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# Clock model interpolation and symmetry breaking in O(2) models

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## Abstract and Figures

The $q$-state clock model is a classical spin model that corresponds to the Ising model when $q=2$ and to the $XY$ model when $q\to\infty$. The integer-$q$ clock model has been studied extensively and has been shown to have a single phase transition when $q=2$,$3$,$4$ and two phase transitions when $q>4$.We define an extended $q$-state clock model that reduces to the ordinary $q$-state clock model when $q$ is an integer and otherwise is a continuous interpolation of the clock model to noninteger $q$. We investigate this class of clock models in 2D using Monte Carlo (MC) and tensor renormalization group (TRG) methods, and we find that the model with noninteger $q$ has a crossover and a second-order phase transition. We also define an extended-$O(2)$ model (with a parameter $\gamma$) that reduces to the $XY$ model when $\gamma=0$ and to the extended $q$-state clock model when $\gamma\to\infty$, and we begin to outline the phase diagram of this model. These models with noninteger $q$ serve as a testbed to study symmetry breaking in situations corresponding to quantum simulators where experimental parameters can be tuned continuously.
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Clock model interpolation and symmetry breaking in O(2)
models
Leon Hostetler,𝑎,𝑏,Jin Zhang,𝑐Ryo Sakai,𝑐Judah Unmuth-Yockey,𝑑Alexei
Bazavov𝑏,𝑎 and Yannick Meurice𝑐
𝑎Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA
𝑏Department of Computational Mathematics, Science and Engineering, Michigan State University,
East Lansing, Michigan 48824, USA
𝑐Department of Physics and Astronomy, The University of Iowa, Iowa City, Iowa 52242, USA
𝑑Fermilab, Batavia, Illinois 60510, USA
E-mail: hostet22@msu.edu
The 𝑞-state clock model is a classical spin model that corresponds to the Ising model when 𝑞=2
and to the 𝑋𝑌 model when 𝑞→ ∞. The integer-𝑞clock model has been studied extensively
and has been shown to have a single phase transition when 𝑞=2,3,4and two phase transitions
when 𝑞 > 4.We deﬁne an extended 𝑞-state clock model that reduces to the ordinary 𝑞-state clock
model when 𝑞is an integer and otherwise is a continuous interpolation of the clock model to
noninteger 𝑞. We investigate this class of clock models in 2D using Monte Carlo (MC) and
tensor renormalization group (TRG) methods, and we ﬁnd that the model with noninteger 𝑞has
a crossover and a second-order phase transition. We also deﬁne an extended-𝑂(2)model (with
a parameter 𝛾) that reduces to the 𝑋𝑌 model when 𝛾=0and to the extended 𝑞-state clock
model when 𝛾→ ∞, and we begin to outline the phase diagram of this model. These models
with noninteger 𝑞serve as a testbed to study symmetry breaking in situations corresponding to
quantum simulators where experimental parameters can be tuned continuously.
The 38th International Symposium on Lattice Field Theory, LATTICE2021 26th-30th July, 2021
Zoom/Gather@Massachusetts Institute of Technology
Speaker
©Copyright owned by the author(s) under the terms of the Creative Commons
Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/
arXiv:2110.05527v1 [hep-lat] 11 Oct 2021
Clock model interpolation Leon Hostetler
1. Introduction
Implementation of quantum ﬁeld theoretical models on a quantum computer (either digital or
analog) requires some form of discretization, either directly in the ﬁeld variables or in the expansion
of the Boltzmann weight in the path integral. 𝑞-state clock models with the Z𝑞symmetry serve
as viable discrete approximations for models with continuous Abelian symmetries. To understand
the applicability of such discretization schemes we have recently proposed [1] a class of models,
called the extended 𝑞-state clock models, that interpolates the conventional 𝑞-state clock models to
noninteger values of 𝑞. As the 𝑞-state clock models reduce to the 𝑂(2)model in the limit 𝑞→ ∞,
we have also constructed a class of extended-𝑂(2)models where a symmetry breaking term with
a tunable coupling 𝛾allows one to smoothly interpolate between the 𝑂(2)model (𝛾=0) and the
extended 𝑞-state clock model (𝛾→ ∞) [1].
We present our results on studying these models in various limits with Monte Carlo (MC) and
tensor renormalization group (TRG) methods and, based on these ﬁndings, suggest a possible phase
diagram of the two-dimensional extended-𝑂(2)model in the (𝑞, 𝛽, 𝛾)parameter space.
2. A new class of models
We deﬁne the extended-𝑂(2)model by adding a symmetry breaking term 𝛾Í𝑥cos(𝑞𝜑𝑥)to
the action of the classical 𝑂(2)model:
𝑆ext-𝑂(2)=𝛽
𝑥, 𝜇
cos(𝜑𝑥+ˆ𝜇𝜑𝑥) − 𝛾
𝑥
cos(𝑞 𝜑𝑥) −
𝑥
cos(𝜑𝑥𝜑),(1)
where 𝜑𝑥are angular variables that encode the spin direction and (ℎ, 𝜑)are the magnitude and
direction of the external magnetic ﬁeld.
In the limit 𝛾→ ∞ the dominant spin directions are the ones that maximize the second term
in Eq. (1), i.e. 𝑞𝜑 𝑥=2𝜋 𝑘 ,𝑘Z. Thus, for integer 𝑞this limit reduces the action in Eq. (1) to
the one of the 𝑞-state clock model, while for noninteger 𝑞we consider the 𝛾→ ∞ limit to be the
deﬁnition of the extended 𝑞-state clock model where the spins take the directions
0𝜑(𝑘)
𝑥=2𝜋𝑘
𝑞<2𝜋. (2)
These “allowed” directions divide the unit circle into d𝑞earcs of which d𝑞e − 1have measure 2𝜋/𝑞
and the small remainder has measure
˜
𝜙2𝜋1b𝑞c
𝑞,(3)
as illustrated in Fig. 1. Here b. . . cdenotes rounding down to the nearest integer, and d. . .edenotes
rounding up to the nearest integer.
For numerical simulations we consider the extended 𝑞-state clock model (i.e. 𝛾→ ∞) where
the angles 𝜑𝑥are allowed to take only the discrete values deﬁned in Eq. (2):
𝑆ext-𝑞=𝛽
𝑥, 𝜇
cos(𝜑𝑥+ˆ𝜇𝜑𝑥) −
𝑥
cos(𝜑𝑥𝜑).(4)
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Clock model interpolation Leon Hostetler
2π/q
2π/q
2π/q
2π/q
2π/q
2π1bqc
q˜
φ
Figure 1: Arrows indicate the allowed spin orientations for the extended 𝑞-state clock model when 𝑞=5.5.
Figure taken from [1].
At noninteger 𝑞the action (4) is no longer invariant under the operation 𝜑mod (𝜑+
2𝜋/𝑞, 2𝜋b𝑞c/𝑞)(in index notation 𝑘mod (𝑘+1,b𝑞c)), i.e. the Z𝑞symmetry is explicitly
broken. However, there is a residual Z2symmetry with respect to the operation 𝜑2𝜋𝜑˜𝜑
(in index notation 𝑘→ b𝑞c 𝑘).
The partition function is
𝑍=
{𝜑𝑥}
𝑒𝑆ext-𝑞,(5)
and the observables that we compute to study the critical behavior are the internal energy
h𝐸i=h−
𝑥, 𝜇
cos(𝜑𝑥+ˆ𝜇𝜑𝑥)i =𝜕
𝜕𝛽 ln 𝑍 , (6)
the speciﬁc heat
𝐶=𝛽2
𝑉
𝜕h𝐸i
𝜕𝛽 =𝛽2
𝑉(h𝐸2i−h𝐸i2),(7)
the magnetization
h®
𝑀i=𝜕
𝜕®
ln 𝑍=*
𝑥®𝜎𝑥+,®𝜎𝑥=(cos 𝜑𝑥,sin 𝜑𝑥),(8)
and the magnetic susceptibility
𝜒®
𝑀=1
𝑉
𝜕h®
𝑀i
𝜕®
=1
𝑉h®
𝑀·®
𝑀i−h®
𝑀i·h®
𝑀i,(9)
where 𝑉is the total number of sites on the lattice.
In a ﬁnite system accessible to Monte Carlo simulations the spontaneous magnetization deﬁned
in Eq. (8) averages out to 0 if there is no external magnetic ﬁeld. For this reason we use proxy
magnetization observables
h| ®
𝑀|i =*
𝑥®𝜎𝑥+and 𝜒|®
𝑀|=1
𝑉h| ®
𝑀|2i − h| ®
𝑀|i2,(10)
as is often done in Monte Carlo simulations, e.g. Ref. [2].
3
Clock model interpolation Leon Hostetler
2.0
1.5
1.0
0.5
0.0
hEi/V
q= 4.1
q= 4.2
q= 4.3
q= 4.5
q= 4.5
q= 4.6
q= 4.7
q= 4.8
q= 4.9
q= 5.0
0.0
0.5
1.0
1.5
C
0.0 0.5 1.0 1.5 2.0
β
0.0
0.2
0.4
0.6
0.8
1.0
h| ~
M|i/V
0.0 0.5 1.0 1.5 2.0
β
0.0
0.2
0.4
0.6
χ|~
M|
Figure 2: Monte Carlo results from a heatbath algorithm for the extended 𝑞-state clock model on a 4×4
lattice with 4< 𝑞 5. The four panels show the energy density h𝐸i/𝑉, speciﬁc heat 𝐶, proxy magnetization
density h| ®
𝑀|i/𝑉and magnetic susceptiblity 𝜒|®
𝑀|. Statistical error bars are omitted since they are smaller than
the line thickness. Dashed lines, where they occur, indicate regions where we have data but the uncertainty
is not fully under control. Plots taken from [1].
3. Monte Carlo Results
We performed Monte Carlo simulations for the two-dimensional extended 𝑞-state clock model
without an external magnetic ﬁeld on 4×4lattice. The updating was performed with a version
of the heatbath algorithm from Ref. [3] modiﬁed appropriately to handle noninteger values of 𝑞.
In this initial round we mainly focused on scanning the parameter space and did not pursue larger
lattices with Monte Carlo since the local updating algorithms suﬀer signiﬁcant slowing down in the
ordered phase at large values of the inverse temperature 𝛽.
Typically, we started from a random conﬁguration (hot start) and performed 215 sweeps for
equilibration followed by a production run with 230 sweeps. The measurements were performed
once in 28sweeps, giving us a time series of length of 222 for averaging and error analysis. For
error propagation we used the jackknife method with 26jackknife bins as described in Ref. [3].
Results for 4.1𝑞5.0are shown in Fig. 2. The four panels show the energy density and
the speciﬁc heat deﬁned in Eqs. (6) and (7) and the proxy magnetization and susceptibility deﬁned
in Eq. (10). We see a double-peak structure in the speciﬁc heat. As 𝑞4from above, the peak at
large-𝛽moves toward 𝛽=. As 𝑞5from below, the speciﬁc heat becomes that of the ordinary
5-state clock model. In the susceptibility, a double-peak structure appears at larger lattice sizes. In
general, we ﬁnd that the thermodynamic curves vary smoothly for 𝑛 < 𝑞 𝑛+1where 𝑛is an
integer. When 𝑞goes from 𝑛to 𝑛+𝜖, the thermodynamic curves change abruptly since the number
of spin orientations is increased by one. At 𝑞=𝑛+𝜖, the speciﬁc heat exhibits a double-peak
structure with the second peak at very large 𝛽. As 𝑞is increased further, this second peak moves
toward small 𝛽, until at 𝑞=𝑛+1, the thermodynamic curves of the integer-(𝑛+1)-state clock
4
Clock model interpolation Leon Hostetler
0.0 0.5 1.0 1.5 2.0
β
100
101
102
103
τint
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
Figure 3: The integrated autocorrelation time 𝜏𝑖𝑛𝑡 (deﬁned in Eq. (11)) in the extended 𝑞-state clock model
for 4< 𝑞 5on a 4×4lattice using a heatbath algorithm. Here, the observable used is the energy density.
For integer 𝑞,𝜏𝑖𝑛𝑡 is well-behaved and grows moderately near the critical points. For noninteger 𝑞,𝜏𝑖𝑛𝑡 grows
rapidly at large 𝛽resulting in a critical slowing down of the Monte Carlo method. Note the log scale on the
vertical axis. Connecting lines are included to guide the eyes. Plot taken from [1].
model are recovered.
In the small-𝛽(high temperature) regime, all allowed angles of the extended 𝑞-state clock
model are essentially equally accessible. In this regime, the model is dominated by the approximate
Zd𝑞esymmetry, and it behaves approximately like a d𝑞e-state clock model. In the large-𝛽(low
temperature) regime, the model is dominated by the residual Z2symmetry, and it becomes a
rescaled Ising model.
At large-𝛽, all the spins tend to magnetize along one speciﬁc direction among the d𝑞eallowed
directions. However, when 𝑞Z, the Zd𝑞esymmetry is broken, and these directions are not all
equivalent. To understand this, we can consider the case 𝑞=5.5with the allowed spin directions
illustrated in Fig. 1. For example, a conﬁguration magnetized in the direction 2𝜋/𝑞is not equivalent
to one magnetized in the direction 0. In the ﬁrst case, the probability of the spins ﬂipping to a new
direction is low because the neighboring directions are far away. In the second case, the probability
of the spins ﬂipping to a new direction is relatively high because of the small angular distance
between directions 0and ˜
𝜙. Thus, we ﬁnd that at large 𝛽, the conﬁguration space separates into
two thermodynamically distinct sectors, and the Markov chain has trouble adequately sampling both
sectors. The resulting Monte Carlo slowdown is illustrated by a sudden increase of the integrated
autocorrelation time in the intermediate-𝛽regime as shown in Fig. 3.
For an observable 𝑂, an estimator of the integrated autocorrelation time is given by [3]
˜𝜏𝑂,𝑖𝑛𝑡 (𝑇)=1+2
𝑇
𝑡=1
𝐶(𝑡)
𝐶(0),(11)
where 𝐶(𝑡)=h𝑂𝑖𝑂𝑖+𝑡ih𝑂𝑖ih𝑂𝑖+𝑡iis the correlation function between the observable 𝑂measured
at Markov times 𝑖and 𝑖+𝑡. The integrated autocorrelation time 𝜏𝑂,𝑖 𝑛𝑡 is estimated by ﬁnding a
5
Clock model interpolation Leon Hostetler
0.0
0.5
1.0
1.5
2.0
2.5
3.0
C
q= 4.1V= 4 ×4
V= 8 ×8
V= 16 ×16
V= 32 ×32
V= 64 ×64
V= 128 ×128
q= 4.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
β
0.0
0.5
1.0
1.5
2.0
2.5
3.0
C
q= 4.9
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
β
q= 5.0
232527
L
0
2
4
peak height
232527
L
0
1
2
3
peak height
Figure 4: The speciﬁc heat of the extended 𝑞-state clock model from TRG for 𝑞=4.1, 4.5, 4.9, and 5.0.
Each panel shows results for lattice sizes ranging from 4×4to 128 ×128. In all four examples, there is a
double-peak structure (for 𝑞=4.1the second peak is at 𝛽75). Insets show that the second peak grows
logarithmically with the linear system size 𝐿=𝑉when 𝑞is noninteger. Plots taken from [1].
window in 𝑇for which ˜𝜏𝑂,𝑖𝑛𝑡 (𝑇)is nearly independent of 𝑇.
To appropriately sample the conﬁguration space we had to use large statistics. The eﬀect of
autocorrelation in our results was mitigated by discarding 28heatbath sweeps between each saved
measurement. The saved measurements were then binned (i.e. preaveraged) with bin size 216 before
calculating the means and variances. This approach was adequate for the 4×4lattice, but the Monte
Carlo slowdown makes it diﬃcult to study larger lattices, and is a strong motivation for using TRG.
4. TRG Results
We studied the extended 𝑞-state clock model using a tensor renormalization group (TRG)
method. This method does not suﬀer from the slowdown experienced with the Monte Carlo
method, and so the model could be studied on much larger lattices and in the thermodynamic
limit, allowing us to perform ﬁnite-size scaling and to characterize the phase transitions. The TRG
results were validated by comparison with exact and Monte Carlo results on small lattices. A full
description of the method is given in [1].
In Fig. 4, we show the speciﬁc heat from TRG for 𝑞=4.1, 4.5, 4.9, and 5.0 at volumes ranging
from 4×4to 128 ×128. In general, there are two peaks in the speciﬁc heat. When 𝑞=5.0, the
two peaks show little dependence on volume for lattice sizes larger than 32 ×32. This is consistent
with the two BKT transitions [4] of the ordinary 5-state clock model. When 𝑞Z, we see that
the ﬁrst peak shows similar behavior, which indicates that the ﬁrst peak is associated with either a
crossover or a phase transition with an order larger than two. In contrast, the second peak grows
logarithmically with volume, as shown in the insets for 𝑞=4.5,4.9. This indicates that the second
6
Clock model interpolation Leon Hostetler
Figure 5: The small-𝛽peak of the magnetic sus-
ceptibility as a function of 𝛽for diﬀerent values of
the external magnetic ﬁeld . The peak height con-
verges to a ﬁnite constant value when the external
ﬁeld is taken to zero, indicating that there is no phase
transition associated with this peak. In this example,
𝑞=4.3. Plot taken from [1].
Figure 6: We plot 𝜒𝑀/𝐿1.75 versus 𝐿(𝛽𝛽𝑐)in
the vicinity of the large-𝛽peak for diﬀerent lattice
sizes. The curves collapse to a single universal curve
providing strong evidence that this peak is associated
with a critical point of the Ising universality class.
In this example, 𝑞=4.3, we ﬁx 𝐿15/8=40, and
𝛽𝑐9.3216 is given by Eq. (12). Plot taken from [1].
peak is associated with a second-order phase transition. However, to conclusively characterize the
phase transitions associated with these two peaks in the extended 𝑞-state clock model, we study the
magnetic susceptibility with a weak external ﬁeld in the thermodynamic limit.
In Fig. 5, we show an example of the magnetic susceptibility in the thermodynamic limit and
in the vicinity of the ﬁrst peak for 𝑞=4.3and several diﬀerent values of the external magnetic ﬁeld.
As the external ﬁeld is decreased, the peak height converges to a constant 𝜒𝑀1.2. Since 𝜒𝑀
does not diverge as 0, there is no phase transition associated with this peak. This is true for all
fractional 𝑞, and so for fractional 𝑞, the ﬁrst peak in the speciﬁc heat is associated with a crossover
rather than a true phase transition.
Whereas the ﬁrst peak in the magnetic susceptibility converges to a constant value when the
external ﬁeld is taken to zero, the second peak diverges. When the peak heights 𝜒
𝑀of the
magnetic susceptibility for small values of are plotted and a power-law extrapolation to =0is
performed, we ﬁnd that 𝜒
𝑀0.93318(16). This gives a magnetic critical exponent 𝛿=14.97(4),
which is consistent with the value 𝛿=15 associated with BKT and Ising transitions. However, a
BKT transition should be accompanied with a continuous critical region, so the divergent large-𝛽
peak of 𝜒𝑀must be an Ising critical point. In the Ising universality class, 𝜒𝑀∼ |𝛽𝛽𝑐|𝛾𝑒𝐿7/4,
where 𝛾𝑒=7/4=1.75 is a universal critical exponent. There is a universal function relating
𝜒𝑀/𝐿1.75 and 𝐿(𝛽𝛽𝑐)with ﬁxed 𝐿15/8. In Fig. 6, we plot 𝜒𝑀/𝐿1.75 versus 𝐿(𝛽𝛽𝑐)for
various lattice sizes around the large-𝛽peak of 𝜒𝑀for 𝑞=4.3. We see that the data collapse
to a single curve, and this is strong evidence that the large-𝛽peak is a critical point of the Ising
universality class. Here, the critical point 𝛽𝑐of the extended 𝑞-state clock model is approximated
by the critical point of a rescaled Ising model,
𝛽𝑐'
ln 1+2
1cos ˜
𝜙.(12)
The small angular distance ˜
𝜙depends on 𝑞and is deﬁned in Eq. (3).
7
Clock model interpolation Leon Hostetler
q
23456
β
disordered
Z2ordered
Zqordered
2nd order trans.
BKT trans.
Critical phase
Crossover
β
?
23456
q
γ
γ= 0
γ=
Figure 7: (Left) The phase diagram of the extended 𝑞-state clock model as determined by a TRG study
of the magnetic susceptibility. For integer 𝑞, the model reduces to the ordinary clock model, which has
a second-order phase transition when 𝑞=2,3,4and two BKT transitions when 𝑞5. For noninteger
𝑞, there is a crossover at small 𝛽, and a second-order phase transition at larger 𝛽.(Right) We outline the
3-dimensional phase diagram of the extended-𝑂(2) model. When 𝛾=0, the model reduces to the 𝑋𝑌 model
for all values of 𝑞. The 𝑋𝑌 model has a single BKT transition separating a disordered phase at small 𝛽from
a critical phase at large 𝛽. In the 𝛾=plane, the extended-𝑂(2) model reduces to the extended 𝑞-state
clock model. The phase structure at ﬁnite-𝛾will be addressed in future work. Figures taken from [1].
5. Phase diagram
For integer 𝑞, the extended 𝑞-state clock model reduces to the well-known ordinary clock
model [5]. For 𝑞=2,3,4, there is a disordered phase and a Z𝑞symmetry-breaking phase separated
by a second-order phase transition. For 𝑞5, there are two BKT transitions [4]. There is a
disordered phase at small-𝛽, and a Z𝑞symmetry-breaking phase at large-𝛽with a critical phase at
intermediate 𝛽. For noninteger 𝑞, the extended 𝑞-state clock model has a double-peak structure in
both the speciﬁc heat and the magnetic susceptibility. Using TRG, we have shown that the small-𝛽
peak is associated with a crossover, and the large-𝛽peak is associated with a phase transition of the
Ising universality class. Thus, we get the phase diagram shown in the left panel of Fig. 7.
In the extended-𝑂(2) model, with action given by Eq. (1), the phase diagram is three-
dimensional with parameters 𝛽,𝑞, and 𝛾. The 𝛾=0plane of this model is the 𝑋𝑌 model for
all values of 𝑞, and the 𝛾=plane is the extended 𝑞-state clock model. Thus, an outline of the
phase diagram of this model is given in the right panel of Fig. 7. In future work, we will study this
model with ﬁnite 𝛾.
6. Summary and Outlook
We deﬁned an extended-𝑂(2) model by adding a symmetry breaking term 𝛾cos(𝑞𝜑𝑥)to the
action of the two-dimensional 𝑂(2)model. In the 𝛾 limit, the spins are forced into the
directions 0𝜑(𝑘)
𝑥=2𝜋𝑘/𝑞 < 2𝜋with 𝑘Z. We take this limit as the deﬁnition of the extended
𝑞-state clock model since in this limit, when 𝑞is integer, the ordinary 𝑞-state clock model is
recovered. In this work, we studied the extended 𝑞-state clock model for noninteger 𝑞using Monte
Carlo and tensor renormalization group methods. We found that there are two peaks in both the
8
Clock model interpolation Leon Hostetler
speciﬁc heat and the magnetic susceptibility. The small- 𝛽peak is associated with a crossover, and
the large-𝛽peak is associated with an Ising phase transition. Thus we obtained the phase diagram
of the extended 𝑞-state clock model and began to outline the 3-parameter phase diagram of the
extended-𝑂(2) model. The full phase diagram of the extended-𝑂(2) model will be discussed in
future work.
Interpolations among Z𝑛clock models have been realized experimentally using a simple
Rydberg simulator [6]. In this experimental work, Z𝑛(𝑛2) symmetries emerge by tuning
continuous parameters—the detuning and Rabi frequency of the laser coupling, and the interaction
strength between Rydberg atoms. Such work paves the way to quantum simulation of lattice ﬁeld
theory with discretized ﬁeld variables. Several concrete proposals for discretizing ﬁelds using
similar conﬁgurable arrays of Rydberg atoms were recently put forward [7].
Acknowledgments
We thank Gerardo Ortiz, James Osborne, Nouman Butt, Richard Brower, and members of the
QuLAT collaboration for useful discussions and comments. This work was supported in part by the
U.S. Department of Energy (DOE) under Awards No. DE-SC0010113 and No. DE-SC0019139.
The Monte Carlo simulations were performed at the Institute for Cyber-Enabled Research (ICER)
at Michigan State University.
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Dualities and the phase diagram of the -clock model
• G Ortiz
• E Cobanera
• Z Nussinov
G. Ortiz, E. Cobanera and Z. Nussinov, Dualities and the phase diagram of the -clock model, Nucl. Phys. B 854 (2012) 780.