INTRODUCTION TO LATTICE GEOMETRY THROUGH GROUP ACTION

INTRODUCTION TO LATTICE GEOMETRY THROUGH GROUP ACTION

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English
271 Pages

Description

Group action analysis developed and applied mainly by Louis Michel to the study of N-dimensional periodic lattices is the central subject of the book. Different basic mathematical tools currently used for the description of lattice geometry are introduced and illustrated through applications to crystal structures in two- and three-dimensional space, to abstract multi-dimensional lattices and to lattices associated with integrable dynamical systems. Starting from general Delone sets the authors turn to different symmetry and topological classifications including explicit construction of orbifolds for two- and three-dimensional point and space groups. Voronoï and Delone cells together with positive quadratic forms and lattice description by root systems are introduced to demonstrate alternative approaches to lattice geometry study. Zonotopes and zonohedral families of 2-, 3-, 4-, 5-dimensional lattices are explicitly visualized using graph theory approach. Along with crystallographic applications, qualitative features of lattices of quantum states appearing for quantum problems associated with classical Hamiltonian integrable dynamical systems are shortly discussed. The presentation of the material is presented through a number of concrete examples with an extensive use of graphical visualization. The book is aimed at graduated and post-graduate students and young researchers in theoretical physics, dynamical systems, applied mathematics, solid state physics, crystallography, molecular physics, theoretical chemistry, ...


 


1 Introduction 1


2 Group action. Basic definitions and examples 5


2.1 The action of a group on itself . . . . . . . . . . . . . . . . . 12


2.2 Group action on vector space . . . . . . . . . . . . . . . . . . 16


3 Delone sets and periodic lattices 25


3.1 Delone sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25


3.2 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31


3.3 Sublattices of L . . . . . . . . . . . . . . . . . . . . . . . . . . 35


3.4 Dual lattices. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38


4 Lattice symmetry 43


4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43


4.2 Point symmetry of lattices . . . . . . . . . . . . . . . . . . . . 43


4.3 Bravais classes . . . . . . . . . . . . . . . . . . . . . . . . . . 46


4.4 Correspondence between Bravais classes and lattice point


symmetry groups . . . . . . . . . . . . . . . . . . . . . . . . . 50


4.5 Symmetry, stratification, and fundamental domains . . . . . . 52


4.5.1 Spherical orbifolds for 3D-point symmetry groups . . 57


4.5.2 Stratification, fundamental domains and orbifolds for


three-dimensional Bravais groups . . . . . . . . . . . 62


4.5.3 Fundamental domains for P4/mmm and I4/mmm . 63


4.6 Point symmetry of higher dimensional lattices. . . . . . . . . 69


4.6.1 Detour on Euler function . . . . . . . . . . . . . . . . 70


4.6.2 Roots of unity, cyclotomic polynomials, and companion


matrices . . . . . . . . . . . . . . . . . . . . . . . . . 71


4.6.3 Crystallographic restrictions on cyclic subgroups of


lattice symmetry . . . . . . . . . . . . . . . . . . . . . 72


4.6.4 Geometric elements . . . . . . . . . . . . . . . . . . . 73


5 Lattices and their Voronoï and Delone cells 81


5.1 Tilings by polytopes: some basic concepts . . . . . . . . . . . 81


5.1.1 Two- and three-dimensional parallelotopes . . . . . . 83


5.2 Voronoï cells and Delone polytopes . . . . . . . . . . . . . . . 84


5.2.1 Primitive Delone sets . . . . . . . . . . . . . . . . . . 88


5.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89


5.4 Voronoï and Delone cells of point lattices . . . . . . . . . . . 90


5.4.1 Voronoï cells . . . . . . . . . . . . . . . . . . . . . . 90


5.4.2 Delone polytopes . . . . . . . . . . . . . . . . . . . . 91


5.4.3 Primitive lattices . . . . . . . . . . . . . . . . . . . . 91


5.5 Classification of corona vectors . . . . . . . . . . . . . . . . . 93


5.5.1 Corona vectors for lattices . . . . . . . . . . . . . . . 94


5.5.2 The subsets S and F of the set C of corona vectors . 95


5.5.3 A lattice without a basis of minimal vectors . . . . . 99


6 Lattices and positive quadratic forms 101


6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101


6.2 Two dimensional quadratic forms and lattices . . . . . . . . . 102


6.2.1 The GL2(Z) orbits on C+(Q2) . . . . . . . . . . . . . 102


6.2.2 Graphical representation of GL2(Z) transformation on


the cone of positive quadratic forms . . . . . . . . . . 104


6.2.3 Correspondence between quadratic forms and Voronoï


cells . . . . . . . . . . . . . . . . . . . . . . . . . . 109


6.2.4 Reduction of two variable quadratic forms . . . . . . 110


6.3 Three dimensional quadratic forms and 3D-lattices . . . . . . 112


6.3.1 Michel’s model of the 3D-case . . . . . . . . . . . . . 113


6.3.2 Construction of the model . . . . . . . . . . . . . . . 116


6.4 Parallelohedra and cells for N-dimensional lattices. . . . . . . 122


6.4.1 Four dimensional lattices . . . . . . . . . . . . . . . . 128


6.5 Partition of the cone of positive-definite quadratic forms . . . 130


6.6 Zonotopes and zonohedral families of parallelohedra . . . . . 134


6.7 Graphical visualization of members of the zonohedral family 136


6.7.1 From Whitney numbers for graphs to face numbers


for zonotopes . . . . . . . . . . . . . . . . . . . . . . . 147


6.8 Graphical visualization of non-zonohedral lattices. . . . . . . 148


6.9 On Voronoï conjecture . . . . . . . . . . . . . . . . . . . . . . 152


7 Root systems and root lattices 153


7.1 Root systems of lattices and root lattices . . . . . . . . . . . 153


7.1.1 Finite groups generated by reflections . . . . . . . . . 156


7.1.2 Point symmetry groups of lattices invariant


by a reflection group . . . . . . . . . . . . . . . . . . 159


7.1.3 Orbit scalar products of a lattice ; weights of a root


lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 162


7.2 Lattices of the root systems . . . . . . . . . . . . . . . . . . . 164


7.2.1 The lattice In . . . . . . . . . . . . . . . . . . . . . . 164


7.2.2 The lattices Dn, n ≥ 4 and F4 . . . . . . . . . . . . . 164


7.2.3 The lattices D∗


n, n ≥ 4 . . . . . . . . . . . . . . . . . 166


7.2.4 The lattices D+


n for even n ≥ 6 . . . . . . . . . . . . 167


7.2.5 The lattices An . . . . . . . . . . . . . . . . . . . . . 167


7.2.6 The lattices A∗


n . . . . . . . . . . . . . . . . . . . . . 169


7.3 Low dimensional root lattices . . . . . . . . . . . . . . . . . . 169


8 Comparison of lattice classifications 171


8.1 Geometric and arithmetic classes . . . . . . . . . . . . . . . . 174


8.2 Crystallographic classes . . . . . . . . . . . . . . . . . . . . . 176


8.3 Enantiomorphism . . . . . . . . . . . . . . . . . . . . . . . . . 179


8.4 Time reversal invariance . . . . . . . . . . . . . . . . . . . . . 181


8.5 Combining combinatorial and symmetry classification . . . . 183


9 Applications 189


9.1 Sphere packing, covering, and tiling . . . . . . . . . . . . . . . 189


9.2 Regular phases of matter . . . . . . . . . . . . . . . . . . . . 192


9.3 Quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . 194


9.4 Lattice defects . . . . . . . . . . . . . . . . . . . . . . . . . . 194


9.5 Lattices in phase space. Dynamical models. Defects. . . . . . 196


9.6 Modular group . . . . . . . . . . . . . . . . . . . . . . . . . . 202


9.7 Lattices and Morse theory . . . . . . . . . . . . . . . . . . . . 206


9.7.1 Morse theory . . . . . . . . . . . . . . . . . . . . . . . 207


9.7.2 Symmetry restrictions on the number of extrema . . . 208


A Basic notions of group theory with illustrative examples 211


B Graphs, posets, and topological invariants 223


C Notations for point and crystallographic groups 229


C.1 Two-dimensional point groups . . . . . . . . . . . . . . . . . . 230


C.2 Crystallographic plane and space groups . . . . . . . . . . . . 231


C.3 Notation for four-dimensional parallelohedra . . . . . . . . . . 231


D Orbit spaces for plane crystallographic groups 235


E Orbit spaces for 3D-irreducible Bravais groups 243


Bibliography 251


Index 259



 

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Introduction to Louis Michel’s lattice geometry through group action
B. Zhilinskii
CURRENT NATURAL SCIENCES
B. Zhilinskii
Introduction to Louis Michel’s lattice geometry through group action
C U R R E N T N A T U R A L S C I E N C E S EDP Sciences/CNRS Éditions
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et
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This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data bank. Duplication of this publication or parts thereof is only permitted under the provisions of the French Copyright law of March 11, 1957. Violations fall under the prosecution act of the French Copyright law.
ISBN EDP Sciences: 9782759817382 ISBN CNRS Éditions: 9782271087393
Contents
Preface
1
2
3
4
Introduction
Group action. Basic definitions and examples 2.1 The action of a group on itself . . . . . . . . . . . . . . . . . 2.2 Group action on vector space . . . . . . . . . . . . . . . . . .
Delone sets and periodic lattices 3.1 Delone sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Sublattices ofL. . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Dual lattices. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lattice symmetry 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Point symmetry of lattices . . . . . . . . . . . . . . . . . . . . 4.3 Bravais classes . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Correspondence between Bravais classes and lattice point symmetry groups . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Symmetry, stratification, and fundamental domains . . . . . . 4.5.1 Spherical orbifolds for 3Dpoint symmetry groups . . 4.5.2 Stratification, fundamental domains and orbifolds for threedimensional Bravais groups . . . . . . . . . . . 4.5.3 Fundamental domains forP4/mmmandI4/mmm. 4.6 Point symmetry of higher dimensional lattices. . . . . . . . . 4.6.1 Detour on Euler function . . . . . . . . . . . . . . . . 4.6.2 Roots of unity, cyclotomic polynomials, and companion matrices . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Crystallographic restrictions on cyclic subgroups of lattice symmetry . . . . . . . . . . . . . . . . . . . . . 4.6.4 Geometric elements . . . . . . . . . . . . . . . . . . .
vii
1
5 12 16
25 25 31 35 38
43 43 43 46
50 52 57
62 63 69 70
71
72 73
iv
5
6
7
Introduction to lattice geometry through group action
Lattices and their Voronoï and Delone cells 5.1 Tilings by polytopes: some basic concepts . . . . . . . . . . 5.1.1 Two and threedimensional parallelotopes . . . . . 5.2 Voronoï cells and Delone polytopes . . . . . . . . . . . . . . 5.2.1 Primitive Delone sets . . . . . . . . . . . . . . . . . 5.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Voronoï and Delone cells of point lattices . . . . . . . . . . 5.4.1 Voronoï cells . . . . . . . . . . . . . . . . . . . . . 5.4.2 Delone polytopes . . . . . . . . . . . . . . . . . . . 5.4.3 Primitive lattices . . . . . . . . . . . . . . . . . . . 5.5 Classification of corona vectors . . . . . . . . . . . . . . . . 5.5.1 Corona vectors for lattices . . . . . . . . . . . . . . 5.5.2 The subsetsSandFof the setCof corona vectors 5.5.3 A lattice without a basis of minimal vectors . . . .
. . . . . . . . . . . . .
Lattices and positive quadratic forms 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Two dimensional quadratic forms and lattices . . . . . . . . . 6.2.1 TheGL2(Z)orbits onC+(Q2). . . . . . . . . . . . . 6.2.2 Graphical representation ofGL2(Z)transformation on the cone of positive quadratic forms . . . . . . . . . . 6.2.3 Correspondence between quadratic forms and Voronoï cells . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Reduction of two variable quadratic forms . . . . . . 6.3 Three dimensional quadratic forms and 3Dlattices . . . . . . 6.3.1 Michel’s model of the 3Dcase . . . . . . . . . . . . . 6.3.2 Construction of the model . . . . . . . . . . . . . . . 6.4 Parallelohedra and cells for Ndimensional lattices. . . . . . . 6.4.1 Four dimensional lattices . . . . . . . . . . . . . . . . 6.5 Partition of the cone of positivedefinite quadratic forms . . . 6.6 Zonotopes and zonohedral families of parallelohedra . . . . . 6.7 Graphical visualization of members of the zonohedral family 6.7.1 From Whitney numbers for graphs to face numbers for zonotopes . . . . . . . . . . . . . . . . . . . . . . . 6.8 Graphical visualization of nonzonohedral lattices. . . . . . . 6.9 On Voronoï conjecture . . . . . . . . . . . . . . . . . . . . . .
Root systems and root lattices 7.1 Root systems of lattices and root lattices . . . . . . . . . . . 7.1.1 Finite groups generated by reflections . . . . . . . . . 7.1.2 Point symmetry groups of lattices invariant by a reflection group . . . . . . . . . . . . . . . . . . 7.1.3 Orbit scalar products of a lattice ; weights of a root lattice . . . . . . . . . . . . . . . . . . . . . . . . . .
81 81 83 84 88 89 90 90 91 91 93 94 95 99
101 101 102 102
104
109 110 112 113 116 122 128 130 134 136
147 148 152
153 153 156
159
162
Contents
8
9
7.2
7.3
Lattices of the root systems . . . . . . . 7.2.1 The latticeIn. . . . . . . . . . 7.2.2 The latticesDn,n4andF4. 7.2.3 The latticesD,n4. . . . . n + 7.2.4 The latticesDfor evenn6 n 7.2.5 The latticesAn. . . . . . . . . 7.2.6 The latticesA. . . . . . . . . n Low dimensional root lattices . . . . . .
. . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
Comparison of lattice classifications 8.1 Geometric and arithmetic classes . . . . . . . . . . . . . . . . 8.2 Crystallographic classes . . . . . . . . . . . . . . . . . . . . . 8.3 Enantiomorphism . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Time reversal invariance . . . . . . . . . . . . . . . . . . . . . 8.5 Combining combinatorial and symmetry classification . . . .
Applications 9.1 Sphere packing, covering, and tiling . . . . . . . . . . . . . 9.2 Regular phases of matter . . . . . . . . . . . . . . . . . . 9.3 Quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Lattice defects . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Lattices in phase space. Dynamical models. Defects. . . . 9.6 Modular group . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Lattices and Morse theory . . . . . . . . . . . . . . . . . . 9.7.1 Morse theory . . . . . . . . . . . . . . . . . . . . . 9.7.2 Symmetry restrictions on the number of extrema .
. . . . . . . . . . . . . . . . . .
A Basic notions of group theory with illustrative examples
B
C
Graphs, posets, and topological invariants
Notations for point and crystallographic groups C.1 Twodimensional point groups . . . . . . . . . . . . . . . . . . C.2 Crystallographic plane and space groups . . . . . . . . . . . . C.3 Notation for fourdimensional parallelohedra . . . . . . . . . .
D Orbit spaces for plane crystallographic groups
E
Orbit spaces for 3D-irreducible Bravais groups
Bibliography
Index
v
164 164 164 166 167 167 169 169
171 174 176 179 181 183
189 189 192 194 194 196 202 206 207 208
211
223
229 230 231 231
235
243
251
259
Preface
This book has a rather long and complicated history. One of the authors, Louis Michel, passed away on the 30 December, 1999. Among a number of works in progress at that time there were a near complete series of big papers on “Symmetry, invariants, topology” published soon after in Physics Reports [75] and a project of a book “Lattice geometry”, started in collaboration with Marjorie Senechal and Peter Engel [53]. The partially completed version of the “Lattice geometry” by Louis Michel, Marjorie Senechal and Peter Engel is available as a IHES preprint version of 2004. In 2011, while starting to work on the preparation of selected works of Louis Michel [19] it became clear that scientific ideas of Louis Michel developed over the last thirty years and related to group action applications in different physical problems are not really accessible to the young generation of scientists in spite of the fact that they are published in specialized reviews. It seems that the comment made by Louis Michel in his 1980’s talk [70] remains valid till now: “Fifty years ago were published the fundamental books of Weyl and of Wigner on application of group theory to quantum mechanics; since, some knowledge of the theory of linear group representations has become necessary to nearly all physicists. However the most basic concepts concerning group actions are not introduced in these famous books and, in general, in the physics literature.” After rather long discussions and trials to revise initial “Lattice geometry” text which require serious modifications to be kept at the current level of the scientific achievements, it turns out that probably the most wise solution is to restrict it to the basic ideas of Louis Michel’s approach concentrated on the use of group actions. The present text is based essentially on the preliminary version of the “Lattice geometry” manuscript [53] and on relevant publica tions by Louis Michel [71, 76, 72, 73, 74], especially on reviews published in Physics Reports [75], but the accent is made on the detailed presentation of the two and threedimensional cases, whereas the generalization to arbitrary dimension is only outlined.
Chapter
1
Introduction
This chapter describes the outline of the book and explains the interrelations between different chapters and appendices. The specificity of this book is an intensive use of group action ideas and terminology when discussing physical and mathematical models of lat tices. Another important aspect is the discussion and comparison of various approaches to the characterization of lattices. Along with symmetry and topol ogy ideas, the combinatorial description based on Voronoï and Delone cells is discussed along with classical characterization of lattices via quadratic forms. We start by introducing in Chapter 2 the most important notions related to group action: orbit, stabilizer, stratum, orbifold,. . .These notions are il lustrated on several concrete examples of the group action on groups and on vector spaces. The necessary basic notions of group theory are collected in appendix A which should be considered as a reference guide for basic notions and notation rather than as an exposition of group theory. Before starting description of lattices, chapter 3 deals with a more general concept, the Delone system of points. Under special conditions Delone sets lead to lattices of translations which are related to the fundamental physical notion of periodic crystals. The study of the Delone set of points is important not only to find necessary and sufficient conditions for the existence of peri odic lattices. It allows discussion of a much broader mathematical frame and physical objects like aperiodic crystals, named also as quasicrystals. Chapter 4 deals with symmetry aspects of periodic lattices. Point sym metry classification and Bravais classes of lattices are introduced using two dimensional and threedimensional lattices as examples. Stratification of the ambient space and construction of the orbifolds for the symmetry group action is illustrated again on many examples of two and threedimensional lattices. The mathematical concepts necessary for the description of point symmetry of higher dimensional lattices are introduced and the crystallographic restric tions imposed on the possible types of point symmetry groups by periodicity condition are explicitly introduced.