# Investigation of ensemble averaged maximal center gauge

Zeinab Dehghan<EMAIL_ADDRESS>Department of Physics, University of
Tehran  Rudolf Golubich<EMAIL_ADDRESS>Atominstitut, Technische
Universität Wien  Roman Höllwieser<EMAIL_ADDRESS>Department of
Physics, Bergische Universität Wuppertal  Manfried Faber
<EMAIL_ADDRESS>Atominstitut, Technische Universität Wien

###### Abstract

To cure the problem of finding the global maximum of the gauge functional in
maximal center gauge we present the Ensemble averaged Maximal Center Gauge
(EaMCG). It allows for a vortex specific extraction of the quark-antiquark
potential from an ensemble of gauge copies. Evaluating center projected Wilson
loops we compare EaMCG with other center vortex based measurements of the
string tension in SU(2).

quantum chromodynamics; confinement; center vortex model; vacuum structure

††preprint: AIP/123-QED

##  I Introduction

The fundamental role of center vortices in the ground-state vacuum fields
governing the confinement of quarks and chiral symmetry breaking has been
shown in Lattice QCD in the past decades [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13]. The center vortex model for quark-confinement assumes that closed
quantised color-magnetic flux tubes percolating the QCD vacuum are responsible
for a non-vanishing string tension. It explains confinement, topological
charge and chiral symmetry breaking [14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35] and reproduces the expected
behavior of the string tension throughout a wide range of $\beta$-values,
although the detection of center vortices in field configurations can proof
difficult.

The main methods to detect center vortices are based on Maximal Center Gauge
(MCG) [36, 37, 38] via maximizing an appropriate gauge functional
$R_{\text{MCG}}$, usually restricted by an upper and lower limit which we can
consider without loss of generality as $0\leq R_{\text{MCG}}\leq 1$. It is not
possible to numerically identify a global maximum and one can only consider an
ensemble of gauge copies
$\\{\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3},...,\mathcal{C}_{n}\\}$
which can be seen as a partially ordered set
$\mathcal{C}_{1}\leq\mathcal{C}_{2}\leq\mathcal{C}_{3}\leq...\leq\mathcal{C}_{n}$
with respect to $R_{\text{MCG}}$. During an optimization procedure the
physical interpretation of consecutive gauge copies can vary, but differences
should decrease with the gauge functional approaching its maximal value. If
they do not, one speaks of Gribov ambiguities or Gribov problems and care has
to be taken when interpreting such gauge configurations.

We were able to solve the problems by usage of non-trivial center regions [39,
40, 41, 42, 43, 44, 45]. Our solution does not perform the gauge fixing solely
on a microscopic level (taking only single lattice points and the links
attached to them into account) but requires the non-trivial center regions,
macroscopic structures, to guide the gauge fixing procedure.

In this work we present a different approach that extracts the quark-antiquark
potential from an ensemble of gauge copies, the Ensemble averaged Maximal
Center Gauge (EaMCG). Motivated by our findings in [46] we look at a
reasonable large ensemble of MCG copies with respect to the value of the gauge
functional and the string tension. The method starts with the production of
the ensemble and executes on each single gauge copy a gradient climb towards
the nearest local maximum of the gauge functional. After center projection,
average values of Wilson loops $\langle W_{\textrm{CP}}\rangle$ and the quark-
antiquark potentials are determined. This new method is compared in SU(2)
gauge theory to other center gauges such as adjoint Laplacian Landau gauge
(ALLG) and direct Laplacian Landau gauge (DLCG) and to the potentials
extracted from the unprojected (original) field configurations. The expected
scaling behavior as predicted by renormalization group theory is reproduced.

##  II Methods and Formalism: From Center gauges to string tension in lattice-
QCD

We work on periodic lattices with $U_{\mu}(x)\in\text{SU(2)}$ denoting a
gluonic link at lattice site $x$ pointing in direction $\mu$, where Wilson
action

$S_{\text{gluons}}=\beta\sum_{x,\mu<\nu}\left(1-\frac{1}{2}\mathfrak{Re}\text{Tr}(U_{\mu\nu}(x))\right)$
(1)

with $U_{\mu\nu}(x)$ corresponding to a plaquette in the plane spanned by the
directions $\mu$ and $\nu$ is used in a Monte Carlo procedure. For the inverse
coupling $\beta=\frac{4}{g^{2}}$ we choose $\beta=2.3,2.4,2.5,2.6,2.7$.

Center vortices are usually detected in direct maximal center gauge (DMCG)
that uses a gradient climb to identify the gauge transformation
$g(x)\in\text{SU(2)}$ maximizing the gauge functional

$R_{\text{MCG}}=\sum_{x}\sum_{\mu}\mid\text{Tr}[{}^{g}U_{\mu}(x)]\mid^{2}$ (2)

with ${}^{g}U_{\mu}(x)=g(x+\hat{\mu})U_{\mu}(x)g^{\dagger}(x)$. Starting on
multiple random gauge copies this procedure is executed and the gauge copy
with the largest value of the gauge functional is taken for further
processing: The link variables ${}^{g}U_{\mu}(x)$ are projected to the center
degrees of freedom, that is $\pm 1$ for SU(2),

${}^{g}U_{\mu}(x)\rightarrow
Z_{\mu}(x)\equiv\mathrm{signTr}[{}^{g}U_{\mu}(x)].$ (3)

The resulting non-trivial center projected plaquettes are known as
P-plaquettes, $U_{\Box}=-1$, which contain one or three non-trivial links. The
duals of P-plaquettes form closed surfaces in the four dimensional dual
lattice and the dual P-vortices correspond to the closed flux lines evolving
in time. Center projection is the common ground of center detection methods,
only the specific gauge fixing procedures vary. Since
$\text{Tr}U\,\text{Tr}U^{\dagger}=\text{Tr}U_{A}+1$, where

$[U^{A}]_{ij}=\frac{1}{2}\text{Tr}[\sigma_{i}U\sigma_{j}U^{\dagger}]$ (4)

is the adjoint representation, MCG also maximizes

$R_{\text{AL}}=\sum_{x,\mu}\text{Tr}_{A}[{}^{g}U_{\mu}^{A}(x)],$ (5)

the gauge functional of the Landau gauge in the adjoint representation, which
is blind to center elements. Landau gauge finds the configuration closest to a
trivial field configuration. Hence, the functionals $R_{\text{MCG}}$ and
$R_{\text{AL}}$ may fail to detect vortices in smooth field configurations,
especially in the continuum limit. As was raised by Engelhardt and Reinhardt
[9] in the context of continuum Yang-Mills theory a thin vortex gets singular
in this limit and must fail to approach a smooth configuration.

Gauges not directly suffering from Gribov copies are Laplacian gauges which
require to solve the lattice Laplacian eigenvalue problem. They were first
applied in Laplacian Landau gauge by Vink and Wiese [47] for the detection of
Abelian monopoles and modified to Laplacian center gauge in [48, 49] for
center vortices which requires solving the eigenvalue problem of the lattice
Laplacian in the adjoint representation

$D_{ij}(x,y)=\sum_{\mu}\left(2\delta_{xy}\delta_{ij}-[U^{A}_{\mu}(x)]_{ij}\,\delta_{y,x+\hat{\mu}}-[U^{A}_{\mu}(x-\hat{\mu})]_{ji}\,\delta_{y,x-\hat{\mu}}\right).$
(6)

The three lowest eigenvectors of this Laplacian are combined to matrices
$M(x)$ in adjoint Laplacian Landau gauge (ALLG) which maximizes the functional

$R_{\text{ALLG}}=\sum_{x}\sum_{\mu}\text{Tr}[M^{\dagger}(x)U^{A}_{\mu}(x)M(x+\hat{\mu})]$
(7)

with $M(x)\in SO(3)$ ”on average”, that is, with weakened orthogonality
constrained $\langle M^{\dagger}\cdot M\rangle=1$. Even if the solution of the
eigenvalue problem is a unique procedure, small modifications of the
configuration may lead to different eigenfunctions corresponding to different
positions of vortices. The combination of ALLG followed by DMCG [50, 51] is
referred to as direct Laplacian center gauge (DLCG).

In the Monte-Carlo runs on a $24^{4}$ lattice we did for each
$\beta\in\\{2.3,2.4,2.5,2.6,2.7\\}$ 20 random starts, 3000 initial sweeps and
produced 1000 configurations with a distance of 100 sweeps. For these
$20\,000$ configurations we determined the averages of unprojected Wilson
loops. Due to the large computational needs we did center projection by ALLG,
DLCG and EaMCG for every 100th of these $20\,000$ configurations only. For
EaMCG we did 100 unbiased gauge copies for each of the 200 configurations. In
Fig. 7 we compare the $24^{4}$ data for unprojected configurations with the
same number of analogously produced $32^{4}$ configurations.

### The problem of the gauge functional and its resolution

The gauge functional (2) has a huge number of local maxima and it is
practically impossible to find the absolute maximum. The numeric maximization
of the gauge functional suffers from a severe problem as was shown by
Bornyakov, Komarov, and Polikarpov (BKP) in ref. [52] with simulated
annealing: The largest maxima of the functional underestimate the density of
vortices and therefore also the string tension. Since there are only
relatively few of these maxima it is rather tiresome to find them by simulated
annealing.

Random gauge copies are usually determined by random gauge matrices $g(x)$
with a homogeneous density on the SU(2) group manifold, as produced by the
Box-Muller method [53]. One can get directly to consecutively increasing local
maxima of the gauge functional by a bias in these random gauge transformations
by restricting the gauge transformation to the hemisphere of
SU(2)$\cong\mathbb{S}^{3}$ with positive trace. In the two diagrams in Fig. 1
we show the maximizing histories for two independently created field
configurations (out of 200) and for ten of their random gauge copies. For each
of these 20 field configurations we perform 100 times the following two steps:
First, we maximize the functional by gradient climb. Secondly, we apply the
biased random gauge transformations. For center projected field configurations
of sufficient large lattice size already one configuration can result in
reasonable values of Creutz-ratios

$\sigma\approx\chi_{\textrm{CP}}(R)=-\ln\frac{W_{\textrm{CP}}(R+1,R+1)\;W_{\textrm{CP}}(R,R)}{W_{\textrm{CP}}(R,R+1)\;W_{\textrm{CP}}(R+1,R)},$
(8)

for small Wilson loops $W_{\textrm{CP}}(R,T)$ of size $R\times T$, which are
averages over the center projected configurations of the ensemble. In Fig. 1
we show $\chi_{\textrm{CP}}-R_{\text{MCG}}$ pairs for the evolution through
the biased gauge fixing iterations. A nearly linear relation between
$\chi_{\textrm{CP}}$ and $R_{\text{MCG}}$ can be observed. With increasing
gauge functional we observe a linear decrease of the string tension,
underestimating the expected value of $\sigma=0.0350(12)$ [54].

Figure 1: In the two diagrams we show the maximizing histories of two
independent field configurations on a $24^{4}$ lattice for ten random gauge
copies each. Interchanging hundred times gradient climbs and biased random
gauge transformations we get hundred points in the
$\chi_{\textrm{CP}}-R_{\text{MCG}}$ plane connected by a line. Observe the
nearly linear relation between Creutz ratio $\chi_{\textrm{CP}}$ and gauge
functional $R_{\text{MCG}}$ and especially the final low value of
$\chi_{\textrm{CP}}$, much lower than the expected string tension
$\sigma=0.035$.

With this method we did not find any configuration where the maximum of the
gauge functional did not dramatically underbid the string tension. On the
other hand, DMCG [6, 55] predicts the string tension very successfully using a
few gauge copies and choosing the one with the highest value of the functional
for further analysis. This demonstrates the physical relevance of the gauge
functional and implies that it is the prescription for global maximization
which is problematic. In DLCG [50, 51] the mentioned problems were
circumvented using the eigenfunctions of the adjoint Laplacian operator to
select a smooth gauge field before maximizing the gauge functional in a
gradient climb. In this article we investigate a different approach: ensemble
averaged maximal center gauge (EaMCG). We produce random gauge copies with the
correct $\mathbb{S}^{3}$-weight, approach the next local maximum by the
gradient method and take the average of an ensemble of such random gauge
copies. The idea is that not the best local maximum alone carries the physical
information but the average over all local maxima does: maxima with a higher
value of the gauge functional result in a reduced string tension, but they do
not dominate the ensemble. The same holds for lower valued maxima, possibly
overestimating the string tension. This method is strengthened by density
plots of 20 000 $R_{\text{MCG}}/\chi_{\textrm{CP}}(2)$ pairs in Fig. 2 for
$\beta\in\\{2.3,2.4,2.5,2.6,2.7\\}$. For all $\beta$-values we see a clear
linear correlation as expected from Fig. 1. The average slope of the
distributions agreees in both figures. The $\beta$-dependence of the elliptic
distributions is a running coupling effect. With increasing $\beta$, higher
Creutz ratios can be resolved, but give broader distributions due to larger
statistical errors.

Figure 2: Correlations between Creutz ratios $\chi_{\textrm{CP}}(2)$ and the
values of the maximized gauge functionals $R_{\text{MCG}}$ for center
projected configurations for $\beta\in\\{2.3,2.4,2.5,2.6,2.7\\}$. On a
$24^{4}$-lattice we produced the density plots with 20 random starts and 10
configurations with a distance of 10000 MC iterations and 100 gauge copies
each. For each $\beta$ the average values are marked by a star.

The average values of the distributions are marked by a “$\star$” and give
first estimates of the string tension. From the type of distributions shown in
Fig. 2 it gets understandable that the gauge configurations with extremely
high functional (2) do not really contribute to the average. According to Eq.
(8) Creutz ratios are determined from almost symmetric Wilson loops which are
polluted by higher excitations and give therefore less confidential values for
the string tension than the potential, extracted from maximally asymmetric
Wilson loops with good signals as we will describe in the next section.

### Extracting the string tension from Wilson loop data with the 1-exp-fit

We extract the static quark-antiquark potential from asymmetric $R\times T$
Wilson loops of an ensemble of configurations

$\langle
W(R,T)\rangle=\sum^{\infty}_{i=0}c_{i}\mathrm{e}^{-\varepsilon_{i}(R)T}.$ (9)

Only for the lowest states the amplitudes $c_{i}$ and energies
$\varepsilon_{i}$ are resolvable. In Fig. 3 the logarithms of Wilson loops are
shown at $\beta=2.5$ comparing a 3-exp fit of original Wilson loops, a 2-exp
fit of ALLG center projected loops, a 1-exp fit of DLCG based loops and 1-exp
fit of EaMCG averaged loops. The curvature of original Wilson loops at small
$T$ is due to excited states that are almost eliminated in DLCG and EaMCG.

Figure 3: A comparison of logarithms of Wilson loop data is depicted at
$\beta=2.5$ for original and projected configurations using different gauge
fixing procedures. Observe that excited states contribute for large $R$
essentially for unprojected configurations, less for ALLG and very little for
DLCG and EaMCG.

The larger the excitation energy, the more are excited states exponentially
suppressed with increasing $T$. Finally, when the Wilson loops remain
resolvable, only the ground state survives. To determine the ground state
energy $\varepsilon_{0}(R)$ we introduce for each $R$ an initial $T_{i}$ and
define a one-exponential fitting function (1-exp fit) for Wilson loops with
$T\geqslant T_{i}$ as

$\langle W(R,T)\rangle=c(R,T_{i})e^{-\varepsilon_{0}(R,T_{i})T}.$ (10)

By removing small $T$s we observe that $\varepsilon_{0}(R,T_{i})$ gets nearly
stable in a fit range $T_{i}\approx T_{\text{opt}}$, the plateau region.
$T_{\text{opt}}$ is a function of $R$ and therefore the plateau region has to
be investigated for each $R$ separately.

Keeping numerical issues under control proofs quite difficult: The Wilson loop
averages from a single configuration are disturbed by noise, hence multiple
configurations are averaged into $\langle W(R,T)\rangle$. The data is still
disturbed by systematic errors resulting from unwanted higher excitations and
statistical errors with increasing loop sizes: a decrease of the systematic
error is accompanied by an increase of the statistical error. Therefore, the
optimal $T_{i}$ value, $T_{\text{opt}}$, we determine increasing $T_{i}$ from
the first local minimum of an error quantifier [46],

$\text{Err}=\frac{2}{3}\langle\Delta_{\delta
i}\rangle+\frac{1}{3}\langle\Delta_{\text{err}}\rangle$ (11)

with $\langle\Delta_{\delta i}\rangle$ being the average change of the fit
variable $\varepsilon_{0}(R,T_{i})$ from previous to present $T_{\text{opt}}$
and from present to next $T_{\text{opt}}$ and
$\langle\Delta_{\text{err}}\rangle$ being the average error of the respective
three data points $\varepsilon(R,T_{i-1})$, $\varepsilon(R,T_{i})$ and
$\varepsilon(R,T_{i+1})$. Fig. 4 depicts some examples for the $T_{i}$
dependence of $\varepsilon_{0}(R,T_{i})$ and the chosen $T_{\text{opt}}$ for
$\beta=2.5$ and some $R$ for unprojected SU(2) configurations, ALLG, DLCG and
EaMCG. For unprojected configurations and also for ALLG the excited states
contribute at small distances and $\varepsilon_{0}(R,T_{\text{opt}}(R))$ is
therefore approached from above. We observed in Fig. 3 that center projection
for DLCG and EaMCG suppresses the excited states, the variations of
$\varepsilon_{0}$ are therefor much smaller.

Figure 4: Some examples of the $T_{i}$ dependence of
$\varepsilon_{0}(R,T_{i})$ are shown. The final values at $T_{\text{opt}}$ are
depicted by horizontal lines. For original Wilson loops and ALLG,
$\varepsilon_{0}(R,T_{\text{opt}}(R))$ is approached from above for
$T_{i}\rightarrow T_{\text{opt}}$ due to the decaying contributions of excited
states. Observe that for DLCG and EaMCG the variations of
$\varepsilon_{0}(R,T_{i})$ are much smaller since the lines in the lower part
of Fig. 3 are remarkably straight.

Once, $\varepsilon_{0}(R,T_{\text{opt}})$ is determined, we can identify it
with the potential

$V_{W}(R)=\varepsilon_{0}(R,T_{\text{opt}}(R))$ (12)

extracted from Wilson loops.

##  III Results and Discussion

###  III.1 Center Dominance and Precocious Linearity

In this section, we compare the potentials extracted from original
configurations with those which we get after center projections, especially
with the main subject of this article with ensemble averaged maximal center
gauge, EaMCG. To extract the physical parameters of the gluonic quark-
antiquark string we fit the potentials $V_{W}(R)$ of Eq. (12) with a Cornell
type of potential

$V(R)=v_{0}+\sigma R-\dfrac{c}{R}$ (13)

with three parameters, self energy $v_{0}$, string tension $\sigma$ and
Coulomb coefficient $c$. For original and ALLG configurations we need all
three parameters. For DLCG and EaMCG we observe the feature of precocious
linearity; i.e. the fact that the Coulombic term is strongly suppressed.
Therefore we choose $c=0$ and start the fit at some $R_{i}$.

Figure 5: For $\beta=2.5$ on a $24^{4}$ lattice, potentials determined with
different methods are shown and compared to Ref [54]. To ease comparisons, the
potentials of EaMCG and DLCG are shifted by the self energy $v_{0}$ of the
potential extracted from unprojected configurations.

For $\beta=2.5$ we show in Fig. 5 the potential $V_{W}(R)$ of unprojected
configurations, the fit potential $V(R)$, its contributions
$v_{0}-\dfrac{c}{R}$ and $v_{0}+\sigma R$ together with the data of Ref. [54].
We compare them with the potentials and their fits resulting from the three
center projection methods, ALLG, DLCG and EaMCG. ALLG has a sizeable Coulomb
contribution, differing in this respect from DLCG and EaMCG. ALLG leads to
confidential values in the same region as $V_{W}(R)$. The potential is more
difficult to resolve than for DLCG and EaMCG. ALLG has a smaller linear term
than DLCG, where this term agrees nicely with the linear term of the
unprojected configurations. Over the whole $R$ region the EaMCG values are
very well resolvable and start at $R_{i}=3$ to predict a stable value for the
string tension $\sigma$. It is interesting to observe in Fig. 5 that, despite
the linear term is larger than for DLCG and the unprojected configurations,
the potential runs at large $R$ nicely parallel to the unprojected potential.
Since the Coulomb factor of the unprojected configurations is nearly
proportional to $\pi/12$ and thus mainly due to string shape fluctuations this
may be an indication that on the finite periodic lattice, in average, vortices
do follow the shape fluctuations of the gluonic string. This behavior is
reproduced for all tested values of $\beta$ as can be seen in Fig. 6.

Figure 6: For $\beta\in\\{2.3,2.4,2.6,2.7\\}$ on a $24^{4}$ lattice, the
different methods to extract potentials and string tension are compared. For
better visual comparison, the potentials of EaMCG and DLCG are shifted by the
self energies of potentials extracted from original configurations.

The parameters of the potentials are summarized in table 1. The indicated
errors are the statistical errors only. The systematic errors are difficult to
quantify, but they are at least of the same size as the statistical errors as
we can see comparing the data.

method | $v_{0}$ | $\sigma$ | $c$
---|---|---|---
$\beta=2.3$ |  |  |
unprojected, $24^{4}$ | $0.492(8)$ | $0.1496(27)$ | $0.214(6)$
unprojected, $24^{4}$, fix $c$ | $0.5289(43)$ | $0.1431(21)$ | $\pi/12$
unprojected, $32^{4}$ | $0.495(5)$ | $0.1490(16)$ | $0.216(3)$
$Z_{2}$, ALLG | $0.557(66)$ | $0.1508(193)$ | $0.053(48)$
$Z_{2}$, DLCG | $0.0496(18)$ | $0.1247(6)$ | –
$Z_{2}$, EaMCG | $0.0237(13)$ | $0.1412(4)$ | –
$\beta=2.4$ |  |  |
unprojected, $24^{4}$ | $0.557(15)$ | $0.0730(25)$ | $0.284(20)$
unprojected, $24^{4}$, fix $c$ | $0.5472(41)$ | $0.0741(12)$ | $\pi/12$
unprojected, $32^{4}$ | $0.552(10)$ | $0.0728(17)$ | $0.271(13)$
$Z_{2}$, ALLG | $0.547(22)$ | $0.0608(56)$ | $0.112(16)$
$Z_{2}$, DLCG | $0.0233(17)$ | $0.0676(3)$ | –
$Z_{2}$, EaMCG | $-0.0070(21)$ | $0.0811(4)$ | –
$\beta=2.5$ |  |  |
unprojected, $24^{4}$ | $0.543(5)$ | $0.0371(8)$ | $0.266(6)$
unprojected, $24^{4}$, fix $c$ | $0.5410(16)$ | $0.0373(5)$ | $\pi/12$
unprojected, $32^{4}$ | $0.552(5)$ | $0.0350(8)$ | $0.276(6)$
$Z_{2}$, ALLG | $0.413(3)$ | $0.0299(7)$ | $0.081(2)$
$Z_{2}$, DLCG | $0.01423(9)$ | $0.03550(2)$ | –
$Z_{2}$, EaMCG | $0.0040(14)$ | $0.0427(3)$ | –
$\beta=2.6$ |  |  |
unprojected, $24^{4}$ | $0.524(4)$ | $0.0191(6)$ | $0.258(5)$
unprojected, $24^{4}$, fix $c$ | $0.5256(12)$ | $0.0189(3)$ | $\pi/12$
unprojected, $32^{4}$ | $0.518(3)$ | $0.0200(5)$ | $0.251(4)$
$Z_{2}$, ALLG | $0.302(2)$ | $0.0197(5)$ | $0.054(2)$
$Z_{2}$, DLCG | $0.00655(11)$ | $0.01946(2)$ | –
$Z_{2}$, EaMCG | $0.0013(8)$ | $0.0242(2)$ | –
$\beta=2.7$ |  |  |
unprojected, $24^{4}$ | $0.479(3)$ | $0.0133(5)$ | $0.219(5)$
unprojected, $24^{4}$, fix $c$ | $0.4951(31)$ | $0.0118(7)$ | $\pi/12$
unprojected, $32^{4}$ | $0.489(2)$ | $0.0116(3)$ | $0.234(3)$
$Z_{2}$, ALLG | $0.211(1)$ | $0.0097(2)$ | $0.034(2)$
$Z_{2}$, DLCG | $0.00822(34)$ | $0.00988(6)$ | –
$Z_{2}$, EaMCG | $0.0042(2)$ | $0.01406(6)$ | –
Table 1: The parameters, string tension $\sigma$, self energy $v_{0}$ and
Coulomb factor $c$, of the fitting potential (13) are compared for the various
methods.

###  III.2 Scaling behavior

The string tension is expected to be proportional to the square of the lattice
spacing $a$. For the SU(2) gauge group in the quenched approximation [56]
asymptotic freedom predicts in two-loop order

$a^{2}=\dfrac{1}{\Lambda_{L}^{2}}(\dfrac{6\pi^{2}\beta}{11})^{\dfrac{102}{121}}\exp({-\dfrac{6\pi^{2}}{11}\beta}),$
(14)

where $\Lambda_{L}$ is the standard lattice scale parameter. In Fig. 7 we
compare the string tensions extracted from unprojected and center projected
configurations of Table 1 with the asymptotic freedom prediction. The main
diagram shows the results of our calculations on a $24^{4}$ lattice. In the
insets we show fits to the asymptotic behavior
$a^{2}\propto\exp({-\dfrac{6\pi^{2}}{11}\beta})$, in the upper inset for EaMCG
and in the lower inset for unprojected configurations on a $32^{4}$ lattice
which scale much better than $24^{4}$ data. We can clearly see in the insets
that EaMCG and unprojected show the same scaling behavior and that EaMCG
scales better than ALLG and DLCG.

Figure 7: The dependence of string tensions extracted from unprojected and
center projected configurations on $\beta$ are compared with the predictions
of asymptotic freedom. For each of the methods the lattice scale paramter
$\Lambda_{L}$ is adjusted separately to the data with $\beta\geqslant 2.5$.

##  IV Conclusion

Center gauge methods suffer from the large differences between the center
projected and unprojected configurations. These differences appear especially
in the region of center vortices. Maximizing the center blind gauge functional
(2) favors therefore to overlook vortices and to underestimate the string
tension. This property is the basis of an approximate linear relation between
the gauge functional and the string tension as shown in Figs. 1 and 2. This is
a clear indication that the request for a global maximization of the gauge
functional is not only difficult but erroneous.

Previously, there were two suggestions to overcome this failure of the global
maximization condition. The first idea [50, 51] was to preselect a region of
smooth gauge configurations where we maximize the gauge functional. With the
three lowest eigenfunctions of the center blind lattice Laplacian in the
adjoint representation we defined in ALLG a smooth gauged transformed
configuration to start the maximization procedure by a gradient climb and to
arrive at DLCG. In [39, 41, 43] we suggested a completely different, but
computationally rather involved approach, where we proposed to use global
structures, non-trivial center regions, to guide the gauge fixing procedure.
These two successful methods indicate that the failure of Direct Maximal
Center Gauge was in the prescription to determine the global maximum of the
gauge functional. In the new method suggested in this article, ensemble
averaged maximal center gauge (EaMCG), we are using the physical information
contained in the huge ensemble of local maxima of the functional. The gluon
string between a static quark-antiquark pair with large separation fluctuates
leading to a Coulombic contribution proportional to $\pi/12$ [57]. Projected
vortices follow these fluctuations leading to an increase of the P-vortex area
compared to a straight string and result in a slightly larger string tension
than DLCG. The potential extracted from these areas follows nicely the large
distance behavior of the string on a finite periodic lattice.

Recently, an improved method for computing the static quark-anti-quark
potential in lattice QCD was investigated in [58, 59, 60], which is not based
on Wilson loops, but formulated in terms of (temporal) Laplace trial state
correlators, formed by eigenvector components of the covariant lattice Laplace
operator. This approach seems very promising and an application to center
vortex configurations as well as a generalization of the methods introduced in
this article to $SU(3)$ is left for future investigations.

## Acknowledgements

The authors gratefully acknowledge the Vienna Scientific Cluster (VSC) for
providing supercomputer resources. R.H. was supported by the MKW NRW under the
funding code NW21-024-A. We thank Sedigheh Deldar for valuable discussions.

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# Heat dissipation mechanisms in hybrid superconductor-semiconductor devices
revealed by Joule spectroscopy

A. Ibabe  Departamento de Física de la Materia Condensada, Universidad
Autónoma de Madrid, Madrid, Spain  Condensed Matter Physics Center (IFIMAC),
Universidad Autónoma de Madrid, Madrid, Spain  G. O. Steffensen  Departamento
de Física Teórica de la Materia Condensada, Universidad Autónoma de Madrid,
Madrid, Spain  Condensed Matter Physics Center (IFIMAC), Universidad Autónoma
de Madrid, Madrid, Spain  I. Casal  Departamento de Física de la Materia
Condensada, Universidad Autónoma de Madrid, Madrid, Spain  Condensed Matter
Physics Center (IFIMAC), Universidad Autónoma de Madrid, Madrid, Spain  M.
Gómez  Departamento de Física de la Materia Condensada, Universidad Autónoma
de Madrid, Madrid, Spain  Condensed Matter Physics Center (IFIMAC),
Universidad Autónoma de Madrid, Madrid, Spain  T. Kanne  Center for Quantum
Devices, Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
J. Nygård  Center for Quantum Devices, Niels Bohr Institute, University of
Copenhagen, Copenhagen, Denmark  A. Levy Yeyati  Departamento de Física
Teórica de la Materia Condensada, Universidad Autónoma de Madrid, Madrid,
Spain  Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de
Madrid, Madrid, Spain  Instituto Nicolás Cabrera, Universidad Autónoma de
Madrid, Madrid, Spain  E. J. H. Lee<EMAIL_ADDRESS>Departamento de Física
de la Materia Condensada, Universidad Autónoma de Madrid, Madrid, Spain
Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid,
Madrid, Spain  Instituto Nicolás Cabrera, Universidad Autónoma de Madrid,
Madrid, Spain

###### Abstract

Understanding heating and cooling mechanisms in mesoscopic superconductor-
semiconductor hybrid devices is crucial for their application in quantum
technologies. Owing to the poor thermal conductivity of typical devices,
heating effects can drive superconducting-to-normal phase transitions even at
low applied bias, observed as sharp conductance dips through the loss of
Andreev excess currents. Tracking such dips across magnetic field, cryostat
temperature, and applied microwave power, which constitutes Joule
spectroscopy, allows to uncover the underlying cooling bottlenecks in
different parts of a device. By applying this technique, we analyze heat
dissipation in devices based on full-shell InAs-Al nanowires and reveal that
superconducting islands are strongly susceptible to heating as their cooling
is limited by the rather inefficient electron-phonon coupling, as opposed to
grounded superconductors that primarily cool by quasiparticle diffusion. Our
measurements indicate that powers as low as 50-150 pW are able to fully
suprpress the superconductivity of an island. Finally, we show that applied
microwaves lead to similar heating effects as DC signals, and explore the
interplay of the microwave frequency and the effective electron-phonon
relaxation time.

††preprint: APS/123-QED

Hybrid superconductor-semiconductor nanostructures have attracted great
attention in the past decade as a platform for the development of novel
quantum devices [1, 2], with focus on both topological Majorana fermions [3,
4, 5], and the study of conventional Andreev bound states [6, 7, 8, 9], among
others. Additionally, of great interest are mesoscopic superconducting
islands, comprised of tunnel coupled isolated regions with finite charging
energy, which have been used to define charge qubits [10, 11], to detect
topological transitions [12, 13, 14], and are promising for building
artificial Kitaev chains [15, 16, 17]. Common for these applications is a
sensitivity to heightened temperature and non-equilibrium distributions of
quasiparticles, which could introduce both quasiparticle (QP) poisoning and
qubit decoherence [18, 19, 20, 21, 22, 23, 24, 25]. The dynamics of such out-
of-equilibrium QPs in superconductors, also referred to as ”hot-electrons”
[26, 27, 28, 29] are also responsible for the observed micro-cooling in INSIN
and SIN devices [30, 31, 32, 33, 34] interesting for the pursuit of nanoscale
thermodynamic elements beyond their detrimental effects in quantum
computation. Despite great advances both in the fabrication and
characterization of hybrid superconductor-semiconductor devices, the
associated heating and relaxation mechanisms and their dependence on device
geometry are still not well understood. Indeed, depending on the device and
its environment, cooling can occur via different mechanisms, such as excited
quasiparticles, limited at low temperatures by the superconducting gap,
coupling to substrate phonons, which is strongly suppressed by the weak
electron-phonon coupling at low temperatures, or emission of photons [35, 36],
which requires an AC-generating coupling. Here, we study such effects for the
case of hybrid nanowire devices with superconducting islands, which, as of
their electric isolation, also display more limited cooling channels than in
open hybrid systems, and therefore, could potentially be more susceptible to
heating.

While charge transport is rather straightforwardly measured through the
application of voltage or current biases, the transport of thermodynamic heat
is more subtle and requires both a well-controlled heat source and effective
thermometers [37, 38]. By applying a DC current across a device, Joule power
is deposited at resistive elements and, at given power thresholds, can drive
superconducting parts to the normal state, as the local temperature reaches
the critical superconducting temperature, $T=T_{c}$. In devices with highly-
transmitting junctions, such superconductor-to-normal transitions can be
observed as a rapid drop of the superconducting excess current, measurable via
lock-in techniques [39, 40]. This, in total, yields both an electrically
operated heat source and a single-shot thermometer at $T=T_{c}$, which can be
employed to investigate heating effects, a technique dubbed ”Joule
spectroscopy” [41]. Furthermore, if $T_{c}$ is tunable, for example via an
external magnetic field, this single-shot measurement can be extended to a
wider range of temperatures and powers.

Figure 1:  $\mid$ Joule heating in mesoscopic superconducting islands. a,
Electron micrograph (false color) of a device similar to device A. A
superconducting island of length $L\approx 0.85$ $\mu$m and two
superconducting leads (purple) are defined by wet etching two segments of the
Al shell of an InAs-Al nanowire. Side gates $V_{g,L}$ $V_{g,P}$, and $V_{g,R}$
are fixed at 5 V for the measurements. An external magnetic field, $B$, is
applied with a small angle with respect to the NW axis, $x$. b, Schematics of
the temperature distribution along the device due to Joule heating. $T_{bath}$
is taken as the cryostat temperature, whereas $T_{j}$ (where $j$ = $I,L,R$)
are the temperatures at the island and on the left and right leads. Our
thermal model (lower panel) indicates the main cooling mechanism for each
superconductor: quasiparticle diffusion for the leads (governed by the
parameter $R_{lead,i}$), and electron-phonon coupling $\propto\Sigma U$ for
the islands. c, $I(V)$ (red line) and $dI/dV(V)$ (black and blue lines) curves
taken at $B=0$. Three $dI/dV$ dips are observed with increasing $V$, as
indicated by the arrows. The dips signal superconductor-to-normal metal
transitions of the island at $V_{dip,I}\approx 0.5$ mV and the leads,
$V_{dip,L}\approx$ 7.5 mV and $V_{dip,R}\approx$ 10 mV. The $I(V)$ curve shows
a suppression of the excess current, $\Delta I_{exs}$, when the island turns
normal. Inset: $dI/dV(V)$ measurement plotted against voltage in log scale.
The black (blue) $dI/dV$ curves were taken with $dV=$ 5$\mu$V (200 $\mu$V) to
better resolve the low- (high-) bias dips (see Methods for more detail).
Figure 2:  $\mid$ Heat dissipation mechanisms revealed by Joule spectroscopy.
a, $dI/dV(V)$ as a function of magnetic field for device A, taken at a
cryostat temperature of 20 mK. Owing to the Little-Parks effect, the positions
of the three $dI/dV$ dips oscillate with applied $B$. The dependence of the
higher bias dips, $V_{dip,L/R}(B)$, which relate to the superconductor-to-
normal transitions of the two full-shell InAs-Al leads, is directly
proportional to $T_{c}(B)$ as underscored by the fits to the Abrikosov-Gork’ov
theory (green and purple dashed lines, labeled as qp, referring to
quasiparticle diffusion). By contrast, the magnetic field dependence of the
lower bias dip, ($V_{dip,I}$), in b, is clearly distinct (the green dashed
line is an attempt to fit the experimental data to $V_{dip,I}(B)\propto
T_{c}(B)$). A good agreement with the experimental data is obtained for
$V_{dip,I}(B)\propto\sqrt{T_{c}^{6}(B)}$ (red dashed line, labeled as e-ph,
referring to electron-phonon coupling).

In this work, we investigate the relevant heat dissipation mechanisms in
hybrid superconductor-semiconductor nanowire devices and reveal that grounded
and floating superconductors display distinct heat bottlenecks. We focus on
devices based on full-shell InAs-Al nanowires incorporating a mesoscopic
superconducting island and present data corresponding to two devices in the
main text. The first one, referred to as device A, was fabricated by wet
etching two segments of approximately $200$ nm of the Al shell, thus defining
an island of length $L\approx 0.85$ $\mu$m and two superconducting full-shell
InAs-Al leads. Fig. 1a displays a scanning electron micrograph of a device
lithographically similar to device A (see Methods for a detailed description
of the fabrication and of the different devices). Side gates $V_{g,L}$ and
$V_{g,R}$ tune independently the transparency of each nanowire junction, and
the plunger gate, $V_{g,P}$, was fixed at 5 V for all measurements. As Joule
spectroscopy relies on detecting changes in the excess current, we have
carried out measurements in the open regime to maximize $I_{exs}$.
Specifically, we fix $V_{g,L}$ = $V_{g,R}$ = 5 V, obtaining a normal
conductance $G_{N}\approx 4.5\times 2e^{2}/h$ and $I_{exs}\approx$ 95 nA. We
note that, in this regime, charging effects are also suppressed and for this
reason they are neglected in the remaining of our analysis. Finally, the
application of an external magnetic field, $B$, which has a small angle of
$\approx 3^{\circ}$ with respect to the nanowire in device A, provides a
powerful knob in our study. Indeed, the modulation of the superconducting
critical temperatures, $T_{c}$, of the island and of the leads by the Little-
Parks effect [42, 43] will allow us to distinguish between different
mechanisms for heat dissipation in our device.

Let us start by discussing the impact of Joule heating on the temperature of
hybrid superconductor-semiconductor devices. At the core of this problem lies
the thermal balance between the amount of heat dissipated by the Joule effect
and the existing cooling power in the device. Importantly, both heating and
cooling powers may be distributed rather unevenly, as is the case for devices
with superconducting islands. It is therefore useful to analyze the thermal
balance at each of the superconductors separately:

$P_{h,j}=P_{c,j}(T_{j},T_{bath}),$ (1)

where $P_{h,j}$ and $P_{c,j}(T_{j},T_{bath})$, respectively, refer to the
Joule power deposited on, and the cooling power available at superconductor
$j$ (with $j=I,L,R$ corresponding to the island, left lead and right lead).
Note that the latter depends both on the electronic temperature of the
superconductor, $T_{j}$, and the bath temperature, $T_{bath}$, which we take
to be equal to the cryostat temperature.

The heating considered here arises from the relaxation of hot quasiparticles
generated in the superconductors by the flow of dissipative electrical
currents. For the sake of simplicity, we first discuss this effect for the
case of a highly-transmitting $S-S$ junction. For
$V>(\Delta_{L}+\Delta_{R})/e$, transport is well described by
$I=V/R+I_{exs,L}+I_{exs,R}$, where $\Delta_{L}$ and $\Delta_{R}$ are the
superconducting gaps of the left and right leads, and $R$ is the normal
resistance. $I_{exs,L}$ ($I_{exs,R}$) refers to the excess current stemming
from Andreev reflections on the left (right) lead. As a result, the dissipated
Joule power will have contributions from both Ohmic and excess current terms
above. Interestingly, while the former is distributed evenly between left and
right leads, the excess current contributions are not. This owes to the fact
that the Andreev reflections underlying $I_{exs}$ generate quasiparticles only
on the opposite side of the junction. Hence, the Joule power deposited on the
left (right) lead is given by $P_{h,L}=V^{2}/2R+I_{exs,R}V$ ($L\leftrightarrow
R$). By extending the same logic to the geometry of device A (see Supplemental
Material for more details), we get:

$\displaystyle I$
$\displaystyle=\frac{V}{R}+\frac{1}{2}\left(2I_{exc,I}+I_{exc,L}+I_{exc,R}\right),$
(2) $\displaystyle P_{h,I}$
$\displaystyle=\frac{1}{2}\frac{V^{2}}{R}+\frac{V}{2}\left(I_{exs,L}+I_{exs,R}\right),$
(3) $\displaystyle P_{h,L/R}$
$\displaystyle=\frac{1}{4}\frac{V^{2}}{R}+\frac{V}{2}I_{exs,I},$ (4)

where we assumed that the resistances of the two nanowire junctions are equal,
$R_{L}=R_{R}=R/2$, leading to approximately equal voltage drops across each
junction, $V/2$. Note that the relations above satisfy the detailed balance,
$IV=P_{h,I}+P_{h,L}+P_{h,R}$, and demonstrate that the Joule power deposited
on the superconductors can be different.

At thermal equilibrium, the power deposited on superconductor $j$ has to match
$P_{c,j}(T_{j},T_{bath})$, the strength of which increases with $T_{j}$. This
means that if at a given instant $P_{h,j}>P_{c,j}(T_{j},T_{bath})$, then
$T_{j}$ rises to restore the balance in Eq. (1). In this way, upon increasing
the Joule power, an inhomogeneous temperature distribution builds up in the
device (Fig. 1b), with hotter zones forming in regions with less efficient
heat dissipation and/or a higher Joule power deposition. Clearly, for high
enough Joule powers, the temperature can reach the superconducting critical
temperature at a given superconductor, i.e., $T_{j}=T_{c,j}$, driving its
transition to the normal state. This occurs at a critical voltage,
$V_{dip,j}$, and leads to the full suppression of Andreev processes and
$I_{exs,j}=0$. Such a drop in current results in a sharp dip in differential
conductance, $dI/dV$, at $V_{dip,j}$, which is measured in the experiment. We
detect three such $dI/dV$ dips for device A (Fig. 1c): one at relatively low
bias ($V_{dips,I}\approx 0.5$ mV), which we ascribe to the superconducting
island as it is more thermally isolated, and two appearing at higher bias
($V_{dip,L}\approx$ 7.5 mV and $V_{dip,R}\approx$ 10 mV), attributed to the
left and right leads. A partial suppression of the excess current, $\Delta
I_{exs}$, which correlates with the low-bias dip, is clearly identified in the
$I(V)$ curve in red, with $I_{exs}$ finally going to zero for $V\gtrsim$ 10 mV
(not shown).

By applying Eqs. (3) and (4), we estimate the Joule powers required for
driving the island and the leads of device A to the normal state,
$P_{dip,I}\approx$ 60 pW and $P_{dip,L}\approx$ 5 nW and $P_{dip,R}\approx$ 9
nW, respectively. Such a difference of around two orders of magnitude already
suggests that the underlying heat dissipation mechanisms must be different.
Following previous works [40, 41], we ascribe quasiparticle diffusion as the
dominant mechanism for the superconducting leads. Solving the corresponding
heat diffusion equation yields the cooling power at $T_{c}$ for the leads,

$P_{dip,j}=P_{c,j}(T_{c,j})=\Lambda\frac{k_{B}^{2}T_{c,j}^{2}}{e^{2}R_{lead,j}},$
(5)

with $R_{lead,j}$ being the normal resistance of the leads, and
$\Lambda=2.112$ [41]. By contrast, owing to the presence of the nanowire
junctions, the superconducting island can not cool down efficiently via
quasiparticles. Instead, we assume that the heat dissipation predominantly
occurs by electron-phonon coupling,

$P_{dip,I}=P_{c,I}(T_{c,I},T_{bath})=\Sigma
U\left(T_{c,I}^{n}-T_{bath}^{n}\right),$ (6)

where $\Sigma$ is a material-dependent parameter, $U$ is the volume of the
island, and $n$ is expected to range from 4 to 6, according to theory [44, 26,
45, 46, 47, 48, 49]. Note that, due to their limited impact, other possible
heating and cooling terms are neglected at this moment.

Figure 3:  $\mid$ Electron-phonon coupling as the dominant heat dissipation
mechanism in superconducting islands. $dI/dV(V)$ as a function of the external
magnetic field for device B at different cryostat temperatures: a, 250 mK, b,
600 mK, c, 900 mK, and d, 1.2 K. In this dataset, a single $dI/dV$ dip related
to the island is observed, as the leads in this device are normal (Cr/Au, see
inset in panel b). For the measurement taken at the lowest temperature (panel
a), we plot the best fits to $V_{dip,I}\propto T_{c}$ (green dashed line,
labeled as qp) and $V_{dip,I}\propto\sqrt{(T_{c}^{6}-T_{bath}^{6}})$ (red
dashed line, labeled as e-ph), using AG theory. $T_{bath}$ is taken as the
cryostat temperature. For the other panels, we plot $V_{dip,I}(B)$ calculated
using the same fit parameters extracted from a and the corresponding
$T_{bath}$.

To test our assumptions regarding the dominant heat dissipation mechanisms for
the superconducting island and leads, we study the dependence of the three
$dI/dV$ dips as a function of a magnetic field applied nearly parallel to the
nanowire axis, as shown in Fig. 2. Owing to the Little-Parks effect, the
magnetic field gives rise to $T_{c,j}(B)$ oscillations which translate into
characteristic $V_{dip,j}(B)$ dependences for the distinct dissipation
mechanisms. For example, quasiparticle diffusion yields $V_{dip,j}$ directly
proportional to $T_{c,j}$, as concluded from Eqs. (4) and (5). We thus fit the
high-bias dips in Fig. 2a to $V_{dip,L/R}(B)\propto T_{c,L/R}(B)$ using
Abrikosov-Gorkov (AG) theory [50, 41]. The excellent observed agreement
reinforces our interpretation of the dominant cooling mechanism for the leads
at $T_{c}$. Note that the fact that the two high-bias dips are not identical
is due to asymmetries in the device, e.g., in $R_{lead,j}$. The $B$-field
behavior of the low-bias dip, on the other hand, does not follow the same
dependence, as evidenced by the green dashed line in Fig. 2b. Instead, we fit
the experimental data to
$V_{dip,I}(B)\propto\sqrt{T_{c,I}(B)^{n}-T_{bath}^{n}}$, which is obtained
from Eqs. (3) and (6) by neglecting the excess current terms in $P_{h,I}$
(which correspond to approximate $25\%$ of total $P_{h,I}$ in device A, and
may explain the lower $V_{dip,I}$ observed in lobe $0$ compared to lobe $1$ in
Fig. 2b, see supplemental material). A good agreement is achieved for $n=6$
(red dashed line), with $T_{c,I}(B)$ obtained from AG theory and $T_{bath}=0$,
thus supporting our assumption of electron-phonon coupling as the main cooling
mechanism in mesoscopic islands.

To further scrutinize heat dissipation in the islands, we have studied a
second device, whose geometry incorporates two main modifications. First, for
this device B, we employ normal metal leads, which simplifies our analysis of
the superconductor-to-normal transition of the island as the excess current
terms in Eq. (3) disappear. The device is fabricated by etching the Al shell
at the ends of the nanowire and evaporating the Cr/Au leads, thus defining a
$L\approx 1.4$ $\mu$m-long island (see schematics in the inset of Fig. 3b).
Second, the device is fabricated with a thinner InAs-Al nanowire which enters
the destructive regime of the Little-Parks effect. This is useful as the
differences between the quasiparticle diffusion and electron-phonon coupling
dependences for $V_{dip,I}(B)$ grow with shrinking $T_{c,I}$ as already
observed for device A. To maximize the excess current, measurements were taken
again in the open regime (where $V_{g}$ = 10 V is the voltage applied to the
global back gate, $G_{N}\approx 1.6\times 2e^{2}/h$, $I_{exs}\approx$ 25 nA).
As device B only contains one superconducting element, a single $dI/dV$ dip is
observed, consistent with the full suppression of $I_{exs}$ when the island
turns normal. Fig. 3 displays $V_{dip,I}(B)$ measurements taken at different
cryostat temperatures, $T_{bath}=$ 250, 600, 900 and 1200 mK. Note that here
$V_{dip,I}\approx 1.5$ mV and $P_{dip,I}\approx$ 150 pW at $B=0$ and
$T_{bath}=250$ mK. We fit the experimental data in Fig. 3a to
$V_{dip,I}(B)\propto T_{c,I}(B)$ (green dashed line), and to
$V_{dip,I}(B)=\beta\sqrt{T_{c,I}(B)^{6}-T_{bath}^{6}}$ (red dashed line),
where $\beta=\sqrt{2R\Sigma U}$ is a fitting parameter, $T_{bath}$ = 250 mK,
and $T_{c,I}(B)$ is obtained by AG theory. An excellent agreement is obtained
with the latter, from which we estimate $\Sigma\approx 4.3\times 10^{9}$
W/m3K6 by taking $R$ = 8.1 k$\Omega$, and $U\approx 4\times 10^{-21}$ m3 as
the volume of the Al shell in the island. The estimated $\Sigma$ is in good
agreement with previous works [51, 47]. In the other panels, we plot
$V_{dip,I}(B)$ calculated using $\beta$ extracted above, and the corresponding
cryostat temperature as $T_{bath}$ (dashed lines). The very good agreement for
all measurements strongly supports our interpretation of electron-phonon
coupling as the main cooling mechanism in islands at $T_{c}$. Note that slight
discrepancies in the measured and calculated $V_{dip,I}$ appear for higher
$T_{bath}$, most clearly seen for the measurement taken at 1.2 K. This could
be related, e.g., to the impact of other weaker cooling/heating mechanisms
neglected in our analysis, and/or to temperature-dependent electron-phonon
coupling parameters, e.g., $\Sigma$ and $n$ [49].

Figure 4:  $\mid$ Heating effects with high-frequency AC signals. a,
$dI/dV(V)$ of device A as a function of the nominal rms amplitude, $V_{rms}$,
of a $f=$ 3.492 GHz AC signal applied to a nearby antenna (see Methods for
more detail). b, Theoretical simulation of $dI/dV(V)$ as a function of the
amplitude of an AC voltage signal, $V_{AC}$, superimposed to the DC bias, $V$.
c, Impact of the AC signal frequency on the heating of the island. $dI/dV(V)$
as a function of $V_{rms}$ for $f=$ 1 MHz (left panel) and 3.492 GHz (right
panel). For the lower frequency, a splitting of the low-bias $dI/dV$ dip is
observed. d, $B$-field dependence of the microwave-induced superconductor-to-
normal transition of the island at zero DC voltage. To improve the visibility
of the transition, we plot the numerical derivative of the zero-bias
conductance, $G=dI/dV(V=0)$, with respect to $V_{rms}$, i.e., $dG/dV_{rms}$
(see main text and SM for more detail). The white dashed line represents a fit
to $\gamma^{-1}V_{dip,I}(B)$, where $V_{dip,I}(B)$ is extracted from Fig. 2b,
and the AC coupling factor $\gamma$ is $\approx$ 0.2.

Finally, we address heating effects stemming from the application of microwave
radiation. To this end, we have studied the response of device A in the
presence of high-frequency AC signals applied to an antenna located a few
millimeters away from the nanowire. By sweeping the frequency of the signal at
a given nominal power, we detect resonances at which a stronger effect on the
dips is observed (see Supplemental Material). In Fig. 4a, we plot the impact
of the AC signal amplitude, $V_{rms}$, at one of such resonances ($f=3.492$
GHz) on the superconductor-to-normal transitions of the islands and leads.
Interestingly, we observe that all three dips move to lower voltages with
increasing $V_{rms}$, signaling that a progressively lower DC Joule power is
required for turning the superconductors normal, due to heating from the AC
signal. To model the observed behavior we assume that the microwave is
absorbed by circuit elements converting it into an effective AC voltage
source, yielding a total bias, $V_{tot}=V+V_{AC}\sin(2\pi ft)$. For
frequencies much faster than the thermalization rate, $1/\tau_{th}$ a net
Joule power, nominally given by $V^{2}/R+V_{AC}^{2}/2R$ when neglecting
heating from excess current, is dissipated by the AC signal even though its
voltage averages to zero over time. We simulate this using Eqs. (2-4),
assuming cooling by quasiparticle diffusion in leads and by phonons in the
island with parameters from DC fittings, and using a smoothed step function to
simulate the onset of excess current, and lastly, by using the average power
of an AC cycle to self-consistently solve for the temperatures, $T_{L}$,
$T_{R}$, and $T_{I}$. The results of these calculations are shown in Fig. 4b.
The simulation captures very well the main qualitative features of the
experimental data, including a splitting of the enhanced conductance at low-
bias due to the onset of excess current, supporting that $f\gg 1/\tau_{th}$.In
Fig. 4c, we evaluate the effect of the AC signal frequency on the low-bias
dip. We notice a subtle yet important difference, namely the splitting of the
finite bias $dI/dV$ dip for increasing $V_{rms}$ in the lower frequency ($f=1$
MHz) measurement. Indeed, for frequencies slower or comparable to
thermalization, we would expect $T_{I}$ to oscillate with the AC signal as the
system seeks thermal equilibrium at each point of the AC cycle. In the limit
of $f\ll 1/\tau_{th}$ this would amount to a convolution of the DC result with
$F(V)=1/\text{Re}\left(\sqrt{V^{2}-V_{AC}^{2}}\right)$, resulting in a dip
splitting of $V=V_{dip,I}\pm\absolutevalue{V_{AC}}$ consistent with our
observations.

Assuming a harmonic solution to the heat-balance equation we obtain the
following expression for the thermal relaxation time [52, 53],

$\tau_{th}=\frac{6}{\pi}\frac{n\Sigma}{k_{B}^{2}\nu_{Al}}T^{n-2},$ (7)

with $n$ being the power of Eq. (6), fitted to $n=6$ in experiment, and
$\nu_{Al}=2.15\hskip 2.84544pt10^{47}J^{-1}m^{-3}$ the Fermi density of states
of aluminium. For $T=T_{c}=1.35$ K, appropriate for a dip, we find
$\tau_{th}\approx 10$ ns, consistent with our measurement of fast and slow
dynamics. For further details see the supplemental material.

For the measurement taken at higher $f$, a critical value of the AC signal
amplitude, $V_{rms}^{c}$, can be identified for which the low-bias dip
disappears after merging at $V=0$. This critical amplitude represents the full
suppression of the superconductivity of the island driven solely by AC
heating. To study this in more detail, we take measurements of the
differential conductance at zero-bias, $G=dI/dV(V=0)$, as a function of
$V_{rms}$ and $B$. At $V_{rms}^{c}$, Josephson and Andreev processes are
suppressed, leading to a drop in $G$. This critical point is better visualized
by plotting $dG/dV_{rms}(V_{rms})$, as shown in Fig. 4d, since this quantity
approaches zero following the transition of the island. Strikingly, we observe
that the $B$-field dependence of $V_{rms}^{c}$ is qualitatively identical to
that of the DC voltage dip. This is underscored by the fit of the experimental
data to $\gamma^{-1}V_{dip,I}(B)$, shown as dashed line, which is carried out
by employing $V_{dip,I}(B)$ from Fig. 2b and yields an AC coupling factor,
$\gamma\approx$ $0.2$, which would be $0.5$ for a perfect AC voltage source in
the $f\gg 1/\tau_{th}$ regime. We therefore demonstrate that high-frequency AC
signals give rise to Joule heating in a similar manner as in the DC case.

We now briefly comment on neglected heating and cooling terms in our analysis.
First, electron-phonon coupling is also present in the leads, but is estimated
to be $\sim$ 100 times less efficient than quasiparticle diffusion and is
therefore negligible. Also, the temperature gradients across the junctions,
given by $T_{L/R}-T_{I}$ contribute to the cooling of the island and heating
of the leads, as thermally activated quasiparticles cross the junction. This
contribution is insubstantial for the leads, but can contribute to a cooling
of about $10$% of the phonon cooling for the island at $T_{I}=T_{c,I}$. For
simplicity, we chose not to include this term, as its addition does not affect
the quality of fits, and can therefore not be separated in experiment from
phonon cooling, and would simply contribute to a slight rescaling of measured
$\Sigma$. For more details on this see supplemental material. Future
experiments will be directed at addressing these more subtle effects.

To conclude, we have performed a detailed study of heating effects in hybrid
superconductor-semiconductor nanodevices by Joule spectroscopy. Our
measurements show that the temperature of superconductors in a hybrid device
can increase significantly with the Joule power deposited from DC electrical
currents and/or microwave signals. Importantly, the temperatures of grounded
and floating superconductors can differ greatly, being determined by their
corresponding heat bottleneck, namely quasiparticle diffusion and electron-
phonon coupling. We note that, as a direct consequence, mesoscopic
superconducting islands display a much stronger susceptibility towards
heating. Altogether, our observations suggest that non-equilibrium conditions
and/or the presence of AC signals (including noise) can substantially impact
the response of hybrid superconductor-semiconductor devices, especially in
geometries with floating superconductors, by leading to increased local
temperatures that can greatly exceed $T_{bath}$. While these effects have been
rather overlooked in relation to hybrid devices, they can have implications
for e.g., in geometries proposed for the realization of topological
superconducting phases and related qubits [54, 55, 12, 13, 14, 15]. We note
that while our experiment focuses on extreme conditions at which the
superconducting elements of a device transition to the normal state, heating
effects are in fact even more pronounced for lower (AC/DC) applied powers as
both quasiparticle diffusion and electron-phonon coupling become less
efficient with decreasing electron temperature.

When finalizing the preparation of this manuscript, we became aware of this
related work [56].

Methods

Sample fabrication and measured samples: The devices discussed in the main
text are based on full-shell InAs-Al nanowires and incorporate a mesoscopic
superconducting island. By fitting the Little-Parks oscillations of the
devices (see SM for more details), we estimate the inner diameter of the core
InAs nanowire to be $\approx$ 150 nm ($\approx$ 110 nm) for device A (B), and
the thickness of the Al shell to be $\approx$ 8 nm.

The nanowires were deterministically transferred using a micro-manipulator
from the growth chip to silicon substrates for further fabrication.
Specifically, intrinsic (degenerately-doped) Si substrates with 300 nm-thick
SiO2 dielectrics were employed for device A (B).

Standard e-beam lithography (EBL) was used to define masks for wet etching
segments of the epitaxial shell. Oxygen plasma descumming at 100 W for 75 s
was carried out before immersing the samples in AZ326 MIF developer
(containing 2.38% tetramethylammonium hydroxide, TMAH) for 75 s at room
temperature. For device A, two segments of $\approx$ 200 nm of the Al shell
were etched, defining the superconducting island and two superconducting
leads. By contrast, for device B, we removed completely the Al shell at the
ends of the nanowire to form the island, and to allow the fabrication of
normal metal leads at a subsequent process. Electrical contacts and side gates
were fabricated by standard EBL techniques, followed by metallization by
e-beam evaporation at pressures of $\sim 10^{-8}$ mbar. Ion milling is carried
out prior to the evaporation to remove the native oxide of the Al shell (InAs
nanowire) for device A (B). The evaporated electrical contacts consisted of
superconducting Ti (2.5 nm)/Al (240 nm) for device A, and normal Cr (2.5
nm)/Au (180 nm) for device B. Note that, for device B, the Cr/Au leads are
fabricated $\approx$ 200 nm away from the superconducting island, and that the
doped Si substrate was used as a global back gate.

The main features discussed in this work have been observed in at least 4
devices similar to device A, and 2 similar to device B.

Measurements: Our experiments were carried out in two different cryogenic
systems: a dilution refrigerator with a base temperature of 20 mK, which was
employed for measuring device A, and a 3He insert with a base temperature of
250 mK, employed in the measurement of device B. A description of the
cryogenic filtering in these setups can be found in the Supplemental Material.
The cryostat temperatures were measured by ruthenium oxide thermometers
attached to the 3He pot and the mixing chamber of the above systems.

All data related to device A was obtained by performing two-terminal voltage-
bias transport measurements using standard lock-in techniques, whereby a
voltage, $V$, is applied with a small low-frequency ($f=117$ Hz) AC
excitation, $dV$, and the current, $I$, and differential conductance, $dI/dV$
are measured. We have employed different amplitudes of the lock-in excitation
depending on whether the measurement targeted resolving the low-bias dip only
($dV=$ 5 $\mu$V), or also the high-bias dips ($dV=$ 100-200 $\mu$V), as the
contrast of the latter is much weaker. (Note: the $dV$ values listed above are
the nominal ones, i.e., before subtracting the voltage drop on the cryogenic
filters, as we explain below).

For device B, we have experimented with both two-terminal voltage-bias and
current-bias measurements (see SM for further details). We have observed that
the experimental data obtained with these two different schemes are completely
equivalent, e.g., when comparing measurements plotted: (i) as a function of
the applied voltage-bias or of the voltage drop across the device as a result
of an applied current-bias, or (ii) as a function of the current resulting
from an applied voltage-bias or of the applied current-bias. In the main text,
we have opted to display the two-terminal current-bias measurements, owing to
a reduced electrical noise in this configuration for our 3He insert setup,
which translated into sharper $dI/dV$ dips. For this reason, $V$ in Fig. 3
refers to the total voltage drop across device B resulting from an applied
current-bias, $I$. These measurements were taken with a lock-in excitation
equal to $dI=1$ nA at $f=117$ Hz.

Microwave radiation from a signal generator (Rohde & Schwarz SMW200A) was
applied to an on-chip antenna located a few millimeters from device A. The
coupling between the devices and the impedance of the microwave antenna is not
precisely characterized. Therefore, the given $V_{rms}$ values in the main
text are nominal values obtained as $V_{rms}\propto 10^{P/20}\sqrt{P_{0}}$
with $P_{0}=1\text{mW}$ and $P$ the nominal power in dBm units.

Data processing. The voltage drop on the total series resistance of our
experimental setups, which is primarily due to cryogenic filters (2.5
k$\Omega$ per experimental line), has been subtracted for plotting the data
shown in Figs. 1-4.

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Acknowledgments

We acknowledge funding by EU through the European Research Council (ERC)
Starting Grant agreement 716559 (TOPOQDot), the FET-Open contract AndQC, by
the Danish National Research Foundation (DNRF 101), Innovation Fund Denmark,
the Carlsberg Foundation, and by the Spanish AEI through Grants No.
PID2020-117671GB-I00 and TED2021-130292B-C41, through the “María de Maeztu”
Programme for Units of Excellence in R&D (CEX2018-000805-M) and the ”Ramón y
Cajal” programme grant RYC-2015-17973.

Author contributions

A.I. fabricated the devices. A.I., I.C., M.G., and E.J.H.L. performed the
measurements. G.O.S. and A.L.Y. developed the theory. G.O.S. performed the
theoretical calculations. T.K. and J.N. developed the nanowires. All authors
discussed the results. A.I., G.O.S., I.C., M.G., A.L.Y., and E.J.H.L, analyzed
the experimental data and wrote the manuscript with input from all authors.
E.J.H.L. proposed and guided the experiment.