Emergence of U(1) symmetry in the 3D XY model with Zq anisotropy
Emergence of U(1) symmetry in the 3D XY model with Zq anisotropy
Jie Lou,1 Anders W. Sandvik,1 and Leon Balents2
Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215
Department of Physics, University of California, Santa Barbara, CA 93106-4030
(Dated: October 21, 2018)
We study the three-dimensional XY model with a Zq anisotropic term. At temperatures T < Tc
this dangerously irrelevant perturbation is relevant only above a length scale Λ, which diverges as a
power of the correlation length; Λ ∼ ξaq . Below Λ the order parameter is U(1) symmetric. We derive
the full scaling function controlling the emergence of U(1) symmetry and use Monte Carlo results
to extract the exponent aq for q = 4, . . . , 8. We find that aq ≈ a4(q/4)
2, with a4 only marginally
larger than 1. We discuss these results in the context of U(1) symmetry at “deconfined” quantum
critical points separating antiferromagnetic and valence-bond-solid states in quantum spin systems.
PACS numbers: 75.10.Hk, 75.10.Jm, 75.40.Mg, 05.70.Fh
A salient feature of the recently proposed theory of
”deconfined” quantum critical points, which separate
Néel and valence-bond-solid (VBS) ground states of an-
tiferromagnets on the square lattice, is the emergence
of U(1) symmetry [1]. The VBS is either dimerized on
columns or forms a square pattern with plaquettes of
four strongly entangled spins [2, 3]. In both cases there
are four degenerate patterns and, thus, Z4 symmetry is
broken. However, as the critical point is approached the
theory predicts a length scale Λ, diverging faster than the
correlation length, Λ ∼ ξa, a > 1, below which the dis-
tinction between columnar and plaquette VBS states dis-
appears. The nature of the VBS state is manifested only
when coarse-graining on length-scales l > Λ, whereas for
l < Λ the Z4 symmetry is unbroken and is replaced by an
emergent U(1) symmetry characterizing the fluctuations
between columnar and plaquette order.
Quantum Monte Carlo simulations [4] of an S = 1/2
Heisenberg model with four-spin couplings have recently
provided concrete evidence for a continuous Néel–VBS
transition, and also detected U(1) symmetry in the VBS
order-parameter distribution P (Dx, Dy), where Dx and
Dy are VBS order parameters for horizontal and vertical
dimers. There is no trace of the expected Z4 anisotropy
in the VBS phase—the distribution is ring shaped—
although the finite-size scaling of the squared order pa-
rameter shows that the system is long-range ordered.
This can be interpreted as the largest studied lattice size
L = 32 < Λ. A ring-shaped distribution was also found
in simulations of an SU(N) generalization of the S = 1/2
Heisenberg model [5]—possibly a consequence of proxim-
ity of this system to a deconfined quantum-critical point.
In order to better understand the U(1) features of these
VBS states, and to guide future studies of them, we here
exploit a classical analogy. In the three-dimensional XY
model including a Zq-anisotropic term,
H = −J
(i,j)
cos(θi − θj)− h
cos(qθi), (1)
the anisotropy is dangerously irrelevant for q ≥ 4 [6, 7, 8,
9, 10], i.e., the universality class is that of the isotropic
XY model but the perturbation is relevant for T < Tc
above a length-scale Λ. In the closely related q-state clock
model, the anisotropy is dangerously irrelevant for q ≥
5. While numerical studies [11, 12, 13] have confirmed
the irrelevance of the anisotropy at Tc, the associated Λ
has, to our knowledge, not been extracted numerically,
except for an analysis of the 3-state antiferromagnetic
Potts model, which corresponds to Z6 [9, 14].
Here we report results of Monte Carlo simulations for
4 ≤ q ≤ 8 on periodic-boundary lattices with N = L3
sites and L up to 32. In addition to Metroplis single-spin
updates, we also use Wolff cluster updates [15] to reduce
critical slowing down. We sample the order-parameter
distribution P (mx,my), where
cos(θi), my =
sin(θi). (2)
The standard order parameter can be defined as
〈m〉 =
dmyP (mx,my)
m2x +m
dθr2P (r, θ). (3)
We will compare this with an order parameter 〈mq〉 which
is sensitive to the angular distribution;
〈mq〉 =
dθr2P (r, θ) cos(qθ). (4)
While the finite-size scaling of 〈m〉 is governed by the
correlation length ξ, 〈mq〉 should instead be controlled by
the U(1) length scale Λ [9], becoming large for a system
of size L only when L > Λ. Fig. 1 shows magnetization
histograms at h/J = 1 for Z4 and Z8 systems with L = 4
and 32. The angular distribution P (θ) =
drrP (r, θ)
is also shown. The average radius of the distribution is
the magnetization 〈m〉, which decreases with increasing
L. The anisotropy, on the other hand, increases with
http://arxiv.org/abs/0704.1472v2
FIG. 1: (Color online) P (mx,my) at h/J = 1 for q = 4, 8,
L = 4, 32. The temperature T/J = 2.17 for Z4 and 1.15 for
Z8; both less than Tc/J ≈ 2.20. The size of the histograms
corresponds to mx,y ∈ [−1, 1]. Angular distributions P (θ)
with θ ∈ [0, 2π] are shown above each histogram.
L. This is particularly striking for Z8, where the L = 4
histogram shows essentially no angular dependence, even
though T is very significantly below Tc, whereas there
are 8 prominent peaks for L = 32. Thus, in this case
the U(1) length scale 4 < Λ < 32. For the Z4 system T
is much closer to Tc but still some anisotropy is seen for
L = 4; it becomes much more pronounced for L = 32.
It is instructive to examine a spin configuration with
mx ≈ my, i.e., θ ≈ π/4. Fig. 2 shows one layer of a Z4
system with L = 10 below Tc. The spins align predomi-
nantly along θ = 0 and θ = π/2, with only a few spins in
the other two directions. Clearly there is some cluster-
ing of spins pointing in the same direction—the system
consists of two interpenetrating clusters. Essentially, the
configuration corresponds to a size-limited domain wall
between θ = 0 and θ = π/4 magnetized states.
Hove and Sudbø studied the q-state clock model and
performed a course graining at criticality [13]. They
found that the structure in the angular distribution di-
minished with the size of the block spins for q ≥ 5, as
would be expected if the anisotropy is irrelevant. Here
we want to quantify the length scale Λ at which the
anisotropy becomes relevant for T < Tc. Consider first
what would happen in a course graining procedure for
a single spin configuration of an infinite system in the
ordered state very close to Tc. The individual spins will
of course exhibit q preferred directions, as is seen clearly
in Fig. 2, i.e., there would be q peaks in the probability
distribution of angles θi. Constructing block spins of l
spins, we would expect the angular dependence to first
FIG. 2: (Color online) Spins in one layer of the Z4 model
with L = 10 at h/J = 1, T/J = 1.9 < Tc. Here mx ≈ my,
corresponding to θ ≈ π/4 in P (r, θ). Arrows are color-coded
according to the closest Z4 angle; nπ/2, n = 0, 1, 2, 3.
become less pronounced because of the averaging over
spins pointing in different directions (again, as is seen in
Fig. 2). Sufficiently close to Tc we would expect the dis-
tribution to approach flatness. However, since we are in
an ordered state, one of the q preferred angles eventually
has to become predominant, and thus one peak in the
histogram will start to grow. This happens at l ≈ Λ. We
cannot simulate the infinite system and instead we carry
out an analogous procedure as a function of the lattice
size L, sampling a large number of configurations. We
calculate the order parameters 〈m〉 and 〈mq〉, defined in
Eqs. (3,4), and analyze them using
〈m〉 = L−σf(tL1/ν), (5)
〈mq〉 = L
−σg(tL1/νq). (6)
Here (5) is the standard finite-size ansatz with σ = β/ν,
and the XY exponents are β ≈ 0.348 and ν ≈ 0.672 [17].
Eq. (6) is an intuitive generalization of (5), which was
proposed and used also in Ref. [9], but we can actually
also derive the scaling function g(X) exactly.
Let us consider the scaling behavior of the order-
parameter distribution P (~m). It depends upon the sys-
tem size L and the size of scaling operators that perturb
the critical theory. Specifically, we consider the temper-
ature deviation t = Tc − T and the presumed irrelevant
q-fold anisotropy strength h. By conventional scaling ar-
guments, we expect
P (~m;L, t, h) = Lσ/2P̂ (Lσ ~m, tL1/ν , H = hL3−∆q), (7)
where ∆q > 3 is the scaling dimension of the irrelevant
anisotropy. The prefactor above is determined from nor-
malization of the probability distribution. In the scaling
regime, |t| ≪ 1, L ≫ 1, so H is small. When the first
two arguments are O(1), P̂ can be well-approximated by
taking H = 0 [with “corrections to scaling” of O(H), i.e.
suppressed by L3−∆q for a large system]. At H = 0,
the distribution is fully XY symmetric, and the integral
in Eq. (4) vanishes. Thus, in this regime 〈mq〉 is small,
O(H), and should be considered as arising from correc-
tions to scaling. This simply reflects the irrelevance of
the anisotropy at the critical point.
Because the anisotropy is dangerously irrelevant, a
larger contribution, however, emerges when tL1/ν ≫ 1,
i.e. L ≫ ξ ∼ t−ν . In this limit, the system can be re-
garded as possessing long-range XY order, and the only
significant fluctuations are the global fluctuations of the
XY phase θ. This is biased by the anisotropy. The scale
κ of the total anisotropy (free) energy can be estimated
by its typical magnitude within an XY correlation vol-
ume, hξ3−∆q , multiplied by the number of correlation
volumes, (L/ξ)3, i.e. κ = hL3ξ−∆q . Note that although
the energy per correlation volume is small (due to the ir-
relevance of anisotropy at the critical point), the number
of correlation volumes becomes very large and more than
compensates for this smallness for L/ξ sufficiently large.
From this argument, we see that for L/ξ ≫ 1, the dis-
tribution of angles θ is just determined from a Boltzmann
factor for a single XY spin with the q-fold anisotropy
energy ∼ −κ cos qθ. Furthermore, for L/ξ ≫ 1, the
magnitude |~m| ≈ 〈m〉 is approximately non-fluctuating.
Thus the distribution factors into the form P (|m|, θ) =
〈m〉−2δ(|m| − 〈m〉)P (θ), with
P (θ) =
eκ cos(qθ). (8)
Here Z =
dθ eκ cos(qθ) is the single-spin partition func-
tion. It is then straightforward to obtain from Eq. (4)
〈mq〉 = 〈m〉
I1(κ)
I0(κ)
, (9)
where In is the modified Bessel function of order n. Os-
hokawa obtained a similar expression in a different way,
but we disagree with his scaling variable. Comparing this
with the scaling form in Eq. (6), we see that νq = ∆qν/3
(aq = ∆q/3), κ = h(tL
1/νq )3νq , and
g(X) ∝
I1(h̃X
I0(h̃X
. (10)
Here h̃ should be viewed as a non-universal scale factor.
From the above discussion, one sees that this form is
valid for L/ξ ≫ 1 but L/ξq arbitrary. For L/ξ of O(1) or
smaller, L/ξq ≪ 1 (implying κ,X ≪ 1), and the scaling
form for 〈mq〉 becomes small and of order the expected
correction to scaling in the critical regime.
In Fig. 3 we show results for the two order parame-
ters for systems with q = 4, 5, 6. We have studied several
values of h/J and here show results for a different value
for each q. We have extracted Tc using finite-size scaling
of 〈m〉 with Eq. (5) and the XY exponents. This works
very well for all q, confirming the irrelevance of h. The
magnetization for T < Tc is seen to decrease marginally
with increasing q in Fig. 3. The Zq order-parameter 〈mq〉
changes more drastically, being strongly suppressed close
to Tc for large q. This is expected, as 〈mq〉 should van-
ish for all T in the XY limit q → ∞. For Z4, the 〈mq〉
2.1 2.15 2.2 2.25
1.6 1.8 2 2.2 2.4
1.6 1.8 2 2.2 2.4
q=4, h/J=1
q=5, h/J=10
q=6, h/J=5
FIG. 3: (Color online) The XY order parameter 〈m〉 (solid
curves) and the Zq order parameter 〈mq〉 (dashed curves)
vs temperature for q = 4, 5, 6. The system sizes are L =
8, 10, 12, 14, 16, 24, and 32. The curves become sharper (in-
creasing slope) around Tc (indicated by vertical lines). The
ratios h/J used are indicated on the graphs.
curves for different L cross each other, with the crossing
points moving closer to Tc as L increases. This is consis-
tent with the above discussion of course-graining: In the
ordered state close to Tc, 〈mq〉 should first, for small L,
decrease with increasing L as the q-peaked structure in
P (θ) diminishes due to averaging over more spins. For
larger L, 〈mq〉 starts to grow with L as the length-scale
Λ is exceeded. This behavior is more difficult to observe
directly for q = 5, 6 because 〈mq〉 is small and dominated
by statistical noise close to Tc where the curves cross.
Fig. 4 shows finite-size scaling of the Zq order-
parameter 〈mq〉, using the hypothesis (6) and the XY
value for σ. Adjusting νq = aqν for each q we find sat-
isfactory data collapse using a4 = 1.07(3), a5 = 1.6(1),
a6 = 2.4(1), and, not shown in the figure, a8 = 4.2(3).
These results are consistent with the form aq = a4(q/4)
in qualitative agreement with the ǫ-expansion by Os-
hikawa, which gave aq → q
2/30 for large q [9]. However,
in the ǫ-expansion there are significant deviations from
the q2 form in the range of q values considered here. Our
a6 is also smaller than the value ≈ 3.5 obtained on the
basis of the 3-state Potts antiferromagnet [9].
In Fig. 4 we also show the scaling function (10). It does
not match exactly the collapsed data, but the agreement
-4 -2 0 2 4
-6 -3 0
-4 -3 -2 -1 0
q=4, h/J=1
q=5, h/J=10
q=6, h/J=5
=0.72
=1.05
=1.60
FIG. 4: (Color online) Scaling of the Zq magnetization. We
use σ = 0.52 for all q, and νq as indicated in the plots. The
colors of the curves correspond to L as in Fig. 3. The dashed
curves are the predicted scaling functions with h̃ and prefac-
tors adjusted to fit the data approximately.
improves as q increases. As we have discussed above,
the scaling function represents the dominant behavior
for T < Tc, but exactly at Tc this contribution vanishes
and the critical-point scaling form 〈mq(Tc)〉 ∼ L
3−∆q be-
comes dominant. For q = 4, ∆4−3 is small; our estimate
is ∆4 − 3 = 0.21(9), in good agreement with previous es-
timates of the scaling dimension [8, 10]. Thus it is clear
that very large systems would be required for this contri-
bution to become invisible on the scale used in our graph.
It is also clear that for t < 0, close to t = 0, there will
be a similarly significant correction to the asymptotically
dominant scaling function. As q increases, we have seen
that ∆q = 3aq increases rapidly, and we thus expect sig-
nificantly smaller correction to scaling. For q = 6 the
agreement is already seen to be quite good, considering
that our lattices are not very large.
To conclude, we relate our results to the quantum VBS
states discussed in the introduction. Returning to Fig. 2,
associating θi ≈ 0 arrows with two adjacent horizontal
dimers on even-numbered columns and θi ≈ π/2 with
vertical adjacent dimers on even rows, 〈θ〉 = 0, π/2 cor-
respond to columnar VBS states. A plaquette is a super-
position of horizontal and vertical dimer pairs, whence a
plaquette VBS corresponds to 〈θ〉 = π/4 [3]. Rotating
the arrows by 90◦ corresponds to translating or rotating
a VBS. Either a columnar or plaquette VBS should ob-
tain in the infinite-size limit, but close to a deconfined
quantum-critical point, for L < Λ, the system fluctuates
among all mixtures of plaquette and columnar states.
This corresponds to a ring-shaped VBS order-parameter
histogram. In numerical studies of quantum antiferro-
magnets [4, 5] no 4-peak structure was observed in the
angular distribution, and hence it is not clear what type
of VBS finally will emerge (although a method using open
boundaries favors a columnar state in [4]). It seems un-
likely that the U(1) symmetry should persist as L → ∞.
In the classical Z4 model we never observe a perfectly
U(1)-symmetric histograms far inside the ordered phase,
in contrast to Refs. [4, 5]. On the other hand, aq is larger
for q > 4, and in Fig. 1 we have shown a prominently
U(1)-symmetric histogram for the Z8 model deep inside
the ordered phase. Thus, the exponent a may be larger
for the Z4 quantum VBS than a4 ≈ 1 obtained here for
the classical Z4 model. There is of course no reason to
expect them to be the same, as the universality class of
deconfined quantum-criticality is not that of the classical
Z4 model [1, 4]. Future numerical studies of VBS states
and deconfined quantum-criticality can hopefully reach
sufficiently large lattices to extract the U(1) exponent
using the scaling method employed here.
We would like to thank Kevin Beach, Masaki Os-
hikawa, Andrea Pelissetto, and Ettore Vicari for useful
discussions and comments. This research is supported by
NSF Grant No. DMR-0513930.
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