# Compression theory for inhomogeneous systems

Doruk Efe Gökmen  Institute for Theoretical Physics, ETH Zurich, 8093 Zurich,
Switzerland  Sounak Biswas  Institut für Theoretische Physik und Astrophysik,
Universität Würzburg, 97074 Würzburg, Germany  Sebastian D. Huber  Institute
for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland  Zohar Ringel
Hebrew University, Racah Institute of Physics, Jerusalem, 9190401, Israel
Felix Flicker  School of Physics and Astronomy, Cardiff University, Cardiff
CF24 3AA, United Kingdom  Maciej Koch-Janusz  Department of Physics,
University of Zurich, 8057 Zurich, Switzerland  James Franck Institute, The
University of Chicago, Chicago, Illinois 60637, USA

###### Abstract

The physics of complex systems stands to greatly benefit from the qualitative
changes in data availability and advances in data-driven computational
methods. Many of these systems can be represented by interacting degrees of
freedom on inhomogeneous graphs. However, the irregularity of the graph
structure and the vastness of configurational spaces present a fundamental
challenge to theoretical tools, such as the renormalization group, which were
so successful in characterizing the universal physical behaviour in critical
phenomena. Here we show that compression theory allows to extract relevant
degrees of freedom in arbitrary geometries, and develop efficient numerical
tools to build an effective theory from data. We demonstrate our method by
applying it to a strongly interacting system on an Ammann-Beenker
quasicrystal, where it discovers an exotic critical point with broken
conformal symmetry.

## Introduction –

Dramatic improvements in data availability, stemming from both experiments and
simulation, enable the exploration of increasingly complex physical systems. A
glut of raw data does not, however, equate understanding, particularly when
its processing easily exceeds our computational resources. A key objective is
to distill data into a succinct theory in terms of appropriate collective
variables that uncovers and summarizes the essence of the system.
Renormalization Group (RG) approaches in statistical physics provide a
systematic path towards that goal Wilson (1971); Kadanoff (1966). However,
both identifying the relevant degrees of freedom (DOFs), as well as executing
the mathematical procedure deriving the effective theory Ma  _et al._ (1979);
Iglói and Monthus (2005) is often extremely challenging in inhomogeneous
systems when prior intuition is scarce.

Many complex systems are, alas, inhomogeneous, or even owe their properties
precisely to the lack of translation invariance. Problems as disparate as
biological tissue mechanics Fletcher  _et al._ (2014); Alt  _et al._ (2017);
Farhadifar  _et al._ (2007) and properties of metallic glasses Bernal (1960);
Sheng  _et al._ can be cast as statistical mechanical problems on irregular
graphs. We would like to understand the emergent behaviour of such systems
much in the same fashion RG has allowed us to understand critical phenomena in
translationally invariant systems.

Here we tackle the two challenges of inhomogeneity and vast configurational
spaces of complex systems. Formulating the RG of an inhomogeneous system as a
lossy compression of information Tishby  _et al._ (2001); Gordon  _et al._
(2021) on a graph allows us to define the procedure in a manner that is
independent of geometry. We develop efficient numerical tools based on
contrastive learning to perform it on high-dimensional data Belghazi  _et al._
(2018); Poole  _et al._ (2019); van den Oord  _et al._ (2019).

Our algorithm explicitly constructs new effective DOFs $\mathcal{H}^{i}$ on a
local subgraph $\mathcal{V}^{i}$. This is achieved by a coarse graining
transformation $\mathbf{\Lambda}^{i}$ _locally_ chosen in region $i$ to
maximise the following mutual information Gökmen  _et al._ (2021a);
Lenggenhager  _et al._ (2020); Koch-Janusz and Ringel (2018)

$\arg\max
I(\mathcal{H}^{i}:\mathcal{E}^{i})=\mathbf{\Lambda}^{i}:\,\,\,\mathcal{V}^{i}\mapsto\mathcal{H}^{i},$
(1)

where the environment $\mathcal{E}^{i}$ is defined using the graph distance
(Fig. 1).

The variational principle in Eq. 1 provides a powerful substitute for
intuition: effective DOFs are individually designed by the statistics of their
_local_ environment. This is essential for moving beyond translation
invariance. While the coarse graining $\mathbf{\Lambda}^{i}$ erases
microscopic fluctuations, its local optimisation in turn allows to retain the
distinct qualitative characteristics that emerge in different spatial regions
across an inhomogeneous system. This can even be reflected in a non-uniform
cardinality of compressed variables $\mathcal{H}^{i}$, as illustrated in Fig.
1.

Figure 1: Constructing effective degrees of freedom in inhomogeneous systems.
Distinct systems like tissues (left, in green) and colloidal suspensions
(right, in blue) can be abstracted into a set of vector degrees of freedom
$\mathcal{V}^{i}$ (here shown by rectangles, and $i=1,2,3$) living on an
irregular graph with local structure. The final component of the vectors are
shown by coloured boxes to indicate different types of degrees of freedom,
unique to each system. To derive a compressed representation of such systems,
it is essential to tailor the coarse graining transformation
$\mathbf{\Lambda}^{i}$ for each local neighbourhood $i$. This is achieved by
an information theoretic variational principle, where
$\mathbf{\Lambda}^{i}:\mathcal{V}^{i}\mapsto\mathcal{H}^{i}$ maximises the
mutual information $I\left(\mathcal{H}^{i}:\mathcal{E}^{i}\right)$. This
allows the compressed variables $\mathcal{H}^{i}$ to capture the emergent
long-range physics according to the statistics of surrounding distant
environment $\mathcal{E}^{i}$. Local optimisation can produce compressed
variables with varying cardinality across the system, here illustrated by
vectors $\mathcal{H}^{i}$ of varying number of components. Finally, the
connectivity of the emergent supergraph is determined through the correlations
of the new variables.

We illustrate the power of this approach by constructively proving an open
conjecture regarding the presence of discrete scale invariance (DSI) Sornette
(1998); Young  _et al._ (2020); Sommers  _et al._ (2023) of a strongly
correlated statistical model on a quasicrystal Lloyd  _et al._ (2022). We
explicitly construct macroscopic variables self-similar to the microscopic
ones, and perfectly compatible with the hierarchical structure of the
quasiperiodic tiling. Our graph-based tool leverages the computational backend
of the RSMI-NE package Gökmen  _et al._ (2021a, b), allowing for efficient
exploration of the large-scale behavior of the model.

Figure 2: Self-similarity of the Ammann-Beenker tiling, and the coarse
graining blocks. (a) A microscopic dimer configuration (small black links) on
the AB tiling’s edges, with an overlaid AB _superlattice_ , self-similar to
the microscopic one. The effective DOF at a supervertex of a given (colour
coded) valence will be obtained by coarse graining the dimer configuration in
the surrounding region $\mathcal{V}$ of a shape dictated by the inflation
rules and shown as a polygon of a matching colour. (b) The inflation
(deflation) $\sigma^{2(-2)}$ of the elementary rhombi and squares generating
the tiling, with parts of the polygonal domains indicated in colour. Coarse
graining all such polygonal patches executes a deflation $\sigma^{-2}$ of the
original AB lattice, yielding the superlattice shown. Figure 3: Finding
effective clock variables. (a) Coarse graining transformation
$\mathbf{\Lambda}$ mapping Monte Carlo configurations on $\mathcal{V}$ into
bitstrings $\mathcal{H}$ on supervertices of $\sigma^{-2}$ deflated tiling.
(b, f) The length of the bitstring $\mathcal{H}^{8(3)}$ is determined by the
saturation of mutual information at 4 (2) bits at 8 (3)-supervertices. Each
bit $\mathcal{H}_{k}$ is decided by the sign of the linear transformation
$\Lambda_{k}\cdot\mathcal{V}$. The respective optimal filters
$\mathbf{\Lambda}$ in (c, g) carry a representation of the local spatial
symmetries of corresponding supervertices, namely $\mathsf{C}_{8}$ and mirror.
(d, h) The probability distributions $P(\mathcal{H}^{8(3)})$ occupy the space
of codes sparsely, and form abstract clock variables. (e) In particular,
$\mathcal{H}^{8}$ forms a closed 8-loop, where each state has exactly two
neighbours with Hamming-distance 1. (i) The representations of the local
symmetries on filters induce transitions between adjacent clock-states,
enabling to identify abstract clock-states with spatial directions along the
links of the quasiperiodic lattice (see main text).

## The system –

The Ammann-Beenker (AB) construction gives quasiperiodic tilings of the plane
utilizing two distinct plaquettes: a rhombus and a square Grunbaum and
Shephard (1986). Like their famous cousins, the Penrose tilings Penrose
(1974), AB tilings feature diffraction patterns exhibiting
crystallographically “forbidden” symmetries, here 8-fold Senechal (1996).
Likewise, they can also be generated by a recursive procedure in which an
_inflation_ map $\sigma$ acts on a small seed patch by decomposing the
constituent plaquettes as shown in Fig. 2b, and subsequently rescaling all the
edge lengths by the silver ratio $\delta=1+\sqrt{2}$. A special role is played
by 8-fold coordinated vertices: under inflations all lower coordinated
vertices ultimately become (and stay) 8-vertices. Each 8-vertex is
characterised by an _order_ , _i.e._ the maximal number of inverse
_deflations_ $\sigma^{-1}$ after which it still remains 8-fold coordinated.
The order of an 8-vertex intuitively specifies the maximal size of the local
patch centered on it, within which the lattice appears perfectly 8-fold
symmetric. The quasiperiodic AB lattice is thus invariant under discrete
rescalings. Such discrete rescalings are easily visualized for even order
deflations $\sigma^{2n}$ by drawing a super-lattice connecting 8-fold vertices
(Fig. 2a).

Dimer models enjoy a deceptively simple definition: the microscopic dimers
live on the links of (any) lattice, which can be either occupied or empty. The
key element is a set of hard local constraints: at every vertex where the
links meet, one and only one of the links is occupied. This gives rise to a
surprisingly rich phenomenology. Dimer models on regular lattices have been
studied extensively, in part due to their relevance to high-$T_{c}$
superconductivity Rokhsar and Kivelson (1988), but have since been shown to
support topological order and fractionalisation Moessner and Sondhi (2001);
Moessner  _et al._ (2001) and exotic critical points Senthil  _et al._ (2004).
The classical version is closely related to the quantum one Alet  _et al._
(2005, 2006) and has deep connections to combinatorics Kenyon and Okounkov
(2005); Kasteleyn (1961); Cohn  _et al._ (2001) and the study of random
surfaces Kenyon  _et al._ (2006); Kenyon and Okounkov (2007).

Recent work has begun to explore the interplay of (strongly-correlated) dimer
physics and quasiperiodicity. Particularly, AB tilings, in contrast to Penrose
tilings Flicker  _et al._ (2020), host perfectly matched dimer configurations
in the thermodynamic limit (_i.e._ with a vanishing density of defects), and
numerically computed dimer correlations exhibit a quasi power-law decay with a
complex spatial structure Lloyd  _et al._ (2022). Moreover, the combinatorial
proof of perfect matching pointed to a hierarchy of self-similar effective
matching problems at different scales.

Taken together these facts suggest a conjecture that not only the AB tilings
themselves, but crucially also the _physics_ of the dimers on the AB tilings,
exhibit DSI Lloyd  _et al._ (2022) – a potentially striking and unusual
example of the relevance of quasiperiodicity for the critical behaviour. A
proof, and a microscopic physical mechanism at the level of the dimer ensemble
was, however, absent. The putative criticality naturally calls for a RG
analysis, but general RG approaches for quasiperiodic systems in $D\geq 2$
dimensions are in their infancy.

## Results –

Figure 4: Emergent dimer exclusion rule and self-similar dimer-dimer
correlations across scales. (a) The probability distribution of microscopic
(_i.e._ $\delta^{0}$) dimers (in greyscale) on an AB patch, conditioned on one
of the links (in orange) hosting a dimer. (b, c) First two columns: the
probabilities $P(\mathcal{H}|\mathcal{H}^{3})$ of the _emergent_ clock
variables on the $\delta^{2}$ and $\delta^{4}$ superlattice (in greyscale),
conditioned on two distinct states of one of the 3-clocks (in orange). The
third column shows distributions conditioned on a state of the central
8-clock. Binding of adjacent clock variables into super-dimers obeying dimer
exclusion constraints is revealed by sharply peaked conditional distributions.
The effective super-dimers reproduce also longer-range dimer-dimer
correlations at both $\delta^{2}$ and $\delta^{4}$ scales. (d, e) Examples of
(a single component of) optimal coarse-graining filters producing the central
8-state clock variable at scales $\delta^{2}$ and $\delta^{4}$. The latter
comprises $2760$ microscopic links.

We address, in turn, two key questions using our compression approach based on
Eq. 1: what are the local effective DOFs, and what is the structure of their
correlations. This is systematically revealed by the analysis of data provided
by our algorithm.

To uncover the emergent DOFs, we need to specify the spatial partition for the
blocks $\mathcal{V}$ first. In the AB tiling there are natural choices, set by
the recursive structure of the AB lattice itself Jagannathan (2004). At each
scale, the AB tiling can be covered by four “classes” of blocks Lloyd  _et
al._ (2022), shown in Fig. 2 in different colours, each deflating to vertices
of differing connectivity in the super-lattice.

In each inequivalent class, the algorithm identifies the emergent DOF as a
$\mathbb{Z}_{n}$ _clock variable_ , with $n$ the connectivity, or class, of
$\mathcal{V}$ in the superlattice. This is revealed as follows: the
variational compression map $\boldsymbol{\Lambda}$ assigns to a Monte Carlo
dimer configuration $\mathcal{V}$ a short binary code $\mathcal{H}$ (Fig. 3a),
the bits being set by applying individual components $\Lambda_{k}$ to
$\mathcal{V}$ (itself a long bitstring of dimer occupations in the block).
Each component of the vector $\mathbf{\Lambda}$ is _a priori_ a general
nonlinear map, parametrized by a neural network, whose output is finally
binarized.

The length of the code is _not_ supplied, but found by sequentially increasing
the number of components in $\boldsymbol{\Lambda}$, and training the
compression of $\mathcal{V}$ to optimally preserve the mutual information with
its environment $\mathcal{E}$. Crucially, the maximal retained information
about $\mathcal{E}$ plateaus with the optimal code-length _depending on the
class_ of block region $\mathcal{V}$. Particularly, for $\mathcal{V}$ in
class-8 the optimal number of components is four, while for class-3 only two
(Fig. 3b,f). Further, nonlinearity of $\boldsymbol{\Lambda}$ networks does not
improve compression: the same amount of information is preserved with only
linear components. Optimal linear maps on the space of dimer configurations on
$V$ are shown for classes 8 and 3 in Figs.3c and g, respectively.

To unravel the physical content of these encodings, we further query the
outputs of our algorithm. The code statistics in Fig. 3d reveal striking
features: of the sixteen 4-bit codes in class-8 only eight are ever assigned,
with half of the codes unused. Yet in Fig. 3b a 3-bit encoding, which has
exactly eight available codes, is suboptimal. Moreover, the frequencies of all
class-8 codes used are the same (Fig. 3d), while for class-3 only two
frequencies are identical (Fig. 3h). These puzzling results indicate that the
optimal compression finds structure beyond merely the number of states of the
DOF, which is essential to correlations with $\mathcal{E}$, and which _cannot_
be encoded with fewer bits.

We thus investigate the codes, and the $\boldsymbol{\Lambda}$ maps. We first
note that the 4-bit codes form a closed 8-cycle, with neighbours differing by
a single bit-flip, and each code having exactly two 1-bit distant neighbours
(Fig. 3e) 111Interestingly, this solves the four dimensional ‘coil in the box’
problem familiar from coding theory.. The uniform frequencies and the cyclic
structure of the code hint at a symmetry.

Indeed, a class-8 patch $\mathcal{V}$ of the AB _lattice_ is locally symmetric
under $\pi/4$ rotations. We observe that under such rotations the components
of the optimal $\boldsymbol{\Lambda}$ map in Fig. 3c change as

$\mathsf{C}_{8}:\,\,\,(\Lambda_{1},\Lambda_{2},\Lambda_{3},\Lambda_{4})\rightarrow(\Lambda_{4},-\Lambda_{3},-\Lambda_{1},-\Lambda_{2}),$
(2)

which is a representation of a generator of the cyclic group $\mathsf{C}_{8}$.
We emphasize that it is the compression map, and consequently the emergent DOF
now carrying a representation of what is _a priori_ a (local) symmetry only of
the AB lattice.

Similar analysis can be performed for other classes of $\mathcal{V}$, which
have a mirror symmetry. In particular, under its action for the class-3 patch
in Fig. 3g we have

${\rm
mirror}:\,\,\,(\Lambda_{1},\Lambda_{2})\rightarrow(\Lambda_{2},\Lambda_{1}),$
(3)

explaining equal frequency of the 01 and 10 codes. Hence, we conclude that,
rather than becoming continuous, the emergent DOFs of the dimer system at
$\sigma^{2}$ scale remain discrete, and mimic the local symmetry of the
underlying super-lattice. This holds equally at $\sigma^{4}$ scale, providing
the first indication of DSI.

Having found the emergent DOFs in each class $\mathcal{V}$ individually, we
turn to their correlations, where DSI manifests itself fully. To this end we
simultaneously coarse grain dimer configurations in multiple blocks, which
collectively form an AB superlattice as in Fig. 2a, using the trained
compression maps (Fig. 3c,g).

As noted before, the number of states of each emergent DOF equals the
connectivity of the supervertex it lives on. Since the distribution of each
state’s frequencies reflects the underlying superlattice symmetry, these
internal DOFs can be identified with spatial orientations along the edges of
the superlattice. For example, since mirror symmetry w.r.t. the axis
connecting the 8- and 3-vertices in Fig. 3i relates the code frequencies of
the 3-vertex codes 01 and 10 (Fig. 3h), the remaining state 11 is the one
pointing towards the 8-vertex.

We probe the correlations by conditioning on the state of one of the vertices.
In Fig. 4b,c, fragments of $\sigma^{-2}$ and $\sigma^{-4}$ superlattices are
shown, with the state of the conditioning variable, identified with a
direction, in orange, while the conditional distribution of DOFs at the other
vertices in greyscale. Remarkably, this distribution is very strongly
correlated, effectively forcing occupation of some states, and excluding
others. To wit, when the 3-vertex DOF points towards the 8-vertex, the
distribution $P(\mathcal{H}|\mathcal{H}^{3})$ of the latter is sharply peaked
in the matching direction, while _no other neighbour_ of the 3-vertex points
towards it (allowing, for example, the identification of the 8-vertex code
1011 with a spatial orientation in Fig. 3i). Conversely, when the 3-vertex DOF
points towards one of its other neighbours, it is “matched” by the latter,
while the 8-vertex DOF distribution has zero weight _precisely and only_ in
the direction towards that 3-vertex.

Examining all such correlations we arrive at a striking conclusion: the
effective DOFs in $\mathcal{V}$’s throughout the lattice are paired with _one
and only one_ of their neighbours into emergent “super-dimers” on the edges of
the superlattice. The exclusion of certain clock variable orientations in
Figs. 3a-e is a precise reflection of the hard dimer-constraints, which these
super-dimers obey. Moreover, comparison of further correlations to those of
the microscopic dimers in Fig. 4a reveals that not just the local-dimer
constraints, but also longer-range correlations are reproduced correctly. The
physics of the microscopic dimer model on the AB lattice is thus replicated to
a high degree of accuracy at the $\delta^{2}$ scale and, again, at the
$\delta^{4}$ scale (where “locking” is even sharper, see Fig. 4c), thereby
demonstrating DSI across three scales.

The quasiperiodicity of the AB lattice and the strong interactions of the
dimer model conspire to _re_ create self-similar DOFs at a higher scale,
giving rise to DSI, which we uncover guided by the outputs of our algorithm A
parallel work Biswas and Parameswaran (2023) gives a microscopic
interpretation of these emergent super-dimers as alternating dimer paths with
respect to a certain reference configuration, and studies the criticality
numerically.

We emphasize the dual computational and conceptual aspect of this result: each
compression map $\boldsymbol{\Lambda}$ at the $\sigma^{4}$ scale is a highly
structured function of approximately $10^{3}$ microscopic dimer occupations
($\sim 2^{10^{3}}$ configurations), effectively impossible to guess or analyze
by hand only, and yet providing sharp and concise physical insights about
DOFs, symmetries, and correlations. We have, in effect, reached a point where
ML techniques can not only assist, but facilitate progress in theoretical
physics.

Our results demonstrate that using lossy compression effective DOFs and their
theory can be extracted from the structure of information inherent in raw
high-dimensional data, and that this approach excels in systems with irregular
geometry. These characteristics are essential in e.g. complex biological
sytems and in disordered materials.

## Acknowledgements –

D.E.G., and S.D.H. gratefully acknowledge financial support from the Swiss
National Science Foundation. S.B. acknowledges support by the European
Research Council under the European Union Horizon 2020 Research and Innovation
Programme via Grant Agreement No. 804213-TMCS. Z.R. acknowledges support from
ISF grant 2250/19. M.K.-J. gratefully acknowledges financial support from the
European Union’s Horizon 2020 programme under Marie Sklodowska-Curie Grant
Agreement No. 896004 (COMPLEX ML).

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  * Gökmen  _et al._ (2021-2023) Doruk Efe Gökmen, Zohar Ringel, Sebastian D. Huber, and Maciej Koch-Janusz, “RSMI-NE/RSMI-NE,”  (2021-2023).

Supplemental materials

##  Appendix A Methods

## Real-space mutual information based coarse-graining –

The method used to construct the effective degrees of freedom (DOFs) is a
generalisation of the compression theoretic approach first introduced by some
of the authors in Koch-Janusz and Ringel (2018) in translation invariant
systems. The key insight is that the random variables, such as the coarse-
grained area $\mathcal{V}$ and the environment $\mathcal{E}$, in variational
principle in Eq. 1 can be defined in any metric space, and are not restricted
to regular lattices. Here we use the graph distance which provides a
meaningful measure for spatial length scale on graphs with a local structure.

The second issue that the above construction solves is that the coarse-grained
lattice structure is defined by the correlations of the emergent DOFs
themselves. Dealing with the changing coarse-grained lattice is the main
problem in RG approaches on disordered systems Iglói and Monthus (2005). Taken
together, these two properties indicate that the approach defined by Eq. 1
provides a mathematical definition of coarse graining valid in inhomogeneous
systems with local structure.

In concrete terms, the procedure is defined as follows. Consider a system of
microscopic DOFs is described by a (large dimensional) random variable
$\mathcal{X}$ distributed according to some joint probability distribution
$P\left(\mathcal{X}\right)$. A coarse-graining (CG) of a partition
$\mathcal{X}=\bigcup_{i}\mathcal{V}^{i}$ into new variables
$\mathcal{X}^{\prime}=\bigcup_{i}\mathcal{H}^{i}$ is then defined as a
conditional probability distribution
$P(\mathcal{X}^{\prime}|\mathcal{X})=\prod_{i}P_{\mathbf{\Lambda}^{i}}(\mathcal{H}^{i}|\mathcal{V}^{i})$
where $\mathcal{X}^{\prime}=\bigcup_{i}\mathcal{H}^{i}$ and where the product
is over individual CG of a blocks variables
$\mathcal{V}^{i}\mapsto\mathcal{H}^{i}$. We emphasise there is a distinction
between the spatial patch $V$, and a configurations of DOFs supported on it
this patch (a random variable) $\mathcal{V}$, which we glossed over in the
main text. The correct meaning is implied by the context. The patch $V$ can be
chosen as any local subgraph, for instance a topological ball defined using
graph distance, or another set dictated by the structure of the problem, such
as the tiles we used in the AB example (Fig. 2a).

Maximisation of the real-space mutual information (RSMI)

$I_{\mathbf{\Lambda}^{i}}(\mathcal{H}^{i}:\mathcal{E}^{i})=\mathbb{E}_{(\mathcal{H}^{i},\mathcal{E}^{i})}\left[\log
P(\mathcal{H}^{i},\mathcal{E}^{i})-\log
P(\mathcal{H}^{i})P(\mathcal{E}^{i})\right]$

between $\mathcal{H}^{i}$ and its distant environment $\mathcal{E}^{i}$
provides a variational principle for the CG map $\mathbf{\Lambda}^{i}$ to
distill the most relevant long-range features Lenggenhager  _et al._ (2020);
Gordon  _et al._ (2021). The compression is enforced by limiting the
information capacity of $\mathcal{H}^{i}$ using a predetermined number of
encoding bits, thereby providing an approximation to the information
bottleneck problem Gordon  _et al._ (2021).

This computationally difficult variational principle _can_ nevertheless be
efficiently implemented with differentiable parametric lower bounds on mutual
information. Such bounds are parametrised by a deep NN, and optimised
simultaneously with the parameters $\mathbf{\Lambda}^{i}$ of the coarse-
graining using stochastic gradient descent. This is the RSMI-NE algorithm,
which some of the authors introduced recently Gökmen  _et al._ (2021a, b).

Here we extended this formalism and the RSMI-NE package to systems on
arbitrary static graphs by casting the configurations into vectors according
to the fixed coordinate system defined by the graph. The new graph-enabled
RSMI-NE code using the NetworkX backend developers (2014-2023) is available
publicly Gökmen  _et al._ (2021-2023).

## Ansatz –

We parametrised the coarse graining
$P_{\mathbf{\Lambda}^{i}}(\mathcal{H}^{i}|\mathcal{V}^{i})$ with an inner-
product ansatz $\mathcal{H}^{i}_{k}:={\rm
sign}\left(\mathbf{\Lambda}^{i}_{kj}\mathcal{V}^{i}_{j}\right)$, parametrised
by a vector of linear NNs $\mathbf{\Lambda}^{i}=(\Lambda_{k}^{i})$. The index
$k$ runs over the components of a coarse-grained variable, and the index $j$
refers to the spatial positions in region $\mathcal{V}$, where the indexing is
defined with respect to a fixed labelling of vertices in the graph. Though we
considered scalar DOFs in the original system, vector DOFs
$\mathcal{V}^{i}_{jl}$ can also be handled by increasing the rank of the
coarse grainer by one: $\Lambda^{i}_{kjl}$. Our code supports also more
general mappings that do not have this multilinear structrure, and are instead
parametrised by general NNs.

For mapping binary variables we used annealed Gumbel-softmax reparametrisation
with exponential relaxation rate $5\times 10^{-3}$. The critic function in the
variational RSMI lower-bound is implemented using a separable architecture

$f(\mathcal{H},\mathcal{E})=u(\mathcal{H})^{\rm T}v(\mathcal{E})$

where $u$ and $v$ are two-layer deep NNs with hidden dimension 16 and output
dimension 8 (the latter is contracted in the inner product of the two
networks).

We trained the neural networks using stochastic gradient descent with learning
rate $10^{-3}$ using 50000 sample dimer configurations, generated via the
directed-loop Monte Carlo algorithm on the AB graph indicated with the gray
links in Fig. S2 (containing 26177 nodes). The sample dataset is supplied to
the RSMI-NE algorithm in minibatches of size 1000 and 120 epochs of the entire
dataset.

The coarse-grained block variable $\mathcal{V}$ at a given scale $\delta^{n}$
is defined on the $\sigma^{n}$ inflated tiles $V$ shown with different colours
in Fig. 2a. The environment regions $E$, are defined as a shell with radius
given by a fixed graph-distance from the centre of $V$. In particular for
$\delta^{2}$, $E$ is defined by an inner radius $L_{E_{\rm i}}=9$ and outer
radius $L_{E_{\rm o}}=24$, whereas for $\delta^{4}$ we used $L_{E_{\rm
i}}=40$, $L_{E_{\rm o}}=64$, as shown in Fig. S1. In Fig. S2 we show examples
of corresponding $\sigma^{-4}$ coarse-graining filters optimised for these
regions.

Figure S1: Block and environment regions. Highlighted in blue are examples of
the coarse-graining blocks $V$, and their annular environment regions $E$ used
for the 3- (a) and 8-vertices (b) at the largest scale considered (_i.e._
$\delta^{4}$). The microscopic lattice, and the $\sigma^{-2}$ superlattice are
shown. The centers of the “kite” and “star” shaped regions $V$ are at the
8-vertices whose positions form the $\sigma^{-4}$ superlattice. Figure S2:
Optimal $\sigma^{-4}$ coarse-graining transformations. (a) 8-supervertex
filters. (b) 3-supervertex filters.

##  Appendix B The odd scales

Our analysis of the coarse graining transformations of the dimer model on the
AB tiling _did not_ find evidence for a discrete scale invariant description
in terms of super-dimer variables under all rescalings, but only for _even_
order ones (_i.e._ under deflations $\sigma^{-2k}$). This is in contrast to
the AB tiling itself (_i.e._ just the AB lattice), which is invariant under
any order of deflation.

In addition, our method finds quantitatively and qualitatively distinct
behaviour at odd orders $\sigma^{-1}$ and $\sigma^{-3}$. The maximal mutual
information $I_{\Lambda}(\mathcal{H}:\mathcal{E})$ attained for the coarse
graining at a 3-supervertex is non-monotonic, exhibiting, within error, two
distinct values characterizing the even and odd scales (with the odd scales’
information reduced to almost half), as shown in Fig. S3.

Furthermore, the optimised coarse graining does not yield a well-defined
three-state clock variable at odd scales. Indeed, for even scales the
optimisation robustly yields a well-defined set of three clusters
(corresponding to the three clock states) even in the distribution of pre-
activations $\mathbf{\Lambda} \cdot\mathcal{V}$, while at odd scales the
distribution of pre-activations lacks any such clear structure. We emphasize
that this is _not_ an optimization issue: computationally, $\sigma^{-4}$
coarse graining is a more challenging problem than $\sigma^{-3}$ (because of
much large size of random variables involved).

Figure S3: Mutual information across different scale transformations. The
maximal MI for the coarse graining at a 3-supervertex. For odd order rescaling
transformations $\sigma^{-1,-3}$, the information attained by the compression
is systematically lower compared to the even ones $\sigma^{-2,-4}$.


# Reflectance and fluorescence characteristics of PTFE coated with TPB at
visible, UV, and VUV as a function of thickness

J. Haefner,aafootnotetext: Corresponding author. A. Fahs  J. Ho  C. Stanford
R. Guenette  C. Adams  H. Almazán  V. Álvarez  B. Aparicio  A.I. Aranburu  L.
Arazi  I.J. Arnquist  F. Auria-Luna  S. Ayet  C.D.R. Azevedo  K. Bailey  F.
Ballester  J.M. Benlloch-Rodríguez  F.I.G.M. Borges  S. Bounasser  N. Byrnes
S. Cárcel  J.V. Carrión  S. Cebrián  E. Church  C.A.N. Conde  T. Contreras
F.P. Cossío  A.A. Denisenko  E. Dey  G. Díaz  T. Dickel  J. Escada  R. Esteve
R. Felkai  L.M.P. Fernandes  P. Ferrario  A.L. Ferreira  F.W. Foss  E.D.C.
Freitas  Z. Freixa  J. Generowicz  A. Goldschmidt  J.J. Gómez-
Cadenasbbfootnotetext:  NEXT Spokesperson.  R. González  J. Grocott  K. Hafidi
J. Hauptman  C.A.O. Henriques  J.A. Hernando Morata  P. Herrero-Gómez  V.
Herrero  P. Ho  Y. Ifergan  B.J.P. Jones  M. Kekic  L. Labarga  L. Larizgoitia
P. Lebrun  D. Lopez Gutierrez  N. López-March  R. Madigan  R.D.P. Mano  J.
Martín-Albo  G. Martínez-Lema  M. Martínez-Vara  A.P. Marques  Z.E. Meziani
R. Miller  K. Mistry  J. Molina-Canteras  F. Monrabal  C.M.B. Monteiro  F.J.
Mora  J. Muñoz Vidal  K. Navarro  P. Novella  A. Nuñez  D.R. Nygren  E. Oblak
M. Odriozola-Gimeno  B. Palmeiro  A. Para  M. Querol  A.B. Redwine  J. Renner
I. Rivilla  J. Rodríguez  C. Rogero  L. Rogers  B. Romeo  C. Romo-Luque  F.P.
Santos  J.M.F. dos Santos  A. Simón  M. Sorel  J.M.R. Teixeira  J.F. Toledo
J. Torrent  A. Usón  J.F.C.A. Veloso  T.T. Vuong  J. Waiton  J.T.
Whiteccfootnotetext: Deceased.

###### Abstract

Polytetrafluoroethylene (PTFE) is an excellent diffuse reflector widely used
in light collection systems for particle physics experiments. In noble element
systems, it is often coated with tetraphenyl butadiene (TPB) to allow
detection of vacuum ultraviolet scintillation light. In this work this
dependence is investigated for PTFE coated with TPB in air for light of
wavelengths of 200 nm, 260 nm, and 450 nm. The results show that TPB-coated
PTFE has a reflectance of approximately 92% for thicknesses ranging from 5 mm
to 10 mm at 450 nm, with negligible variation as a function of thickness
within this range. A cross-check of these results using an argon chamber
supports the conclusion that the change in thickness from 5 mm to 10 mm does
not affect significantly the light response at 128 nm. Our results indicate
that pieces of TPB-coated PTFE thinner than the typical 10 mm can be used in
particle physics detectors without compromising the light signal.

##  1 Introduction

Polytetrafluoroethylene (PTFE), often referred to by the brand name Teflon, is
a plastic with excellent diffuse reflectance and low cost. These properties
make it commonly used in particle physics experiments (e.g., [1, 2, 3]) for
light collection. The NEXT experiment uses PTFE as a diffuse reflector around
the drift volume of its high pressure gaseous xenon time projection chamber
(TPC) to increase light collection [1], in an attempt to search for neutrino-
less double beta decay. The measurements described here have informed the
design decisions for the upcoming phases of NEXT, including NEXT-100 [4]
(where the thickness of PTFE has been halved relative to previous detectors)
and a future tonne-scale experiment [5].

While PTFE is excellent for light collection, its radiopurity can be a concern
for low-background experiments [6], especially at large scale, where a
significant quantity of material is needed. Moreover, PTFE can absorb gaseous
xenon [7] leading to a loss of active detector mass, and outgas contaminants
in the high purity detector medium. Minimizing the quantity of PTFE in such
experiments is therefore an attractive solution, provided that the light
collection remains high.

A significant number of particle physics experiments that use PTFE, especially
for rare event searches, utilize liquid or gaseous noble element detectors.
These detectors typically collect the scintillation light, produced when a
particle interacts in the detection medium. Noble elements scintillate
primarily in the vacuum ultraviolet (VUV) range. At these wavelengths, common
photosensors —such as silicon photomultipliers (SiPMs) and photomultiplier
tubes (PMTs)— generally have very low sensitivity and wavelength shifters are
often used to shift the VUV light to the visible range. Similarly to many
comparable experiments, the NEXT experiment uses tetraphenyl butadiene (TPB)
to shift the 175 nm light produced in xenon to the visible $\sim$420 nm
(blue). The light response properties of PTFE in these regimes, both at VUV
and in blue, are thus of great interest to many particle physics experiments.

A variety of studies have been carried out in the past to investigate
reflectance of PTFE, such as the reflectance of PTFE at 175 nm at a single 5
mm thickness [8], reflectance in liquid xenon for thicknesses from 1 mm to 9.5
mm [9], reflectance of PTFE thicknesses from 5 mm to 10 mm for visible and UV
light [13], and reflectance as a function of angle [10].

The light response of PTFE coated with a wavelength shifter such as TPB is
less explored, although there have been several studies on the fluorescence of
TPB itself [15, 14]. In the case of TPB-coated PTFE, fluorescence must also be
accounted for and pure reflectance no longer gives a reasonable description of
the overall light response behavior in the UV. In this work, the variation of
light response of TPB-coated PTFE is investigated for a variety of wavelengths
ranging from VUV to blue, as a function of PTFE thickness using four distinct
methods.

The first two methods involve measuring TPB-coated PTFE disks in commercial
spectrophotometers. These are devices which are commonly used for materials
characterization in which light of one or several wavelengths is incident upon
a given sample to measure a variety of optical properties, such as
fluorescence or reflectance. In Section 3, reflectance spectra of TPB-coated
and uncoated PTFE disks of different thicknesses are measured using a
universal measurement spectrophotometer and in Section 4, fluorescence spectra
of the same disks are measured using a fluorescence spectrometer.

The remaining two methods involve enclosed boxes made of PTFE of different
thicknesses, where the reflectance is studied by measuring the light intensity
as a function of the distance of our light source and detector plane from the
end of the box. These measurements are made in two ways where Section 5
describes measurements in air with an LED light sources at 260 nm and 450 nm,
and Section 6 describes measurements in an argon gas chamber measuring
scintillation light at 128 nm from alpha decay events from a 210Po source.

##  2 Sample Preparation

In order to investigate the reflectance of PTFE, several
$12~{}\mathrm{in}\times 24~{}\mathrm{in}$ sheets of different thicknesses were
purchased. All sheets for the primary measurements were sourced from Altec
Plastics.111https://altecplastics.com/ The different thicknesses are
$\frac{3}{8}$ in = 9.5 mm, $\frac{5}{16}$ in = 7.9 mm, $\frac{1}{4}$ in = 6.4
mm, and $\frac{3}{16}$ in = 4.8 mm, referred to as 10 mm, 8 mm, 6 mm, and 5 mm
respectively. Additional sheets were procured from Industrial
Plastics222https://www.industrialplasticsinc.com/ (thicknesses of 9.8 mm, 7.7
mm, 7.2 mm and 5 mm ) and ePlastics333https://www.eplastics.com/ (thicknesses
of 9.5 mm, 6.5 mm, 5 mm, and 3.4 mm) in order to assess variation between
different sheets and manufacturers. This information is summarised in Table 1.
All the parts used for the studies, such as the disks in Sections 3 and 4 as
well as the boxes in 5 and 6 were cut from these same sheets.

Table 1: Summary of the various PTFE sheet thicknesses (in mm) studied by manufacturer. Altec Plastics | Industrial Plastics | ePlastics
---|---|---
9.5 | 9.8 | 9.5
7.9 | 7.7 | 6.5
6.4 | 7.2 | 5
4.8 | 5 | 3.4

All the pieces cut from the PTFE sheets were treated and cleaned in one day in
the same way based on methods commonly used in particle physics experiments
[9]. One side of each piece was sanded by hand with 300-, then 1000-, and
finally 2000-grit sandpaper to make the surface uniform and matte. Each piece
was then cleaned in an ultrasonic bath in a solution of deionized (DI) water
and Alconox for 15 minutes, followed with an ultrasonic bath run in pure DI
water for 15 minutes and the pieces were air-dried overnight. Following an
initial run of measurements described in the next sections of all the bare
(uncoated) PTFE pieces, the pieces were evaporation-coated with a layer of
TPB. The thickness of this layer was measured by a crystal monitor, but since
the monitor can only measure at one point, it does not necessarily represent a
uniform layer. Across the different evaporations, the thickness values read by
the crystal monitor ranged from $25.3~{}\mu$m to $26.3~{}\mu$m. In order to
check for the level of non-uniformities in the evaporator, a coating test was
performed with glass slides at a variety of positions in the same plane in the
evaporator, where the weights of the slides were measured before and after
evaporation to the precision level of $0.1$ milligrams. In the area of the
evaporator where the PTFE pieces are coated for the studies, a maximum
difference of $23.2~{}\mu$m to $40.5~{}\mu$m ($42\%$ difference) was observed
in the coating thickness from the center of the area to the edge. To convert
from the mass measurement to thickness, a uniform layer of TPB was assumed, as
the surface is small enough ( 1 in). This could correspond to a variance in
reflectance of up to 15%-20%, depending on the wavelength [14]. In order to
reduce the level of non-uniformities observed with the test slides, the PTFE
pieces were positioned in a way to be perpendicular to the TPB crucible, as
opposed to in the plane parallel to the top of the evaporator as was done with
the test slides. All the pieces of PTFE were placed equidistant from the TPB
crucible, meaning that the coating on each piece should be identical. On the
disks, due to their small diameter and small area investigated, non-
uniformities are not expected to have a significant impact on the reported
reflectances. For the box method, concerns of non-uniformity are addressed by
the systematic errors. All PTFE pieces were coated in the same week to
minimize any potential relative degradation between the pieces. As UV light is
known to degrade TPB coatings [16], and our previous study also suggested a
possible degradation of light response for exposed uncoated PTFE [13], all
samples were stored in a cabinet wrapped in aluminium foil, protected from
light.

Figure 1 shows the different PTFE pieces used in the measurements presented
here. The disks, of 7.6 cm diameter, are used for measurements in the
spectrophotometers (Sections 3 and 4) and the box with inner dimensions
$7~{}\mathrm{cm}\times 7~{}\mathrm{cm}\times 25~{}\mathrm{cm}$ are used for
reflectance measurements (Sections 5 and 6).

Figure 1: Left: 7.6 cm diameter PTFE disks used for measurements in the
spectrophotometers. Right: An example of a PTFE box used for reflectance
measurements. The boxes have inner dimensions of $7~{}\mathrm{cm}\times
7~{}\mathrm{cm}\times 25~{}\mathrm{cm}$.

##  3 Universal measurement spectrophotometer

The first method for determining reflectance makes use of Cary 7000 Universal
Measurement Spectrophotometer (UMS) coupled to an Agilent diffuse reflectance
integrating
sphere,444https://www.agilent.com/cs/library/flyers/public/5991-1717EN_PromoFlyer_UV_DRA.pdf
shown in Figure 2. The UMS allows to measure the reflectance of a given sample
relative to some reference, as a function of wavelength. The measurements
presented here are taken relative to a Labsphere Certified Reflectance
Standard (a Spectralon® puck) with $99\%$ uniform reflectance in the 400 to
500 nm range, and $>95\%$ reflectance in the 250 to 400 nm range.

Figure 2: The Cary 7000 Spectrophotometer with the internal components of the
integrating sphere exposed. Figure 3: Example of reflectance measurements from
the Universal Measurement Spectrophotometer for different uncoated Altec PTFE
thicknesses (colored lines) relative to a certified reflectance Spectralon
reference, which corresponds to 100% reflectance on the vertical axis. The
purple and blue dashed lines represent the wavelengths that are used for
comparisons with other methods.

The UMS is used to measure the reflectance of the both the uncoated and TPB-
coated PTFE disks of varying thicknesses, from all the manufacturers. Figure 3
shows an example of the measurements of the reflectance of the different Altec
PTFE disk thicknesses relative to the Spectralon standard. Since this is a
relative measurement, reflectance values over 100% indicate the sample is more
reflective than the Spectralon standard. The error bars are estimated from
taking twenty measurement runs of each disk. As seen in Figure 3, the
differences between thicknesses only rise to the few percent level at most,
but the variations are dependant on wavelength. Significant variability was
observed in reflectance across disks of the same thickness, especially when
the disks were cut from different sheets.

###  3.1 Results from the universal measurement spectrophotometer

A summary of all the different reflectances measured with the UMS for the TPB-
coated PTFE disks is presented in Figure 4. The results are for 450 nm light
in order to directly compare with the second method, where an LED of this
wavelength is used (see Section 5, 7). No results for UV wavelength are
presented in this section, as the output signal of the UMS is interpreted as
being purely due to reflectance, and thus is not accurate for the disks coated
with TPB where there is significant fluorescence at that wavelength. For the
uncoated PTFE disks, there is no clear trend for reflectance in thickness, so
the results are consistent with previous work [13].

Figure 4: Reflectance measurements from the Universal Measurement
Spectrophotometer for different thicknesses of TPB-coated PTFE (thicknesses
are shown on the horizontal axis) at 450 nm. Measurements were taken relative
to the certified $99\%$ reflectance Spectralon reference, and then converted
to absolute reflectance. The results for different PTFE suppliers are shown.
The measurement values and uncertainties are described in Section 3.1.

Since the measurements with the TPB-coated disks could only be performed once
to avoid potential damage to the coating when inserting the disk in the
integrating sphere, the data points shown in Figure 4 come from a single
measurement. The error bars were estimated by repeating the measurement of a
single uncoated disk with corresponding thickness twenty times. The magnitude
of the error is not dependent on thickness, so this error is largely a
systematic uncertainty from the UMS.

At 450 nm, for TPB-coated PTFE, the differences in reflectance between
thicknesses measured are of the order of a $\sim$1–2%, indicating that the
reflectance of TPB-coated PTFE for blue light is not strongly dependent on the
thickness of the material. For all the TPB-coated PTFE samples, the
reflectance does not appear to monotonically increase with thickness, giving
an indication that the primary factor for the observed differences is
differences in sheet quality rather than variation with thickness. In
addition, the reflectance of the ePlastics samples is consistently lower,
indicating again that sheet quality would have a larger impact than thickness.
Little change is observed between coated and uncoated Altec PTFE relative to
our error bars, and this is as expected: the TPB should not significantly
impact reflectance at 450 nm. These results agree with previous work, where
the reflectance of uncoated PTFE was found not to depend on thickness in both
[9] and [13].

##  4 Fluorescence spectrophotometer

The second method for determining light response uses a Cary Eclipse
Fluorescence Spectrometer
(FS),555https://www.agilent.com/cs/library/brochures/5990-7788EN_Cary_Eclipse_Brochure.pdf
shown in Figure 5. The FS allows for the measurement of the fluorescence
spectrum of a material for a range of wavelength inputs.

Figure 5: Left: Picture of the Cary Eclipse Fluorescence Spectrophotometer.
Right: The inside of the Fluorescence Spectrophotomer, with labels for the key
components, where the sample is a disk of PTFE.

The fluorescence of the same TPB-coated disks described in section 3 are
measured using the FS. The error on these measurements are estimated by
repeating the measurements in the FS 13 times for each of the four disks.
Here, the FS is used to measure the wavelength at a fixed input wavelength of
200 nm. This is the lowest wavelength from the FS that still had a sufficient
signal output for the purposes of comparison. The spectra for the four
measured Altec Plastics disks for 200 nm light are shown in Figure 6. Longer
wavelengths (between 420 nm and 450 nm) were also studied in the FS, and it
was confirmed that no fluorescence of TPB is observed for the blue light.

###  4.1 Results from the fluorescence spectrophotometer

When excited by UV light, TPB is expected to produce a fluorescence peak at
around 430 nm [14]. The measured spectra from the TPB-coated Altec PTFE shown
in Figure 6 clearly exhibit the expected peak. The measurements indicate that
variations in fluorescence are not significant from 5-8 mm for the selected
samples of Altec PTFE coated with TPB, but that the 10 mm thick disk
fluoresces slightly more than the rest. The fluorescence of the 10 mm
thickness is at most 15%–20% higher. This difference is likely not due to PTFE
thickness, as the same result was not seen in the UMS measurements in Section
3, but potentially to a variation in thickness of the TPB coating as a
variation in WLSE of up to 14% can be seen in Ref. [14] when the variation in
TPB thickness is up to 0.4 $\mu$m. This difference in TPB coating thickness
was caused by a combination of factors: firstly, there is variation across
each individual disk, which led to a measured difference of up to 10% in
intensity, in addition to coating variation across different evaporator runs,
addressed in Section 2.

Figure 6: Fluorescence spectrum measurements from the Fluorescence
Spectrometer for different thicknesses of TPB-coated Altec PTFE (blue, orange,
green, and red lines) and for an uncoated 10 mm thick disk (shown in purple).
Uncertainties come from standard deviation of measurements taken 13 times for
each disk.

The peak at 530 nm has been observed as part of the TPB spectrum with light
excited by 200 nm before and is a result of the interaction between
diffraction effects in the FS and our sample, but not the TPB or PTFE [18].
This hypothesis is supported by the fact that the peak is also visible for an
uncoated PTFE disk, demonstrating that it is not caused by the TPB.
Furthermore, a sample of plain white paper also shows that peak, discarding
the origin being the PTFE itself. The only sample which did not produce such a
peak was the Spectralon reference, which is engineered to have no fluorescence
for any wavelength.

In conclusion, the FS measurements allow us to investigate exclusively the
visible light that has been shifted by the TPB, rather than all the different
wavelengths of light like in the UMS. These measurements have shown that the 5
mm, 6 mm and 8 mm disks of PTFE fluoresce near identical amounts of light,
while the 10 mm disk produces a larger signal. The increase in signal from the
10 mm disk could be caused by variations in TPB thickness, rather than the
thickness of the PTFE, either by evaporations, or by spot to spot variations
on each disk. Either way, this underlines the importance of the box
measurements in section 5.

##  5 PTFE boxes in air

One significant disadvantage of the spectrophotometer methods (both UMS and
FS) described in Sections 3 and 4 is that the small area of the piece that is
measured (approximately 0.2 cm2) makes the measurements highly sensitive to
potential variation across the piece or even across the sheet from which they
were cut. Furthermore, the measurements do not allow extension into the VUV as
the lowest wavelengths studied for the UMS is 260 nm and 200 nm for the FS. An
alternative method described in this section, referred to as the _box method_
, uses a hollow rectangular prism of PTFE with a total interior surface area
of 749 cm2, illustrated in Figure 7. This method allows to average potential
reflectance variations within the larger PTFE samples by extracting the
reflectance using detailed simulations of the setup. In Section 6, this method
is also used in an argon environment to extend the studies to VUV wavelength.

Figure 7: Schematic side-view of the box setup. The inside dimensions of the
box are
$7~{}\mathrm{cm}~{}\times~{}7~{}\mathrm{cm}~{}\times~{}25~{}\mathrm{cm}$. The
board holding the LED and SiPMs system is inserted through the open end of the
box with a motorised rod. The amount of light collected is measured as a
function of the distance of the board insertion to extract the reflectance of
the PTFE box.

###  5.1 Box method setup

The box method uses several rectangular boxes with inner dimensions of
$7~{}\mathrm{cm}\times 7~{}\mathrm{cm}\times 25~{}\mathrm{cm}$ made of PTFE
pieces of a given thickness, which were cut from the same Altec sheets used in
the disk measurements. The boxes are open at one end to allow the insertion of
a printed circuit board (PCB) with an LED and 4 silicon photomultipliers
(SiPMs) that are sensitive to blue and UV light. The board is mounted on a
motorized sliding rod such that the insertion distance can be changed quickly
and precisely between measurements. A cross section of the setup is
illustrated in Figure 7. Measurements of the light collected as a function of
the distance of insertion are used to extract the reflectance for the uncoated
case, whereas in the TPB-coated case, we measure the overall light signal
(coming from reflection and fluorescence of the TPB). This process is further
elaborated upon in Section 5.2. Other than minor improvements such as
motorizing the system, this setup is largely identical to the experimental
setup described in previous work [13]. In particular the dimensions of the
boxes, the LEDs and the SiPMs are the same. However, in this work, the
measurements are taken for insertion distances ranging from 0 cm to 16 cm at
intervals of 2.3 cm compared to the previous work’s 0 cm to 21 cm in 3 cm
intervals. This change was made due to the physical improvements, such as
motorizing the setup.

The systematic errors of these measurements were determined by measuring the
variance in signal for different orientations of the box, to average out
potential variations of the PTFE and/or TPB efficiencies for different parts.
The box was made of 5 pieces, 4 identical rectangular pieces and 1 square
shaped piece, with dimensions outlined above. When estimating the systematic
errors of each box (uncoated and TPB-coated), the box is rotated in the 4
possible orientations relative to the PCB board. After this, the back piece
was removed and placed on the other end, with the PCB now being inserted from
the other side. Measurements in the 4 different orientations were also
repeated for this reversed box configuration, summing to 8 total
configurations being measured to estimated systematics variations at each
insertion distance. Finally, in order to minimize temperature variations,
measurements were taken within a single week, in a lab controlled to have
temperature variations under $1^{\circ}$C.

###  5.2 Extraction of the reflectance value

In order to estimate the reflectances from the box measurements, the
measurements are compared to detailed simulations of the setup with different
reflectance values. The simulations are performed using Geant4 [12], where the
box geometry includes the PTFE with a diffuse reflectance component only. The
simulations are described in more details in previous work [13].

The simulations provide the total number of LED photons that reach the SiPMs
based on the simulated PTFE reflectances. The simulation results for the
different reflectances are fitted to the data where the reflectance value that
best fits that data set is extracted. In order to allow comparison of the data
taken with the SiPMs (in V$\cdot$ns) to the simulation (in photon counts), it
is assumed that there is a constant and uniform conversion factor between the
number of photons collected and the signal read out by the SiPMs, referred to
as the scale factor parameter ($\alpha$), which is left free to vary in the
fit. This $\alpha$ parameter incorporates effects such as the intensity of the
LED and the efficiency of the SiPMs and should not change between measurements
of different reflectances. Hence, the fit is performed simultaneously on all
the data sets for the different reflectances using the same $\alpha$
parameter.

In the case of TPB-coated boxes in UV light, fluorescence becomes significant
contributor to the overall light signal. To quantify this and to better
compare different boxes, the previously described simulation is fit to the
data in the same way, but the output metric becomes reflectance* instead of
pure reflectance. Reflectance* is a combination of contributions from the
reflected light and from the fluorescence light, and can be used to compare
the overall light intensity, which retains a significant dependence on the
PTFE reflectance.

###  5.3 Results of the box method

Measurements of reflectances using the boxes were taken for the previously
described thicknesses of 5 mm, 6 mm, 8 mm, 10 mm of Altec PTFE. Figure 8 shows
the results obtained with the 450 nm LED before the boxes were TPB-coated on
the left, and shows the results obtained with the 260 nm LED for TPB-coated
boxes on the right. This is done in an attempt to directly compare the impact
of the PTFE thickness only, as once the 200 nm light has interacted with the
TPB, the light reaching the PTFE is mostly at 430 nm and can be more directly
compared to the blue LED shining on bare PTFE. For the 450 nm LED case, the
fit to simulation provides a reflectance associated with each thickness. As
already discussed, a direct reflectance value cannot be easily extracted for
TPB-coated PTFE, as the fluorescence of the TPB adds to the reflectance and
here reflectance* is computed.

Figure 8: Top plot: Measured signal (points) as a function of insertion
distance for uncoated (left) and TPB-coated (right) PTFE boxes measured in
air. On the left the setup used a blue (450 nm) LED, while on the right a UV
(260 nm) LED is used. The lines represent the results of the simulations
described in Section 5.2 which the points are then best fit to and the values
in the legend provide the extracted reflectances(*) for each fit with the fit
errors. The vertical axis gives the amount of light, measured in V$\cdot$ns by
the SiPMs. The errors on the data points include statistical uncertainties of
the SiPM pulses, as well as systematic errors regarding the rotations of the
boxes. The errors on the simulations are statistical only [13]. Bottom: These
plots show the residuals between the data points and their corresponding
simulated points (simulation minus data) for each of the insertion distances.
Figure 9: Comparison of reflectance values for uncoated and TPB-coated PTFE of
different thicknesses measured with the box method for 450 nm and of
reflectances* for 200 nm. The values for the coated measurements come from
Figure 8. Note that the comparison between blue and UV light is only relative,
as the values extracted for UV cannot directly translate to reflectance
values.

In Figure 8, it can be seen that the simulations reproduce the data relatively
well. The results for the 450 nm LED are consistent with those previously
reported in [13], though the previous measurements were with a different PTFE
manufacturer (ePlastics). Differences in the MC fit reflectances of a few
percent (2% at most) are observed for the uncoated Altec PTFE pieces, and 5%
in the coated case.

To ensure that there is no additional effects produced by the TPB coating, a
direct comparison of reflectance, using the blue (450 nm) LED, is performed on
uncoated and TPB-coated PTFE boxes. Figure 9 shows that there is excellent
agreement between the reflectances values extracted for uncoated and TPB-
coated PTFE at 450 nm, indicating that the TPB does not, as expected, produce
fluorescence at this wavelength. These directly agree with the conclusions of
the UMS measurements in Section 3.

For 260 nm light, Figure 8 shows some difference in signal strength with
thickness. The 10 mm PTFE has the largest signal strength, followed by 8 mm, 6
mm and 5 mm, which are all close to one another (5 mm and 6 mm being
approximately the same). This matches the results seen in the FS from Section
4, where the fluorescence of the 10 mm PTFE was larger than the rest, and 5
mm, 6 mm, and 8 mm were all close to one another. These results are summarized
in Figure 9, which shows the reflectance* values for different thicknesses of
PTFE, compared to the results at 450 nm. Viewed in terms of this metric, while
the curves are distinct (with the exception of 5 mm and 6 mm), the difference
in reflectance* value does not reach more than 5%. The reason the 10 mm box
has a much larger reflectance* is likely due to the TPB coating and variations
in runs of the evaporator, rather than the PTFE. This is similar reasoning as
the measurement in the FS, found in Section 4.

In contrast to the UMS and FS measurements, the box measurements seem to
indicate a stronger trend of reflectance with thickness. This demonstrates the
value of having the two distinct methods. However, even for the box
measurements the reflectance and reflectance* values have a difference of
under 5% for different thicknesses. In the measurements with the blue LED, the
results are consistent with the results from the UMS, where coated and
uncoated boxes have reflectance values that agree within error. In the
measurements with the UV LED, the results agree with the results of the FS,
where the 10 mm box stands out in comparison to the rest, but the magnitude of
this difference remains small for NEXT. This could be caused by the fact that
10 mm pieces were coated together, and if that coating deposited more TPB,
both the 10 mm box and disk would have higher reflectance. This relationship
will be further investigated in a future work.

##  6 PTFE boxes in argon

All measurements so far have been for light at a minimum of 200 nm, but in a
typical gaseous xenon or argon experiment, the wavelength produced by
scintillation would be even shorter (175 nm and 128 nm, respectively).
Measuring the reflectance using the box method (described in Section 5) in
gaseous argon with an alpha source, would thus allow us to verify the results
of our light response measurements for TPB-coated PTFE with argon
scintillation light.

###  6.1 Argon chamber setup

A setup identical to that described in Section 5 is used for the measurement
of TPB-coated PTFE boxes in argon. The LED source on the SiPM board is
replaced with a 210Po source alpha emitter with an activity of 5 nCi (185
events $\cdot$ s-1) supplied by Spectrum Techniques. The measurements were
conducted in a sealed chamber filled with gaseous argon at approximately 1 bar
of pressure at room temperature, shown in Figure 10. It is expected that the
alpha particle will travel about 6 cm from the source. Due to technical
limitations of the setup, measurements of the light signals for the four boxes
described in Section 5 were taken at only two insertion distances, 6.0 cm and
13.7 cm.

Figure 10: Left: Picture of the argon chamber setup for a measurement. The
main body of the chamber is removed so that the PTFE box and PCB holder are
visible. Right: Picture of closed argon chamber, ready for measurements in a
gaseous argon environment with radioactive source.

Approximately 6500 triggers are collected for each boxes at the two insertion
distances, in 20 minutes runs. For each triggered pulse, full waveforms for
all four SiPMs are saved. The small number of events relative to the source
activity indicates that a large majority of alpha particles emitted by the
210Po source are not detected, mostly due to limitations from the trigger rate
of the scope to read out and save full waveforms for all four SiPMs.

All events with nonzero signals in all four SiPM channels are considered to be
alpha decays. For a given configuration (insertion distance + PTFE thickness),
the measured signal is taken to be the sum of the median integrated signals
(in all four channels) for all selected events in that configuration. The
lower (upper) error bar corresponds to the 16${}^{\text{th}}$
(84${}^{\text{th}}$) percentile of the distribution of summed integrated
signals, which result to be asymmetric. For a Gaussian distribution of event
energies, this would correspond to the standard mean and standard deviation of
the distribution.

###  6.2 Results from PTFE boxes in argon

Measurements of the signal as a function of distance for the Altec Plastics
boxes were taken for the four thicknesses and are shown in Table 2.

The replacement of the LED with an alpha source makes it more challenging to
assess the reflectance as a function of thickness with simulations in the same
way as Section 5. Many additional factors need to be taken into account in the
simulations, including but not limited to: the purity of the argon and the
nitrogen contamination for light quenching affects, the argon pressure, and
the quantum efficiency of TPB and SiPMs at the different wavelengths
generated. Due to this difficulty, the method described above is not able to
extract reflectance (or reflectance*) values, but is rather used to cross
check that conclusions hold at 128 nm.

Another limitation of the setup was the degree of nitrogen contamination. Due
to some issues with the vacuum system, namely the vinyl tubing used to get the
gaseous argon to the measurement setup, some level of air contamination
occurred. Without a purity monitor it is unclear to what degree the
contamination would change the signal but it is very likely a contributing
factor to the error in both the size and number of signals. Because of this,
these results should be taken qualitatively. In a future work, measurements
into the degree of Oxygen and Nitrogen contamination would be required.

Table 2: A summary of median signal sizes (A.U.) of alpha decay events in TPB-coated PTFE boxes placed inside a gaseous argon enviroment (128 nm). Errors on each point range from the 16${}^{\text{th}}$ to 84${}^{\text{th}}$ percentile. Insertion Distance | Sample thickness
---|---
| 5 mm | 6 mm | 8 mm | 10 mm
6 cm | $34.1^{+5.8}_{-4.7}$ | $32.1^{+5.2}_{-4.2}$ | $30.1^{+4.8}_{-4.0}$ | $29.7^{+4.5}_{-3.7}$
13.7 cm | $28.8^{+3.5}_{-3.7}$ | $34.2^{+5.4}_{-4.5}$ | $29.3^{+4.0}_{-3.5}$ | $31.4^{+4.5}_{-4.2}$

##  7 Comparison between methods

Four distinct measurement methods for determining the light response of TPB-
coated PTFE as a function of thickness were presented above. Using the UMS
provided measurements relative to a Spectralon reference, this allowed us to
confirm that the TPB does not significantly impact the reflectance in the blue
wavelengths. Measurements with the FS allowed us to directly characterize the
impact of TPB on the response of PTFE to UV light, and to verify that there is
only a weak dependence on thickness. Using the box methods allowed us to
integrate together larger surface areas, and verify that our results still
hold in a noble gas environment for argon scintillation light (128 nm).

In order to compare between these distinct methodologies, the respective
output values are normalized by the corresponding value for the thickest disk
(10 mm). This makes it possible to display the variation in reflectance,
fluorescence, and response to LED and argon scintillation light in one place,
as shown in Figure 11. In addition, a summary of the results are shown in
Table 3. All of the results essentially point towards there being very little
variation of TPB-coated PTFE light response with thickness in the range
investigate (from 5 mm to 10 mm). The results all show reasonable agreement,
although each draws out distinct features relating to the particularities of
the measurement methods.

Table 3: A summary of the various results throughout the paper, all for plastics sourced from Altec Plastics. Descriptions of individual measurements can be found in the text. Measurement in argon taken at 6 cm insertion. Reflectance and reflectance* should not be directly compared. Method |  Input wavelength | Coated | Units | Sample thickness
---|---|---|---|---
10 mm | 8 mm | 6 mm | 5 mm
UMS | 450 nm | y | Refl. % | $91.7\pm 0.4$ | $92.7\pm 0.5$ | $91.0\pm 0.4$ | $93.0\pm 0.3$
FS | 200 nm | y | Norm. peak | 1.0 | 0.92 $\pm$ 0.02 | 0.90 $\pm$ 0.03 | 0.90 $\pm$ 0.04
Box | 450 nm | n | Refl. % | $94.5_{-0.51}^{+0.73}$ | $94.0_{-1.00}^{+0.34}$ | $92.5_{-0.56}^{+0.47}$ | $92.5_{-0.51}^{+0.37}$
Box | 450 nm | y | Refl. % | $94.0_{-0.86}^{+0.49}$ | $93.0_{-1.09}^{+0.46}$ | $92.5_{-0.61}^{+1.06}$ | $92.5_{-1.21}^{+0.86}$
Box | 260 nm | y | Refl.* % | $90.5_{-1.10}^{+0.41}$ | $91.0_{-1.07}^{+0.87}$ | $92.5_{-0.55}^{+0.62}$ | $95.5_{-0.65}^{+0.56}$
Ar | 128 nm | y | Arb. | $29.7^{+4.5}_{-3.7}$ | $30.1^{+4.8}_{-4.0}$ | $32.1^{+5.2}_{-4.2}$ | $34.1^{+5.8}_{-4.7}$
Figure 11: Summary of all Altec PTFE results, combined in a single plot. For
each distinct measurement method (UMS, FS, box in air and in argon) all data
points are normalized by the corresponding value for the 10 mm box, allowing
comparison between different methods. Some methods indicate minor differences
between them, but overall results are consistent within error bars.

##  8 Conclusions and discussion

Measurements of the variation in reflectance of TPB-coated PTFE with
variations in thickness and wavelength (summarized in Figure 11) were
presented. It was found that the variation in reflectance at 450 nm for PTFE
of thicknesses from 5 mm to 10 mm is measurable, ranging from 91% to 93% for
our primary supplier, Altec Plastics. It was found that variation as a
function of thickness at this wavelength for TPB-coated PTFE is very small,
with an average variation of 2% and no clear pattern with thickness,
indicating a significant impact coming from the quality of the PTFE sheets
themselves. Variations between suppliers were more significant, on average
around 4%.

Similar consistency is found between the level of fluorescence light observed,
with the samples under 10 mm all agreeing with each other to within error bars
and the 10 mm approximately 14% higher than the others. This increase is
likely not caused due to the thickness of PTFE, but rather variation in the
thickness of the TPB coating.

The measurements with PTFE boxes further reinforce these findings. The
measurements in air clearly indicate that the different samples, while
distinguishable, do not vary significantly. With the level of homogeneity
observed in the NEXT White detector, NEXT-100 should be able to reach its
target energy resolution [17]. Furthermore, the measurements in gaseous argon
give a good initial indication that our findings continue to hold for VUV
light (128 nm), and in a detector environment at room temperature.

As the challenges faced by the NEXT experiment are common to many rare event
searches (large mass of PTFE for light collection, need for low background),
we anticipate these results will be valuable to the community. In particular,
while PTFE can vary from vendor to vendor, our results indicate that reducing
thickness can be beneficial for room temperature experiments. This is further
confirmed by the consistency of the results for PTFE coated with TPB, and in
an argon environment.

## Acknowledgments

The NEXT Collaboration acknowledges support from the following agencies and
institutions: the European Research Council (ERC) under Grant Agreement No.
951281-BOLD; the European Union’s Framework Programme for Research and
Innovation Horizon 2020 (2014–2020) under Grant Agreement No. 957202-HIDDEN;
the MCIN/AEI of Spain and ERDF A way of making Europe under grants
RTI2018-095979 and PID2021-125475NB , the Severo Ochoa Program grant
CEX2018-000867-S and the Ram’on y Cajal program grant RYC-2015-18820; the
Generalitat Valenciana of Spain under grants PROMETEO/2021/087 and
CIDEGENT/2019/049; the Department of Education of the Basque Government of
Spain under the predoctoral training program of non-doctoral research
personnel; the Portuguese FCT under project UID/FIS/04559/2020 to fund the
activities of LIBPhys-UC; the Pazy Foundation (Israel) under grants 877040 and
877041; the US Department of Energy under contracts number DE-AC02-06CH11357
(Argonne National Laboratory), DE-AC02-07CH11359 (Fermi National Accelerator
Laboratory), DE-FG02-13ER42020 (Texas A&M), DE-SC0019054 (Texas Arlington) and
DE-SC0019223 (Texas Arlington); the US National Science Foundation under award
number NSF CHE 2004111; the Robert A Welch Foundation under award number
Y-2031-20200401. Finally, we are grateful to the Laboratorio Subterr’aneo de
Canfranc for hosting and supporting the NEXT experiment.

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