br,2br+e1,2+(1 ↔2)
+V b†
r,1br,1b†
r+e2,1br+e2,1+(1 ↔2)o
+(12) ↔(13) ↔(23),(3.1)
subject to the equivalent large-Sconstraint
Πr≡X
l∈vr
b†
lbl−S=0.(3.2)
Using the radial gauge expression (2.4) for the bosonic field
with z=6, this constraint can be written as in (2.6), which in
Fourier space reduces to
X
η
cηδρη(k)=0.(3.3)
Contrary to the case of the square lattice, we have now three
fields δρηand one constraint, leaving two independent field
variables. The Hamiltonian is made of three terms equivalent
to the square lattice Eq. (2.5), upon replacing (12) ↔(13) ↔
(23).
As we did for the square lattice QDM, we derive here the
resulting field theory to quadratic order. The calculation is
similar to the one carried out in Sec. II C, keeping the con-
straints implicit. It is also possible to explicitly resolve the
constraints and obtain an action in terms of two independent
real scalar fields, as in Sec. II D, but for brevity, we omit the
details of the calculation and only present the final result at
the end of this section.
The terms in the cubic Hamiltonian are equivalent to com-
bining the square lattice terms for the (12), (13), and (23) com-
ponents, see Eqs. (2.13), (2.14), and (2.15), up to a factor of
4/9 due to the fraction S/6 replacing S/4 in Eq. (2.4):
S=Zdτ
X
k,µ
iδρµ(k)∂τΦµ(−k)+JS 2
18 X
r,µ h1−cos(φµ(r, τ))i+X
k,µ,ν
Dµν
2δρµ(k)δρν(−k)
,(3.4)
where
D=2
2(J−V)+(2V−J)(c2
2+c2
3)−Jc1c2−Jc1c3
−Jc1c22(J−V)+(2V−J)(c2
1+c2
3)−Jc2c3
−Jc1c3−Jc2c32(J−V)+(2V−J)(c2
1+c2
2),(3.5)
and we labelled φµthe argument of the cosine term involving
the phase fields Φνwith ν,µ.
A. Action without instantons
When we expand about one given minimum,
JS 2
18 X
µ=1,2,3
[1 −cos φµ(r, τ)]'JS 2
36 X
µ=1,2,3
φµ(r, τ)2,(3.6)
we see that integrating out the fields φµrequires some care,
since they are not all independent of one another.
Following the same steps as for the square lattice dimer
model, it is convenient to introduce in Fourier space the fields
φ1(k)≡eik(e2+e3)/2˜
φ1(k)=2[c3Φ2(k)−c2Φ3(k)]
φ2(k)≡eik(e3+e1)/2˜
φ2(k)=2[c1Φ3(k)−c3Φ1(k)]
φ3(k)≡eik(e1+e2)/2˜
φ3(k)=2[c2Φ1(k)−c1Φ2(k)].
(3.7)
These are most conveniently expressed as φµ=ZµνΦν, where
Zµν =2εµνλcλ(εµνλis the fully antisymmetric tensor and sum-
mation is implied).
Notice that Zis a nonzero traceless antisymmetric matrix.
It has one zero eigenvalue, PνZµνcν=0, and the other two
must be non-vanishing and opposite, ±ζ:
2ζ2=trZ2=−2(c2
1+c2
2+c2
3), ζ =iqc2
1+c2
2+c2
3.(3.8)
The two non-vanishing eigenvectors define the physical space,
and the null one is the gauge degree of freedom. One can
therefore construct a projector onto the physical space as
−Z2/(c2
1+c2
2+c2
3).
The cosine term in the Hamiltonian reduces to
JS 2
36 X
µ
φµ(k)φµ(−k)=JS 2
36 φT(k)φ(−k) (3.9)
=JS 2
36 ΦT(k)ZT(k)Z(−k)Φ(−k)
=−JS 2
36 ΦT(k)Z2(k)Φ(−k)
=X
µν
Mµν
2Φµ(k)Φν(−k),
where we used the fact that Z(−k)=Z(k) and ZT=−Z,
8
0.0
0.5
1.0
1.5
2.0
R M X R ΓX
Photon frequency ω(k)/JS
Wave vector
V/J=1
0.5
0
FIG. 4. Photon dispersion relation of the large-SQDM on the cubic
lattice. The spectrum is gapless at the R=(π, π, π) point; near this
point, the dispersion is quadratic at the RK point and linear away
from it. The spectrum has two non-degenerate branches away from
the RK point. The lower branch develops minima at the X=(π, 0,0)
points for lower values of V, which drives an ordering transition at
V=0.
and we defined
M=−JS 2
18 Z2
=2JS 2
9
c2
2+c2
3−c1c2−c1c3
−c1c2c2
1+c2
3−c2c3
−c1c3−c2c3c2
1+c2
2.(3.10)
Integrating out the fields Φµthen gives
1
2X
µ,ν M−1
µν ∂τδρµ(k)∂τδρν(−k),(3.11)
so we can write the full quadratic action without instantons as
S=1
2ZdτX
k,µ,ν hM−1
µν ∂τδρµ(k)∂τδρν(−k)+Dµνδρµ(k)δρν(−k)i.
(3.12)
The dispersion can be obtained from the eigenvalues of
MD =−2JS 2Z2D/9, after projecting out the unphysical
modes that do not satisfy the constraint Pµcµδρµ=0. This
could be done formally by adding an infinite Lagrange mul-
tiplier, but in fact there is no need to do so because the only
unphysical mode is trivially the zero mode of Z2D– as we
had previously observed in the square lattice QDM. The two
non-vanishing eigenvalues are
ω2=4JS 2
9nJh2(c2
1+c2
2+c2
3)−(c4
1+c4
2+c4
3)i(3.13)
+2Vhc4
1+c4
2+c4
3−(c2
1+c2
2+c2
3)+c2
1c2
2+c2
1c2
3+c2
2c2
3i
±2|J−V|qc4
1c4
2+c4
1c4
3+c4
2c4
3−c2
1c2
2c2
3(c2
1+c2
2+c2
3)o.
This dispersion is plotted for three values of V/Jin Fig. 4. It
is interesting to note that ω2|J=V∝(c2
1+c2
2+c2
3)2and the two
bands are degenerate at the RK point, with vanishing minima
at (π, π, π) and symmetry related points, and quadratic disper-
sion around them. Expanding near such minima, we find
ω2'2S2
9J(J−V)k2,(3.14)
where kis the (small) vector distance from the minimum, giv-
ing a speed of light
c=S
3p2J(J−V).(3.15)
As mentioned earlier, one could have alternatively resolved
the constraint explicitly, writing the three fields δρµin terms
of two independent real scalar fields haand resolving the cor-
responding inter dependence of the three fields φµ:
c1φ1+c2φ2+c3φ3=0.(3.16)
For instance, one can do so via the relation φ1=−(c2/c1)φ2−
(c3/c1)φ3and δρµ=PaRµahawith
R=2
c2−c3
c10
0c1.(3.17)
After a few lines of algebra, one arrives at the action
S=1
2ZdτX
k,a,bhf
M−1
ab ∂τha(k)∂τhb(−k)+e
Dabha(k)hb(−k)i
(3.18)
where e
D=RTDRand
f
M=JS 2
18c2
1 c2
1+c2
2c2c3
c2c3c1
1+c2
3!.
One can easily verify that the action (3.18) gives indeed the
same dispersion as Eq. (3.13).
B. Instantons
The instanton contributions are calculated similarly to the
square lattice case, by performing the Villain transformation
(2.29) on φµin (3.4) and integrating out the smooth fields
φµ,hµ. The result is a 3D Coulomb gas action for the inte-
ger charges qµ,
S=(2π)2
2X
ω,kX
µ,ν
qµ(−ω, −k)"18ω2
JS 2M−1+D#−1
µν
qν(ω, k),
(3.19)
with the standard unscreened long-range Coulomb interac-
tion. However, the constraint (3.16) on the fields φµimplies
the equivalent constraint
c1q1+c2q2+c3q3=0 (3.20)
on the instanton configurations qµ(ω, k) allowed in the low-
energy sector. In (3 +1) dimensions, such instantons are irrel-
evant and can be safely neglected.
9
V/J
10−0.25
staggeredU(1)RVBcubecolumnar
FIG. 5. S→ ∞ phase diagram of the cubic lattice quantum dimer
model. (To our knowledge, there are no conclusive studies of the
S=1 QDM on the cubic lattice, to which one could compare the
large Sresults of this work.)
C. Large-Sphase diagram
Using the method described in Sec. II E, we can obtain the
ground state of the QDM Hamiltonian (3.1) in the limit S→
∞. The results are summarised in Fig. 5. Similarly to the
square lattice case, we observe an extended U(1) RVB liquid
phase: in three dimensions, this phase is anticipated to survive
at S=1. For V>J, the photon modes become unstable at
the (π, π, π) point, leading to staggered order. Likewise, the
instability of the (π, 0,0) points for V<0 drives a transition
into an RVB solid phase with isolated, resonating cubes. For
V<−J/4, this phase gives way to columnar order.
IV. TRIANGULAR LATTICE
It is interesting to consider the case of the triangular lattice
QDM immediately after the cubic one. It has sixfold connec-
tivity and is tripartite, and one can view it as a cubic lattice
projected along one of the [111] directions. Following the
notation in Fig. 6, the solid dots belong to one cubic sublat-
tice, the open circles to the other, and the dotted circles are
sites that belong to both cubic sublattices but get projected
onto one another. The rhombic plaquettes of the triangular
lattice correspond to the three independent faces of a cube,
and therefore one can precisely identify flippable plaquettes
and plaquette flipping terms between the two models. This
projective view allows us to draw a complete correspondence
between the two QDMs. Formally, the large-Spath integral
approach presented in this paper proceeds identically, down
to the numerical prefactors, for the triangular and cubic cases
and we end up with the same two-component field theory [29],
with action given (to quadratic order) by Eq. (3.12) and dis-
persion given by Eq. (3.13). The correspondence holds only
so long as we express the positions and wave vectors formally
as rand k, and the basis vectors as e1,e2and e3.
To study the triangular lattice QDM, one then needs to sub-
stitute k=(kx,ky) and a given choice of base vectors, for ex-
ample e1=(√3,1)/2, e2=(−√3,1)/2, and e3=(0,−1),
illustrated in Fig. 6. This is, however, beyond the scope of the
present paper. For S=1, we expect a Z2liquid phase to be
stable around the RK point [30]; its fate, however, is unclear in
the large-Slimit. If the description is able to capture it at all, it
can only be after accounting for instantons and understanding
e1
e2
e3
FIG. 6. The triangular lattice illustrating a choice of base vectors
e1,2,3. Its three sublattices are shown as solid dots, open circles, and
dotted circles. Some bonds of the lattice appear as dashed rather
than solid lines, in accordance with the correspondence to a projected
cubic lattice discussed in the main text.
how their role differs in bipartite and non-bipartite lattices – a
task which promises to pose non-trivial challenges.
V. HONEYCOMB LATTICE
In this section, we consider the QDM on the honeycomb
lattice, illustrated in Fig. 7, and we present only the approach
in which the constraint is resolved explicitly. Contrary to the
cases considered so far, the primitive cell of the lattice con-
tains two distinct sites (shown as solid dots and open circles in
the figure). With the choice of lattice vectors e1=(√3,1)/2,
e2=(−√3,1)/2, and e3=(0,−1) in Fig. 7, we can write the
bosonic representation of the Hamiltonian as
HD=X
r∈An−J b†
r,1b†
r+e1−e3,2b†
r+e2−e3,3(5.1)
×br+e1−e3,3br+e2−e3,1br,2+h.c.
+V b†
r,1br,1b†
r+e1−e3,2br+e1−e3,2b†
r+e2−e3,3br+e2−e3,3
+V b†
r+e1−e3,3br+e1−e3,3b†
r+e2−e3,1br+e2−e3,1b†
r,2br,2o.
One has to write two separate (large-S) constraints, one for
each sublattice:
X
l∈vr∈A
b†
lbl−S=0,X
l∈vr∈B
b†
lbl−S=0.(5.2)
The rest of the calculation follows rather straightforwardly
from the square lattice case, barring some added algebraic dif-
ficulties, and is presented for completeness in App. B. The
argument of the cosine leads us to introduce the field
φ(k)=2ihΦ1(k)(s2c3−s3c2)+ Φ2(k)(s3c1−s1c3)
+ Φ3(k)(s1c2−s2c1)i
10
e1
e2
e3
FIG. 7. Honeycomb lattice showing the lattice vectors e1,2,3defined
in the text.
and the following convenient resolution of the constraint in
terms of a scalar field h(k):
δρ1(k)=−2i(s2c3−s3c2)h(k)
δρ2(k)=−2i(s3c1−s1c3)h(k)
δρ3(k)=−2i(s1c2−s2c1)h(k).(5.3)
To quadratic order, one arrives then at the action
S=ZdτX
rnih(r, τ)∂τφ(r, τ)+2JS 3
27 1−cos(φ(r, τ))o
+ZdτX
k
D(k)
2h(k, τ)h(−k, τ),(5.4)
where
D(k)=8JS
3s4
12 +s4
23 +s4
31
−16JS
3s12 s23c12 c23 +s23 s31c23 c31 +s31 s12c31c12
+16VS
3hs12 s23(c12 c23 −s12 s23)
+s23 s31(c23 c31 −s23 s31)
+s31 s12(c31 c12 −s31 s12)i,(5.5)
where we introduce for convenience
sµν =sin[k(eµ−eν)/2] =sµcν−sνcµ
cµν =cos[k(eµ−eν)/2] =cµcν+sµsν.(5.6)
Expanding about one given minimum,
2JS 3
27 [1 −cos (φ(r, τ))]'JS 3
27 φ(r, τ)2,(5.7)
and integrating over φ, we arrive at the action
S=1
2ZdτX
k"27
2JS 3∂τh(k, τ)∂τh(−k, τ)
+D(k)h(k, τ)h(−k, τ)#,(5.8)
0.0
0.2
0.4
0.6
0.8
1.0
MΓK M
Photon frequency ω(k)/JS2
Wave vector
V/J=1
1/2
−1/8
FIG. 8. Photon dispersion relation of the large-SQDM on the hon-
eycomb lattice. The spectrum is gapless at the Γpoint. Another
minimum develops for small Vat the Kpoints; however, a plaquette
ordering transition occurs before this minimum would become un-
stable.
and the corresponding dispersion ω2(k)=2JS 3D(k)/27. This
dispersion is plotted for three values of V/Jin Fig. 8. It is
gapless at the Γpoint for all values of Jand V. An instability
develops for V>Jwhereby ω2becomes negative for small k.
For V.J, the long-wavelength dispersion is linear:
ω'√2J(J−V)S2
3k.(5.9)
As the value of Vis lowered, secondary minima appear at the
Kpoints in the Brillouin zone which drive the system through
an instability for V<−J/8.
Expanding D(k) about its minimum at Γ = (0,0),
D[(kx,ky)] '3S(J−V)(k2
x+k2
y)
+9S
16 (7V−J)(k4
x+k4
y+2k2
xk2
y),
we obtain the action:
S=1
2Zdτd2r"27
2JS 3(∂τh)2+3S(J−V)(∇h)2
+9S
16 (7V−J)(∇2h)2#+. . . (5.10)
Contrary to the square lattice QDM, see Eq. (2.27), we find
that rotational symmetry is preserved near the band minimum
of the honeycomb lattice QDM for J,V, at least up to quartic
order. This is expected, since the discrete symmetry of the
lattice upon 2π/3 rotations implies that the first symmetry-
breaking term allowed in the action must be of sixth order in
k.
The ground state of the S→ ∞ model was also obtained as
a function of V/Jby comparing the energy of several ordered
and resonating phases suggested in the literature [31,32] un-
der the Hamiltonian (5.1). The resulting phase diagram is
11
S
S=1
V/J1−0.228(2)
staggeredcolumnar plaquette
S→ ∞
V/J
0.1−1/6
columnar plaq.
U(1)RVB
staggered
FIG. 9. Phase diagram of the honeycomb lattice quantum dimer
model at S=∞(this work) and S=1 [32].
shown in Fig. 9, together with the S=1 phase diagram ob-
tained in Ref. [32]. As in the previous cases, the large-Sphase
diagram predicts a columnar ordered, a plaquette ordered, an
RVB liquid, and a staggered ordered phase, which are also
sketched in Fig. 9. It is interesting to note that the RVB liquid
phase instability for V<Joccurs below the RVB–plaquette
transition point (cf. Figs. 8and 9); this suggests that the latter
may in fact be a first order transition. At finite S, instantons
are expected to gap out the U(1) liquid phase in two dimen-
sions, leading to its immediate collapse, see Sec. II D 2. The
other three phases are all observed at S=1 with critical V/J
similar to the large-Sresult.
VI. DIAMOND LATTICE
The diamond lattice is composed of two interpenetrating
face centred cubic (fcc) lattices, as shown in Fig. 10. Sublat-
tice Ais connected to sublattice Bby the vectors
e1=1
√3(1,1,1)e2=1
√3(1,−1,−1)
e3=1
√3(−1,1,−1)e4=1
√3(−1,−1,1),(6.1)
which we chose to define the unit length in the system. Notice
that Pµeµ=0. The lattice is fourfold coordinated and the
smallest lattice loop over which a dimer move can take place
is a hexagon. Each hexagonal plaquette involves lattice bonds
of three out of the four types, and therefore there are four in-
equivalent plaquette terms, involving respectively µ=1,2,3,
µ=1,2,4, µ=1,3,4, and µ=2,3,4.
The QDM Hamiltonian is the sum of four copies of the hon-
eycomb lattice Hamiltonian (5.1) and we refrain from writing
it here explicitly for convenience (see Sec. Vand App. B).
Expressing the bosonic operators in the radial gauge (2.4) and
e1
e2
e3
e4
FIG. 10. Unit cell of the diamond lattice showing the lattice vectors
e1,...,4introduced in the text. The two interpenetrating fcc lattices are
shown as solid and open dots, respectively. A hexagonal plaquette is
shown shaded by way of example.
expanding to quadratic order in δρ leads to the action
S=ZdτX
k,µ
iδρµ(k, τ)∂τΦµ(−k, τ)
+ZdτX
r,α
JS 2
81−cos(φα(r, τ))
+ZdτX
k,µ,ν
Dµν(k)
2δρµ(k, τ)δρν(−k, τ),(6.2)
where αruns over the inequivalent plaquette sublattices,
Dµν(k) is the interaction matrix that follows from expanding
the Hamiltonian terms −2J√ρρρρρρ and Vρρρ to quadratic
order, and φis the lattice curl of Φaround each plaquette. In
Fourier space, we have
φα(k)=2iX
βγδ
εαβγδΦβ(k)sγcδ,(6.3)
where εαβγδ is the totally antisymmetric tensor, cµ=
cos(keµ/2), and sµ=sin(keµ/2).
Since the two sublattices of the diamond lattice are inequiv-
alent, there are two separate (large-S) constraints on ρ, one for
each sublattice:
X
l∈vr∈A
b†
lbl−S=0,X
l∈vr∈B
b†
lbl−S=0.(6.4)
In Fourier space, these can be written as
X
µ
cµδρµ(k)=X
µ
sµδρµ(k)=0; (6.5)
since we have two constraints for the four fields δρµ, there are
two independent fields left. Resolving the constraints explic-
itly, however, leads to rather intractable algebra; therefore, we
only present the implicit approach of Sec. II C.
12
0.0
0.2
0.4
0.6
0.8
1.0
X U L ΓX W K Γ
Photon frequency ω(k)/JS2
Wave vector
V/J=1
0
v∗
FIG. 11. Photon dispersion relation of the large-SQDM on the di-
amond lattice. The spectrum is gapless at the Γpoint and has two
non-degenerate branches away from the RK point. Another mini-
mum develops for small Vnear the Kpoints; however, a plaquette
ordering transition occurs before this minimum would become un-
stable at V/J=v∗≈ −0.694.
Ignoring instantons (for the reasons discussed in Sec. III B),
the middle term of Eq. (6.2) can be expanded about a given
minimum of the cosine. After Fourier transforming and sub-
stituting (6.3), we obtain
S=ZdτX
k,µ
iδρµ(k, τ)∂τΦµ(−k, τ)
+ZdτX
k,µ,ν
Mµν
2Φµ(k, τ)Φν(−k, τ)
+ZdτX
k,µ,ν
Dµν(k)
2δρµ(k, τ)δρν(−k, τ); (6.6)
obtaining the matrices Mand Dfrom the QDM Hamiltonian
is straightforward but algebraically tedious and is presented in
App. C.
As in Sec. II C, the fields Φµcan now be integrated out to
obtain the action
S=1
2ZdτX
k,µ,ν hM−1µν ∂τδρµ(k, τ)∂τδρν(k, τ)
+Dµν(k)δρµ(k, τ)δρν(−k, τ)i.(6.7)
The dispersion ω2(k) is given by the eigenvalues of the matrix
MD. Two of these eigenvalues are identically zero: these cor-
respond to the unphysical modes ruled out by the constraints
(6.5). The remaining two eigenvalues can be worked out by
straightforward but rather lengthy algebra; the resulting dis-
persion is plotted along high-symmetry directions for three
values of V/Jin Fig. 11.
Away from the RK point, the resulting two photon modes
are normally non-degenerate; near k=0, however, they agree
S
S=1
V/J
10.75(2)
staggered
U(1)
RVB
ordered R
S→ ∞
V/J
−1/7
ordered R U(1)RVB staggered
FIG. 12. Phase diagram of the diamond lattice quantum dimer model
at S=∞(this work) and S=1 [14].
to leading order:
ω2'S4
6J(J−V)k2,(6.8)
giving a speed of light
c=S2pJ(J−V)/6.(6.9)
This can be contrasted to the quantum Monte Carlo result for
S=1 in Refs. [13,14], namely c'√0.8J(J−V). We expect
higher-order corrections in the 1/S expansion to substantialy
reduce this discrepancy [16].
The ground state of the S→ ∞ model was obtained as
a function of V/Jby comparing the large-Sminimum en-
ergy of ordered and resonating phases suggested in the lit-
erature [13,14,33]. The resulting phase diagram is shown
in Fig. 12, together with the S=1 phase diagram obtained
in Ref. [14]. Unlike the previous cases, no resonating solid
phase is identified in either limit: both phase diagrams consist
of staggered ordered, U(1) liquid and ordered R[13] phases.
The liquid phase is far larger for S→ ∞ than at S=1: this is
consistent with the intuition that soft dimers (and spins) favour
fluctuating phases.
VII. RELATION TO OTHER FIELD THEORETIC
APPROACHES
For quantum dimers models on bipartite lattices, field-
theoretic approaches were already known in the literature.
These include height mappings in two dimensions [8] and
U(1) gauge theory descriptions in three dimensions [9]. In
this section, we briefly review them and discuss how they re-
late to our large-Srepresentation. For simplicity, we focus on
the square and cubic lattices.
A. Height mapping on the square lattice
In S=1 dimer models, a height mapping can be con-
structed by assigning a height ˜
hto all plaquettes of the lat-
tice, starting from a reference plaquette with ˜
h=0. Going
13
clockwise around a vertex in the Asublattice (see Fig. 1), the
height field changes by −1/4 on crossing a bond without a
dimer and by +3/4 on crossing one with a dimer. Likewise,
going clockwise around a vertex in the Bsublattice, ˜
hchanges
by −3/4 and +1/4 on bonds with and without dimers, respec-
tively. These numbers are the difference between the occupa-
tion of a given bond and the average occupation of all bonds,
1/4. Therefore, this mapping can be generalised to the large-S
case by requiring that
∆˜
h=±δρ, (7.1)
where the sign depends on the direction in which the bond is
traversed, as described above.
In order to coarse-grain this height description, it is useful
to define the so-called magnetic field representation,
~
B(r)=Bx,By,0,Bµ(r)=(−1)x+yδρµ(r),(7.2)
in which the height mapping (7.1) can be expressed as
Bµ(e+rµ/2) =˜
h[r+(eµ+εµνeν)/2]−˜
h[r+(eµ−εµνeν)/2].(7.3)
In reciprocal space, the signs appearing in (7.2) have the ef-
fect of shifting the gapless point of the dispersion (2.20) from
(π, π) to (0,0), that is, coarse-graining will capture the long-
wavelength, low-frequency features of the gapless modes. In
particular, (7.3) can be Fourier transformed to give
Bµ(k)=2iεµν sin(kν/2)˜
h(k); (7.4)
comparing with (2.22) immediately gives
˜
h(k)=h[k+(π, π)] =⇒˜
h(r)=(−1)x+yh(r),(7.5)
where his the single-component height field introduced in
Sec. II D. This concludes the reconstruction of the height map-
ping for classical (nondynamical) dimer models.
To recover quantum dynamics, the conjugate variables Φµ
and φcan be reexpressed as
~
Φ(r)=˜
Φx,˜
Φy,0,˜
Φµ(r)=(−1)x+yΦµ(r) (7.6)
˜
φ(r)=(−1)x+yφ(r),(7.7)
which leads to Berry phase terms of the form
iX
k,µ
δρµ(k)∂τΦµ(−k)=iX
k
~
B(k)·∂τ~
Φ(−k) (7.8)
iX
k
h(k)∂τφ(−k)=iX
k
˜
h(k)∂τ˜
φ(−k).(7.9)
Integrating ˜
φout and coarse-graining (that is, focusing on
small k) then yields the quadratic action (2.27).
B. Coulomb gauge theory on the cubic lattice
The key idea of the mapping discussed above is turning the
lattice gauge field and its divergence-free condition (2.3) into
a coarse-grained, true vector field with ∇ · ~
B=0. Indeed, for
small k, (7.4) reduces to
~
B(k)=ik ×~
h(k)⇐⇒ ~
B=∇ × ~
h,(7.10)
where ~
h=(0,0,˜
h). Similarly, we now want to construct
a divergence-free magnetic field ~
Bon the cubic lattice by
coarse-graining δρµ. To achieve this, we express our field in
terms of a magnetic vector potential: ~
B=∇ × ~
A.
The construction again starts by defining
Bµ(r)=(−1)x+y+zδρµ(r); (7.11)
similarly to the square lattice case, this amounts to shifting
the gapless point of the photon dispersion from (π, π, π) to the
origin. The reciprocal space constraint (3.3) now becomes
X
µ
2isin(kµ/2)Bµ(k)=0.(7.12)
This constraint can be resolved similarly to (7.4) by introduc-
ing another vector field ~
A:
Bµ(k)=2iεµνλsin(kν/2)Aλ(k); (7.13)
at small k, this reduces to B=∇ × A, as desired.
We now turn the conjugate variables Φand φinto vector
fields by writing
~
Φµ(r)=(−1)x+y+zΦµ(r) (7.14)
~
φµ(r)=(−1)x+y+zφµ(r).(7.15)
Eq. (3.7) can be written as
~
φµ(k)=2iεµνλsin(kν/2)~
Φλ(k) (7.16)
and the Berry phase can be integrated by parts to obtain
iX
k,µ
δρµ(k)∂τΦµ(−k)=iX
k
∂τ~
A(k)·~
φ(−k).(7.17)
Expanding to quadratic order in φaround a given minimum of
the cosine, we finally obtain the action
S=ZdτX
khi∂τ~
A(k)·~
φ(−k)+JS 2
36 ~
φ(k)·~
φ(−k)
+e
Dµν
2Aµ(k)Aν(−k)i,(7.18)
where e
Dis given in terms of the matrix Ddefined in (3.5) by
e
Dµν(k)=4εµκρ ενλσsin(kκ/2) sin(kλ/2)Dρσ[k+(π, π, π)].
(7.19)
Finally, we integrate out φand expand e
Daround q=0 to
obtain a coarse-grained action in terms of Aonly. Away from
the RK point, the leading order terms give
S ' ZdτX
k9
JS 2∂τ~
A(k)·∂τ~
A(−k)+2(J−V)~
B(k)·~
B(−k).
(7.20)
14
Similarly to ordinary quantum electrodynamics, we can iden-
tify ∂τ~
Aas the electric field; that is, (7.20) is the action of a
linearly dispersing U(1) gauge theory. The speed of light is
given by
c=S
3p2(J−V)J,(7.21)
in agreement with (3.15). At the RK point, the B2term van-
ishes; to leading order, the action becomes
S ' ZdτX
k9
JS 2∂τ~
A(k)·∂τ~
A(−k) (7.22)
−J
2k×~
B(k)·k×~
B(−k).
VIII. CONCLUSION
We proposed a general route to obtain the field-theoretic
action for microscopic Hamiltonians with hard constraints,
based on a slave boson representation of the relevant degrees
of freedom and their constraints, combined with a large-S
path integral formulation.
We used it to systematically derive Lagrangians for bipar-
tite QDMs in 2D and 3D from the corresponding microscopic
Hamiltonians. We find good agreement with known results in
the literature; namely, calculations up to quadratic order yield
a stiffness for the square lattice QDM at the RK point equal to
1/4 compared to the exact result π/18 [5,12]; and they yield
a speed of light in the gapless phase of the diamond lattice
QDM c=√J(J−V)/6S2compared to the numerical result
c(S=1) '√0.8J(J−V) [13,14].
Our approach applies straightforwardly to the nonbipartite
case of the QDM on the triangular lattice, where we observe
an intriguing analytical relation to the formalism of the cubic
lattice, which will be interesting to explore in future work.
ACKNOWLEDGEMENTS
We thank Baptiste Bermond, John Chalker, and Roderich
Moessner for useful discussions. A.M.T. is supported by the
Condensed Matter Physics and Materials Science Division,
in turn funded by the U.S. Department of Energy, Office of
Basic Energy Sciences, under Contract No. DE-SC0012704.
G.G. was supported by the U.S Department of Energy, Office
of Science, Basic Energy Sciences as a part of the Computa-
tional Materials Science Program. This work was supported,
in part, by the Engineering and Physical Sciences Research
Council (EPSRC) Grant No. EP/M007065/1 (C.C. and G.G.)
and Grant No. EP/P034616/1 (C.C.). Statement of compli-
ance with the EPSRC policy framework on research data: this
publication reports theoretical work that does not require sup-
porting research data.
Appendix A: Instanton measure on the square lattice
Instantons appear naturally in the compact gauge theory as
stationary trajectories of the action with φchanging by 2πon
a given site between τ=±∞. Unlike the point-like instantons
described in Sec. II D 2, these objects are smooth as a func-
tion of time, and thus have a nontrivial instanton core. The
action can then be expanded to quadratic order around such
solutions, similarly to the case h=φ≡0 shown in Sec. II D 1.
The resulting fluctuation determinant will appear in the prob-
ability of instantons as a preexponential factor.
We thus have to construct such a stationary instanton so-
lution. Using (2.24,2.29) with a single instanton event at
r=τ=0, we get the following quadratic action in terms
of Fourier components:
S=Z(dω)(d2k)ωh(k, ω)φ(−k,−ω)
+D0(k)
2h(k, ω)h(−k,−ω)
+M
2"φ(k, ω)−2πi
ω#×c.c,(A1)
where we introduce M=J S 2/8 for brevity and i/ω is the
Fourier transform of the Heaviside function in τ. Since the
different Fourier components are decoupled in this action, we
can minimise with respect to them separately, resulting in the
stationary action
h0(k, ω)=2πiqM
ω2+MD0(k)
φ0(k, ω)=2πiq
ω
MD0(k)
ω2+MD0(k).(A2)
In real space, this corresponds to a point-like instanton de-
scribed in Sec. II D 2 together with a power-law decaying “in-
stanton core.” We should also note that (A2) is not a station-
ary trajectory under the original action, but it becomes one if
the M(1 −cos φ) potential term is replaced with a continued
parabolic potential, V(φ)=M
2minn(φ−2πn)2: Indeed, the
only point where φ0reaches πis r=τ=0, where the cusp in
the potential is recovered by the external instanton charge. In
the following, we will thus use V(φ).
The action can now be expanded to quadratic order in δh
and δφ around both the trivial trajectory h=φ≡0 and the
instanton trajectory (A2). δhcan easily be integrated out in
both cases, giving the following actions in δφ:
δS0=1
2Z(dω)(d2k) ω2
D0(k)+M!δφ(k, ω)δφ(−k,−ω)
(A3)
δSi=δS0−πM
∂τφ0(r=τ=0) [δφ(r=τ=0)]2,(A4)
where the additional term in (A4) corresponds to the cusp of
the continued parabolic potential at φ=π, only reached at
r=τ=0. The most important difference between the two
actions is that δSihas the zero mode ψ=∂τφ0, corresponding
15
to the continuous time-translation symmetry of the setup. For
such modes, the usual contribution to the partition function,
e−Scl (det K)−1/2, is replaced by
dτrhψ|ψi
2πe−Scl (det ˜
K)−1/2,(A5)
where Scl is the action due to the stationary instanton, given by
(2.31), and ˜
Kis the fluctuation kernel of (A4) restricted to the
non-zero modes. Suppose |ψiis proportional to a basis vector
(this can always be achieved using a unitary transformation
on the kernel K). Then, ˜
Kis the principal minor of Kthat
excludes the row and column of ψ. det ˜
Kis thus a cofactor of
the full kernel, and we have
det ˜
K=h˜
ψ|K−1|˜
ψidet K(A6)
by the cofactor formula for matrix inversion, where |˜
ψiis the
normalised zero mode |ψi/phψ|ψi.
In a matrix language, Kis the sum of the kernel K0appear-
ing in (A3) and a dyad −λ|vihv|, where |vicorresponds to the
δ-function at r=τ=0 implied in (A4) in an arbitrary basis.
For such a matrix, we have the following:
(i) det K=det K0(1 −λhv|K−1
0|vi).
(ii) If 1 −λhv|K−1
0|vi=0, K−1
0|viis an eigenvector of Kwith
eigenvalue 0.
(iii) K−1=K−1
0+
λK−1
0|vihv|K−1
0
1−λhv|K−1
0|vi.
The first statement can be shown by inserting factors of
K1/2
0K−1/2
0into the definition of K; the other two are straight-
forward to verify. Now, det ˜
Kfollows as
det ˜
K=*˜
ψ0K−1
0+
λK−1
0|vihv|K−1
0
1−λhv|K−1
0|vi˜
ψ0+×
(1 −λhv|K−1
0|vi) det K0(A7)
=h˜
ψ0|λK−1
0|vihv|K−1
0|˜
ψ0idet K0
=λhv|K−2
0|videt K0.(A8)
In the first line, we substitute statements (i) and (iii) into
(A6); in the second, we note that 1 −λhv|K−1
0|vi=0 in our
case, so only the second term of K−1gives any contribution
[34]. Finally, we use that |˜
ψ0iis the normalised zero mode, so
by statement (ii), it must be K−1
0|vi/qhv|K−2
0|vi. Altogether,
the measure of the instanton solutions relative to that of the
instanton-free solution, (det K0)−1/2, is
dτshψ|ψi
2πλhv|K−2
0|vie−Scl .(A9)
In the (k, ω) basis, K0is positive definite and diagonal,
K0(k, ω)=ω2/D0(k)+M,λ=2πM/[∂τφ0(0,0)], and
v(k, ω)=1 (the Fourier transform of a δ-function at the ori-
gin). It is easy to verify that 1 −λhv|K−1
0|vi=0, that is, K
indeed has a zero mode; furthermore, K−1
0|vi ∝ ∂τφ0, as ex-
pected. Substituting into (A9) then gives that the measure of
instanton solutions of a given sign is µI, where I=e−S0and
µ=sMπZ(d2k)ω(k); (A10)
at the RK point on the square lattice, µ=JS 3/2√π/8.
Appendix B: Details of the calculation for the honeycomb lattice
Following from Sec. V, we express the bosonic operators in
the radial gauge in the path integral formulation of the model.
For r∈A:
br,η =qρη(r+eη/2) exp hiΦη(r+eη/2)i(B1)
≡rS
3+δρη(r+eη/2) exp hiΦη(r+eη/2)i.
For r∈B, the expressions are equivalent except for a change
in sign, +eη/2→ −eη/2. Notice that, by thinking of ρη(r+
eη/2) and Φη(r+eη/2) as functions defined on the midpoints
of the bonds, there is no ambiguity nor redundancy in the no-
tation.
The constraints in Eq. (5.2) can then be written (choosing
for concreteness r∈A) as
X
η
δρη(r+eη/2) =0 (B2)
δρ1(r+e1/2) +δρ2(r+e1−e2/2)
+δρ3(r+e1−e3/2) =0.(B3)
Taking the Fourier transform with respect to sublattice A, the
constraints take on a more symmetric form:
X
η
e−ikeη/2δρη(k)=0,X
η
eikeη/2δρη(k)=0,(B4)
where in the second line we divided out an overall factor e−ike1.
More conveniently, we can add and subtract them to obtain
X
η
cηδρη(k)=0,X
η
sηδρη(k)=0,(B5)
where again we used the shorthand notation cη=cos(keη/2)
and sη=sin(keη/2). With three field variables δρη(k) and two
constraints, we expect only one degree of freedom.
As we did for the square lattice, it is convenient not to at-
tempt to resolve the constraint directly but rather consider first
the argument of the cosine term in the Hamiltonian,
˜
φ(r)= Φ1(r+e1/2) −Φ3(r+e1−e3/2) + Φ2(r+e1−e3+e2/2)
−Φ1(r+e2−e3+e1/2) + Φ3(r+e2−e3/2) −Φ2(r+e2/2),(B6)
16
and its Fourier transform
˜
φ(k)= Φ1(k)e−ike1/2−Φ3(k)e−ik(e1−e3/2) + Φ2(k)e−ik(e1−e3+e2/2)
−Φ1(k)e−ik(e2−e3+e1/2) + Φ3(k)e−ik(e2−e3/2) −Φ2(k)e−ike2/2
=eike32i[Φ1(k)(s2c3−s3c2)+ Φ2(k)(s3c1−s1c3)+ Φ3(k)(s1c2−s2c1)].(B7)
The nicely symmetric expression in Eq. (B7) required a few
lines of algebra and the property of the lattice vectors e1+
e2+e3=0. As we inferred above from the constraints on
the δρηfields, there is only one real scalar degree of freedom,
and as before it is convenient to do away with the phase factor
[effectively, use plaquette centres as reference points for φ(r)]
and define
φ(k)=e−ike3˜
φ(k) (B8)
=2ihΦ1(k)(s2c3−s3c2)+ Φ2(k)(s3c1−s1c3)
+Φ3(k)(s1c2−s2c1)i.
[Notice the importance of the factor of iin preserving the con-
dition of Fourier transform of a real field, φ∗(k)=φ(−k) due
to the antisymmetric behaviour of the sine function.]
Finally, we look for a conjugate field h(k) such that the
Berry phase in the path integral can be written as ih(k)∂τφ(−k).
Namely, we look for h(k) that satisfies
h(k)∂τφ(−k)=X
η
δρη(k)∂τΦη(−k).(B9)
Substituting the expression for φ(k), Eq. (B8), into the equa-
tion above, we obtain
δρ1(k)=−2i(s2c3−s3c2)h(k)
δρ2(k)=−2i(s3c1−s1c3)h(k)
δρ3(k)=−2i(s1c2−s2c1)h(k).(B10)
One can then straightforwardly verify that the introduction of
the field h(k) automatically resolves the constraints,
X
η
cηδρη∝hc1(s2c3−s3c2)+c2(s3c1−s1c3)
+c3(s1c2−s2c1)ih(k)=0 (B11)
X
η
sηδρη∝hs1(s2c3−s3c2)+s2(s3c1−s1c3)
+s3(s1c2−s2c1)ih(k)=0.(B12)
We are thus in the position to write the full large-Saction of
the system, including both the Berry phase and Hamiltonian
contributions, in terms of the fields h(k) and φ(k) only.
To obtain the Gaussian field theory for the honeycomb lat-
tice QDM, it is convenient to rewrite cos(φ)=1−[1 −cos(φ)]
and notice that the term in square brackets contains only
quadratic and higher-order contributions. Correspondingly,
we can write
√ρρρρρρ cos(φ)'√ρρρρρρ −S3
27 1−cos(φ),(B13)
and focus on expanding to second order the first term on the
right hand side, as well as the Vρρρ terms in the Hamiltonian
in Eq. (5.1). Linear terms in δρη(r+eη/2) vanish upon sum-
ming over rbecause of the dimer constraint. After quite some
algebra, one arrives at the following contributions to quadratic
order:
JS
3nδρ1(k)δρ1(−k)s2
23 +δρ2(k)δρ2(−k)s2
31
+δρ3(k)δρ3(−k)s2
12o(B14)
−JS
3nδρ1(k)δρ2(−k)+c.c.c23c31
+δρ2(k)δρ3(−k)+c.c.c31c12
+δρ3(k)δρ1(−k)+c.c.c12c23 o
+VS
3nδρ1(k)δρ2(−k)+c.c.c23c31 +s23 s31
+δρ2(k)δρ3(−k)+c.c.c31c12 +s31 s12
+δρ3(k)δρ1(−k)+c.c.c12c23 +s12 s23 o,
where we introduced
sµν =sin[k(eµ−eν)/2] =sµcν−sνcµ
cµν =cos[k(eµ−eν)/2] =cµcν+sµsν(B15)
for convenience. Substituting the expressions (B10) of δρη(k)
in terms of h(k), and ignoring trivial constants, we obtain the
action given in Eq. (5.4) in Sec. V.
Appendix C: Details of the calculation for the diamond lattice
The interaction matrix Dµν can be obtained in much the
same way as for the honeycomb lattice, see Appendix. B. For
plaquettes in the µ=1,2,3 sublattice, we obtain the contribu-
tion
JS
4nδρ1(k)δρ1(−k)s2
23 +δρ2(k)δρ2(−k)s2
31 +δρ3(k)δρ3(−k)s2
12
−δρ1(k)δρ2(−k)+δρ2(k)δρ1(−k)c23c31 −δρ2(k)δρ3(−k)+δρ3(k)δρ2(−k)c31 c12
−δρ3(k)δρ1(−k)+δρ1(k)δρ3(−k)c12c23 o
17
+VS
4nδρ1(k)δρ2(−k)+δρ2(k)δρ1(−k)c23c31 +s23 s31 +δρ2(k)δρ3(−k)+δρ3(k)δρ2(−k)c31c12 +s31 s12
+δρ3(k)δρ1(−k)+δρ1(k)δρ3(−k)c12c23 +s12 s23 o,(C1)
where we again introduced cµν =cos[k(eµ−eν)/2] =cµcν+sµsνand sµν =sin[k(eµ−eν)/2] =sµcν−cµsν. The other three
sublattices of plaquettes give rise to equivalent contributions with 123 ↔234 ↔341 ↔412. Now, the matrix Dcan be written
explicitly as
D=JS
2J+VS
2V(C2)
J=
s2
23 +s2
34 +s2
43 −(c13c23 +c14 c24)−(c12 c32 +c14c34)−(c12 c42 +c13c43 )
−(c23c13 +c24 c14)s2
34 +s2
41 +s2
13 −(c21c31 +c24 c34)−(c21 c41 +c23c43)
−(c32c12 +c34 c14)−(c31 c21 +c34c24)s2
41 +s2
12 +s2
24 −(c31c41 +c32 c42)
−(c42c12 +c43 c13)−(c41 c21 +c43c23)−(c41 c31 +c42c32 )s2
12 +s2
23 +s2
31
(C3)
V=
0c13c23 −s13 s23 +
c14c24 −s14 s24
c12c32 −s12 s32 +
c14c34 −s14 s34
c12c42 −s12 s42 +
c13c43 −s13 s43
c23c13 −s23 s13 +
c24c14 −s24 s14
0c21c31 −s21 s31 +
c24c34 −s24 s34
c21c41 −s21 s41 +
c23c43 −s23 s43
c32c12 −s32 s12 +
c34c14 −s34 s14
c31c21 −s31 s21 +
c34c24 −s34 s24
0c31c41 −s31 s41 +
c32c42 −s32 s42
c42c12 −s42 s12 +
c43c13 −s43 s13
c41c21 −s41 s21 +
c43c23 −s43 s23
c41c31 −s41 s31 +
c42c32 −s42 s32
0
.(C4)
We can express the matrices Jand Vmore concisely by in-
troducing the matrix Cwith entries cµν, the diagonal matrix
Rwith entries Rµµ =Pνcos[k(eµ−eν)] −2, and defining
Ω = Pµ<ν s2
µν:
J= ΩI+R − C2+2C(C5)
V=2C2−6C−R,(C6)
where Iis the identity matrix.
Ignoring instantons, the middle term of Eq. (6.2) can be
expanded about a given minimum of the cosine, giving
JS 3
64 X
α,r
φ2
α(r, τ)=JS 3
64 X
α,k|φα(k, τ)|2=(C7)
JS 3
64 X
k,µνα Zαµ(k)Zαν (−k)Φµ(k, τ)Φν(−k, τ),
where Zµν =2iεµνλκsλcκ, see Eq. (6.3). Defining the matrix
M=JS 3
32 ZT(−k)Z(k)=JS 3
32 Z2,(C8)
the quadratic action can now be written in the form (6.6).
The dispersion ω2(k) is given by the eigenvalues of the ma-
trix MD. It is important to note that Zµνcν=Zµν sν=0 by
the definition of Z. Consequently, MC =0, which greatly
simplifies the form of MD. All in all, one has to diagonalise
the matrix
JS 4
64 ΩJZ2+(J−V)ZRZ.(C9)
It follows from the definition of Zthat cµand sµ(understood
as 4-dimensional vectors) are eigenvectors of this matrix with
zero eigenvalue. These correspond to the unphysical modes
ruled out by the constraint (6.5).
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[34] Note that, since 1 −λhv|K−1
0|vi=0 due to the presence of the
zero mode, both terms in (A7) are ill-defined. However, both
are finite for any other λ, so we can formally evaluate it else-
where and take the limit of λgoing to its physical value. The
divergence due to 1 −λhv|K−1
0|vi=0 cancels, so the correct
limit is indeed given by (A8).
