Multi-Higgs U(1) Lattice Gauge Theory in Three Dimensions
Multi-Higgs U(1) Lattice Gauge Theory in Three Dimensions
Tomoyoshi Ono and Ikuo Ichinose
Department of Applied Physics, Graduate School of Engineering,
Nagoya Institute of Technology, Nagoya, 466-8555 Japan
Tetsuo Matsui
Department of Physics, Kinki University, Higashi-Osaka, 577-8502 Japan
(Dated: September 19, 2021)
We study the three-dimensional compact U(1) lattice gauge theory with N Higgs fields numeri-
cally. This model is relevant to multi-component superconductors, antiferromagnetic spin systems
in easy plane, inflational cosmology, etc. For N = 2, the system has a second-order phase transition
line c̃1(c2) in the c2(gauge coupling)−c1(Higgs coupling) plane, which separates the confinement
phase and the Higgs phase. For N = 3, the critical line is separated into two parts; one for c2 . 2.25
with first-order transitions, and the other for c2 & 2.25 with second-order transitions.
PACS numbers: 11.15.Ha, 05.70.Fh, 74.20.-z, 71.27.+a, 98.80.Cq
There are many interesting physical systems involving
multi-component (N -component) matter fields. Some-
times they are associated with exact or approximate sym-
metries like “flavor” symmetry. In some cases, the large-
N analysis[1] is applicable and it gives us useful informa-
tion. But the properties of the large-N systems may dif-
fer from those at medium values of N that one actually
wants to know. Study of the N -dependence of various
systems is certainly interesting but not examined well.
Among these “flavor” physics, the effect of matter
fields upon gauge dynamics is of quite general inter-
est in quantum chromodynamics, strongly correlated
electron systems, quantum spins, etc.[2] In this let-
ter, we shall study the three-dimensional (3D) U(1)
gauge theory with multi-component Higgs fields φa(x) ≡
|φa(x)| exp(iϕa(x)) (a = 1, · · · , N). This model is of gen-
eral interest, and knowledge of its phase structure, order
of its phase transitions, etc. may be useful to get better
understanding of various physical systems. These sys-
tems include the following:
N -component superconductor: Babaev[3] argued that
under a high pressure and at low temperatures hydro-
gen gas may become a liquid and exhibits a transition
from a superfluid to a superconductor. There are two
order parameters; φe for electron pairs and φp for pro-
ton pairs. They may be treated as two complex Higgs
fields (N = 2). In the superconducting phase, both φe
and φp develop an off-diagonal long-range order, while
in the superfluid phase, only the neutral order survives;
lim|x|→∞〈φe(x)φp(0)〉 6= 0.
p-wave superconductivity of cold Fermi gas: Each
fermion pair in a p-wave superconductor has angular mo-
mentum J = 1 and the order parameter has three com-
ponents, Jz = −1, 0, 1. They are regarded as three Higgs
fields (N = 3). As the strength of attractive force be-
tween fermions is increased, a crossover from a super-
conductor of the BCS type to the type of Bose-Einstein
condensation is expected to take place[4].
Phase transition of 2D antiferromagnetic(AF) spin
models: In the s = 1/2 AF spin models, a phase tran-
sition occurs from the Neel state to the valence-bond
solid state as parameters are varied. Senthil et al.[5]
argued that the effective theory describing this transi-
tion take a form of U(1) gauge theory of spinon (CP 1)
field za(x) (|z1|2 + |z2|2 = 1). In the easy-plane limit
(Sz = 0), |z1|2 = |z2|2 = 1/2 and so they are expressed
by two Higgs fields as za = exp(iϕa)/
2 (N = 2)[6].
Effects of doped fermionic holes (holons) to this AF
spins are also studied extensively. The effective theory
obtained by integrating out holon variables may be a
U(1) gauge theory with N = 2 Higgs fields (with non-
local gauge interactions). Kaul et al.[7] predicts that
such a system exhibits a second-order transition, while
numerical simulations of Kuklov et al.[8] exhibit a weak
first-order transition. This point should be clarified in
future study.
Inflational cosmology: In the inflational cosmology[9],
a set of Higgs fields is introduced to describe a phase
transition and inflation in early universe. Plural Higgs
fields are necessary in a realistic model[10].
The following simple consideration “predicts” the
phase structure of the system. Among N phases ϕa(x)
of the Higgs fields, the sum ϕ̃+ ≡
a ϕa couples to the
gauge field and describes charged excitations, whereas
the remaining N − 1 independent linear combinations
ϕ̃i(i = 1, · · · , N − 1) describe neutral excitations. The
latter N − 1 modes may be regarded as a set of N − 1
XY spin models. As the N = 1 compact U(1) Higgs
model stays always in the confinement phase[11], we ex-
pect N − 1 second-order transitions of the type of the
XY model.
Smiseth et al.[12] studied the noncompact U(1) Higgs
models. A duality transformation maps the charged sec-
tor into the inverted XY spin model. Thus they pre-
dicted that the system exhibits a single inverted XY
transition and N − 1 XY transitions. Their numerical
study confirmed this prediction for N = 2.
For N = 2, Kragset et al.[13] studied the effect
of Berry’s phase term in the N = 2 compact Higgs
model. They reported that Berry’s phase term sup-
http://arxiv.org/abs/0704.1323v2
0.89 0.9 0.91 0.92 0.93 0.94 c
1 -1-0.5 0 0.5 1 1.5 2
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FIG. 1: (a) System-size dependence of specific heat C for
N = 2 at c2 = 0.4. (b) Scaling function η(x) for Fig.1(a).
presses monopoles (instantons) and changes the second-
order phase transitions to first-order ones.
In this letter, we shall study the multi-Higgs models
by Monte Carlo simulations. We consider the simplest
form, i.e., the 3D compact lattice gauge theory without
Berry’s phase; the Higgs fields φxa are treated in the
London limit, |φxa| = 1. The action S consists of the
Higgs coupling with its coefficient c1a (a = 1, . . . , N) and
the plaquette term with its coefficient c2,
x+µ,aUxµφxa +H.c.
x,µ<ν
(U †xνU
x+ν,µUx+µ,νUxµ + H.c), (1)
where Uxµ[= exp(iθxµ)] is the compact U(1) gauge field,
µ, ν(= 1, 2, 3) are direction indices (we use them also as
the unit vectors).
We first study the N = 2 case with symmetric cou-
plings c11 = c12 ≡ c1. We measured the internal energy
E ≡ −〈S〉/L3 and the specific heat C ≡ 〈(S − 〈S〉)2〉/L3
in order to obtain the phase diagram and determine the
order of phase transitions, where L3 is the size of the
cubic lattice with the periodic boundary condition.
In Fig.1(a), we show C at c2 = 0.4 as a function
of c1. The peak of C develops as the system size
is increased. The results indicate that a second-order
phase transition occurs at c1 ≃ 0.91. By applying the
finite-size-scaling (FSS) hypothesis to C in the form of
C(c1, L) = L
σ/νη(L1/νǫ), where ǫ = (c1 − c1∞)/c1∞ and
c1∞ is the critical coupling at L → ∞, we obtained
ν = 0.67, σ = 0.16, and c1∞ = 0.909. In Fig.1(b) we
plot η(x), which supports the FSS.
The above results for N = 2 are consistent with the
“prediction” given above. The sum ϕ̃x+ ≡ ϕx1+ϕx2 cou-
ples with the compact gauge field and generates no phase
transition[11], while the difference ϕ̃x− ≡ ϕx1 − ϕx2 be-
haves like the angle variable in the 3D XY model. The
3D XY model has a second-order phase transition with
the critical exponent ν = 0.666...[14]. Our value of ν
obtained above is very close to this value. However, it
should be remarked that the simple separation of vari-
ables in terms of ϕ̃± is not perfect due to the higher-
order terms in the compact gauge theory. Nonetheless,
0 0.5 1 1.5 2
FIG. 2: Instanton density ρ for N = 2 at c2 = 0.4 as a
function of c1.
0.5 1 1.5 2 2.5 3 3.5 4 c
confinement
Higgs
�Fsecond order
�Fcrossover
FIG. 3: Phase diagram for N = 2. There are two phases, con-
finement and Higgs, separated by second-order phase transi-
tion line. There also exists a crossover line in the confinement
phase separating dense and dilute instanton-density regions.
our numerical studies strongly suggests that the phase
transition for N = 2 belongs to the universality class of
the 3D XY model.
It is instructive to see the behavior of the instanton
density ρ. We employ the definition of ρ in the 3D
U(1) compact lattice gauge theory given by DeGrand and
Toussaint[15]. ρ in Fig.2 decreases very rapidly near the
phase transition point. This indicates that a “crossover”
from dense to dilute instanton “phases” occurs simulta-
neously with the phase transition. In other words, the
observed phase transition can be interpreted as a con-
finement(small c1)-Higgs(large c1) phase transition.
In Fig.3, we present the phase diagram forN = 2 in the
c2-c1 plane. There exists a second-order phase transition
line separating the confinement and the Higgs phases.
There also exists a crossover line similar to that in the
3D N = 1 U(1) Higgs model[11].
Let us turn to the N = 3 case. Among many pos-
sibilities of three c1a’s, we first consider the symmetric
case c11 = c12 = c13 ≡ c1. One may expect that there
are two (N − 1 = 2) second-order transitions that may
coincide at a certain critical point. Studying the N = 3
case is interesting from a general viewpoint of the critical
phenomena, i.e., whether coincidence of multiple phase
transitions changes the order of the transition. We stud-
ied various points in the c2 − c1 plane and found that
the order of transition changes as c2 varies. In Fig.4, we
show E and ρ along c2 = 1.5 as a function of c1. Both
quantities show hysteresis loops, which are signals of a
first-order phase transition. In Fig.5, we present C at
c2 = 3.0. The peak of C at around c1 ∼ 0.48 develops
as L is increased, whereas E shows no discontinuity and
hysteresis. Therefore, we conclude that the phase transi-
tion at (c2, c1) ∼ (3.0, 0.48) is second order. In Fig.6(a),
we present the phase diagram of the symmetric case for
0.548 0.55 0.552 0.554 0.556 c
1 0.548 0.55 0.552 0.554 0.556
0.015
0.025
(a) (b)
FIG. 4: (a) Internal energy E and (b) instanton density ρ for
N = 3 at c2 = 1.5 and L = 16. Both exhibit hysteresis loops
indicating a first-order phase transition at c1 ≃ 0.551.
0.46 0.48 0.5 0.52 0.54 c
1 0.475 0.48 0.485 0.49 0.495
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(a) (b)
FIG. 5: (a) Specific heat for N = 3 at c2 = 3.0. (b) Close-up
view near the peak. The peak develops as L increases.
N = 3, where the order of transition between the confine-
ment and Higgs phases changes from first (smaller c2) to
second order (larger c2). In Fig.6(b) we present C along
c1 = 0.2, which shows a smooth nondeveloping peak. ρ
decreases smoothly around this peak. These results in-
dicate crossovers at c2 ≃ 1.5.
Then it becomes interesting to consider asymmetric
cases, e.g., c11 6= c12 = c13. This case is closely related to
a doped AF magnet. φ2 and φ3 correspond there to the
CP 1 spinon field in the deep easy-plane limit, whereas
φ1 corresponds to doped holes. This case is also relevant
to cosmology because the order of Higgs phase transition
in the early universe is important in the inflational cos-
mology. Furthermore, one may naively expect that once
a phase transition to the Higgs phase occurs at certain
temperature T , no further phase transitions take place at
lower T ’s even if the gauge field couples with other Higgs
0.5 1 1.5 2 2.5 3 3.5
�Fsecond order
�Ffirst order
�Fcrossover
Higgs
confinement
0 0.5 1 1.5 2 2.5
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c2 c10
(a) (b)
FIG. 6: (a) Phase diagram for the N = 3 symmetric case.
The phase transitions are first order in the region c2 . 2.25,
whereas they are second order in the region c2 & 2.25.
There exists a tricritical point at around (c2, c1) ∼ (2.25, 0.5).
Crosses near c2 = 1.5 line show crossovers. (b) Specific heat
for N = 3 at c1 = 0.2. It has a system-size independent
smooth peak at which a crossover takes place.
0.3 0.35 0.4 0.45 0.5 c
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0.34 0.345 0.35 0.355
c11 0.460.48 0.5 0.520.540.56
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(b) (c)
FIG. 7: (a) Specific heat of the c1 = (1, 2, 2) model (N=3) at
c2 = 1.0. (b,c) Close-up views of C near (b) c11 ∼ 0.35 and
(c) c11 ∼ 0.52.
bosons. However, our investigation below will show that
this is not the case.
Let us consider the case c12 = c13 = 2c11, which we call
the c1 = (1, 2, 2) model, and focus on the case c2 = 1.0.
As shown in Fig.7(a), C exhibits two peaks at c11 ∼ 0.35
and 0.52. Figs.7(b),(c) present the detailed behavior of C
near these peaks, which show that the both peaks develop
as L is increased. We conclude that both of these peaks
show second-order transitions. This result is interpreted
as the first-order phase transition in the symmetricN = 3
model is decomposed into two second-order transitions in
the c1 = (1, 2, 2) model.
Let us turn to the opposite case, c12 = c13 = 0.5c11,
i.e., the c1 = (2, 1, 1) model at c2 = 1.0. One may expect
that two second-order phase transitions appear as in the
previous c1 = (1, 2, 2) model. However, the result shown
in Fig.8 indicates that there exists only one second-order
phase transition near c11 ∼ 1.08. The broad and smooth
peak near c11 ∼ 0.85 shows no L dependence and we
conclude that it is a crossover. This crossover is similar
to that in the ordinary N = 1 gauge-Higgs system as we
shall see by the measurement of ρ below.
The orders of these transitions are understood as fol-
lows: In the c1 = (1, 2, 2) model, as we increase c11,
the two modes φxa(a = 2, 3) with larger c1a firstly be-
come relevant and the model is effectively the symmetric
N = 2 model. The peak in Fig.7(b) is interpreted as the
second-order peak of this model. For higher c11’s, the
gauge field is negligible due to small fluctuations, and
the effective model is the N = 1 XY model of φx1. It
gives the second-order peak in Fig.7(c). Similarly, in the
c1 = (2, 1, 1) model, φx1 firstly becomes relevant. The ef-
fective model is the N = 1 model, which gives the broad
peak in Fig.8 as the crossover[11]. For higher c11’s, the
effective model is the N = 2 symmetric model of φx2, φx3
and Uxµ, giving the sharp second-order peak in Fig.8.
In Fig.9, we present ρ of the c1 = (1, 2, 2) and (2, 1, 1)
models as a function of c11. ρ of the c1 = (1, 2, 2) model
decreases very rapidly at around c11 ∼ 0.35, which is the
0.7 0.8 0.9 1 1.1 1.2
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FIG. 8: Specific heat of the c1 = (2, 1, 1) model at c2 = 1.0.
0.25 0.3 0.35 0.4 0.45 0.5 0.55
c110 0.5 1 1.5
(a) (b)
c1=(1,2,2)
model
c1=(2,1,1)
model
FIG. 9: Instanton density ρ at c2 = 1.0 in the (a) c1 = (1, 2, 2)
model and (b) c1 = (2, 1, 1) model.
phase transition point in lower c11 region. On the other
hand, at the higher phase transition point, c11 ∼ 0.52, ρ
shows no significant changes. This observation indicates
that the lower-c11 phase transition is the confinement-
Higgs transition, whereas the higher-c11 transition is a
charge-neutral XY -type phase transition.
On the other hand, ρ of the c1 = (2, 1, 1) model de-
creases rapidly at around c11 ∼ 0.85, where C exhibits a
broad peak. This indicates that the crossover from the
dense to dilute-instanton regions occurs there just like in
the N = 1 case[11]. No “anomalous” behavior of ρ is ob-
served at the critical point c11 ∼ 1.1, and therefore the
phase transition is that of the neutral mode.
We have also studied the symmetric case for N = 4, 5
at c2 = 0. Both cases show clear signals of first-order
transitions at c1 ≃ 0.89(N = 4), 0.86(N = 5). On the
other hand, at c2 = ∞, the gauge dynamics is “frozen” to
Uxµ = 1 up to gauge transformations, so there remain N -
fold independent XY spin models, which show a second-
order transition at c1 ≃ 0.46. Thus we expect a tricritical
point for general N > 2 at some finite c2 separating first-
order and second-order transitions.
Let us summarize the results. ForN = 2 there is a crit-
ical line c̃1(c2) of second-order transitions in the c2 − c1
plane, which distinguishes the Higgs phase (c1 > c̃1) and
the confinement phase (c1 < c̃1). This result is consis-
tent with Kragset et al.[13]. For N = 3 there is a similar
transition line, but the region 0 < c2 < c2c ≃ 2.25 is of
second-order transitions while the region c2c < c2 is of
first-order transitions. To study the mechanism of gener-
ation of these first-order transitions, we studied the asym-
metric cases and found two second-order transitions [in
the c1 = (1, 2, 2) model] or one crossover and one second-
order phase transition [in the c1 = (2, 1, 1) model]. The
former case implies that two simultaneous second-order
transitions strengthen the order to generate a first-order
transition. Chernodub et al.[16] reported a similar gen-
eration of an enhanced first-order transition in a related
3D Higgs model with singly and doubly charged scalar
fields. We stress that the above change of the order is
dynamical because (1) It depends on the value of c2, (2)
Related 3D models, the CPN−1 and N -fold CP 1 gauge
models, exhibit always second-order transitions (See the
last reference of Ref.[2]).
We thank Dr.K. Sakakibara for useful discussion.
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ρ2a = 1.
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http://arxiv.org/abs/cond-mat/0611536
http://arxiv.org/abs/hep-ph/0605035
