An Exactly Solvable Quantum Four-Body Problem Associated with the - PowerPoint PPT Presentation

An Exactly Solvable Quantum Four-Body Problem Associated with the Symmetries of an Octacube Steven Jackson UMass Boston, Mathematics Maxim Olshanii UMass Boston, Physics

Introduction

In memory of Marvin Girardeau Oct 3, 1930 - Jan 13, 2015

Plan

Every instance of an integrable one-dimensional many-body system with zero-range two-body interactions can be traced to a multidimensional kaleidoscope Example: 4 hard- core bosons on a line A 3 H. Nishiyama

Kaleidoscopes are the systems of mirrors where the seams between the mirrors are do not seem to be there. � � “Inside kaleidoscope” A 2 San Francisco

It is proven that the existing list of kaleidoscopes, or reflection groups, ~ ~ ~ ~ ~ ~ ~ ~ ~ A N , B N , C N , D N ; G 2 , F 4 , E 6 , E 7 , E 8 ; I 2 (n), H 3 , H 4 , classical exceptional � crystallographic = non-crystallographic = closed mirror chamber one mirror must be missing is complete. � “Inside kaleidoscope” A 2 San Francisco

Plan ~ C 3, 4, … ~ B 2, 3, … ~ Affine D 4, 5, … ~ reflection groups A 2,3,… ~ I 1 ~ ~ E 6-8 G 2 ~ F 4

Plan Solvable z t ~ a s simplex-shaped n A C 3, 4, … ~ e h t e quantum billiards B 2, 3, … B ~ Affine D 4, 5, … ~ reflection groups A 2,3,… ~ I 1 E n ~ ψ n ( z ) ~ E 6-8 G 2 ~ F 4 Gutkin-Sutherland, Emsiz-Opdam-Stokman

Plan ~ C 3, 4, … ~ B ~ McGuire D Girardeau ~ Lieb A 2,3,… ~ I 1 ~ ~ E G ~ Yang F Gaudin E n Solution for ψ n ( x 1 , x 2 , … ) same-mass hard-cores on a circle and in a box

Plan Solvable z t ~ a s simplex-shaped n A C 3, 4, … ~ e h t e quantum billiards B B ~ Affine D ~ reflection groups A 2,3,… ~ with non-forking I 1 E n Coxeter diagrams ~ ψ n ( z ) ~ ~ E G 2 G 2 ~ F 4 E n Solvable systems ψ n ( x 1 , x 2 , … ) of hard-cores in a box ~ (for A , on a circle) Original result

Also need finite reflection groups, both for technical reasons and for future projects z t a s n A e h B 2,3,… = C 2,3,… Solvable t e B open-simplex-shaped A 2,3,… Finite D 4,5,… quantum billiards reflection groups E 6-8 F 4 ψ E ( z ) H 2-4 I 1 G 2 I 2 ( m ≥ 7)

Also need finite reflection groups, both for technical reasons and for future projects z t a s n A B 2,3,… = C 2,3,… B 2,3,… = C 2,3,… e Solvable h t e B open-simplex-shaped A 2,3,… Finite D quantum billiards reflection groups E F 4 with non-forking ψ E ( z ) Coxeter diagrams H 2-4 I 1 G 2 I 2 ( m ≥ 7) ψ E ( x 1 , x 2 , … ) Solvable systems of hard-cores on a line Original result

Affine reflection groups → → solvable billiards (short summary of known results and new results)

Alcove of an affine reflection group as a solvable quantum hard-wall billiard ^ � ψ ( r ) = ∑ (-1) P [ g ] exp[( gk ) r ] , g where g = an element of the finite ~ alcove of G nucleus G of the ~ full affine group G , � P [ g ] = parity of g, � ~ k ∈ lattice reciprocal to the lattice G . After Gutkin-Sutherland, Emsiz-Opdam-Stokman (covers Robin’s boundary conditions, includes completeness)

Alcove of an affine reflection group as a solvable quantum hard-wall billiard Original result � Integrals of motion in involution = invariant polynomials (Chevalley polynomials) of the non-affine nucleus , with coordinates replaced by momenta (in the billiard coordinate system). A hint to a Bethe Ansatz <=> Liouville’s integrability connection

An example of a billiard solving Above, we used G 2, the symmetry group of a hexagon, , as an example . → → G 2

Non-forking affine reflection groups → solvable particle systems

d particles on a line in a box d -dimensional billiard � � ￢ x i = y i / √ m i � � m 1 m 1 m 2 m 2 ∞ ∞ � � � inter-particle contact � ( d -1)-faces � particle-wall contact � � � � � left-mid and mid-right two ( d -1)-faces at α contacts in a an angle consecutive triplet m 2 ( m 1+ m 2+ m 3 ) � α = arctan[ ] � m 1 m 3 � � contacts in two two ( d -1)-faces unrelated consecutive doublets at 90°

A solvable particle system associated with the affine reflection ~ group F 4

Our subject of is F 4 , the symmetry group of an octacube, , a unique to 4D Platonic solid, with no 3D analogue, and its many-body realization.

Our subject of is F 4 , the symmetry group of an octacube, , a unique to 4D Platonic solid, with no 3D analogue, and its many-body realization. The “Octacube” and its designer, Adrian Ocneanu, PennState

The “Octacube” and its designer, Adrian Ocneanu, PennState

Rhombic dodecahedron, the 3D cousin of the octacube

Rhombic dodecahedron, the 3D cousin of the octacube

Rhombic dodecahedron, the 3D cousin of the octacube

Rhombic dodecahedron, the 3D cousin of the octacube

Rhombic dodecahedron, the 3D cousin of the octacube

Rhombic dodecahedron, the 3D cousin of the octacube

Rhombic dodecahedron, the 3D cousin of the octacube

www.tintouen.fr

Repeat the steps above with two tesseracts and you will get an octacube. But unlike in 3D, in 4D you will get a Platonic solid.

The rhombic dodecahedron and the octacube are the 3D and 4D members of a family, that goes through all numbers of dimensions: in every dimension, the resulting polyhedron tiles the corresponding space

Building a particle system ~ from the F 4 Coxeter diagram

“ [T]he angel of geometry and the devil of algebra share the stage, illustrating the difficulties of both. ” Hermann Weyl Alcoves of reflection groups (and many other geometric objects) are cataloged using Coxeter diagrams

Building a particle system ~ from the F 4 Coxeter diagram π / angles between the generating mirrors of a reflection group ~ F 4 3 3 3 3 4 4 3 3 generating mirrors of the reflection group

Building a particle system ~ from the F 4 Coxeter diagram ~ F 4 3 3 3 3 4 4 3 3 m 0 - m 1 m 1 - m 2 m 2 - m 3 m 3 - m 4 m 4 - m 5 m 3 ( m 2+ m 3+ m 4 ) m 1 ( m 0+ m 1+ m 2 ) arctan[ ] = π / arctan[ ] = π / m 2 m 4 m 0 m 2 m 4 ( m 3+ m 4+ m 5 ) m 2 ( m 1+ m 2+ m 3 ) arctan[ ] = π / arctan[ ] = π / m 3 m 5 m 1 m 3 m 0 , m 1 , m 2 , m 3 , m 4 , m 5 > 0 m 1 m 4 m 0 m 2 m 3 m 5 Original result

Building a particle system ~ ~ from the F 4 Coxeter diagram ~ F 4 3 3 4 3 m 0 - m 1 m 1 - m 2 m 2 - m 3 m 3 - m 4 m 4 - m 5 m 3 ( m 2+ m 3+ m 4 ) m 1 ( m 0+ m 1+ m 2 ) 3 4 arctan[ ] = π / arctan[ ] = π / m 2 m 4 m 0 m 2 m 4 ( m 3+ m 4+ m 5 ) m 2 ( m 1+ m 2+ m 3 ) 3 3 arctan[ ] = π / arctan[ ] = π / m 3 m 5 m 1 m 3 m 0 , m 1 , m 2 , m 3 , m 4 , m 5 > 0 m 1 m 4 m 0 m 2 m 3 m 5 Original result

Building a particle system ~ from the F 4 Coxeter diagram ~ F 4 3 3 4 3 m 0 - m 1 m 1 - m 2 m 2 - m 3 m 3 - m 4 m 4 - m 5 Single solution: m 0 = ∞ , m 1 =6 m , m 2 =2 m , m 3 = m, m 4 =3 m , m 5 = ∞ m 2 m 3 m 6 m Original result

Results m 2 m 3 m 6 m L Periodicity cell: octacube (24 octahedral 3-faces at all signs and permutations of (±1, ±1, 0, 0)) � Energy spectrum: � π 2 ħ 2 E n1,n2,n3,n4 = [2 n 1 ( n 1 + n 2 + n 3 + n 4 ) 6 m L 2 + n 22 + n 32 + n 42 + n 2 n 3 + n 2 n 4 + n 3 n 4 ] n 1 =1, 2, 3, … n 2 =1, 2, 3, … Weyl’s law n 3 = n 2 +1, n 2 +2 , n 2 +3, … exact n 4 = n 3 +1, n 3 +2 , n 3 +3, … Original result

Results m 2 m 3 m 6 m L Ground state energy: � 13 π 2 ħ 2 E 1,1,2,3 = 2 m L 2 ~ F 4 � 3 3 3 4 3 4 3 Ground state wavefunction: consists of 1152 plane waves (the same for any other eigenstate) � Original result

Results m 2 m 3 m 6 m L Four integrals of motion in involution: � I l ( p 1 , p 2 , p 3 , p 4 ) = ( p 1 + p 2 ) l + ( p 1 - p 2 ) l + ( p 1 + p 3 ) l + ( p 1 - p 3 ) l + ( p 1 + p 4 ) l + ( p 1 - p 4 ) l + ( p 2 + p 3 ) l + ( p 2 - p 3 ) l + ( p 2 + p 4 ) l + ( p 2 - p 4 ) l + ( p 3 + p 4 ) l + ( p 3 - p 4 ) l , � l = 2, 6, 8, 12, invariant with polynomials particle momenta billiard momenta of F 4 p 1 -1 -1 -1 -1 p 1 3 m � p 2 -1 1 1 1 p 2 m ≡ p 3 0 -2 1 1 p 3 2 m p 4 0 0 -3 1 p 4 6 m Remark: I 2 ∝ E Original result Remark: A N -1 ->fermionic momentum moments

Summary

Summary ~ Extablished a map between affine reflection groups with non-forking Coxeter diagrams and exactly solvable quantum hard-core few-body problems on a line; � ~ Worked the F 4 (symmetry of an octacube, ) to the end. The resulting integrable four-body system consists of four hard-cores with mass ratios 6:2:1:3, ; m 2 m 3 m 6 m L � ~ For F 4 , found all four integrals of motion: Chevalley polynomials of square roots of particle kinetic energies.

Joint work with Maxim Olshanii UMass Boston Physics

Numerous discussions with: Marvin Girardeau (U Arizona) Vanja Dunjko (UMB) Felix Werner (ENS) Jean-Sébastien Caux (U Amsterdam) Alfred G. Noël (UMB) Dominik Schneble (Stony Brook)