Spin solid phases of spin 1 and spin 3/2 antiferromagnets on a cubic
  lattice
Spin solid phases of spin 1 and spin 3/2 antiferromagnets on a cubic lattice.
Karol Gregor and Olexei I. Motrunich
Department of Physics, California Institute of Technology, Pasadena, CA 91125
(Dated: October 30, 2018)
We study spin S = 1 and S = 3/2 Heisenberg antiferromagnets on a cubic lattice focusing on
spin solid states. Using Schwinger boson formulation for spins, we start in a U(1) spin liquid phase
proximate to Neel phase and explore possible confining paramagnetic phases as we transition away
from the spin liquid by the process of monopole condensation. Electromagnetic duality is used to
rewrite the theory in terms of monopoles. For spin 1 we find several candidate phases of which the
most natural one is a phase with spins organized into parallel Haldane chains. For spin 3/2 we find
that the most natural phase has spins organized into parallel ladders. As a by-product, we also
write a Landau theory of the ordering in two special classical frustrated XY models on the cubic
lattice, one of which is the fully frustrated XY model. In a particular limit our approach maps to
a dimer model with 2S dimers coming out of every site, and we find the same spin solid phases in
this regime as well.
PACS numbers:
I. INTRODUCTION
A simple, nontrivial, and physically common example
of a regular system of quantum objects is a collection of
spins on a lattice. This is easiest to analyze if the in-
teractions do not compete and all prefer the same spin
state; the resulting phases have been known for a long
time and include ferromagnetic and Neel states. A much
richer situation of current interest is when interactions
compete. The frustration together with quantum fluctu-
ations can destroy the magnetic order and produce spin
solid or spin liquid phases. In a spin solid, spins combine
into larger singlet objects such as valence bonds which
form an ordered pattern on a lattice. Such phases have
been found in nature,1,2,3 and also in numerical studies
of model Hamiltonians.4,5,6 A spin liquid, on the other
hand, is a featureless paramagnet, which can be crudely
viewed as a quantum superposition of many valence bond
configurations, thus the name “resonating valence bonds”
(RVB) state. So far there are only few experimental can-
didates, but on the theoretical side the existence of spin
liquids in many varieties and our understanding of them
is well established (see Ref. 7 for a recent collection of
references and also a very recent example of the so-called
Coulomb phase in 3d, which is the spin liquid relevant to
the present work).
In this paper we look for natural spin solid phases of
spin 1 and spin 3/2 on a cubic lattice. A direct study of
spin Hamiltonians that can stabilize such phases is diffi-
cult but can be done in some cases with Quantum Monte
Carlo. Which phases are realized will of course depend on
the specific model: For example, Refs. 4,5 found valence
bond solids in spin 1/2 systems with ring exchanges on
the square and cubic lattices. Refs. 6,8 found spin solid
phases for a spin 1 model with biquadratic interaction
on the anisotropic square lattice, but only magnetically
ordered phases on the isotropic square and cubic lattices.
Here we follow instead a more phenomenological
approach.9,10,11,12 A systematic and commonly used
route to achieve this, and the one we start with, is to
generalize the spins to a representation of higher symme-
try group, here taken to be SU(N).9,10 The problem can
be solved exactly in the N → ∞ limit and one can con-
sider fluctuations around this limit to get long distance
properties of the system. This approach, while difficult to
connect with the actual microscopic SU(2) spin system,
nevertheless gives us some guidance about what phases
to expect and gives us a form of the effective field theory.
Here it results in a gauge theory which naturally exhibits
deconfined (liquid) and confined (solid) phases, and we
expect that if a microscopic spin system has such phases,
they should be described by this theory.
FIG. 1: The most natural spin solid phase for S = 1 on the
cubic lattice. The thick lines denote links with large spin-spin
correlations suggesting that the spins organize into Haldane
chains along one lattice direction.
One of the spin liquid phases expected in 3d is the
so-called Coulomb phase. It is a compact U(1) gauge
theory coupled to matter in the deconfined phase, where
the matter fields (spinons) are gapped, gauge field (emer-
gent photon) is gapless, and monopoles (which arise due
http://arxiv.org/abs/0704.0821v1
FIG. 2: The most natural spin solid phase for S = 3/2 on the
cubic lattice. The drawn bold lines denote links with large
spin-spin correlations suggesting that the spins organize into
ladders.
to compactness) are gapped. In addition, importantly,
there are spin Berry phases that lead to the presence of a
background charge in the gauge theory formulation. This
makes the confined phases nontrivial in that they break
lattice symmetries and therefore correspond to various
spin solids. The transition occurs because the monopoles
condense, and the theory can be equivalently analyzed in
terms of them by employing standard electro-magnetic
duality. The background charge causes monopoles to
acquire a phase when they hop around a plaquette.12
This leads to a nontrivial monopole condensation pat-
tern, which then corresponds to a spin solid phase. In
2d the physics is similar, except that the monopoles are
instantons and they always proliferate, so there is no
Coulomb spin liquid. This approach was first used by
Read and Sachdev10 on the square lattice. The spin solids
for spin 1/2 on the cubic lattice were analyzed in Ref. 12
and near several different Coulomb spin liquids in Ref. 13.
Ref. 14 was led (in a different context) to a gauge theory
with background charges on a diamond lattice which was
attacked using analogous techniques.
For the spins on the cubic lattice, the analysis depends
only on the spin magnitude. Any case can be mapped
onto S = 0, 1/2, and 1 in 2d and S = 0, 1/2, 1, and
3/2 in 3d. Only the spin 1/2 case was considered so
far, but these results cannot be transferred to the other
spins since each requires a separate analysis. This is the
task of the present work. We find that the most nat-
ural phases for spin 1 and 3/2 are the ones shown on
Figures 1 and 2. In the S = 1 case the spins organize
into Haldane chains. This is easiest to understand in the
standard picture where we break spin 1 into two spin
1/2’s and form singlets with spin 1/2’s of spins on either
side. Similarly, in the S = 3/2 case we break spin 3/2
into three spin 1/2’s and form singlets on the bonds of
the ladders. Several approaches that we have taken and
used in different parameter regimes suggest the same spin
solid states, which gives us confidence that these phases
are very natural in the two cases.
II. SCHWINGER BOSONS, DUAL
REFORMULATION, AND A BASIC PHASE
DIAGRAM
A. Schwinger bosons
We begin by briefly reviewing the standard technique
of large N for spins.9,10 This maps (approximately) our
spin system into a theory of spinons coupled to a U(1)
gauge field in the presence of static background charges.
Our main work is the analysis of this theory, while the
purpose of the review here is to establish the connection
with the properties of the original spin system.
The basic steps in the derivation are as follows. We
generalize the SU(2) spin to SU(N) spin and denote it by
Sβα(i). We write the spins in terms of Schwinger bosons:
Sβα(i) = b
α(i)b
β(i) sublattice A ,
Sβα(j) = −b̄β†(j)b̄α(j) sublattice B , (1)
where the b, b̄’s are bosonic operators that transform un-
der the fundamental representation of SU(N) if the index
is on the top and under its conjugate if the index is on
the bottom. To get the Hilbert space of the spins we
need to restrict the boson occupations as
b†α(i)b
α(i) = nc ,
b̄α†(j)b̄α(j) = nc , (2)
where nc corresponds to the spin length. The SU(N)
spin Hamiltonian is
〈i,j〉
Sβα(i)S
β (j) , (3)
which reduces to the SU(2) Heisenberg spin model for
spin S when N = 2 and nc = 2S.
Next we write the system in the path integral pic-
ture, imposing the constraints (2) by Lagrange multi-
pliers. The spin interaction contains quartic terms; to
get action that is quadratic in the boson fields, we use
Hubbard-Stratonovich transformation and obtain
b†α(i)
+ iλ(i)
bα(i)− iλ(i)nc
b̄α†(j)
+ iλ(j)
b̄α(j)− iλ(j)nc
〈i,j〉
|Qij |2 −Q∗ijbα(i)b̄α(j) + h.c. (4)
The path integral goes over b, b̄, Q, λ.
We can now integrate out the b’s. The resulting ex-
pression will have coefficient N in front of it. At large
N it can be approximated by its saddle point value.
Our departing point is such a “mean field” with uni-
form Qr,r+m̂(τ) = Q̄ and λ(r, τ) = λ̄ and assuming
gapped b spectrum; this represents a Coulomb spin liq-
uid, which is a stable phase in three dimensions. The
effective theory is obtained by considering the fluctua-
tions of the fields, Qr,r+m̂(τ) = [Q̄ + qm(r, τ)]e
iαm(r,τ)
and λ(r, τ) = λ̄ + iα0(r, τ). Here r runs over all sites of
the cubic lattice and m̂ = x̂, ŷ, ẑ denotes one of the direc-
tions in 3d. The amplitude fields qm are massive, and so
are the fields αm and α0 near the wavevector (0, 0, 0). On
the other hand, the fields αm and α0 near the wavevector
(π, π, π) are massless and describe the gauge field (pho-
ton) of the Coulomb phase, am ≡ α(π,π,π)m , aτ ≡ α(π,π,π)0 .
For details of the derivation, see the original Ref. 10 (our
notation is slightly different compared to these papers,
which use a two-site unit cell labeling instead).
As emphasized in Refs. 10,11, we also have to con-
sider effect of Berry phases, which is crucial for the un-
derstanding of the spin solid states. A very convenient
encapsulation of the low-energy degrees of freedom and
the Berry phases is provided by the following re-latticized
Euclidean action:15,16
Daiµe
−Sa−SB , (5)
Sa = −β
i,µ<ν
cos(∇µaν −∇νaµ) ,
SB = i
ηiaiτ .
Here we have a compact U(1) gauge field a residing on
the links of a (3+1)d space-time lattice and described by
the action term Sa. The SB term comes from detailed
consideration of the Berry phases, and ηi is 2S on one
sublattice of the spatial lattice and −2S on the other
one. In the Hamiltonian language this has a simple in-
terpretation as a background charge of value 2S on one
sublattice and −2S on the other one:
H = u
E2rm − κ
r,m<n
cos(∇man −∇nam) , (6)
(∇ ·E)r = ηr = ±2S , (7)
where Em are electric fields residing on the links of the
3d cubic lattice and conjugate to am. Thus we obtained a
compact U(1) gauge theory in the presence of background
charge.17,18,19
Throughout, we will assume the spinons are gapped
and are integrated out. Note that even though we start
in the Coulomb phase where the gauge field is decon-
fined, the above action also provides access to confining
paramagnetic phases, and this will be our main focus.
To sum up, we will be describing spin solid phases that
are proximate to the simple Coulomb phase; the latter
with the specified Berry phases encoded in the staggered
background charge is in turn appropriate in the vicinity
of the conventional Neel phase.
Since we will continue with (5) we need a way to con-
nect the variables there to the original spin variables.
This is done as follows. The nearest neighbor spin-spin
correlation 〈Sr ·Sr′〉 is proportional to the bond variable
|Qrr′|2. To get the connection between the fluctuation
of the magnitude of Q and the gauge fields we have to
also keep the massive amplitude fields qm in the above
derivation when integrating out the b’s. One finds that
the (π, π, π) component couples to the gauge fields in
the action as follows: δS = iγ
(π,π,π)
m (∂maτ − ∂τam),
with some coupling parameter γ. On the other hand, in
the derivation of the path integral from the Hamiltonian
formulation of the gauge theory, the electric field is cou-
pled to the gauge field in the same way, i.e., via a term
Em(∂maτ − ∂τam) in the action. Thus the electric
field gives the fluctuation of the staggered nearest neigh-
bor spin-spin correlation function.
B. Electro-magnetic duality
We now proceed to the analysis of the model (5). We
are interested in the confining phases, which will neces-
sarily break lattice symmetries for spins S = 1/2, 1, 3/2
studied here. The confinement occurs due to condensa-
tion of monopoles. Therefore we would like to express
the theory in terms of them. This can be done by the
standard electro-magnetic duality. The duality maps the
theory of a compact U(1) gauge field without charges into
a theory of a noncompact gauge field coupled to charges
– the monopoles of the original theory. The noncompact-
ness comes from the fact that we have dropped the elec-
tric charges in the original theory; had we retained them,
we would have obtained a compact dual gauge field whose
monopoles would correspond to the original charges. The
new variables reside on the lattice dual to the original lat-
tice. The background charge of the original theory gives
rise to a static dual magnetic flux emanating out of the
center of each cube as drawn in Figure 3. This flux al-
ternates in sign from one cube to the next and frustrates
the monopole hopping. Therefore we obtain a theory of
monopoles with frustrated hopping that are coupled to
the dual noncompact gauge field.12 The duality can be
done explicitly with various approximations clearly dis-
played as is written in Appendix A.
Explicitly, the partition function is
DL e−Sdual , (8)
Sdual =
∑ (∂L)2
λ cos(L + L0 −∇θ) , (9)
where L is the dual gauge field, (∂L)µν = ∇µLν −∇νLµ
is the four dimensional curl, L0 is the frustration that re-
sults from the original background charge and ultimately
z even
z odd
/32πS
FIG. 3: a) Original background electric charges 2S and −2S
on the two sublattices give rise to the static dual magnetic
fluxes as seen by the monopoles. b) Gauge choice for L0 that
realizes these fluxes modulo 2π.
from Berry phases, and θ is the monopole field. A con-
venient choice of L0 that produces the appropriate static
fluxes is shown in Figure 3; all subsequent work is done
in this gauge.
The advantage of the dual formulation is that it has no
sign problem and can be in principle studied by Monte
Carlo. A sketch of the phase diagram is on Figure 4. In
the bottom left side of the diagram, the monopoles are
gapped and the system is in the deconfined phase, which
correspond to the Coulomb spin liquid in the spin model.
At large enough λ and 1/β, the monopoles condense.
They can condense in various patterns which translate
to various spin solid phases of the original model. Dual-
ity relates the original field theory (5) to the large λ part
of the dual theory. It is hard to analyze the transition in
the large λ limit. Instead we look at three different places
in this phase diagram. At 1/β = ∞ the system becomes
frustrated XY model. First we analyze the phase tran-
sition looking for ordering of the XY spins as we cross
the phase boundary to the ordered phase. This gives us
the most likely monopole condensation patterns near the
transition. Next we look at the classical ground state
of the XY model in the upper right corner of the phase
diagram as approached from 1/β = ∞. Finally we look
near the same point but in the limit λ ≫ 1/β.
FIG. 4: Sketch of the expected phase diagram for the dual
action Eq. (9).
III. ANALYSIS 1,2: FRUSTRATED XY MODEL
AT 1/β = ∞
A. Outline of the Analysis
In this section we describe in general terms the anal-
ysis in the 1/β = ∞ limit where the dual action Eq. (9)
reduces to a frustrated XY model. We look at the phase
transition and the classical ground state.
In the Analysis 1, we consider the transition in the
spirit of the Landau theory. We identify the relevant low-
energy fields, write the most general quartic potential
consistent with the symmetries, and study it in mean
field. The approach is the same for each spin S, but
the details are unique in each case and are contained
in Subsections III B and III C for spin 1 and spin 3/2
respectively (spin 1/2 was considered using this approach
in Ref. 12).
More explicitly, the mean field derivation is done as
follows. The mean field theory of XY spins is described
by the continuous soft spin action
|∂τΦR|2 −
〈RR′〉
(tRR′Φ
RΦR′ + c.c.) +
V (|φR|2)
with some potential V (|ΦR|2) = r0|ΦR|2 + u0|ΦR|4 + · · ·.
After crossing the transition from the disordered side, the
system enters a phase with non-zero ΦR that minimize
this action. The initial step is to minimize the kinetic
energy. This will turn out to have a three-dimensional
manifold of minima for spin 1 and four-dimensional one
for spin 3/2. One then expands around these minima
and writes all terms to a given order that are allowed by
symmetry. In both cases the degeneracy is lifted at the
fourth order. We will find that for spin 1 there are three
independent quartic terms and we manage to draw a gen-
eral phase diagram of the Landau theory. For spin 3/2,
there are five such terms and the parameter space is too
rich for us to describe the phase diagram completely. In
this case we confine ourselves to examining the potential
obtained from the most natural microscopic fourth order
term and determining its mean field phase.
In the Analysis 2, we find the classical ground state of
the frustrated XY model – the state in the upper right
point of the phase diagram Figure 4 – by a direct mini-
mization of the hard-spin action (9). We use the following
method: For some system size, we start with a random
configuration of spins. We pick a random spin and min-
imize its (local) energy and repeat this process until the
total energy converges. Different starting configurations
will lead to different final energies, because sometimes the
system gets stuck in some local minima. We repeat this
procedure for many starting configurations and also for
different system sizes. We then select the configurations
with the same lowest energy, which gives the absolute
minimum of the potential. The case of spin 3/2, which
corresponds to a fully frustrated XY model, was already
considered some time ago in Ref. 20, and our method
produces results in agreement with that work.
For both spin 1 and spin 3/2, we find that the clas-
sical ground state coincides with the most natural state
identified in the mean field theory near the transition.
This suggests that there is only one XY-ordered phase
along the 1/β = ∞ line in Fig. 4, which could in princi-
ple be tested in Monte Carlo studies of the corresponding
frustrated XY models.
We have described how to find the phases of the dual
action in the 1/β = ∞ limit. However we are interested
in the phases of the original spin model. To make the
connection we calculate the energies and staggered curls
of the monopole currents in the dual model and relate
them to variables in the original spin problem. These
variables are the plaquette energy and the bond expecta-
tion value respectively. This allows us to determine the
spin solid patterns.
The mapping of the first variable, the energy, is sim-
ple. Energy simply maps to energy. In the dual model
we can calculate the energy for each bond, which is
ǫ = 2Re(tRR′Φ
RΦR′). The center of a bond of the dual
lattice coincides with the center of a plaquette of the orig-
inal lattice, and so the calculated energy is the plaquette
energy of the original model.
The connection of the staggered curls of the monopole
current to the original bond variables is established
as follows. The monopole current is given by JM =
2Im(tRR′Φ
RΦR′). In terms of the original gauge the-
ory, Eq. (5), just as the electric current produces mag-
netic field, the magnetic current produces electric field.
The resulting electric field is given by the analog of Biot-
Savart law. However, approximately, if we have a loop of
the magnetic current, the electric field it produces in the
center is proportional to the circulation of the current,
which is what we call the curl of the monopole current.
As we described in the preceding section, the electric field
is proportional to the staggered fluctuation of the near-
est neighbor spin-spin correlation function, therefore the
claimed connection. We will use this extensively in the
detailed treatment of spin 1 and spin 3/2 below.
B. Results: Spin 1
1. Analysis 1: Phase transition of the XY model
Now we turn to finding the phases for spin 1. We
choose the gauge shown on Figure 3. In this case the
hopping amplitudes in (10) are given by
tR,R+x̂ =
(−1)z
3 + i(−1)x+y
, (11)
tR,R+ŷ =
(−1)z
3− i(−1)x+y
, (12)
tR,R+ẑ = 1 . (13)
The band structure has three minima and hence the space
of ground states of the kinetic energy is three dimen-
sional. Convenient choice of the basis is the following:
(−1)x+y+z − (−1)x+y
3 + i[(−1)z +
Ψ2 = i
(−1)x+y+z − (−1)x+y
3− i[(−1)z +
Ψ3 = −
(−1)y + i(−1)x√
A general kinetic energy ground state can be written as
Φ(R) = φ1Ψ1(R) + φ2Ψ2(R) + φ3Ψ3(R) (14)
with complex fields φ1,2,3. This degeneracy will be lifted
by nonlinear terms. To find out how, we would like to
write the Landau theory for the φ’s, including all terms
that are allowed by symmetry. Thus we need to find how
the φ’s transform under the lattice symmetries.
The generators of the symmetries are the translations
by one lattice spacing in the x,y,z directions, 90 degree
rotations around the x,y,z axes (it suffices to consider two
out of three rotations), and mirror reflections. Note that
the fluxes seen by the monopoles (and encoded in the
complex phases of the hopping amplitudes tRR′) change
sign under unit translations. The original spin problem
is translationally invariant, and this is represented in the
dual action (10) as follows. The fluxes remain unchanged
if the t’s are also conjugated after the translation, and
there is a gauge transformation that brings such mod-
ified t’s to the original themselves. The action of the
symmetry on the field Φ is then a combined application
of the translation of the coordinates, conjugation, and
gauge transformation. Similar considerations apply for
the 90 degree rotations performed here about the dual
lattice axes. After carrying through this analysis, the
transformation properties of φ’s are remarkably simple:
Tx Ty Tz Rx Ry Rz
φ1 → – + + + φ∗3 φ∗2
φ2 → + – + φ∗3 + φ∗1
φ3 → + + – φ∗2 φ∗1 +
In this table, “+” or “−” stands for φi → φ∗i or
φi → −φ∗i respectively. We see that φ1 can be loosely
associated with the x direction, φ2 with y, and φ3 with z.
We should also point out that under mirror symmetries
in the dual lattice planes the fields transform simply
φi → φi.
There is only one invariant term at the quadratic level:
V (2) = m(|φ1|2 + |φ2|2 + |φ3|2) , (15)
where m is a constant. There are three independent al-
lowed terms at the quartic level, and the most general
quartic potential can be written in the form
V (4) = u(|φ1|2 + |φ2|2 + |φ3|2)2
+ v(|φ1|4 + |φ2|4 + |φ3|4)
+ w(φ∗21 φ
2 + φ
3 + φ
1 + c.c.) , (16)
where u, v, w are constants.
To find the phases of the Landau theory, we simply
need to minimize this potential. Before we start describ-
ing the phases, however, it is useful to introduce bilinears
of the fields. The reason is that these are gauge indepen-
dent objects whereas the form of Ψ1,2,3 and hence the
transformation properties of φ1,2,3 are gauge dependent.
We consider the following bilinears:
B0 = |φ1|2 + |φ2|2 + |φ3|2 ,
(|φ1|2 + |φ2|2 − 2|φ3|2) ,
F2 = |φ1|2 − |φ2|2 ,
Dx = φ
3φ2 + φ
2φ3 ,
Dy = φ
1φ3 + φ
3φ1 ,
Dz = φ
2φ1 + φ
1φ2 ,
Nx = i(φ
3φ2 − φ∗2φ3) ,
Ny = i(φ
1φ3 − φ∗3φ1) ,
Nz = i(φ
2φ1 − φ∗1φ2) .
The B0 and the groups of F ’s, D’s and N ’s form ir-
reducible representations of dimensions 1, 2, 3, and 3
respectively. The transformation properties of these bi-
linears are displayed in the following table
Tx Ty Tz Rx Ry Rz
B0 + + + + + +
F1 + + + − 12F1 +
F2 − 12F1 −
F2 + + +
Dx + − − + Dz Dy
Dy − + − Dz + Dx
Dz − − + Dy Dx +
Nx − + + + Nz Ny
Ny + − + Nz + Nx
Nz + + − Ny Nx +
We should also add that all bilinears transform
trivially under mirror symmetries in the dual lattice
planes.
We next calculate the energies and the staggered
curls of the monopole currents, which, as described
in Subsec. III A, are related to the plaquette ener-
gies and bond variables of the original spin prob-
lem. To repeat, the energy is given by ǫµ(R) =
2Re(tR,R+µ̂Φ
RΦR+µ̂), and the monopole current is given
by Jµ(R) = 2Im(tR,R+µ̂Φ
RΦR+µ̂). The staggered curl
of the monopole current is what the name suggests, for
example, fz ≡ (−1)x+y+z[Jx(R) + Jy(R + x̂) − Jy(R) −
Jx(R+ ŷ)].
The energies and staggered curls of the monopole cur-
rents are bilinears in φ and thus can be expressed in terms
of the B0, . . . , Nz. They are
− 2(−1)y+zDx
3 [(−1)yNy + (−1)zNz] , (17)
fx = 3
− 4(−1)xNx . (18)
The components in the other directions are obtained from
these by the appropriate rotations using the table, which
for all bilinears except for F ’s gives the same result as the
obvious permutation of indices. More generally, while the
numerical coefficients in these expressions are obtained
from the bare monopole hopping problem, the overall
structure of the contributing terms is dictated by the
symmetries – one only needs to remember that ǫx and fx
are associated with scalars residing on respectively pla-
quettes and bonds of the original spin lattice and also
that the rotations and mirrors quoted here are about the
axes and planes passing through the dual lattice sites.
With the above results in hand, we now turn to analyz-
ing phases of the Landau theory. The phase diagram is
obtained simply by minimizing the potential (15)+(16)
and is shown in Figure 5. The different phases are de-
scribed in the following. In each case the ground state
has finite degeneracy; we display few such states and the
others are obtained from them by obvious permutations;
we display nonzero bilinears, the energies, and the stag-
gered curls of the monopole currents for the first listed
state.
FIG. 5: Phase diagram of the Landau theory for spin 1 ob-
tained by minimizing the potential (15)+(16) for m < 0 and
u > 0 (the latter choice is made for concreteness). In the
”Quartic unstable” region on the left the potential to quartic
order is asymptotically negative and we would have to include
sixth order terms to stabilize it. The cross denotes the pa-
rameter point obtained by simply expanding the microscopic
potential |Φ|4 in terms of the slowly varying fields φ1,2,3.
Phase 1. There are three degenerate states. The
values in one of them are
φ1 = 1, φ2 = φ3 = 0; (19)
B0 = 1; F1 =
, F2 = 1; (20)
ǫx = 2, ǫy = ǫz = 1; (21)
fx = 4
3; fy = fz = −2
3. (22)
The bond variables are drawn on the original spin lattice
in Figure 1; they suggest that the spins are organized
into Haldane chains along the x direction. The values of
plaquette energies are consistent with this: the plaque-
ttes in the xy and xz planes are the same and differ from
the plaquettes in the yz plane, ǫz = ǫy 6= ǫx.
Phase 2. There are six degenerate states. The values
in one of them are
φ1 = 0, φ2 = 1, φ3 = ±i; (23)
B0 = 2; F1 = − 1√3 , F2 = −1; Nx = 2; (24)
ǫx = 2, ǫy = ǫz = 3+ 2
3(−1)x; (25)
fx = −4[
3 + 2(−1)x], fy = fz = 2
3. (26)
The corresponding drawing of the bond variables on the
original spin lattice is in Figure 6, suggesting that in this
phase the spins combine into singlets and form a colum-
nar dimer state along one direction. Permuting the val-
ues of φ1,2,3 gives six degenerate states that correspond
to six possible ways of placing such columnar solid onto
the cubic lattice.
Phase 3. There are eight degenerate states specified
as follows:
φ1 = 1, φ2 = e
iα2 , φ3 = e
iα3 , (27)
{α2, α3} = ±{2π/3,−2π/3}, ± {2π/3, π/3},
± {π/3, 2π/3}, ± {π/3,−π/3}; (28)
FIG. 6: Phase 2 of spin 1. The thick lines denote the positions
where the bond variables are strongest and dashed lines where
they are weakest. This suggests that the spins organize into
singlets (dimers) and form a columnar order.
B0 = 3; Dx = Dy = Dz = −1;
Nx = Ny = Nz =
3 (29)
ǫx = 4 + 2(−1)y+z + 3[(−1)y + (−1)z], etc., (30)
fx = −4
3(−1)x, etc. (31)
The nearest neighbor spin-spin correlation has higher ex-
pectation value on the sides of the cubes shown in Figure
7, which suggests that this phase corresponds to a box
state. There are eight possible ways of placing such box
state onto the cubic lattice.
FIG. 7: Phase 3 of spin 1. The bond variables have higher
expectation values on the cubes shown.
Phase 4. There are four degenerate states:
φ1 = 1, φ2 = e
iα2 , φ3 = e
iα3 ; (32)
α2 = 0, π; α3 = 0, π; (33)
B0 = 3; Dx = Dy = Dz = 2; (34)
ǫx = 4− 4(−1)y+z, ǫy = 4− 4(−1)z+x,
ǫz = 4− 4(−1)x+y; (35)
fx = fy = fz = 0. (36)
This state breaks lattice symmetries as can be seen from
the plaquette energies. However, because the bond vari-
ables fx,y,z are zero, we do not know a simple interpre-
tation of this phase in terms of the original spins; some
finer characterization than what we use here is needed to
establish this state.
This concludes the discussion of the general phase dia-
gram of the Landau theory including quadratic and quar-
tic terms. Higher-order interactions may stabilize some
other phases, but the presented states are the most nat-
ural ones. The actual lowest-energy state depends on the
parameters u, v, w, unknown apriori. If we are to guess
which of the four phases is the most likely candidate in
the specific frustrated XY model, we can consider the
simplest microscopic quartic potential |Φ|4. When ex-
panded in terms of the continuum fields, we find u = 2,
v = −1, w = −1/2; this point is denoted by the cross in
Figure 5 and lies in the Phase 1, i.e., the Haldane chains
phase.
2. Analysis 2: The ground state of the XY model
Minimizing the classical energy of the hard spin XY
model as described in Sec. III A, we find that the ground
state configurations coincide with the condensate wave-
functions of the phase 1 and hence the state is that of the
phase 1. In particular note that each wavefunction Ψ1,2,3
has the same length |Φ| on all sites. The XY angles of
spins in this gauge in the three ground states are
(0,−30,−30, 0, 60,−90,−90, 60) , (37)
(0, 30, 30, 0,−60, 90, 90,−60) , (38)
(0,−90, 90, 180, 0,−90, 90, 180) , (39)
where the convention is that we vary position on the cube
in the x direction first, then in the y direction, and then
in the z.
3. Discussion and extension to anisotropic system
Some remarks are in order. First, it is useful to note
that the doublet F1,2 can be interpreted as an order
parameter of the Haldane chains phase. Indeed, one
can readily see that the transformation properties of F1
and F2 coincide with those of (Qx +Qy − 2Qz)/
3 and
Qx − Qy respectively, where Qm is the bond variable
in the direction m̂. On the other hand, Nx transforms
as (−1)xQx and similarly for Ny and Nz, so ~N can be
viewed as an order parameter of the valence bond solids
such as the columnar Phase 2 or the box Phase 3. In
the columnar phase, it is suggestive to view each strong
bond in Fig. 6 as representing a singlet formed by two
spin-1’s, which can be also drawn as two spin-1/2 valence
bonds connecting the two sites. However, we should be
cautious with such interpretation, since we can only tell
that the deviations of the bond variables from their mean
value will have the displayed pattern. The actual state
needs to be studied by constructing the corresponding
spin wavefunction. For example, the Haldane phase of a
spin 1 chain is stable to weak dimerization and should be
viewed as a solid formed by single-strength bonds along
the chains, so such distinct possibilities should be kept
in mind.
Let us now assume that the system is in the Phase
1. It is also interesting to ask what happens when we
stretch the lattice in one of the axis directions, say the
z-direction. In this case the Rx and Ry rotations are no
longer symmetries but the other transformations are. At
the quadratic level, the translation symmetries already
prohibit all terms except B0 and F ’s. Then from Rz we
see that only F1 is allowed. Thus at the quadratic level
one new term is allowed. In principle we should look at
the new allowed terms at the quartic level, however we
will assume that this quadratic term is leading but small
compared to the terms that were there before we broke
the symmetry.
We find that if the F1 comes with a positive pre-factor,
out of the three ground states it selects the state with
chains running along the z direction whereas if it comes
with a negative pre-factor it selects the two states with
chains running along the x and y directions.
This has a simple interpretation in terms of spins.
If the coupling in the z direction is stronger than in
the other directions the state with maximum number of
bonds in this directions is selected which is the state with
chains running in the z-direction. In the opposite case,
the states with fewest bonds in the z direction are se-
lected which are the states with chains running in the x
or y directions.
C. Results: Spin 3/2
1. Analysis 1: Phase transition of the XY model
We choose the gauge as shown on Figure 3 with S =
3/2. The hopping amplitudes are
(−1)z
1 + i(−1)x+y
, (40)
(−1)z
1− i(−1)x+y
, (41)
tz = 1 . (42)
The band structure has four minima and hence the space
of the ground states of kinetic energy is four-dimensional.
Unlike the spin 1 case where this space was three-
dimensional and simple basis vectors were found corre-
sponding to the three directions of the physical space,
there is no such form in the spin 3/2 case. The four
wavefunctions that give us relatively simple subsequent
analysis are the following
Ψ1 = (−1)x
cosβ − i(−1)x+y+z sinβ
Ψ2 = i(−1)y
cosβ + i(−1)x+y+z sinβ
1 + i(−1)x+y√
cosβ − i(−1)x+y+z sinβ
1− i(−1)x+y√
cosβ + i(−1)x+y+z sinβ
where
cosβ =
3 + 1
, sinβ =
. (43)
We again write Φ(R) =
i=1 φiΨi(R). The transfor-
mation properties of the slow fields φ1,2,3,4 are derived in
the same manner as in the spin 1 case. The symmetries
Tx : ~φ → τ3σ0 ~φ∗ , (44)
Ty : ~φ → τ0σ0 ~φ∗ , (45)
Tz : ~φ → τ1σ0 ~φ∗ , (46)
Ry : ~φ → τ1e−i
~φ∗ , (47)
Rz : ~φ → e−i
σ1 ~φ∗ . (48)
Here ~φ, ~φ∗ are column vectors, and we introduced two
sets of Pauli matrices: τ matrices that act on the blocs
{1, 2} and {3, 4}, and σ matrices that act within each bloc
(τ0 and σ0 are the corresponding identity matrices). At
the quadratic order there is one invariant term
V (2) = m
|φi|2 . (49)
At the quartic order there are five invariant terms. The
expressions in terms of φ are rather complicated and not
very illuminating, particularly since φ’s depend on the
choice of gauge and the basis. Instead, we will use gauge
invariant bilinears of φ to which we now turn.
There are 16 bilinears and they can be conveniently
organized using tensor product of the introduced two
sets of Pauli matrices, namely φ†τµσνφ with µ, ν =
0, 1, 2, 3. These break up into irreducible representa-
tions of the cubic lattice symmetry group. There are two
one-dimensional, one two-dimensional, and four three-
dimensional representations. The convenient combina-
tions that we use are
B0 = φ
†τ0σ0φ ,
C = φ†τ0σ2φ ,
F1 = φ
†τ0σ1φ ,
F2 = φ
†τ0σ3φ ,
~D = (Dx, Dy, Dz) = φ
†~τσ2φ ,
~N = (Nx, Ny, Nz) = φ
†~τσ0φ ,
Mx = φ
†τ1(−1
σ3)φ ,
My = φ
†τ2(−1
σ3)φ ,
Mz = φ
†τ3σ1φ ,
Kx = φ
σ1 − 1
σ3)φ ,
Ky = φ
†τ2(−
σ3)φ ,
Kz = φ
†τ3σ3φ .
The transformation properties of these bilinears are in
the following table
Tx Ty Tz Rx Ry Rz
B0 + + + + + +
C − − − + + +
F1 + + + − 12F1 +
F2 − 12F1 −
F2 + + +
Dx + − − + Dz Dy
Dy − + − Dz + Dx
Dz − − + Dy Dx +
Nx − + + + Nz Ny
Ny + − + Nz + Nx
Nz + + − Ny Nx +
Mx − + + + Mz My
My + − + Mz + Mx
Mz + + − My Mx +
Kx − + + − −Kz −Ky
Ky + − + −Kz − −Kx
Kz + + − −Ky −Kx −
The energies and staggered curls of monopole cur-
rents in term of these bilinears are
B0 − 2(−1)y+zDx
2 [(−1)yMy + (−1)zMz]
[(−1)yKy − (−1)zKz] , (50)
fx = 2
2(F1 +
3F2) +
8(−1)x√
Nx . (51)
The components in the other directions are obtained from
these by simple rotations of the coordinates. Our general
discussion following similar expressions (17) and (18) in
the spin 1 case apply here as well (for ease of compari-
son, we are using similar labels for objects with identical
transformation properties in the two cases). However, a
word of warning is in order here, which will be explained
in Sec. III C 3 below. Observe, for example, that ~N and
~M have identical transformation properties and therefore
should enter similarly in any expression. The absence of
M ’s in the expression for ǫx and the absence of N ’s in
the expression for fx is due to their different eigenval-
ues under an additional artificial symmetry present in
the frustrated XY model, namely a charge conjugation
symmetry defined later, which is also present in our bare
kinetic term and thus in the above expressions. This
symmetry is not physical in the original spin model and
will not be used here; it is therefore important to note
that the degeneracy of the four slow modes obtained from
the bare kinetic term is protected at the quadratic level
by the physical lattice symmetries.
There are five independent fourth order terms in φ al-
lowed by translation and rotation symmetries:
I1 = B
0 , (52)
I2 = C
2 , (53)
I3 = N
z , (54)
I4 = M
z , (55)
I5 = NxMx +NyMy +NzMz . (56)
As we have said earlier, because the number of invariant
terms is large, we will not attempt to draw the phase
diagram of the Landau’s theory. Instead we look at the
natural microscopic term
V (4) = |Φ|4 = 4
I4 , (57)
where the second equality is obtained after some calcu-
lation keeping only non-oscillatory terms.
This potential does not have any continuous symme-
try left other than the global U(1) transformation of all
fields. In fact the dimensions of the subgroups of SU(4)
that keep the terms I1, . . . , I5 invariant are 15, 7, 6, 0, 0
respectively. The potential (57) achieves global mini-
mum at twelve discrete points. As an illustration, we
consider the following four minima that are associated
with the z direction in the sense to become clear below:
(φ1, φ2, φ3, φ4) =
(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1). (58)
The four states can be related to each other by transla-
tions in the z direction and rotations about the z axis.
Besides B0 = 1, the only nonzero bilinears in these states
are (F2, Nz,Kz) = (1, 1, 1), (−1, 1,−1), (1,−1,−1), and
(−1,−1, 1) respectively.
The energies are
∓ (−1)z
, (59)
± (−1)z
, (60)
, (61)
where the upper sign corresponds to the first and fourth
minima and the lower sign to the other two.
The staggered curls of monopole currents are respec-
tively
fx, fy, fz =
8(−1)z√
, (62)
8(−1)z√
, (63)
6,− 8(−1)
, (64)
6,− 8(−1)
. (65)
The staggered curls are interpreted as the strength
(above some mean) of the expectation value of nearest
neighbor spin-spin correlation function. The above val-
ues imply that the spins organize themselves into ladders
as shown in Figure 2, obtained by drawing say the pos-
itive bonds for the first of the above minima. The four
listed states correspond to the four different positions of
ladders with rungs oriented along the z-axis. The other
eight minima are obtained by 90 degree rotations around
the x and y axes and we will not write the specific values
of the variables. The ladder state is natural for S = 3/2
system, in the picture where spin 3/2 breaks up into three
spin 1/2’s and each of them forms a bond with some other
neighboring spin 1/2.
2. Analysis 2: The ground state of the XY model
We can use the same procedure as in the case of spin
1 to find the classical ground state of the appropriate
XY model. In fact, this was already done in Ref. 20
because this problem is the fully frustrated XY model
(FFXY), which is of interest by itself, and we can use
the available results. We find that the ground state con-
figurations coincide with the condensate wavefunctions
obtained above. Thus, in each of the four displayed states
(58), the microscopic boson field Φ is given precisely by
one of the four wavefunctions Ψ1,...,4. One can see that
|Φ| = 1 on all lattice sites, and the complex phases of Φ
can be interpreted as angles of the hard-spin XY model.
For example, for Φ = Ψ1 the angles are
(−β, π + β, β, π − β, β, π − β,−β, π + β) , (66)
listed in the same order as in Eq. (39). All other ground
states can be obtained by appropriate symmetry trans-
formations. The agreement of the two analyses suggests
that there is only one ordered phases in the FFXY model,
which is also supported by the available Monte Carlo
studies.20,21
3. Remark on charge conjugation symmetry in the FFXY
It is worth to point out that the fully frustrated XY
model has an additional charge conjugation symmetry.
Indeed, since π and −π fluxes are indistinguishable, tRR′
and t∗RR′ are related by a gauge transformation, tRR′ =
eiγRt∗RR′e
−iγR′ , and so the action remains invariant under
the following unitary transformation:
C : ΦR → eiγRΦ∗R . (67)
In terms of the continuum fields, this becomes
C : ~φ → τ2σ2 ~φ∗ . (68)
In particular, the bilinears Nx,y,z are odd under C while
Mx,y,z are even, so if this symmetry is included, the I5
quartic term is not allowed (this is why this term did not
appear in Eq. 57 since both the microscopic |Φ|4 and the
bare quadratic terms in Eq. 10 have this additional sym-
metry). Thus, the complete field theory for the FFXY
model is a φ4 theory with four complex fields and inde-
pendent quartic terms I1,...,4.
One consequence of the charge conjugation symmetry
is that, for example, if we draw the Ψ1 state using neg-
ative values of the staggered curls fx,y,z as opposed to
using positive values which was done in Fig. 2, we would
obtain another set of ladders that go perpendicularly to
the ones displayed and are shifted up by one lattice spac-
ing. To put this in other words, the Ψ1 and Ψ4 states that
can be related by a translation in the z direction followed
by a rotation around the z axis are also related by C. In
this sense, each of the states Eq. (58) does not define a
direction in the x-y plane since the correlations in the x
and y directions are related by the charge conjugation
symmetry.
Tracing back to the original gauge theory formulation,
this symmetry is present in the simplest model Eq. (5) for
S = 3/2 that we wrote down and the corresponding sim-
plest “dimer model” Hamiltonian Eq. (6). Specifically,
the transformation E → 1−E on the links oriented from
one sublattice to the other, or equivalently 1 ↔ 0 in the
dimer language, takes the model corresponding to spin S
to the one corresponding to spin 3−S, while the S = 3/2
case maps back onto itself. This symmetry is useful in
the specific models, but there is no corresponding sym-
metry in the microscopic derivation from the spin model,
and therefore it was not used in the preceding analysis.
Let us look what happens to the ground states when we
add small term that breaks the charge conjugation sym-
metry, the I5, to the potential. Using general arguments
it is easy to check that the twelve minima will shift but
not split, and the twelve-fold degeneracy remains since
all are related to each other by lattice symmetries. Fur-
thermore, each ground state stays translationally invari-
ant along the ladders and perpendicular to the plane of
ladders (otherwise, if this were not true, there would be
more than twelve states). In other words, the states still
have the structure of ladders. However since the charge
conjugation is broken, it is no longer true that the nega-
tive bonds are of the same magnitude as the correspond-
ing positive ones. This makes sense when interpreted
in terms of spins. In the picture where spin 3/2 breaks
up into three spin 1/2 and ladders of valence bonds are
formed, the links that belong to these ladders are differ-
ent from the links without bonds (which also form lad-
ders). For example, the system is entangled along the
former but not along the latter. Thus these two should
not be related by any symmetry.
Explicitly, the four states in Eq. (58) become
(1, δ, 0, 0), (δ, 1, 0, 0), (0, 0, 1, δ), (0, 0, δ, 1), (69)
with appropriate δ obtained from minimization. There
is now an additional non-zero bilinear Mz, and also both
F1 and F2 are non-zero. The expressions for the ener-
gies and staggered curls in the x and y directions are no
longer related, and we can then associate a unique x or y
direction with each of the four states. These are ladders
with rungs oriented in the z direction and are related to
each other by the z translations and rotations.
4. Extension to anisotropic system
As in the spin 1 case, we ask what happens when we
stretch the system along one axis, say the z-direction.
Again, the Rx and Ry rotations are no longer symmetries
but the translations and Rz are. At the quadratic level,
the translation symmetries already prohibit all terms ex-
cept B0 and F ’s. Then from Rz we see that only F1
is allowed. Thus at the quadratic level one new term is
allowed.
We find that if the F1 comes with a positive pre-factor,
out of the twelve ground states it selects four with the
ladders that lie entirely in the x-y plane, whereas if it
comes with a negative pre-factor it selects four states
with the ladders running along the z-direction. Note that
this breaking up into groups of four is a consequence of
the remaining symmetries in the system.
These results have a simple physical interpretation for
the spin system. If the coupling in the z direction is
weaker than in the other directions, the states with fewest
bonds in the z direction are selected which are the states
with the ladders lying in the x-y plane. On the other
hand, if the coupling in the z direction is stronger, the
states with the largest number of bonds in the z direction
are selected, which are the states with ladders oriented
in the z direction.
IV. ANALYSIS 3: MAPPING TO DIMERS AT
λ ≫ 1/β ≫ 1
Here we look at the right hand corner of the phase
diagram Fig. 4 in the regime with λ ≫ 1/β ≫ 1, where
as we will see the system can be mapped to dimers.17,18,19
The analysis proceeds as follows. First we gauge away
the ∇θ in Eq. (9) to obtain
Sdual =
∑ (∂L)2
λ cos(L+ L0) , (70)
Because we assume λ ≫ 1/β ≫ 1 the configurations
that contribute significantly to the partition function can
be written in the form L = −L0 + 2πn + δL where n
is an integer and δL is small. Note that the λ term
does not depend on n and the 1/β term has a gauge
invariance n → n+∇m where m’s are integers on sites.
The partition function can be written as a sum over the
gauge equivalent classes. These classes are in one-to-
one correspondence with the fluxes j = ∂n which are
integers on plaquettes, where ∂n is the four dimensional
curl (∂n)µν = ∂µnν − ∂νmµ.
Consider first configurations with δL = 0. Some con-
figurations of j minimize the action and we denote them
by jgs. As we show below, there is an extensive num-
ber of them in all our cases. The configurations with j
that are not jgs are at energy of at least ∼ 1/β higher.
Now turning on δL, if we show that the typical energy of
excitation in δL around a given j is much smaller then
1/β then we can neglect all configurations which are not
around jgs. We will assume that this is true and show
this self-consistently below.
We define Jgs = −∂L0/(2π) + jgs. We expand the
action to the second order and drop the terms that do
not depend on Jgs, δL to obtain
∑ 4πJgs · (∂δL) + (∂δL)2
(δL)2 . (71)
This is just a gaussian integral. There are two quadratic
terms and the first one has 1/β in front and contains two
derivatives while the second has λ in front and contains
no derivatives. Since we are on a lattice the derivatives
are of order one. Since λ ≫ β, the first term can be
neglected. Next we sum by parts and integrate out the
δL. Before we do this however, we notice that the cou-
pling is ferromagnetic in time direction and L0 has zero
time components and its spatial components do not de-
pend on time. This implies that the jgs and Jgs have
zero time components and their spatial components do
not depend on time. Thus we drop time components and
time derivatives from the action and treat the Jgs and
L0 as three-dimensional. Now we integrate out the δL
and obtain
Seff [J
gs] = − 1
8π2β2λ
(∇× Jgs)2 (72)
Thus, to obtain a ground state, we need to maximize the
sum of the squares of curls of Jgs.
Let us check the consistency of our approach. From
(71), δL ∼ ∇× Jgs/(λβ) and so energy∼ 1/(λβ2). This
needs to be much smaller then 1/β which implies λ ≫
1/β which is what we assumed.
FIG. 8: a) L0/(2π) where S = 1/2, 1, 3/2 is the spin. The link
variables switch orientations under elementary translation in
the x or y direction. b) The fluxes (∇ × L0)/(2π). This
figure is similar to Fig. 3 with 2π’s removed to simplify the
discussion of the dimer ground states.
Now let us turn to the specific cases of spins. Since the
spin 1/2 case has not been considered using this approach
before, we will add it here for completeness. The gauge
choice for L0 and the fluxes∇×L0/(2π) through the faces
of the spatial cubes are shown on Figure 8 with S = 1/2.
It is easy to see that the set of ground states consists of
all configurations with precisely one −5/6 and five 1/6
fluxes Jgs coming out of every site of one sublattice of
the original spin lattice (and coming into every site on
the other sublattice). Jgs = −∇ × L0/(2π) is one such
configuration in the spin 1/2 case, but there are many
more. Associating the −5/6 plaquettes with dimers on
the links of the original spin lattice, the set of the ground
states is thus the set of dimer configurations with one
dimer coming out of every site.
Now turn to the case of spin 1. The fluxes (∇ ×
L0)/(2π) are shown on Figure 8 with S = 1. If we try
Jgs = −∇× L0/(2π), each cube contributes 1/β energy
term proportional to 5(1/3)2 + (5/3)2 = 10/3. However
we can do better. Using L = −L0+2πn, if we pick n = 1
on the upper link on the front face and zero elsewhere on
the cube in Fig. 8, we lower the magnitude of the flux
on the upper face, at the expense of increasing the flux
through the front face. The energy of this cube is then
4(1/3)2 + 2(2/3)2 = 4/3, which is lower. It is easy to
show that this is the lowest we can achieve and that the
ground state configurations have two fluxes of value −2/3
and four fluxes of value 1/3 coming out of every site of
one sublattice of the original spin lattice. Associating the
2/3 links with dimers, the set of the ground states is thus
the set of dimer configurations with two dimers coming
out of every spin site.
Finally, in the S = 3/2 case, it is easy to see that
the ground state configurations have precisely three −1/2
and three 1/2 fluxes coming out of every site of one sub-
lattice of the original cubic lattice. Associating the −1/2
links with dimers, the set of the ground states is thus the
set of dimer configurations with three dimers coming out
of every spin site.
Thus, as claimed, in each case there is an extensive
number of Jgs’s. To find the true ground state, we need
to minimize (72) among these dimer configurations. It is
not hard to show that for the spin 1/2 we get columnar
state, for spin 1 the Haldane chains state of Fig. 1 and
for spin 3/2 the ladder state of Fig. 2.
Finally we note that defining Egs = S/3− Jgs, the set
of Egs is the set of electric fields on links, cf. Eq. (6),
with the property that the magnitude of each is either
zero or one (which can be imposed by minimizing the
energy term
E2); the mapping between such electric
fields and dimers above is the standard one on the cubic
lattice17,18,19. The final ground state selection is obtain
by maximizing
(∇× E)2.
V. CONCLUSIONS
In this paper we looked for spin solid phases in the sys-
tem of spin 1 and 3/2 on the cubic lattice. We wrote the
spins in terms of Schwinger bosons, assumed the uniform
Coulomb spin liquid phase and by process of monopole
condensation transitioned into spin solid phases. Using
the duality we rewrote the system in terms of monopoles
coupled to a noncompact U(1) gauge field, Eq. (9), and
analyzed this theory in three different limits shown in
Figure 4.
In the first two limits the theory becomes a frustrated
XY model. For spin 1 the frustrating flux through every
plaquette is 2π/3, while for spin 3/2 it is π. In the first
approach, using symmetries we wrote the Landau’s the-
ory near the ordering transition. It is a φ4 theory with φ
a complex vector with three components for S = 1 and
four components for S = 3/2. At the quadratic level only
the rotationally invariant mass term is allowed. At the
quartic level there are three allowed terms for spin 1 and
five for spin 3/2. For spin 1 we draw a mean field phase
diagram Figure 5. For spin 3/2 we didn’t attempt it due
to a large number of parameters. In both cases we also
considered the most natural microscopic potential and
found that it selects a state with parallel Haldane chains
of Figure 1 for S = 1 and a state with parallel ladders
of Figure 2 for S = 3/2. These are natural states for the
spin systems to be in, in the picture where spin 1 breaks
up into two and spin 3/2 into three spin 1/2’s and each
such spin 1/2 forms a singlet bond with another spin 1/2
of some neighbor.
In the second approach we looked at the classical
ground states of the frustrated XY models and found
that these actually describe the same phases as the most
natural ones identified near the transition.
In the third approach the theory becomes a dimer
model with 2S dimers coming out of every site. Dimer
configurations with parallel lines for spin 1 and parallel
ladders for spin 3/2 are selected, which is the same re-
sult as in the other two limits suggesting that these are
indeed the most natural valence bond solids in the corre-
sponding spin systems. It would be interesting to look for
such spin solid phases in Quantum Monte Carlo studies
of models on the cubic lattice.5,8
It is also worth noting14 that if we consider our quan-
tum 3d systems at a finite temperature, we obtain simply
the corresponding classical 3d dimer models, e.g., with
the classical energy given by the first term in Eq. (6). Our
results then provide appropriate long-wavelength (dual)
description of the dimer ordering patterns transitioning
out of the so-called Coulomb phase of the classical dimer
models,22,23,24 stressing in particular a composite char-
acter of the naive order parameters for the valence bond
solid phases. It would interesting to explore such 3d clas-
sical dimer models and their transitions further.
APPENDIX A: CLASSICAL U(1) DUALITY WITH
BACKGROUND CHARGE
In this section we derive duality for classical com-
pact U(1) gauge theory.12,25 However we will use a gen-
eral notation of antisymmetric tensors, or differential
forms which are fields of antisymmetric tensors. Thus
the derivation will work not only for the gauge theory,
whose objects are one dimensional, but for general n-
dimensional objects. For n = 0 this is the vortex duality
of the XY model and for n = 1 the duality of the gauge
theory. The further advantage of this derivation is that
the formulas are simpler and more transparent.
First we give the basic notations and properties of
antisymmetric tensors. An n-dimensional antisymmet-
ric tensor ω in d dimensions is a collection of numbers
ωµ1,µ2,...,µn , where µv = 1, . . . , d, which is completely an-
tisymmetric. A differential form ω(~r) is a field of these
tensors.
We define two operations. First is the exterior deriva-
tive ∂. The derivative of ω, denoted ∂ω is the (n+1)-form
(∂ω)µ1,µ2,...,µn+1 =
(−1)p∂µp1ωµp2 ,...,µpn (A1)
where the sum is over all permutations of the n+1 indices
and (−1)p is −1 if the permutation is odd and 1 if it
is even. Thus for example for n = 1, a vector field,
(∂ω)12 = ∂1ω2 − ∂2ω1 and hence this is the curl of a
vector field.
The second operation that we define is the star opera-
tor that takes n-form to (d− n)-form
(∗ω)ν1,...,νd−n =
ǫν1,...,νd−n,µd−n+1,...,µdωµd−n+1,...,µd
where ǫ is the fully antisymmetric tensor in d dimensions
and repeated indices are summed over. For example in
three dimensions for n = 2, (∗ω)1 = 12 (ω23 − ω32). Note
that ∗∗ = (−1)n(d−n).
A common operator is divergence which in this nota-
tion is proportional to ∗∂∗. As easily checked,
(∇ · ω)µ1,...,µn−1 ≡ ∂νων,µ1,...,µn−1 (A3)
= (−1)(n−1)(d−n)(∗∂ ∗ ω)µ1,...,µn−1(A4)
For a vector field this is the standard divergence.
We will work on the lattice. The variables are defined
on discrete points. We will define the coordinates of a
given variable to be those of the center of the object the
variable belongs to. For example the x component of a
one form ω in d = 3 lies on a link pointing in x direction
and it is denoted by ωx(x+1/2, y, z). The ∂ now denotes
the difference operator. For example the curl of the ω
is (∂ω)xy(x + 1/2, y + 1/2, z) = ωy(x + 1, y + 1/2, z) −
ωy(x, y+1/2, z)−ωx(x+1/2, y+1, z)+ωx(x+1/2, y, z).
Finally we will write the integration (summation) by
parts
ω · ∂φ = −
(∇ · ω) · φ+ surface term (A5)
where the dot is the sum over the component by com-
ponent product of two forms of the same n. Note that
∗ω1 · ∗ω2 = ω1 · ω2. Because we use periodic boundary
conditions below, the surface term will be zero.
Now we are ready to turn to the duality. Let a be an
n-form in d dimensions where its variables are defined on
the unit circle. The action is
S = −β
cos(∂a)− i
η · a (A6)
In the first term one takes every component at every
point, takes cosine of it and sums. In the second term
the n-form η denotes the background charge. For the
action considered in this paper, the first term is the Sa
and the second term the SB in (5), while the η is the four
dimensional vector with the time component being ±2S
and the other components being zero.
The duality proceeds by the following steps.
cos(∂a)+i
(∂a−2πp)2+i
(∂a−2π∂−1q′)2+i
J·(∂a−2π∂−1q′)+i
J·∂−1q′
×∆(∇ · J − η) (A7)
All numerical factors are dropped throughout, while the
sign “≈” is used when an approximation is being made
that does not change the qualitative aspects.
In the second line we use the Villain form of the cosine.
In the third line we have written the field p = ∂α+∂−1q′
as a curl of α plus a field of a particular monopole current
configuration q′, ∂−1q′. The ∂−1 denotes a particular
configuration of p that gives the monopole currents - that
satisfies q′ = ∂p. Then we shifted a → a − α. The
summation over α extends the integration of a over the
whole real line. The prime on q′ denotes that fact that
we are summing over fields for which ∂q′ = 0.
The third line can be obtained from the fourth one by
completing the square, shifting J and integrating it out.
In the fifth line, the ∆ denotes that the operator inside
of it is zero. This line is obtained from the fourth one by
integrating (summing) by parts and integrating out the
Next, as shown explicitly below, in our case there are
fields J0 and L0 such that
η = ∇ · J0 (A8)
J0 = (∗P∂L0)/2π (A9)
∂J0 = 0 (A10)
The P shifts a real number by a multiple of 2π so that
the result is in the interval (−π, π].
Using (A8) in (A7) we see ∂ ∗ (J − J0) = 0 and hence
we can write
J = J0 + (∗∂L)/2π (A11)
for some field L. To substitute this into (A7) we notice
the following
J2 = J20 + (∗∂L/2π)2 + 2J0 · ∗∂L/2π ≃ J20 + (∗∂L/2π)2.
The ≃ denotes that these expressions are equal under
integration, which follows from Eq. (A10). Also
∂−1q′ · ∗∂L ≃ − ∗ ∂∂−1q′ · L = − ∗ q′ · L ≡ −Q′ · L,
′·∗P∂L0 = ei∂
′·∗∂L0 ≃ e−iQ
′·L0,
where Q′ ≡ ∗q′, and we have dropped inconsequential
± signs; in the last line, the P can be removed because
the resulting expression, which is in the exponent, differs
from the original one by a multiple of 2π.
With this we can proceed to complete the duality
(∂L)2
′·(L+L0)
(∂L)2
Q·(L+L0−∂θ)
(∂L)2
(L+L0−∂θ−2πp)2
(∂L)2
−λ cos(L+L0−∂θ)
(A12)
In the first line the summation over Q′ is over integer
fields Q′ with zero divergence ∇ · Q′ = 0 - currents. In
the second line we introduced θ that imposes this con-
straint as a Lagrange multiplier and summed by parts.
In the third line we added a small term
Q2/2λ and as-
sumed that it is not going to change the basic behavior of
the system. Then we summed out Q, which introduced
integer p because Q is an integer (this is the Poisson sum-
mation formula). The second term is the Villain form of
cosine. In the last line we approximated it by cosine.
To complete it remains to find J0 and L0. The η has
values ητ (x, y, z, τ + 1/2) = (−1)x+y+z2S and zero for
other components. As easily checked
(J0)τx(x+ 1/2, y, z, τ + 1/2) =
(−1)x+y+z (A13)
and similarly for y and z with other components (other
then the ones obtained by permutation of indices) being
zero. This gives the right η and satisfies ∂J0 = 0. The
L0 can be chosen as on the Fig. 3.
In the final expression (A12) the L is 1-form and hence
a gauge field. The θ is 0-form - a number on a circle -
a matter field. Thus we obtained a noncompact U(1)
gauge theory coupled to scalar fields of monopoles with
frustrated hopping.
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http://arxiv.org/abs/cond-mat/0612126
http://arxiv.org/abs/cond-mat/0702029
