# On the Lowey length of modules of finite projective dimension-II.

Tony J. Puthenpurakal  Department of Mathematics, IIT Bombay, Powai, Mumbai
400 076<EMAIL_ADDRESS>
###### Abstract.

Let $(A,\mathfrak{m})$ be a Gorenstein local ring of dimension $d\geq 1$.
Suppose there exists be a non-zero $A$ module $M$ of finite length and finite
projective dimension such that $\ell\ell(M)$, the Lowey length of $M$, is
equal to $\lambda(M)$, the length of $M$. Then we show that necessarily $A$ is
at worst a hypersurface singularity. We also characterize Gorenstein local
rings having a non-zero module $M$ of finite length and finite projective
dimension with $\ell\ell(M)=\lambda(M)-1$.

###### Key words and phrases:

index, Loewy length

###### 1991 Mathematics Subject Classification:

Primary 13D02; Secondary 13D40

##  1\. introduction

Dear Reader,
While reading this paper it will be a good idea to keep a copy of part 1, [P],
of this paper with you. The notation used in this paper are same as that of
[P].

Let $(A,\mathfrak{m})$ be a Gorenstein local ring of postive dimension and let
$M$ be a non-zero module of finite length and finite projective dimension. In
part 1 of this paper we showed that the Lowey length
$\ell\ell(M)\geq\operatorname{ord}(A)$. This generalizes a result of Avramov,
Buchweitz, Iyengar and Miller, [ABIM, 1.1]. Thus the singularity of $A$
imposes a lower bound on the Lowey length of modules of finite projective
dimension.

In this paper we investigate upper bounds of Lowey lengths of finite length
modules of finite projective dimension. An obvious upper bound is
$\ell\ell(M)\leq\lambda(M)$. Note that if $A$ is regular then
$k=A/\mathfrak{m}$. has finite projective dimension and
$\ell\ell(k)=\lambda(k)=1$. Our first result is

######  Theorem 1.1.

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d\geq 1$.
Suppose there exists an $A$-module $M$ of non-zero finite length module of
finite projective dimension such that $\ell\ell(M)=\lambda(M)$. Then $A$ is at
worst a hypersurface singularity, i.e., the completion $\widehat{A}=Q/(f)$
where $(Q,\mathfrak{n})$ is a complete regular local ring and
$f\in\mathfrak{n}$.

Note that we do not require $A$ to be Gorenstein in Theorem 1.1.

We complement Theorem 1.1 by proving

######  Proposition 1.2.

Let $(A,\mathfrak{m})$ be an abstract hypersurface ring of positive dimension.
Then there exists a finite length $A$\- module $M$ of finite projective
dimension with $\ell\ell(M)=\lambda(M)$.

A natural question is what happens if $\lambda(M)-\ell\ell(M)=1$? Our next
result implies that this still forces strong conditions on $A$. Set
$\operatorname{embdim}(A)$ to be the embedding dimension of $A$, i.e, the
number of minimal generators of $\mathfrak{m}$. We show

######  Theorem 1.3.

Let $(A,\mathfrak{m})$ be a Gorenstein local ring of dimension $d\geq 1$ and
let $M$ be a non-zero finite length module of finite projective dimension. If
$\lambda(M)-\ell\ell(M)=1$ then $\operatorname{embdim}(A)-d\leq 2$ (in
particular $A$ is a complete intersection of codimension $\leq 2$).
Furthermore we have

  1. (1)

Assume $\operatorname{embdim}(A)-d=1$ (the hypersurface ring case). If
$\widehat{A}=Q/(f)$ where $(Q,\mathfrak{n})$ is regular local then
$f\in\mathfrak{n}^{2}\setminus\mathfrak{n}^{3}$.

  2. (2)

Assume $\operatorname{embdim}(A)-d=2$. If $\widehat{A}=Q/(f,g)$ where
$(Q,\mathfrak{n})$ is regular local then
$f,g\in\mathfrak{n}^{2}\setminus\mathfrak{n}^{3}$. Furthermore the images of
$f,g$ in $\mathfrak{n}^{2}/\mathfrak{n}^{3}$ is linearly independent over
$k=A/\mathfrak{m}$.

  3. (3)

If $M$ is not cyclic then necessarily $A$ is regular local.

We complement Theorem 1.3 by giving examples LABEL:examples-thm-2 which shows
that all the cases above occur.

To prove our results we use some elementary techniques which arise in the
study of associated graded rings of $\mathfrak{m}$-primary ideals in Cohen-
Macaulay rings. It is suprising to us that a single module $M$ imposes such
restrictions on the ring $A$.

##  2\. preliminaries

In this section we discuss a few preliminaries that we need.

######  2.1.

_reduction to the case when $A$ is complete:_
Let $\widehat{A}$ be the completion of $A$ with respect to $\mathfrak{m}$. If
$M$ is a finitely generated $A$-module then set
$\widehat{M}=M\otimes_{A}\widehat{A}$ the completion of $M$. Then the
following results are well-known:

  1. (1)

$\lambda_{A}(M)=\lambda_{\widehat{A}}(\widehat{M}).$

  2. (2)

$\ell\ell_{A}(M)=\ell\ell_{\widehat{A}}(\widehat{M})$

  3. (3)

$\dim M=\dim\widehat{M}$.

  4. (4)

$\operatorname{embdim}(A)=\operatorname{embdim}(\widehat{A})$

  5. (5)

$\operatorname{ord}(A)=\operatorname{ord}(\widehat{A})$.

  6. (6)

$\operatorname{projdim}_{A}M=\operatorname{projdim}_{\widehat{A}}\widehat{M}$.

We also note that if $M$ has finite length then it has a natural structure as
an $\widehat{A}$-module and $\widehat{M}\cong M$ as $A$-modules.

We need the following well known result:

######  Proposition 2.2.

Let $(A,\mathfrak{m})$ be a complete local ring and let $E$ be the injective
hull of $k=A/\mathfrak{m}$. Let $M$ be a finite length $A$-module. Set
$M^{\vee}=\operatorname{Hom}_{A}(M,E)$. Then

  1. (1)

$M\cong(M^{\vee})^{\vee}$.

  2. (2)

$\lambda(M^{\vee})=\lambda(M)$.

  3. (3)

$\ell\ell(M^{\vee})=\ell\ell(M)$.

  4. (4)

If $\operatorname{projdim}M<\infty$ then
$\operatorname{injdim}M^{\vee}<\infty$. Furthermore if $A$ is Gorenstein then
$\operatorname{projdim}M<\infty$.

###### Proof.

The first two assertions are well-known, cf [BH, 3.2.12].
(3) Suppose $\mathfrak{m}^{c}M=0$. Let $f\in M^{\vee}$, $m\in M$ and let
$a\in\mathfrak{m}^{c}$. Then

$(af)(m)=f(am)=f(0)=0$

It follows that $\ell\ell(M)\leq\ell\ell(M^{\vee})$.

By same argument we get $\ell\ell(M^{\vee})\leq\ell\ell(M^{\vee\vee})$. But
$M^{\vee\vee}\cong M$. The result follows.

(4) This is well-known. ∎


On the use of the Operator Product Expansion in finite-energy sum rules for
light-quark correlators

Diogo Boito,1,2 Maarten Golterman,3,4$\star$ Kim Maltman5,6 and Santiago
Peris,4

1 Instituto de Física de São Carlos, Universidade de São Paulo, CP 369,
13560-970,

São Carlos, SP, Brazil

2 University of Vienna, Faculty of Physics, Boltzmanngasse 5, A-1090 Wien,
Austria

3 Department of Physics and Astronomy, San Francisco State University,

San Francisco, CA 94132, USA

4 Department of Physics and IFAE-BIST, Universitat Autònoma de Barcelona

E-08193 Bellaterra, Barcelona, Spain

5 Department of Mathematics and Statistics, York University, Toronto, ON
Canada M3J 1P3

6 CSSM, University of Adelaide, Adelaide, SA 5005 Australia

*<EMAIL_ADDRESS>
16th International Workshop on Tau Lepton Physics (TAU2021),
September 27 – October 1, 2021 10.21468/SciPostPhysProc.?

## Abstract

Tau-based finite-energy sum-rule (FESR) analyses often assume that scales
$s_{0}\sim m_{\tau}^{2}$ are large enough that (i) integrated duality
violations (DVs) can be neglected, and (ii) contributions from non-
perturbative OPE condensates of dimension $D$ scale as $(\Lambda_{\rm
QCD}/m_{\tau})^{D}$, allowing the OPE series to be truncated at low dimension.
The latter assumption is not necessarily valid since the OPE series is not
convergent, while the former is open to question given experimental results
for the electromagnetic, $I=1$ vector ($V$), $I=1$ axial vector ($A$) and
$I=1$ $V+A$ current spectral functions, which show DV oscillations with
amplitudes comparable in size to the corresponding $\alpha_{s}$-dependent
perturbative contributions at $s\sim 2-3$ GeV2. Here, we discuss recently
introduced new tools for assessing the numerical relevance of omitted
higher-$D$ OPE contributions. Applying these to the “truncated OPE” strategy
used in Refs. [1, 2] and earlier work by the same authors, we find that this
strategy fails to yield reliable results for the strong coupling from hadronic
$\tau$ decays.

###### Contents

  1. 1 Introduction
  2. 2 The truncated-OPE strategy
  3. 3 Tests of the truncated-OPE strategy on data for $e^{+}e^{-}\to$ hadrons
  4. 4 Conclusion

##  1 Introduction

The determination of the strong coupling $\alpha_{s}$ from hadronic $\tau$
decays is interesting for two important reasons: (i) In principle, a high
precision can be reached, if we “normalize” a determination at the $\tau$ mass
scale by evolving the coupling to the $Z$ mass, and (ii) because of the low
scale set by the $\tau$ mass, it provides a direct test of the running of the
coupling predicted by QCD. However, at the same time, the strong coupling at
the $\tau$ mass, $\alpha_{s}(m_{\tau})$, is rather large (of order $0.3$ in
the $\overline{\rm MS}$ scheme), and non-perturbative effects threaten to
contaminate the extraction of $\alpha_{s}(m_{\tau})$ from experimental data.
It is thus important to introduce methods to quantify such contamination, and
to test the resulting strategies for their reliability. In this talk, we
discuss the “truncated OPE" (tOPE) strategy [3], which has most recently been
used in Refs. [1, 2]. We will demonstrate that this strategy leads to results
with unquantifiable systematic errors at the scale of the current desired
level of precision, and hence should no longer be used.

The determination of $\alpha_{s}(m_{\tau})$ starts by considering finite-
energy sum rules (FESRs) of the form111For a more detailed description of our
use of FESRs, see Ref. [4] and references therein.

$\int_{0}^{\infty}ds\,s^{n}\,\rho(s)=-\frac{1}{2\pi
i}\oint_{|z|=s_{0}}dz\,z^{n}\,\Pi(z)\ ,$ (1)

where $\Pi(z)$ is the (scalar) vacuum polarization obtained from the $V+A$
non-strange $I=1$ or the electro-magnetic (EM) spectral function $\rho(s)$
(with $s=q^{2}>0$), and the contour $|z|=s_{0}$ is a circle in the complex
$q^{2}=z$ plane around the origin with radius $s_{0}$, which, if $\rho(s)$ is
obtained from hadronic $\tau$ decays, is bounded from above by the $\tau$
mass, $s_{0}\leq m_{\tau}^{2}$.

Equation (1) is exact. To proceed, we then approximate

$\Pi(z)=\Pi_{\rm pert.}(z)+\Pi_{\rm OPE}(z)+\Pi_{\rm DV}(z)\ ,$ (2)

where $\Pi_{\rm pert.}(z)$ is the perturbative part, for which a five-loop
expression exists [5], $\Pi_{\rm OPE}(z)$ is the operator-product-expansion
(OPE) part,222For the non-strange channel, the $D=2$ term proportional to the
square of the light quark masses can be safely neglected. We have also checked
that the logarithmic dependence of the $C_{D}$ on $q^{2}$ can be neglected
[6].

$\Pi_{\rm
OPE}(q^{2})=\frac{C_{4}}{(q^{2})^{2}}-\frac{C_{6}}{(q^{2})^{3}}+\frac{C_{8}}{(q^{2})^{4}}+\dots\
,$ (3)

and $\Pi_{\rm DV}(z)$ is the part violating quark-hadron duality, which is not
captured by the OPE [7].

The existence of a duality-violating (DV) part is closely related to the fact
that the OPE is not convergent; rather it is (at best) an asymptotic
expansion. According to expectations, the DV part, which physically represents
the oscillations about perturbative expectations associated with the presence
of resonances visible in the spectral function, decreases exponentially with
$q^{2}$, i.e., non-perturbatively in the OPE expansion parameter $1/q^{2}$. We
note that the weight $z^{n}$ in Eq. (1) picks out the term proportional to
$1/q^{2(n+1)}$ in the OPE—this will be important in what follows.

While the DV part is exponentially suppressed, the data are limited to $s\leq
m_{\tau}^{2}$, and we are forced to consider the possibility that they may not
be negligible. In this respect, it is instructive to consider Fig. 1.

Figure 1: Blow-up of the ALEPH data (red data points) [1] in the large-$s$
region of the $V+A$ non-strange spectral function. What is shown is
$2\pi^{2}\rho_{V+A}-1$, i.e., the dynamical QCD contribution to the spectral
distribution. Black dashed line: perturbation theory, also with the parton-
model contribution subtracted.

The oscillations in the spectral function (red data points) represent the
presence of resonances, and are not captured by the OPE (black dashed curve).
It is clear that DVs, which represent the oscillations of the data around the
dashed curve, are not a small part of the dynamical QCD contribution to the
spectral function in this region. Note that in order to make this comparison,
one should subtract the parton-model contribution to the spectral function, as
it is independent of $\alpha_{s}$ and thus not part of the dynamics produced
by QCD.

##  2 The truncated-OPE strategy

Two different strategies have been developed to deal with the non-perturbative
contamination, i.e., the $D>0$ terms in the OPE and DVs: the tOPE strategy,
and the “DV-model” strategy.333A variant of the tOPE strategy can be found in
Ref. [8]. For the latter, we refer to Boito’s talk at this workshop [9]; the
most recent application of the DV-model strategy can be found in Ref. [10].

The assumptions underlying the tOPE strategy are the following. First, DVs are
neglected, but the dangerous region, where the circular contour on the right-
hand side of Eq. (1) crosses the positive real axis, is suppressed by
combining the weights of Eq. (1) into polynomials with multiple zeroes at
$s=s_{0}$. These multiple zeroes are thus introduced to suppress DVs, at the
intersection of the contour on the right-hand side of Eq. (1) with the
positive real axis. Furthermore, $s_{0}$ is typically chosen equal to
$m_{\tau}^{2}$, in order to keep values of $\alpha_{s}(s_{0})$ appearing in
the fit as small as possible.

In implementations of the tOPE strategy in the literature, these polynomial
have degrees varying between 3 and 7. This implies that OPE terms up to order
$D=16$ contribute to the right-hand side of Eq. (1), and one thus would have
to fit $\alpha_{s}(m_{\tau})$ as well as $C_{4,6,8,10,12,14,16}$, eight
parameters in total. However, the number of independent polynomials of maximal
degree 7 with at least two zeroes at $s=s_{0}$ is smaller than eight, and
thus, necessarily, a number of parameters have to be set equal to zero by
hand.

One such set of polynomials, used in Ref. [2] and referred to as "optimal"
there, is the set

$\displaystyle w_{2n}=1-(n+2)x^{n+1}+(n+1)x^{n+2}\ ,\quad n=1\ ,\dots,\ 5\ ,$
(4)

with $x=s/s_{0}$. These weights have a double zero at $s=s_{0}$ (they are
“doubly pinched”), and probe $\alpha_{s}(s_{0})$ as well as
$C_{6,8,10,12,14,16}$. With five weights, and taking $s_{0}=m_{\tau}^{2}$, one
has five data points, which leads to the choice to set
$C_{12}=C_{14}=C_{16}=0$, so that one has four parameters for a fit to five
data points,444The weights of Eq. (4) do not project on the $D=4$ term in the
OPE. Different, but similar, sets of weights have been considered in Ref. [2]
as well as in our analysis of this strategy [4]. Here we will only discuss the
set (4), as the conclusions from our more extensive study of other sets of
weights is the same. Finally, in the tOPE strategy, one primarily considers
the $V+A$ channel, as DV and OPE-truncation effects are argued to be less
severe in $V+A$ than in the $V$ or $A$ channels separately.

In a first look at the tOPE strategy, let us compare two choices for the OPE
coefficients $C_{12}$, $C_{14}$, and $C_{16}$, which are not part of the fit,
and for which thus a priori values have to be chosen. We compare the choice of
Refs. [1, 2], which effectively sets $C_{12}=C_{14}=C_{16}=0$ (choice 1), with
choice 2:

$C_{12}=0.161~{}\mbox{GeV}^{12}\ ,\quad C_{14}=-0.17~{}\mbox{GeV}^{14}\ ,\quad
C_{16}=-0.55~{}\mbox{GeV}^{16}\ .$ (5)

This choice is equally arbitrary, but equally reasonable. The results of
applying the tOPE strategy with either of these two choices are:

$\begin{array}[]{|c|c|c|c|c|c|}\hline\cr&\alpha_{s}(m_{\tau})&C_{6}~{}(\mbox{GeV}^{6})&C_{8}~{}(\mbox{GeV}^{8})&C_{10}~{}(\mbox{GeV}^{10})&\chi^{2}/\mbox{dof}\\\
\hline\cr\mbox{choice}~{}1&0.317(3)&0.0014(4)&-0.0010(5)&0.0004(3)&1.26/4\\\
\hline\cr\mbox{choice}~{}2&0.295(4)&-0.0130(4)&0.0356(5)&-0.0836(3)&1.09/1\\\
\hline\cr\end{array}$ (6)

The fits were done with FOPT [4], and only the fit errors are shown. The OPE
coefficients in Eqs. (5) and (6) are very reasonable, increasing in absolute
value with the order in the OPE, but consistent with it being an asymptotic
expansion.

Clearly, the choice-1 and choice-2 fits are inconsistent, and lead to values
of $\alpha_{s}(m_{\tau})$ which are about 7% apart; this is about double the
total error quoted in Ref. [2]. Moreover, there is no way to tell which of
these two fits is closer to the truth; in fact, both fits may be wrong.

##  3 Tests of the truncated-OPE strategy on data for $e^{+}e^{-}\to$ hadrons

In order to probe this unsatisfactory state of affairs in more detail, we will
apply the tOPE strategy next to $R$-ratio data obtained from $e^{+}e^{-}\to$
hadrons. The key observation is, of course, that if the tOPE strategy works at
$s_{0}=m_{\tau}^{2}$, as necessitated with data from $\tau$ decays, it should
certainly work at $s_{0}>m_{\tau}^{2}$, where $R$-ratio data are available.

There are, of course, differences with the $\tau$-based approach. First,
$e^{+}e^{-}\to$ hadrons only gives access to $V$ channel data, whereas it is
advocated to apply the tOPE strategy to $V+A$. However, Refs. [1, 2] find that
the tOPE strategy applied to the $V$ channel $\tau$-decay data yields results
consistent with those from $V+A$, with equally good fit qualities. Another
difference is that $R$-ratio data contain an $I=0$ component, in addition to
the $I=1$ component related to the $\tau$-based $V$ spectral function.
However, the $I=0$ component is an $SU(3)$-flavor partner of the $I=1$
component, and it is known that the strange quark mass that breaks $SU(3)$ has
a very small effect on $\alpha_{s}$ [11]. We thus conclude that lessons
learned from applying the tOPE strategy to $R$-ratio data are relevant for the
application to $\tau$-based analyses at as well. Below we will use $R$-ratio
data from Refs. [12, 13].

First, we repeat the fit with weights (4) at $s_{0}\approx m_{\tau}^{2}$, now
using the $R$-ratio data. We find555We omit results for the OPE coefficients
$C_{D>0}$ for this discussion.

$\displaystyle\chi^{2}~{}\mbox{fit}:$
$\displaystyle\alpha_{s}(m_{\tau})=0.308(4)\ ,\qquad p\mbox{-value}=2\times
10^{-15}\ ,$ $\displaystyle\mbox{diagonal~{}fit}:$
$\displaystyle\alpha_{s}(m_{\tau})=0.245(10)\ .$

This is clearly a disaster. We note that a diagonal fit uses a diagonal fit
quality, but takes the full data covariance matrix into account for error
propagation; for more detail on this, see Refs. [6, 14]. Next, let us take
$s_{0}$ larger: if we take $s_{0}=3.6$ GeV2, the $p$-value of the $\chi^{2}$
fit becomes larger than 10%. In fact, at $s_{0}=3.6$ GeV2, we find

$\displaystyle\chi^{2}~{}\mbox{fit}:$
$\displaystyle\alpha_{s}(m_{\tau})=0.264(5)\ ,\qquad p\mbox{-value}=0.41\ ,$
$\displaystyle\mbox{diagonal~{}fit}:$
$\displaystyle\alpha_{s}(m_{\tau})=0.256(12)\ .$

This is a clear improvement, but it yields a very low value for $\alpha_{s}$:
at the $Z$ mass, this would translate into $\alpha_{s}(m_{Z})=0.110$! Other
sets of weights considered in Refs. [1, 2] lead to very similar results. For
more details, we refer to Ref. [14].

One might argue that the tests of Eqs. (3,3) are possibly somewhat
inconclusive. However, the $R$-ratio data allow us to subject the tOPE
strategy to a more stringent test, by considering the $s_{0}$ dependence of
tOPE-based fits. Again, the simple observation is that if the tOPE strategy
works at values of $s_{0}\geq m_{\tau}^{2}$, there should be a good match
between theory and experiment for all values of $s_{0}\geq m_{\tau}^{2}$.

Figure 2: tOPE fits using optimal weights with $s_{0}=s_{0}^{*}=3.6$ GeV2 in
Eq. (1). The red data points show the left-hand side of Eq. (1) for each
weight; the black curves show the fits of the right-hand side.

Figure 2 shows the left-hand side and the right-hand side of Eq. (1) obtained
in tOPE $\chi^{2}$ fits employing optimal weights at a fixed $s_{0}\equiv
s_{0}^{*}=3.6$ GeV2, as a function of $s_{0}$. While superficially, one might
conclude that the agreement between experiment (left-hand side of Eq. (1)) and
theory (right-hand side Eq. (1)) as a function of $s_{0}$ is not unreasonable,
this is not actually the case. In fact, it is not easy to judge the level of
agreement visually, as there are strong correlations, both between the
spectral integrals at different $s_{0}$, between the fitted theory integrals
at different $s_{0}$, and between the theory integrals and the spectral
integrals used to fit the parameters of the theory representation. A careful
look reveals a possible sign of trouble: clearly the slopes of the fit curves
at $s_{0}>s_{0}^{*}$ are rather different than the corresponding slopes in the
data, for the weights $w_{23}$, $w_{24}$ and $w_{25}$.

Whether this difference in slopes is statistically significant or not can be
investigated by considering the double differences

$\Delta^{(2)}(s_{0};s_{0}^{*})=\left[I^{\rm th}_{w}(s_{0})-I^{\rm
exp}_{w}(s_{0})\right]-\left[I^{\rm th}_{w}(s_{0}^{*})-I^{\rm
exp}_{w}(s_{0}^{*})\right]\ ,$ (9)

where $I^{\rm exp}_{w}(s_{0})$ denotes the left-hand side of Eq. (1) for
polynomial weight $w$, and $I^{\rm th}_{w}(s_{0})$ denotes the right-hand side
of Eq. (1). These double differences compare theory (i.e., the fits) with
experiment, relative to a reference value $s_{0}^{*}$. Of course, it is
important, in computing these double differences, to take all correlations,
including those between data and fitted parameters, into account. These double
differences should be consistent with zero for the tOPE strategy to pass this
type of test.

Figure 3: The double differences $\Delta^{(2)}(s_{0};s_{0}^{*})$ as a function
of $s_{0}$, for $s_{0}^{*}=3.6$ GeV2. See text.

Figure 3 shows the double differences for the fits shown in Fig. 2. Clearly,
for at least the weights $w_{23}$, $w_{24}$ and $w_{25}$, they are very far
from consistent with zero, for values of $s_{0}$ on both sides of $s_{0}^{*}$.
(By construction, the double differences vanish exactly at $s_{0}=s_{0}^{*}$.)
From these (and other, see Ref. [14]) $R$-ratio based tests, we conclude that
the tOPE strategy fails. Because of the close similarity between the EM and
$\tau$-based spectral functions, it is clear that also for hadronic $\tau$
decays, the tOPE cannot be trusted to yield reliable results.

##  4 Conclusion

Since the $\tau$ mass is relatively light, one has deal with the question of
possible non-perturbative contamination in any strategy to determine the
strong coupling from hadronic $\tau$-decay data. In order to do this,
assumptions are needed, and these assumptions need to be tested. Here we
considered the truncated-OPE strategy, in which the main assumption is that
higher-order terms in the OPE can be neglected, and thus effectively be set
equal to zero. This constitutes an arbitrary choice, and it is particularly
dangerous, given the asymptotic nature of the OPE, when working with FESRs
involving weights for which unsuppressed contributions of high dimension are
in principle present. We carried out several tests of the tOPE strategy,
quantitatively probing the validity of the assumption made about the OPE.

We found that indeed this assumption cannot be trusted at scales of order the
$\tau$ mass: The tOPE strategy does not pass EM-based self-consistency tests,
described in Sec. 3. Moreover, in Sec. 2 we also showed that it does not pass
$V+A$ $\tau$-based self-consistency tests. Our conclusion is that the tOPE
strategy is not reliable if the goal is to obtain $\alpha_{s}(m_{\tau})$ with
currently competitive accuracy. Values of $\alpha_{s}(m_{\tau})$ obtained with
this approach depend very strongly on arbitrary assumptions made about the
OPE; these assumptions are not based on QCD.

For a much more detailed description and discussion of this work, as well as
many more references, we refer to Refs. [4, 14].

## Acknowledgements

We would like to thank the organizers of the Tau2021 workshop for organizing
this conference series during difficult times. We also thank Alex Keshavarzi,
Daisuke Nomura and Thomas Teubner for making their compilation of $R$-ratio
data available to us.

### Funding information

DB was supported by the São Paulo Research Foundation (FAPESP) Grant No.
2015/20689-9, by CNPq Grant No. 309847/2018-4, and by Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code
001. MG is supported by the U.S. Department of Energy, Office of Science,
Office of High Energy Physics, under Award No. DE-SC0013682. KM is supported
by a grant from the Natural Sciences and Engineering Research Council of
Canada. SP is supported by CICYTFEDER-FPA2017-86989-P and by Grant No. 2017
SGR 1069. IFAE is partially funded by the CERCA program of the Generalitat de
Catalunya.

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# Vogel-Fulcher-Tammann Freezing of a Thermally Fluctuating Artificial Spin
Ice Probed by X-ray Photon Correlation Spectroscopy

S. A. Morley<EMAIL_ADDRESS>School of Physics and Astronomy,
University of Leeds, Leeds LS2 9JT, United Kingdom  D. Alba Venero  ISIS, STFC
Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, United Kingdom  J.
M. Porro  ISIS, STFC Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX,
United Kingdom  S. T. Riley  School of Electronic and Electrical Engineering,
University of Leeds, Leeds LS2 9JT, United Kingdom  A. Stein  Center for
Functional Nanomaterials, Brookhaven National Laboratory, Upton, New York
11973, USA  P. Steadman  Diamond Light Source, Chilton, Didcot OX11 0DE,
United Kingdom  R. L. Stamps  SUPA, School of Physics and Astronomy,
University of Glasgow, Glasgow G12 8QQ, United Kingdom  S. Langridge  ISIS,
STFC Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, United Kingdom
C. H. Marrows<EMAIL_ADDRESS>School of Physics and Astronomy,
University of Leeds, Leeds LS2 9JT, United Kingdom

###### Abstract

We report on the crossover from the thermal to athermal regime of an
artificial spin ice formed from a square array of magnetic islands whose
lateral size, 30 nm $\times$ 70 nm, is small enough that they are
superparamagnetic at room temperature. We used resonant magnetic soft x-ray
photon correlation spectroscopy (XPCS) as a method to observe the time-time
correlations of the fluctuating magnetic configurations of spin ice during
cooling, which are found to slow abruptly as a freezing temperature
$T_{0}=178\pm 5$ K is approached. This slowing is well-described by a Vogel-
Fulcher-Tammann law, implying that the frozen state is glassy, with the
freezing temperature being commensurate with the strength of magnetostatic
interaction energies in the array. The activation temperature,
$T_{\mathrm{A}}=40\pm 10$ K, is much less than that expected from a Stoner-
Wohlfarth coherent rotation model. Zero-field-cooled/field-cooled magnetometry
reveals a freeing up of fluctuations of states within islands above this
temperature, caused by variation in the local anisotropy axes at the oxidised
edges. This Vogel-Fulcher-Tammann behavior implies that the system enters a
glassy state on freezing, which is unexpected for a system with a well-defined
ground state.

###### pacs:

75.40.Gb, 75.50.Tt, 75.75.Jn, 05.40.-a

In the past decade a new species of magnetic metamaterials has emerged: the
artificial spin ices (ASI) Nisoli  _et al._ (2013); Heyderman and Stamps
(2013). They consist of a 2-dimensional array of nanoscale magnetic islands
arranged so that the magnetostatic interactions between the islands are
geometrically frustrated Wang  _et al._ (2006). The size and shape of the
individual islands are designed with the intention that their shape anisotropy
means they act as single-domain Ising-like macrospins, mimicking the atomic
spins of their naturally-occurring 3-dimensional analogs Bramwell and Gingras
(2001); Ramirez  _et al._ (1999); Castelnovo  _et al._ (2008), but confined to
a plane. They are thus realizations of the square ice vertex models solved by
Wu Wu (1967) and Lieb Lieb (1967), in which the exact microstate of the
statistical mechanical system can be inspected using advanced microscopy
methods. Until very recently, all ASIs studied have been athermal–albeit
showing an effective thermodynamics Nisoli  _et al._ (2007, 2010); Lammert
_et al._ (2010); Morgan  _et al._ (2013)–since the shape anisotropy energy
barrier $E_{\mathrm{A}}$ that must be surmounted to flip the magnetic moment
of any one island is orders of magnitude larger than the thermal energy
$k_{\mathrm{B}}T$ that can be reached experimentally. Whilst convenient for
imaging studies, these athermal, arrested systems lack ergodicity and so fail
to explore phase space in the manner of a true statistical mechanical system
to find thermally equilibrated configurations.

Figure 1: Scanning electron micrograph of an artificial spin ice with a
lattice spacing of 240 nm, with Permalloy islands of lateral size 30 nm
$\times$ 70 nm and 8 nm thickness.

However, there are recent reports of thermalized ASIs, made either by heating
the sample close to the Curie point of the material from which the islands are
fabricated in order to drive dynamics Kapaklis  _et al._ (2012), whereupon the
arrested state may be imaged upon cooling Porro  _et al._ (2013); Zhang  _et
al._ (2013), or a one-shot anneal process that occurs during fabrication has
been used in the same way Morgan  _et al._ (2011). Within the last year or
two, studies have been carried out that have dynamically imaged real-time
thermal fluctuations in artificial spin ice in the square Farhan  _et al._
(2013a), kagomé Farhan  _et al._ (2013b); Kapaklis  _et al._ (2014), and
tetris ice geometries Gilbert  _et al._ (2015). Nevertheless the nature of the
crossover from a thermally fluctuating system to the arrested, less ergodic
state has so far received little attention. Here we report on measurements of
an ASI with islands of lateral size 30 nm $\times$ 70 nm Morley  _et al._
(2015), shown in Fig. 1. By studying the time-dependence of their soft x-ray
speckle scattering patterns, the ASI is shown to be thermal at room
temperature and to freeze into a fully arrested state below $\sim 178$ K. The
fluctuation rate follows a Vogel-Fulcher-Tammann law on cooling, implying that
the frozen state is glassy in nature, which is unexpected for a system with a
well-defined ground state.

The ASI that we studied was fabricated using electron beam lithography with a
lattice constant of 240 nm, as shown in Fig. 1. A lift-off process was used,
with 8 nm thick Permalloy (Ni80Fe20) evaporated through the resist mask to
form the islands, followed by a 2 nm Al cap. The substrate was a Si wafer. The
ASI array had a 2 mm $\times$ 2 mm area. Soft x-ray photon correlation
spectroscopy (XPCS) coherent scattering measurements were carried out at the
I10 beamline at Diamond in order to measure the time-time correlations of the
magnetic fluctuations in the ASI. The experiments were carried out using
circularly polarized light at the Fe $L_{3}$ (707 eV) resonance. The scattered
intensity from the ASI was recorded in the reflection geometry (illustrated in
Fig. 2(a)) using a charge coupled device (CCD) camera, mounted at a fixed
scattering angle $2\theta=9.6^{\circ}$, 80 cm from the sample, which was kept
at a temperature of 223 K in order to reduce dark noise. Each image arose from
a 40 ms exposure and the images were separated by the 4 s readout time of the
camera. The transverse coherence length of the beam was calculated to be 14.9
$\upmu$m, and so a 10 $\upmu$m diameter pinhole was mounted 23 cm in front of
the sample. Airy rings were observed when the direct beam was imaged through
the pinhole, shown in Fig. 2(b), confirming the coherence of the beam. The
samples were mounted for measurement on a temperature-controlled stage in the
absence of any applied magnetic field.

Figure 2: (Color online) Coherent soft x-ray scattering measurements. (a)
Schematic of the experimental XPCS setup, showing the incoming x-ray beam and
scattered beams from the ASI array, which is represented by an atomic force
microscopy image. The three main diffraction spots in the row above the
specular reflection are shown here. Since the diffraction spots arise from a
small, disordered region of the sample that has been coherently illuminated,
they contain speckle contrast. (b) Fraunhofer diffraction obtained using the
10 $\upmu$m pin hole and straight-through beam on to the CCD detector.

Since our ASI array has a square unit cell, it produces a square pattern of
diffraction spots. In Fig. 2, the centre of the diffracted speckle pattern
corresponds to $Q_{x}=2\pi/a$, where $a$ is the ASI lattice constant. The
calculated lattice spacing for the ASI measured from the position on the array
detector and the geometry was $239\pm 2$ nm, which agrees with that measured
from SEM, $243\pm 4$ nm. In previous scattering experiments on square ASIs
incoherent light has been used to determine the different sublattice
contributions to the hysteresis measured on different orders of diffraction
Morgan  _et al._ (2012) or to observe half order magnetic ground state peaks
from as-grown samples Perron  _et al._ (2013). When coherent illumination is
used then the diffraction spot decomposes into speckle, arising from the
disorder in the illuminated region of the sample introducing different phase
shifts into the scattered waves Sutton  _et al._ (1991): this speckle is
visible in the detailed views of the diffraction spots shown at the top of
Fig. 2(a). The crux of the XPCS technique is that dynamics in the magnetic
configuration is reflected in changes in the details of the speckle pattern.
This effect has long been used with light at visible wavelengths to study
relaxation and aging phenomena in soft matter Cipelletti  _et al._ (2003).
XPCS experiments utilising hard x-rays have been used to study charge density
waves associated with antiferromagnetic domains in Cr Shpyrko  _et al._
(2007), giving an indirect measurement of magnetic dynamics. Soft x-rays at
the $M_{5}$ resonance of Ho revealed antiferromagnetic domain fluctuations in
a thin film of that metal Konings  _et al._ (2011), whilst the jamming of
spiral magnetic domains in a Y-Dy-Y trilayer was revealed in the stretched
exponential correlations studied using soft X-rays at the Dy $M_{5}$ resonance
Chen  _et al._ (2013). There have also been speckle scattering studies at the
Co $L_{3}$ resonance using small-angle X-ray scattering geometry to study the
effects of disorder on the domain pattern of multilayer perpendicular Co/Pt
films in response to a field Pierce  _et al._ (2007).

As expected, measurements taken at an energy of 700 eV, below the Fe $L_{3}$
resonance, showed no change beyond random noise fluctuations (see
Supplementary Movie 1), since the ASI physical structure is static. On tuning
to the $L_{3}$ resonance at 709 eV, magnetic sensitivity is achieved and the
speckle reconfigures as time passes, since the magnetic state of the sample is
reconfiguring under thermal activity (see Supplementary Movie 2). The XPCS
measurements were carried out at different temperatures to drive the thermal
fluctuations at different rates. In order to quantify the time-dependent
behaviour, we calculated the intensity-intensity temporal autocorrelation
function Cipelletti  _et al._ (2003),

$g_{2}(\mathbf{Q},\tau)=\frac{\langle
I(\mathbf{Q},t^{\prime})I(\mathbf{Q},t^{\prime}+\tau)\rangle_{t^{\prime}}}{\langle
I(\mathbf{Q},t^{\prime})^{2}\rangle_{t^{\prime}}}=1+A\left|F(\mathbf{Q},\tau)\right|^{2},$
(1)

where $I(\mathbf{Q},t^{\prime})$ is the intensity at wave vector $\mathbf{Q}$
at a time $t^{\prime}$, $\tau$ is the time delay, and
$\langle...\rangle_{t^{\prime}}$ indicates a time average.
$F(\mathbf{Q},t^{\prime})$ is the so-called intermediate scattering function,
and $A$ is a measure of the degree of speckle contrast. The $g_{2}$ function
was calculated for each pixel within the speckle pattern of the Bragg peak,
and averaged over all such pixels to give the autocorrelation $g_{2}(\tau)$ at
each temperature, shown normalized to an initial value in Fig. 3(a).

Figure 3: (Color online) XPCS results. (a) Normalized $g_{2}(\tau)$ functions
at various temperatures. The lines are fits to Eq. 2. (b) Relaxation times
$t_{\mathrm{r}}$ as a function of temperature, fitted by a Vogel-Fulcher-
Tammann (VFT) law. Also plotted is the superparamagnetic Néel-Brown (NB) law,
which fits poorly.

These $g_{2}(\tau)$ curves were fitted with a heterodyne model, since the
experimental setup means that the fluctuating magnetic signal is mixed with a
static signal that comes from the structural scattering of the array Sinha
_et al._ (2014). This forms the equivalent of a reference beam in the
measurement, leading to the following modified form of the Kohlrausch-
Williams-Watts Kohlrausch (1854); Williams and Watts (1970) intermediate
scattering function Livet  _et al._ (2006):

$g_{2}(\tau)=1+A\cos(\omega\tau)\exp\left(-(\tau/t_{\mathrm{r}})^{\beta}\right)$
(2)

where $t_{\mathrm{r}}$ is the characteristic relaxation time, $\beta$ is a
stretching exponent, and $\omega$ is an oscillation frequency associated with
the heterodyne mixing. The value of $\omega$ was found to be in the range
$\approx 0.001$-0.002 rads-1, corresponding to a time period, 2$\pi$/$\omega$
= 3,140 - 15,700 s ($\approx 1$-4 hrs), which correlates well with the total
measurement time. The speckle contrast coefficient, $A$, was found to be in
the range 0.02-0.05, which agrees with that found from the contrast of the
Airy pattern Beutier  _et al._ (2007). The fitted value of $\beta$ in all our
measurements was $1.0\pm 0.1$, indicating equilibrated behavior Shpyrko  _et
al._ (2007). The function is rather flat at 180 K, and therefore the
relaxation time must be much longer than the time of measurement ( $\approx 3$
hrs) at that temperature, and so cannot be accurately determined. The
correlation function drops off more and more quickly with increasing
temperature, as can be seen from the initial slope and the drop-off in Fig.
3(a). At room temperature the system fluctuates too quickly and the speckle
appears blurred due to the insufficient time resolution of our CCD
acquisition.

For temperatures between 190 and 250 K, $t_{\mathrm{r}}$ lies in the
experimentally accessible range between these two extremes, shown in Fig. 3b.
The simplest thermal activation behavior in a collection of magnetic
nanoparticles is the Arrhenius-type Néel-Brown (NB) law expected for a non-
interacting superparamagnetic system,
$t_{\mathrm{r}}=\tau_{0}\exp(T_{\mathrm{A}}/T)$, where
$T_{\mathrm{A}}=E_{\mathrm{A}}/k_{\mathrm{B}}$ is an activation temperature
and $\tau_{0}$ is an activation time. As can be seen in the figure, this
cannot be fitted to the data at all well. On the other hand, a Vogel-Fulcher-
Tammann (VFT) law Vogel (1921); Fulcher (1925); Tammann and Hesse (1926)
captures the low temperature detail much more accurately: the data were fitted
with the expression

$t_{\mathrm{r}}=\tau_{0}\exp\left(\frac{T_{\mathrm{A}}}{T-T_{0}}\right),$ (3)

where $T_{0}$ is the freezing temperature. The fit yields
$T_{\mathrm{A}}=40\pm 10$ K and $T_{\mathrm{0}}=178\pm 5$ K.

The defining feature of a VFT law is the freezing temperature $T_{0}$, below
which interactions in the system prevent any relaxation or fluctuation. Whilst
the VFT law has been found to fit many different types of data sets very well,
such as spin glasses, super-cooled organic liquids, metallic liquids, and
glassy (bio)polymer systems Angell (1995); Cyrot (1981), it is nevertheless an
empirical law not based on any underlying microscopic picture. Models that
attempt to provide a VFT-like temperature dependence include–among others–ones
based on a time-dependent percolation process Cyrot (1981), or on the energy
distribution of the depth of coupled traps Bouchaud  _et al._ (1995). In the
spin glass theory of Shtrikman and Wohlfarth (S-W) Shtrikman and Wohlfarth
(1981), the random exchange interactions between the positionally disordered
spins are represented by a mean magnetic interaction field $H_{\mathrm{i}}$.
The magnetostatic interactions in our ASI are real magnetic fields, making the
S-W theory a natural one to adapt to our system. In it, the freezing
temperature $T_{0}$ is determined by the characteristic interaction energy
$E_{\mathrm{i}}$. Using the results from the VFT fit, we find
$E_{\mathrm{i}}=\sqrt{k_{\mathrm{B}}^{2}T_{0}T_{\mathrm{A}}}\approx 1.2\times
10^{-21}$ J.

We can estimate the scale of magnetostatic interactions in our ASI using a
point dipole model to determine the S-W interaction energy:

$E_{\mathrm{i}}=\mu_{0}H_{\mathrm{i}}m_{\mathrm{island}}=\frac{\mu_{0}m_{\mathrm{island}}^{2}}{2\pi
a^{3}},$ (4)

where $m_{\mathrm{island}}=M_{\mathrm{s}}V$ is the magnetic moment of an
island of volume $V$ and the lattice constant $a=240$ nm. Assuming the nominal
island size and the bulk magnetization of Permalloy ($M_{\mathrm{s}}=860$
kA/m) yields $E_{\mathrm{i}}=3.0\times 10^{-21}$ J, which is of the same order
of magnitude as the experimental value but overestimated by about a factor of
three. Part of the overestimate may be due to the extrinsic effect of the
reduction in effective $M_{\mathrm{S}}$ Kapaklis  _et al._ (2014) or $V$ Le
_et al._ (2015) due to oxidation. An intrinsic contribution to the
overestimate is the fact that the total interaction energy arises from the sum
of several frustrated interactions from all the island’s neighbors, leading to
partial cancellation of the energy value given above. Thus, the magnetostatic
interactions in the system are clearly of the right scale to break the
Arrhenius-like behaviour and give rise to a VFT-like freezing of the
fluctuations as $T_{0}$ is approached.

Now we turn to a discussion of the activation temperature $T_{\mathrm{A}}$. We
can estimate the energy barrier $\Delta E$ for Stoner-Wohlfarth coherent
rotation of a single island, again assuming the bulk magnetization of
Permalloy, using $\Delta
E=KV=\ln(f_{0}t_{\mathrm{m}})k_{\mathrm{B}}T_{\mathrm{A}}$, with the shape
anisotropy constant $K=\frac{1}{2}\mu_{0}\Delta{\cal D}M^{2}$, and the
difference in demagnetizing factor along the two relevant directions
$\Delta{\cal D}\approx 0.1$ based on the island geometry Osborn (1945). Here
the measurement time, $t_{\mathrm{m}}$, is taken as two hours, a typical value
for $2\pi/\omega$, and we assume $f_{0}=10$ GHz. This yields $T_{A}\approx
1700$ K. This is far larger than the fitted $T_{\mathrm{A}}$ of 40 K, ruling
out this simple picture.

Figure 4: (Color online) (a) ZFC and FC magnetization measurements under
different applied magnetic probe fields of (i) 50 mT, (ii) 35 mT, (iii) 25 mT
and (iv) 5 mT. The magnetic moment, $m$, has been normalised to the peak in
the ZFC curve for each field ($m_{\mathrm{maxZFC}}$) for comparison. The
average blocking temperature, $T_{\mathrm{B}}$, is also defined by this point
and is indicated by the black arrows ($T_{\mathrm{B}}>400$ K at 5 mT). The
grey arrow indicates the $\sim 40$ K feature in the ZFC data. The field was
applied along $[01]$ direction of the ASI lattice. (b) The temperature
dependence of the coercivity measured along $[01]$. The line is a fit to
Sharrock’s equation, the departure from which is clear below 40 K.

In order to seek other evidence of thermally-activated processes that might
cast light on this large discrepancy, standard zero-field cooled (ZFC) and
field-cooled (FC) protocols were carried out on the samples using SQUID-
vibrating sample magnetometry. First, the sample was heated to 395 K and a
large negative saturating field applied and removed. The sample was then
cooled to 5 K without any applied field. Next, a probe field was applied and
the magnetic moment was measured during heating, which is the lower curve in
each measurement in Fig. 4(a). Last, the sample was measured again during
cooling back down with the probe field still applied. The peak in the ZFC,
close to where the curves bifurcate, is defined as the blocking temperature,
$T_{\mathrm{B}}$, and marked with a black arrow. For example, the data with
probe field $\mu_{0}H=50$ mT has $T_{\mathrm{B}}(50~{}\mathrm{mT})=200\pm 10$
K. The blocking temperature systematically rises for smaller probe fields, and
exceeds our experiemntal limit of 400 K at 5 mT. This behavior can be
extrapolated to the case of zero applied probe field, using
$T_{\mathrm{B}}\propto H^{2/3}$ El Hilo  _et al._ (1992), which gives
$T_{\mathrm{B}}(0~{}\mathrm{mT})=460\pm 20$ K. This is about a factor of four
lower than that expected from the sample parameters and nominal volume. Whilst
reduced magnetization or volume can explain some of this discrepancy, it seems
clear once again that a pure coherent rotation mechanism is unlikely to be
strictly followed.

Nevertheless, there is another feature in the magnetometry data: there is an
initial increase in moment up to 40 K, marked with a grey double-headed arrow,
visible in all the ZFC curves. This signifies a much lower energy scale for
some magnetic relaxation process, one that is strikingly similar to the
$T_{\mathrm{A}}$ found from the VFT fit. As previously shown by Ozatay et al.
Ozatay  _et al._ (2008), the properties of Py can change below 40 K when
native oxidation has occurred. They showed in elements of a similar size that
such oxidation is likely around the unprotected edges and can have significant
effects on the reversal properties, where they observed an upturn in
coercivity below this temperature.

We performed the same measurement and have plotted the coercivity of our
sample as a function of temperature in Fig. 4(b). The same characteristic
features as those seen in Ref. Ozatay  _et al._ , 2008 are present. The line
in the figure is a fit to Sharrock’s equation Sharrock (1994), which describes
the temperature dependence expected for the dynamic coercivity of magnetic
particles. Ozatay et al. showed that for conformally capped islands this
feature disappears and the coercivity can be fit to Sharrock’s equation for
all temperatures, whereas here and for their Py particles with a native oxide
around the edges there is a sharp departure from this at around 40 K. They
attribute this to the modification of the local anisotropy which arises from
the exchange coupling between the antiferromagnetic oxide edges and the
ferromagnetic Py. This results in different local anisotropy axes which create
pinning points for the magnetization, and allow non-uniform reversal modes to
become energetically accessible. This allows the system to sample a
distribution of metastable states only once a temperature of
$T_{\mathrm{A}}=40$ K is exceeded.

To summarize, on cooling an artificial spin ice we have observed a dramatic
lengthening of the relaxation time as measured by magnetic XPCS. The system
slows abruptly as it crosses over from thermal equilibration to an athermal,
frozen state. This crossover can be described by a Vogel-Fulcher-Tammann law,
which is typically used to describe glassy systems. The VFT freezing
temperature $T_{0}\approx 178$ K can be accounted for by the magnetostatic
interaction strength through Shtrikman-Wohlfarth theory. The activation
temperature $T_{\mathrm{A}}\approx 40$ K arising from a fit of this law
implies a much lower energy barrier to reversal than is expected from a
single-domain coherent rotation picture. The value of 40 K appears to be
determined by the onset of fluctuating magnetic states within the islands
coupled to magnetic oxides at the edges of the islands.

The glass-like freezing is remarkable since the square ice system possesses a
well-defined ground state, unlike a conventional spin glass. This raises the
question of the true nature of the glassy state that our artificial spin ice
freezes into. Related VFT-like freezing behaviour has recently been observed
in the pyrochlore spin ice Dy2Ti2O7 Kassner  _et al._ (2015), prompting
speculation about this representing many-body localization of spins in a
translationally-invariant quantum system De Roeck and Huveneers (2014); Yao
_et al._ (2014); Schiulaz  _et al._ (2015). This is unlikely to be the case
here since the artificial systems of the type we have studied consist of
classical macrospins, ruling out explicitly quantum behavior, implying that
some other unusual glass state is obtained in artificial spin ices once
ergodicity is broken.

###### Acknowledgements.

This work was supported by the EPSRC (grant numbers EP/J021482/1,
EP/I000933/1, and EP/L002922/1). Research carried out in part at the Center
for Functional Nanomaterials, Brookhaven National Laboratory, which is
supported by the U.S. Department of Energy, Office of Basic Energy Sciences,
under Contract No. DE-AC02-98CH10886. We would like to thank Diamond Light
Source for beamtine and S. S. Dhesi for the loan of the CCD camera.

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