A Topological Fingerprint for Periodic Crystals 2021-04-09

A Topological Fingerprint for Periodic Crystals
Teresa Heiss, Herbert Edelsbrunner, Alexey Garber, Janos Pach,
Morteza Saghafian
IST Austria
BMC-BAMC, April 9, 2021
Teresa Heiss (IST Austria) Topological Fingerprint for Periodic Crystals BMC-BAMC, April 9, 2021 1 / 20

What is a periodic crystal?
“A [periodic] crystal is a solid composed of atoms, ions, or molecules
arranged in a pattern that is periodic in three dimensions.” [ASTM F1241]
i.e. there exist three linearly independent translations, such that each of
them maps the crystal to itself.

What is our mathematical model for a periodic crystal?
We model a periodic crystal by an infinite periodic point set.
Centers of the atoms = points

Definition (Periodic point set)

Definition (Periodic point set)
((v1, v2, v3), M)
cutout of S

Cutout not unique
A B
C

We model a periodic crystal by a periodic point set
.

We model a periodic crystal by a periodic point set
an equivalence class of periodic point sets.

Understanding space of periodic crystals
Goal: map every crystal to a point in a nice space

Goal: map every crystal to a point in a nice space
A B
C

Goal: map every crystal to a point in a nice space (we call this a crystal
invariant)
A B
C

invariant), s.t. the map is

invariant), s.t. the map is injective

invariant), s.t. the map is injective and continuous.

Definition bottleneck distance
dB,2(S, S0
) := inf
γ:S→S0
sup
s∈S
ks − γ(s)k2

Pairs of periodic point sets that should get a small
distance assigned
Example α:

Pairs of periodic point sets that should get a small
distance assigned
Example β:

invariant), s.t. the map is injective and continuous.

invariant), s.t. the map is injective and continuous. → “fingerprint” [1]

invariant), s.t. the map is injective and continuous. → “fingerprint” [1]
[1] Edelsbrunner, Heiss, Kurlin, Smith, Wintraecken: The density fingerprint of a periodic point set. In: Proceedings of SoCG,
2021

Caveat
Goal: map every periodic crystal to a point in Rn, s.t. the map is
invariant under isometries, injective, continuous.

Caveat
Problem: There are too many equivalence classes of periodic point sets.

Caveat
Problem: There are too many equivalence classes of periodic point sets.
Possible Solution: Make n depend on parameter of crystals

Why do we want a fingerprint function?
• develop understanding for the space of periodic crystals
• compare crystals by comparing fingerprints
• vector representation → machine learning
• . . .

Possible solution: the persistence fingerprint

Persistent homology
What is the “shape” of the point set?

Persistent homology

k-fold covers
Cover1(X, r)

k-fold covers
Cover1(X, r)
Coverk(X, r) := {p ∈ R3 : p ∈ Br (x) for at least k points x ∈ X}

k-fold covers
Cover2(X, r)

k-fold covers
Cover3(X, r)

k-fold covers
Cover4(X, r)

k-fold persistence
Fix k, let r increase:
Coverk(X, r1) ⊆ Coverk(X, r2) ⊆ · · · ⊆ Coverk(X, rn)

k-fold persistence
In the figures above: k=2

k-fold persistence
Dgmk(X) :=
Dgm((Coverk(X, r))r∈R)
. . . k-fold persistence [2]
[2] Herbert Edelsbrunner and Georg Osang. The Multi-cover Persistence of Euclidean Balls. In 34th International Symposium on
Computational Geometry (SoCG 2018), volume 99 of Leibniz International Proceedings in Informatics (LIPIcs), pages
34:1–34:14, Dagstuhl, Germany, 2018.

k-fold persistence
Dgmk(X) :=
Dgm((Coverk(X, r))r∈R)
. . . k-fold persistence [2]
(Dgmk(X))k∈N ∈ DgmN
. . . persistence fingerprint
where Dgm is the set of persistence
diagrams
[2] Herbert Edelsbrunner and Georg Osang. The Multi-cover Persistence of Euclidean Balls. In 34th International Symposium on
Computational Geometry (SoCG 2018), volume 99 of Leibniz International Proceedings in Informatics (LIPIcs), pages
34:1–34:14, Dagstuhl, Germany, 2018.

Definition (Persistence fingerprint function ϕ)
P . . . set of equivalence classes of periodic point sets in R3
Definition
Define the persistence fingerprint function as
ϕ : P → DgmN
[S]' 7→ (Dgmk(S))k∈N

Definition (Persistence fingerprint function ϕ)
P . . . set of equivalence classes of periodic point sets in R3
Definition
Define the persistence fingerprint function as
ϕ : P → DgmN
To avoid multiplicities = ∞, define
Dgmk(S) := lim
j→∞
Dgmper
k (S ∩ Uj )
Vol(Uj )
, (1)
where Uj consists of j × j × j copies of the unit cell U and
where the division and the limit is applied to the multiplicity of each
persistence pair individually.
Note, this definition yields non-integer multiplicities.

Results
Theorem (Invariance)
The persistence fingerprint
ϕ : P → DgmN
is well-defined (i.e. invariant under isometries and choice of unit cell).

Results
Theorem (Invariance)
ϕ : P → DgmN
is well-defined (i.e. invariant under isometries and choice of unit cell).
Theorem (Continuity)
ϕ :

P, inf
isometries
dB,2

→

DgmN
, sup
k
dB,∞

is Lipschitz continuous.

Ideas for Injectivity Results
• Generic injectivity (proof sketch)

• Injectivity for lattices (in R3) (proof, but not polished yet)

• Injectivity for lattices (in R3) (proof, but not polished yet)
• Injectivity for periodic point sets (only some ideas, still open)

Herbert Edelsbrunner, Teresa Heiss, Vitaliy Kurlin, Phil Smith, and
Mathijs Wintraecken.
The density fingerprint of a periodic point set.
In Proceedings of SoCG 2021, 2021.
Herbert Edelsbrunner and Georg Osang.
The Multi-cover Persistence of Euclidean Balls.
In 34th International Symposium on Computational Geometry (SoCG
2018), volume 99 of Leibniz International Proceedings in Informatics
(LIPIcs), pages 34:1–34:14, Dagstuhl, Germany, 2018. Schloss
Dagstuhl–Leibniz-Zentrum fuer Informatik.

A Topological Fingerprint for Periodic Crystals 2021-04-09

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