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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+ˆ𝜇−𝜑𝑥) − 𝛾

𝑥

cos(𝑞 𝜑𝑥) − ℎ

𝑥

cos(𝜑𝑥−𝜑ℎ),(1)

where 𝜑𝑥are angular variables that encode the spin direction and (ℎ, 𝜑ℎ)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 ,𝑘∈Z. 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

0≤𝜑(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

˜

𝜙≡2𝜋1−b𝑞c

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+ˆ𝜇−𝜑𝑥) − ℎ

𝑥

cos(𝜑𝑥−𝜑ℎ).(4)

2

Clock model interpolation Leon Hostetler

2π/q

2π/q

2π/q

2π/q

2π/q

2π1−bqc

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 𝜑→mod (𝜑+

2𝜋/𝑞, 2𝜋b𝑞c/𝑞)(in index notation 𝑘→mod (𝑘+1,b𝑞c)), i.e. the Z𝑞symmetry is explicitly

broken. However, there is a residual Z2symmetry with respect to the operation 𝜑→2𝜋−𝜑−˜𝜑

(in index notation 𝑘→ b𝑞c − 𝑘).

The partition function is

𝑍=

{𝜑𝑥}

𝑒−𝑆ext-𝑞,(5)

and the observables that we compute to study the critical behavior are the internal energy

h𝐸i=h−

𝑥, 𝜇

cos(𝜑𝑥+ˆ𝜇−𝜑𝑥)i =−𝜕

𝜕𝛽 ln 𝑍 , (6)

the speciﬁc heat

𝐶=−𝛽2

𝑉

𝜕h𝐸i

𝜕𝛽 =𝛽2

𝑉(h𝐸2i−h𝐸i2),(7)

the magnetization

h®

𝑀i=𝜕

𝜕®

ℎ

ln 𝑍=*

𝑥®𝜎𝑥+,®𝜎𝑥=(cos 𝜑𝑥,sin 𝜑𝑥),(8)

and the magnetic susceptibility

𝜒®

𝑀=1

𝑉

𝜕h®

𝑀i

𝜕®

ℎ

=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 4.1≤𝑞≤5.0are 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 𝑞→4from above, the peak at

large-𝛽moves toward 𝛽=∞. As 𝑞→5from 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 𝑞∉Z, 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+𝑡i−h𝑂𝑖ih𝑂𝑖+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 𝛽∼75). 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 𝑞∉Z, 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 ℎ. 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(𝛽−𝛽𝑐)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 ℎ𝐿15/8=40, and

𝛽𝑐≈9.3216 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 𝜒𝑀≈1.2. Since 𝜒𝑀

does not diverge as ℎ→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 ℎis taken to zero, the second peak diverges. When the peak heights 𝜒∗

𝑀of the

magnetic susceptibility for small values of ℎare plotted and a power-law extrapolation to ℎ=0is

performed, we ﬁnd that 𝜒∗

𝑀∼ℎ−0.93318(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∼ |𝛽−𝛽𝑐|−𝛾𝑒∼𝐿7/4,

where 𝛾𝑒=7/4=1.75 is a universal critical exponent. There is a universal function relating

𝜒𝑀/𝐿1.75 and 𝐿(𝛽−𝛽𝑐)with ﬁxed ℎ𝐿15/8. In Fig. 6, we plot 𝜒𝑀/𝐿1.75 versus 𝐿(𝛽−𝛽𝑐)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

1−cos ˜

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 𝑞≥5. 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 𝑞≥5, 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 0≤𝜑(𝑘)

𝑥=2𝜋𝑘/𝑞 < 2𝜋with 𝑘∈Z. 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(𝑛≥2) 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.

References

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