N L HARSHMAN et al PHYS REV X 7 041001 2017, interactions form the basis for approximation schemes for the energies can be calculated algebraically and all excited. strongly interacting systems providing a valuable bench states can be expressed as orthogonal polynomials times the. mark for testing numerical methods 37 39 This pos ground state this property is called exact solvability. sibility is of special importance for mass imbalanced 45 46 These solutions are constructed by Bethe. systems where ab initio calculations for strong interaction ansatz like superpositions not of plane waves but of. are especially challenging 40 spherical or hyperspherical harmonics The conditions. We derive the criteria for which sets of imbalanced on the masses can be seen as the requirement that the. masses are solvable and integrable using a geometrical scattering is nondiffractive i e integrable in the Bethe. approach In the hard core limit configuration space is ansatz sense 2 47 49 On the other hand we provide. sectioned into N disconnected sectors one for each order evidence that these models are also integrable in the. of particles After separating out the center of mass and classical Liouvillian sense Classically Liouvillian inte. relative hyperradial d o f the dynamics in each ordering grability means there are N functionally independent. sector of relative angular configuration space can be invariant operators in involution that act continuously on. mapped onto hard wall quantum billiards on a N 2 2N dimensional phase space The quantum versions of. dimensional sphere The domain is a simplex whose shape these operators commute with each other and therefore all. depends on the mass ratios For three particles the six states can be characterized by the spectrum of this set of. ordering sectors are just arcs of a circles and every set of operators for a discussion of ambiguities in defining. masses is therefore integrable by separation of variables integrability in quantum systems see Ref 50 As an. For four particles the 24 ordering sectors are spherical example we construct these operators for the four particle. triangles Quantum billiards in a general spherical case of H3 Further we conjecture that these solvable. triangle cannot be solved by separability nor can higher models are superintegrable meaning they have more. dimensional generalizations to N 2 simplexes on integrals of motion than d o f and even maximally super. N 2 spheres However when a particular ordering integrable with 2N 1 integrals of motion for a more. sector tiles the sphere under reflections then the problem complete discussion see Ref 51 Superintegrable sys. is exactly solvable using something like the method of tems can have a rich mathematical structure like multi. images The possible spherical tilings are classified using separability and exact solvability Further perturbations. Coxeter groups Described in more detail below Coxeter from superintegrable models sometimes remain integrable. groups are point symmetry groups generated by reflections For example the unitary hard core limit of equal mass. They were originally developed for the purpose of analyz particles in a harmonic trap with contact interactions is a. ing symmetric polytopes 41 In hard core contact inter maximally superintegrable system isomorphic to one limit. action models the N 1 dimensional hyperplanes where of the CM model 7 52 53 In the near unitary limit. two particles coincide define planes of reflection If the defined as a first order perturbation from the hard core. particle masses are correct then these coincidence reflec limit maximal superintegrability is broken However the. tion planes generate a Coxeter group system still retains enough symmetry in the near unitary. This logic can be reversed we show that for every finite limit that it can be mapped onto a spin chain model 37 38. connected nonbranching Coxeter group with rank r there is At the other extreme from maximal superintegrability the. a one parameter family of masses for which the dynamics Coxeter group criteria could be used to identify mass. of r 1 ordered particles is integrable These models are imbalanced systems where quantum ergodic dynamics. nontrivial when N 3 we focus on the case N 4 where should be expected In the experimental outlook we. there are three families of solvable masses and the discuss the connections to quantum billiards and the. symmetries in relative configuration space are the same consequences of integrability for thermalization in ultra. as the Platonic solids We give special attention to the cold atomic gases. exceptional Coxeter group H3 of icosahedral symmetries. and therefore this work is closely related to CSM models II MODEL AND SYMMETRY. based on exceptional reflection groups 42 44 The role of. Coxeter groups in providing integrability criteria for this We consider the N particle Hamiltonian with contact. model is perhaps not surprising because they have pre interactions All particles are harmonically trapped with the. viously played an important role in the theory of classical same frequency Written in terms of the particle coordinates. and quantum dynamical systems For example extensions x x1 x2 xN the Hamiltonian has the form. of the CSM model have a closely related classification. scheme 8 and so do Gaudin models 3 H 2 2,mi xi g xi xj 1. Besides applications to mass imbalanced ultracold i 1 i j. atomic gases a motivating interest in this model is that. it sits at the intersection of related notions of integrability In the limit g of hard core contact interactions the. and solvability For sectors with Coxeter tiling symmetry order of the particles is a dynamical invariant 39. INTEGRABLE FAMILIES OF HARD CORE PARTICLES PHYS REV X 7 041001 2017. Configuration space is divided into N ordering sectors by. N N 1 2 impenetrable coincidence hyperplanes one,for each pair of particles The order xp1 xp2 xpN. is labeled by a permutation p fp1 p2 pN g or more, briefly p p1 p2 pN We denote by Xij the coincidence. hyperplane defined by xi xj 0, In Appendix A we show that solving for the eigenstates. of Eq 1 in a particular ordering sector p is equivalent to. solving for the motion of a free quantum particle confined. to an N 2 sphere and trapped inside an angular sector. p bounded by N 1 hard walls We establish this, equivalence by making a mass dependent transformation. T of the position coordinates z Tx and then separating. out the scaled center of mass zN and the scaled relative FIG 1 The relative configuration space for four particles in the. hyperradius simplest case when all masses are the same an example of the. r N one parameter mass family of the Coxeter group A3 The gray. sphere represents an equipotential of the harmonic trap in the. zN mi xi and 2 z2j 2 mass normalized coordinates z1 z2 z3 The six colored disks. M i 1 j 1 that intersect the plane represent the coincidence planes Z12 red. Z13 yellow Z14 green Z23 cyan Z24 blue Z34 magenta. where M is the total mass The remaining relative coor Twelve sectors are visible and are labeled by p where p is the. dinates are the N 2 hyperangles f 1 N 3 g that order of the four particles For two sectors 1234 and 1324 the. cover the sphere S N 2 The transformation to these coor three angles are also labeled. dinates gives the harmonic potential a spherically sym. metric form but the coincidence hyperplanes now, transformed to Zij T Xij break that symmetry For angular sector p is integrable and exactly solvable when. later convenience we denote by ij the unit normal vector the following are satisfied. to the Zij coincidence hyperplane i The sector p tiles the N 2 sphere under reflec. tions across its boundaries The tiling covers the sphere. The specific angular ordering sector p is bounded by. with no gaps or overlaps and distinguishable sides In. the intersection of the sphere with the N 1 coincidence. other words the N 2 N 1 2 angles of a sector, hyperplanes Zp1 p2 Zp2 p3 ZpN 1 pN Each sector p has. ijk and ij kl define a spherical kaleidoscope,N 2 angles ijk of the form. ii The N 1 reflections across the bounding hyper,s planes Zp1 p2 Zp2 p3 ZpN 1 pN generate a finite. mj mi mj mk Coxeter reflection group The N 1 reflection. ijk arctan 3 normals p1 p2 p2 p3 pN 1 pN are the simple roots. of the Coxeter group, All finite reflection groups in all dimensions were. corresponding to the intersections of coincidence planes Zij. classified by Coxeter 41 63 Abstractly a Coxeter group. and Zjk that share a particle and N 3 N 2 2 angles of rank m is a finite group generated by m reflections where a. of ij kl 2 for the intersections of coincidence planes reflection is a group element that squares to the identity. Zij and Zkl that do not share a particle For four particles Every point symmetry group in m dimensions is either a. each ordering sector p ijkl is a spherical triangle Coxeter group or a subgroup of a Coxeter group of rank m. bounded by three great circles see Fig 1 For example the three dimensional point groups familiar. Solving the hyper spherical Helmholtz equation on an from chemical and solid state physics are all subgroups of the. angular sector p with Dirichlet boundary conditions is an Coxeter groups A3 tetrahedral symmetry C3 cubic sym. example of quantum billiards The problem of quantum metry and H 3 icosahedral symmetry or they are sub. and classical billiards in planar triangles is well studied groups of products of lower rank Coxeter groups. 54 62 and the integrability and solvability of the The structure of the reflection group can be encoded by. dynamics depends critically on the domain shape of the the Coxeter diagram which can be branching or non. billiards For example the only three triangular billiards in branching and connected or not connected There is a. a plane that have classically integrable dynamics are the family of N masses that determines a good sector for. three triangles with distinguishable sides that tile the plane every nonbranching and connected Coxeter reflection. under reflections without gaps or overlaps see footnote 3 group with rank N 1 These groups are listed in. of Ref 34 This serves as our guide for the following Table I Only the nonbranching Coxeter groups are rel. result for spherical quantum billiards The dynamics in an evant because in one dimension each pair can have at most. N L HARSHMAN et al PHYS REV X 7 041001 2017, TABLE I Connected nonbranching finite Coxeter reflection 1 1. groups For N particles each rank m N 1 Coxeter group A3 2. defines a one parameter family of masses for which the system is 3. exactly solvable For each group Gm the following data are 0 5 4. provided 64 the Coxeter bracket q1 qm from which one. determines the angles of the integrable sector the number of. reflections 0 in the group which determines the relative angular 0 2 0 4 0 6 0 8 1 0. momentum of the ground state solution and the order G of the 1. group which gives the number of integrable sectors required to C3. tile the sphere Note that there are two series of groups Am and Cm. that provide integrable mass families for any number of particles 0 5. N A series C series H type Others 0 25, 3 A2 I 2 3 C2 I 2 4 H2 I 2 5 I 2 q 0 2 0 4 0 6 0 8 1 0. 0 3 0 4 0 5 0 q H3,G 6 G 8 G 10 G 2q,4 A3 C3 H3,3 3 4 3 5 3 0 25. 0 6 0 9 0 15,0 2 0 4 0 6 0 8 1 0,G 24 G 48 G 120, 5 A4 C4 H4 F4 FIG 2 The integrable mass families for four particles with the. 3 3 3 4 3 3 5 3 3 3 4 3 Coxeter symmetries A3 C3 and H3 The mass fractions i. 0 10 0 16 0 60 0 24 mi M are plotted versus 4 legend is in the top graph The case. G 120 G 384 G 14400 G 1152 where all masses are the same i 1 4 is in mass family A3. Mass family C3 includes two cases where two finite masses are. 6 AN 1 CN 1 the same and H3 includes one,3N 1 4 3N 2. 0 N N 1 2 0 N 1 2,G N 1 G 2N N, R12 R23 and R34 All three generators square to the. identity and the relations, two adjacent pairs Geometrically this enforces that the. coincidence planes of nonadjacent pairs like Zij and Zkl. are orthogonal ij kl 2 We focus our attention on the R12 R34 2 R34 R23 3 R23 R12 q 1 4. connected Coxeter groups because they are relevant. when all masses are finite Disconnected graphs realize. limiting cases of extreme mass imbalances For example. for four particles if the first or fourth particle is much hold for q 3 q 4 and q 5 for A3 C3 and H 3. more massive i e like the Born Oppenheimer case of respectively Generally within each Coxeter group Gm. Ref 26 then the reducible Coxeter groups like I 2 q A1 there is a conjugacy class K Gm of all reflections. could be employed R Gm We denote the number of reflection planes and. For each group in Table I there exists a one parameter the order of K by 0 and denote the normals to these planes. family of mass sequences for which there are integrable by R but remember that only N 1 of these planes and. sectors The bracket notation for the Coxeter group normals are real i e they correspond to the actual. q1 q2 qN 2 determines the sector angle by Eq 3 coincidence planes and normals See Fig 3. where i i 1 i 2 qi In Fig 2 we show the integrable Note that for the Coxeter groups in the A series C series. mass spectra for the three four particle families A3 C3 and and the exceptional group F4 the sector angles are all 2. H3 This can be reversed given a set of N masses in a 3 or 4 Inspecting Eq 3 we see that for these cases the. particular order one could check how close the sector angles masses are all rational fractions of each other For example. derived from the masses come to the angles q1 qN 2 for the group C3 there are a countably infinite number of. that define a rank N 1 Coxeter group rational mass sequences that give integrable sectors The four. For N particles that define a good sector one that tiles the with the lowest rational denominators are given by. N 2 sphere the generators of the Coxeter group can be 3m m 2m 6m 10m 2m 3m 5m 12m 3m 5m 10m. chosen as the m N 1 reflections Rij across the boun and 56m 7m 9m 12m As we discuss below this allows. dary hyperplanes Zij of the sector p 12 N For N 4 the possibility of building integrable systems out of clusters. the Coxeter groups are rank m 3 and are generated by of particles with the same mass. INTEGRABLE FAMILIES OF HARD CORE PARTICLES PHYS REV X 7 041001 2017. the Calogero Moser model with inverse square interactions. in a harmonic trap with zero coupling constant 67 68. However unlike the equal mass solutions the nonequal. mass solutions cannot be considered as restrictions of. fermionic solutions to a single sector, In addition to the ground state Eq 5 the excited state. relative hyperangular wave functions in a Coxeter sector. are also constructed using a Bethe ansatz like superposi. tion of hyperspherical harmonics Hyperspherical harmon. ics are homogeneous polynomials in z i that are eigenstates. of the relative angular momentum L2rel with eigenvalue. FIG 3 The top row a c e depicts the arrangement of the N 3 69 70 The method takes advantage of. 0 reflection planes gray and colored disks and the tiling of the fact that reflections Rij commute with L2rel or in other. sphere into G spherical triangle sectors for A3 C3 and H 3 words the Coxeter group of rank N 1 is a subgroup of. respectively The bottom row b d f shows the coincidence the orthogonal transformations O N 1 see. planes colored disks for specific nonsymmetric choices of Appendix B Like Eq 5 excited solutions are first. masses within the mass families for A3 C3 and H3 respectively constructed over the whole sphere and then restricted to. The disk colors are the same as in Fig 1 The black spherical. the Coxeter sectors By construction the excited states are. triangle tiles the sphere in the top figure and it is similar to the. integrable sector 1234 in the bottom figure This sector is antisymmetric with respect to reflections in the Coxeter. bounded by the planes Z12 red Z23 cyan Z34 magenta group Not all values 0 for the relative angular. which are the generating planes for the Coxeter symmetry momentum support such solutions For the three groups. A3 C3 and H3 the allowed spectra of are,III EXACT SOLVABILITY AND A3 6 3n1 4n2 6a. BETHE ANSATZ INTEGRABILITY, For each Coxeter group there is therefore a one C3 9 4n1 6n2 6b. parameter family of masses such that the complete spec. trum of energy eigenstates can be exactly solved in the H3 15 6n1 10n2 6c. ordering sector 1 N and its inverted sector N 1 The. ground state in each of these Coxeter sectors is non In each case the first number in the sum is 0 the relative. degenerate and its hyperangular wave function can be angular momentum of the ground state and the number of. expressed as reflections in the Coxeter group Then the non negative. Y integers n1 and n2 label the excited states Degeneracies in. 0 z N 0 R z 5 the hyperangular d o f arise when multiple pairs of integers. R K provide the same and the pattern of degeneracies matches. the prediction of Weyl s law for a spherical triangle. where z z1 zN 1 is a unit vector expressed in see below The series of positive integers 3n1 4n2. hyperspherical coordinates and N 0 is a normalizing factor 4n1 6n2 and 6n1 10n2 in Eq 6 corresponds pre. For all R K the function from Eq 5 is reflection cisely to the orders of homogeneous polynomials that have. antisymmetric 0 Rz 0 z and therefore vanishes definite relative angular momentum and are symmetric. on all reflections planes including the coincidence planes under the action of reflections in the groups A3 C3 and H 3. Note that 0 0 z is the lowest degree anti invariant respectively 63 Incorporating the center of mass and. polynomial of the corresponding group 63 The function hyperradial d o f all energy eigenstates are uniquely. from Eq 5 is defined on the entire sphere but its identified by four quantum numbers fn n1 n2 g. restriction to the ordering sectors 1 N or N 1 provides For N 4 and general masses or for Coxeter masses but. that sector s ground state with energy 0 N 2 in an arbitrary order we believe the dynamics within. Exploiting separability a tower of states are laddered from sectors are not integrable in any sense In the case of. the ground state manifold with energies n 2 0 H3 we provide evidence by numerically solving the. N 2 where n is the center of mass excitation and is the spherical Laplacian for Coxeter masses in all sectors We. relative hyperradial excitation see Appendix A use the following method 31 Each spherical triangle is. For N equal masses the Coxeter group is AN 1 and the flattened into an isosceles right triangle The flattening. ground state corresponds to the lowest energy fermionic coordinate transformation distorts the spherical Laplacian. state in a harmonic trap restricted to a sector la into a new operator whose spectrum must be solved inside. Girardeau as expected 65 66 This equal mass solution the triangle with hard wall boundary conditions The. can also be seen as the limiting case of the ground state of spectrum is found by diagonalizing this transformed. N L HARSHMAN et al PHYS REV X 7 041001 2017, quantum ergodicity in the form of Wigner Dyson distri. butions for their eigenvalues, Our numerical results for the spacing statistics open. questions about the transition from integrability to ergo. dicity We perform numerical simulations on a variety of. integrable mass families While the characteristics of the. Coxeter sectors remains stable the other sectors sometimes. look closer or farther from Wigner Dyson distributions as. is already visible in Fig 4 We investigate several pos. sibilities for these intermediate distributions such as. integrable subclusters but we have not arrived at any. conclusive results We also investigate small random. deviations from integrable mass sectors of the order of. 5 For this scale of deviation the formerly integrable. sectors still look far from Wigner Dyson but closer to. Poissonian than the energy level statistics for exact Coxeter. masses Understanding the ragged edge between integra. bility and ergodicity using this model seems to be a. productive avenue for future investigation, To demonstrate that we find all the spectrum from this. procedure we compare our results for N E the total. number of energy eigenvalues below scaled energy E to the. prediction of Weyl s law 71 for a sphere, FIG 4 Unfolded spectrum statistics for H 3 Coxeter masses with 4 4. mass fractions 1 4 0 442 79 2 0 033 81 and 3, 0 080 61 There are only six different sectors because of two equal where E 2mR2 2 E and R and m are arbitrary length. masses m1 m4 and because p1 p2 p3 p4 is congruent to p4 p3 p2 p1 and mass parameters constrained by 2 2mR2 The. by inversion The variable s is the normalized unfolded energy level second term is the correction due to the Dirichlet boundary. difference 20 The integrable sector 1234 is depicted in the top conditions proportional to the boundary length l The area. left graph and agrees with the prediction from Eq 6c The blue of the spherical triangle ijkl is A R2 ijk jkl. lines depict the Poissonian statistics expected for an integrable ij kl Girard s theorem 72 The perimeter is. system the red lines are Wigner Dyson distribution derived from. random matrix theory expected for quantum ergodic systems with. l R ijk jkl ij kl where the vertex angles ijk, time reversal symmetric Hamiltonians The bottom graph shows jkl ij kl satisfy 73. the quality of Weyl s law Eq 7 in the integrable sector 1234 as. well as the nonintegrable sectors cos ijk cos jkl cos ij kl. sin jkl sin ij kl, Laplacian in a basis of exact solutions for the right triangle and cyclic permutations of ijk jkl ij kl and ijk jkl. Details about this procedure and its convergence are ij kl Figure 4 compares numerical solutions to this. described in Appendix C prediction, The level spacing statistics after the standard unfolding. 20 for a set of four particles with H3 mass ratios are IV LIOUVILLE INTEGRABILITY. depicted in Fig 4 The first sector depicted is the integrable AND SUPERINTEGRABILITY. sector whose numerical solution agrees with the prediction. of Eq 6c the other five sectors are for the same masses The separability of the model defined in Eq 1 provides. arranged in other orders For integrable sectors the four functionally independent integrals of the motion for. unfolded energy level statistics are expected to follow a N 3 particles with any masses the center of mass. Poissonian distribution In our case within the Coxeter Hamiltonian Hc m the relative Hamiltonian Hrel the total. sector the extra degeneracies of the system due to super angular momentum squared L2 and the relative angular. integrability distort the distribution so that it is peaked even momentum squared L2rel Three of these integrals. more strongly at zero energy gap 21 What we demon fHc m H rel L2rel g are in involution but L2 does not. strate in Fig 4 is that even for Coxeter mass families the commute with them This set of four integrals of motion. incorrectly ordered sectors show numerical evidence for is sufficient to prove integrability and superintegrability. INTEGRABLE FAMILIES OF HARD CORE PARTICLES PHYS REV X 7 041001 2017. but not maximal superintegrability for N 3 with any I 6 naturally commutes with the total Hamiltonian H The. masses For N 4 this is not enough to prove integrability resulting seven member set is classically functionally. which requires four integrals in involution nor is it enough independent and establishes maximal superintegrability. for superintegrability requiring at least five total conserved for the H3 model The scheme can readily be generalized. quantities and certainly not enough for maximal super to the other two three dimensional reflection groups A3. integrability requiring seven conserved quantities and C3 However no ready generalization to higher. Nonetheless we conjecture that our model is maximally dimensions exists for the Liouvillian sets because a priori. superintegrable for N 3 when in a sector with Coxeter the operators Jm do not commute between themselves. mass ratios Our evidence is the following 1 using the Finding Liouvillian sets for higher dimensional groups is a. method of images 55 74 the classical problem can be subject of future work Identifying and classifying the. shown to support closed orbits indicating maximal super maximal superintegrablty sets and ideally connecting them. integrability 2 the correspondence of our model as a to the known integrals for the Calogero Moser model. limiting case of certain CSM models which are maximally 52 53 76 is another ongoing project. superintegrable 52 53 The general construction of a set of. observables that provide maximal superintegrability for all V EXPERIMENTAL OUTLOOK. reflection groups in any number of spatial dimensions. At the moment three possible experimental applications. requires the methods of algebraic geometry This ongoing. of the models we consider in this article can be foreseen The. project will be the subject of a future publication. first possibility is the straightforward idea of finding a. Let us outline a potential strategy for searching for the. collection of atoms that naturally have the right mass ratios. missing integrals of motion for four particles using H3 as an. and seeking the signatures of integrability in the spectral. example The key role is played by the invariant poly. coherence and thermalization properties of the system Even. nomials of the group qm z1 z2 z3 with order m These are. if the particles are only close to a Coxeter family our. the lowest order homogeneous functionally independent. numerical results for the energy spectrum suggest that traces. polynomials that remain unchanged under any of the group. of integrability should still be present More generally the. transformations They are known and tabulated for all the. Coxeter criteria can be used to measure how far from. reflection groups 75 For H 3 there are three invariant. integrability particular arrangements of imbalanced masses. polynomials qm with order m 2 6 and 10 and up to a. are expected to be or whether there are integrable subclusters. normalization they are constructed as, X possible within a multispecies ultracold atomic gas. qm z1 z2 z3 z1 z2 z3 m 9 In the second scheme if real masses with the correct ratio. f g are not available the atomic mass is controlled using. optical lattices Given sufficient laser power the effective. where f g are the set of vectors describing the six fivefold mass 77 can be tuned from its bare value to almost zero. rotation axes of H3 From the three polynomials q2 q6 and 78 In particular the effective mass can be made 3 times. q10 we define the three operators Jm qm L12 L23 L31 greater than its bare counterpart in a lattice of a depth. Here Lij are the components of the vector of the relative V 0 7ER and 23 times greater for V 0 16ER respec. angular momentum in the ij plane Note that J2 is tively Here ER 2 k2 2m is the so called recoil. proportional to L2rel and so it does not give an additional energy k is the wave vector of light that creates the lattice. integral of motion and m is the bare atomic mass In both cases harmonic. However the operator J6 completes the commuting set confinement represents the most natural experimental. fHc m Hrel L2rel g to a Liouvillian set Since J6 commutes environment unlike the box and ring geometries tradition. with mirror reflections of the H3 group by Schur s lemma ally studied using Bethe ansatz methods. it must act as a multiple of the identity on the antisymmetric In the third scheme described in more detail in Ref 36. states It commutes with the previous three members of the the role of massive particles is played by bosonic solitons in. Liouvillian set and all four can be readily shown to be an atom waveguides 79 81 The solitons are made. functionally independent in the classical sense The five of atoms in two alternating internal states where the. member set fHc m L2 Hrel L2rel J 6 g now establishes intraspecies interaction is attractive and the interspecies. superintegrability for the H3 mass family interaction is repulsive The goal would be to engineer. This set can be further extended to a maximally super clusters of atoms whose combined masses satisfy the. integrable set using the operator J10 and another invariant Coxeter criteria For example the clustering pattern. operator I 6 defined by 3m m 2m 6m has C3 symmetry A mixture of 7Li atoms. with mF 1 and mF 0 in a magnetic field of 855 G, I 6 q6 a 1 a 2 a 3 q6 a1 a2 a3 10 constitutes an example 82. p Any implementation of the models considered in, where aj i zj izj 2 is an annihilation operator our article may constitute an efficient experimental. for the jth component of the relative motion The operator realization of spherical triangular or higher dimensional. N L HARSHMAN et al PHYS REV X 7 041001 2017, simplex shaped quantum billiards 83 The ergodicity of. classical flat triangular billiards is conjectured to strongly. depend on the rationality of the billiard angles 59 62. Numerically such questions about ergodicity are difficult. requiring long propagation for averages to converge to their. infinite time limits A study of the eigenstate to eigenstate. variance of the expectation values of observables, 23 84 85 which is a faithful quantum analogue of FIG 5 Coordinate transformation for three particles with the. classical deviations from ergodicity cf Ref 86 for a A2 masses m1 1 13M m2 9 52M and m 3 4M where. comparison may provide an efficient alternative to M is the total mass a Configuration space in natural particle. classical long time averages Experimentally one may positions x The gray ellipsoid represents an equipotential for the. conjecture an appearance of a memory of initial conditions equal frequency and therefore not equal strength and the red. if the billiard is not ergodic 87 The mass mixtures we green and blue disks represent the X 12 X 23 and X 31 coincidence. consider in our paper could constitute a way to study planes respectively b Configuration space in mass scaled and. rotated coordinates z Jx with a spherical equipotential and. multidimensional classical and quantum hard wall billiards. transformed coincidence planes Z12 Z23 and Z31 c Projection. with continuously tunable geometry a powerful extension of structure in b into relative coordinate z1 z2 plane The. of the existing experimental techniques 88 sector with angle 123 3 is the Coxeter sector. ACKNOWLEDGMENTS p,where ij mi mj mi mj and g ij gij ij This. This work was supported by the Aarhus University scaling transformation y Sx has brought the harmonic. Research Foundation the U S National Science potential into a form with N spherical symmetry but at the. Foundation Grant No PHY 1607221 the Office of cost of desymmetrizing the coincidence planes To describe. Naval Research Grant No N00014 12 1 0400 The the geometry we define the transformed coincidence planes. Danish Council for Independent Research Sapere, Y ij SXij with normals ij The contact interaction then. Aude and the Humboldt Foundation The authors thank. has the form,F F Bellotti M E S Andersen L Rammelm ller and. F Werner for useful conversations X,g ij ij y A2,APPENDIX A COORDINATE. TRANSFORMATIONS AND THE MAP TO The angle ijk between coincidence planes Y ij and Y jk with. QUANTUM BILLIARDS normals ij and jk is now, Here we establish the map from the model Hamiltonian s. from Eq 1 to a free particle in a bounded region on the mj mi mj mk. ijk arctan A3, N 2 sphere Much of this is well known 31 89 but we mi mk. reproduce it here for the readers convenience and to. establish notation Whereas in the equal mass case ijk is always 3 for. The equipotentials of Eq 1 are N dimensional ellip three arbitrary masses it can range from 0 mj much lighter. soids segmented into N sectors by N 1 dimensional than the other two masses to 2 mj much heavier The. hyperplanes Xij defined by the particle coincidences. angle ij kl between coincidence planes Y ij and Y kl that do. xi xj 0 see Fig 5 In the limit g these planes,not share a particle remains 2. are impenetrable The angle between coincidence hyper In the limit g the Hamiltonian from Eq A1. planes Xij and Xjk that share a particle is 3 possible for separates. only N 3 the angle between hyperplanes Xij and Xkl P in hyperspherical coordinates with radius. R2 i y2i The interaction term Eq A2 is proportional. that do no share a particle is 2 possible for only N 4 to 1 R so it is not separable for finite values of g ij but as. As a first step we scale the position coordinates, g ij there is no distinction between 1 R or 1 R2 times. unitless position variables yi mi xi Then the, the sum of delta functions So hyperspherical symmetry of. Hamiltonian from Eq 1 becomes, Eq A1 emerges and there is SO 2 1 dynamical symmetry. N in the total hyperradial coordinate R 89 To develop. H 2 yi physical intuition it is sometimes useful to imagine a single. 2 i 1 yi classical particle bouncing around in this N dimensional. X r r landscape In this mass rationalized geometry the classical. g ij yi y A1 particle trajectory changes its direction of angular momen. tum when it bounces off of a coincidence hyperplane Y ij but. INTEGRABLE FAMILIES OF HARD CORE PARTICLES PHYS REV X 7 041001 2017. it does not change its magnitude of angular momentum in angular potential g ij 0 then would be a non negative. configuration space However total angular momentum integer the eigenfunctions on S N 2 would be the hyper. does not respect the center of mass separability and does spherical harmonics and would be a collective index to. not commute with the relative Hamiltonian or relative label degeneracies 69 70 However since there is an. angular momentum see below so we do not exploit it here angular potential in Eq A6 the hyperangular solutions. Next we rotate the coordinate system z Jy so that the are unknown and we must explicitly solve for including. component z Z is the scaled center of mass Z any possible degeneracies Whatever value takes. i yi mi M and M is the total mass The orthogonal, including noninteger values the relative hyperradial. transformation J with this property is not unique and its function R is the standard solution for the radial. selection determines a particular choice for Jacobi relative factor of an N 1 dimensional isotropic harmonic. coordinates z1 through zN 1 The transformation J also oscillator in hyperspherical coordinates 70 with energy. rotates the coincidence planes Zij JY ij and their normals 2 N 1 2. ij J ij but leaves the angles between planes like ijk N 3 2 2. R A L 2 e 2 A8, and ij kl invariant Since all the normal vectors ij have. zero Z components the Hamiltonian in z coordinates where A 2 N 1 2. separates into H H c m Hrel where Hc m is the We have achieved our desired result this series of. Hamiltonian for a one dimensional harmonic oscillator coordinate transformations has reduced solving the N. in the center of mass Z coordinate and the relative particle Hamiltonian from Eq 1 with equal frequencies. Hamiltonian is and infinite strength contact interactions into solving hard. N 1 X wall quantum billiards in N 2 simplexes on N 2. Hrel 2 zi g ij ij z A4 spheres,2 i 1 zi i j, Finally we go to hyperspherical coordinates in the APPENDIX B CONSTRUCTION OF EXACT. relative space where the relative hyperradius is SOLUTIONS. N Here we construct the wave functions within the Coxeter. 2 z2i y2i Z2 A5 sectors i e sectors of the S N 2 hypersphere defined by the. i 1 i 1 relative hyperangular coordinates that have the right shape to. tile the sphere under reflection For convenience we choose. and there are N 2 angles charting the sphere S N 2 the Coxeter sector to be the ordering sector 12 N so that it is. conventionally chosen as f 1 N 3 g with bounded by the coincidence hyperplanes Z12 through. 0 2 and i 0 The relative Hamiltonian now Z N 1 N These N 1 hyperplanes define the reflections. R12 though R N 1 N that generate the Coxeter group. 1 N 2 1 2 The Coxeter group Gm is generated by m reflections in m. Hrel N 2 2, 2 dimensions As such it can considered as a subgroup of. X g ij O m orthogonal transformations in m dimensions and the. ij z A6 symmetry of the sphere S m 1 To summarize the method. there is a solution to the Hamiltonian in the Coxeter sectors. 12 N and N 21 whenever an irreducible representation. where is the angular part of the Laplacian in relative. irrep of GN 1 that is antisymmetric under all reflections. configuration space As before the relative Hamiltonian. appears in the decomposition of an irrep of O N 1 The. Eq A6 is separable in the limit g ij A general, irreps of O m generally are reducible with respect to. energy eigenstate can be separated into a product of center the subgroup Gm The method of characters can answer the. of mass Z relative hyperradial R and relative question as to whether an irrep of a subgroup appears in the. hyperangular functions decomposition of the irreps of the group When it does. n Z n Z R A7 exist the corresponding states can be constructed using. projection operators and in the case of degeneracies an. where n is the center of mass quantum number the orthonormalization procedure. function n Z is the one dimensional harmonic oscillator The irreducible representations for O m and their. wave function and is the relative hyperradial quantum realizations by hyperspherical harmonics are well known. number At this point is just derived from the judiciously 90 91 and so we just summarize a few facts here for the. parametrized relative hyperangular separation constant readers convenience The subgroup SO m is a Lie group. N 3 and is just an additional label to distinguish with m m 1 2 generators in the Lie algebra We denote. any possible degenerate states for a given If there was no these generators as Lij for i j with i j f1 mg. N L HARSHMAN et al PHYS REV X 7 041001 2017, where Lij generates a rotation in the ij plane The quadratic TABLE II Conjugacy classes of H3 The first column is the. Casimir of SO m is the sum of all of these generators Sch nflies notation for the elements in the class K i The second. squared column gives the angle of rotation i of the element realized in O. 3 The third column is whether it is generated by an even or odd. X number of reflections i 1 All odd elements that are. L2 L2ij rotations can be considered as rotoreflections i e a rotation. hi ji followed by a reflection in the plane perpendicular to the rotation. axis The fourth and fifth columns are the order of the elements in. For SO 3 this is the familiar angular momentum squared the class K i and the number of elements ki in the class. operator with eigenvalues 1 The SO 3 irreps are respectively. labeled by and have degeneracy 2 1 In m 3,Elements Angle Even or odd Order Number. dimensions the operator L2 is the hyperangular momentum. squared operator with eigenvalue m 2 and an irrep E 0 1 1. C5 C45 2 5 5 12,labeled by has degeneracy 69,C25 C35 4 5 5 12. C3 C23 2 3 3 20,S10 S910 5 10 12, Once the representation of total inversion is chosen the S310 S710 3 5 10 12. irreps of SO m also naturally carry a representation of S6 S56 3 6 20. O m For example inversion is represented in the irrep 0 2 15. by multiplication by 1 for O 3,To reduce an O m irrep into the irreps of Coxeter. group Gm the G elements are sorted into conjugacy classes As an example consider the Coxeter group H 3 This. K i with ki elements Each irrep W of Gm has a unique group has ten conjugacy classes summarized in Table II. pattern of characters W K i In particular we are interested The H 3 character A K i for the anti invariant irrep is 1. in the Gm irrep W A of all anti invariant states meaning for the five even conjugacy classes and 1 for the five odd. A K i 1 when K i is a conjugacy class whose elements classes The O m character K i for irrep is. are an even composition of reflections and A K i 1 X. when K i is a class composed of odd compositions Further K i cos m i i B3. each conjugacy class has a character K i in the Gm. reducible O m irrep denoted by When these characters where i is the angle of rotation and i is the reflection. are known then the number of times the Gm irrep A appears parity for the conjugacy class K i Plugging this into. in the decomposition of the O m irrep is 90 Eq B1 we find the pattern of degeneracies given in. Eq 6c of the main text The same method is used in, 1X A Ref 92 to find which O m irreps have symmetric irreps. a k K i K i B1 of the spherical triangle groups in their reduction. i To construct the actual states we use the, projection operator Eq B2 acting on the spherical har. Note that A K i A K i because Coxeter groups are monics Instead of explicitly constructing the 2 1. ambivalent The number a is an integer that counts how 2 1 unitary matrices D g that act on the spherical. many solutions there are with relative angular momentum harmonics Y for each of the 120 elements of H3 we. The projection operator onto the anti invariant irrep A is choose a slightly different method that takes advantage of. given by two facts 1 the spherical harmonics can be written as. polynomials of the relative coordinates and 2 we already. 1 X A have the 3 3 matrices O g O 3 that represent H3 as. PA g D g B2,G g G rotation and reflections, The first step is to express the spherical harmonics in. where D g is the representation of group element g acting terms of the relative coordinates z1 z2 z3 We work with. irreducibly on the d dimensional representation space If the real form of the spherical harmonics defined as. a 1 any vector in the irrep space with a nonzero 8 p. 2N P cos cos 0, projection will be proportional to the solution we seek If. a 1 then a set of orthonormal solutions can be found by 0. Y N 0 P cos 0, projecting multiple vectors and then applying Gram p j j. Schmidt orthogonalization 2N P cos sin j j 0, INTEGRABLE FAMILIES OF HARD CORE PARTICLES PHYS REV X 7 041001 2017. where problem for arbitrary masses a particular rotation J in. s z JSx must be specified For numerical simplicity we. 2 1 j j choose the H type four body coordinates 93 Then the. N rotation J aligns the coordinate plane Z12 with the plane. z1 0 and aligns the coordinate plane Z34 with z2 0 see. Fig 1 The other four coincidence planes are given by the. The real spherical harmonics can be written in terms of the. following equations,relative coordinates,Y z Y cos 1 z3 tan 1 z2 z1 m2 m3 m4 m4 m1 m2. Z13 z1 z2 z3 0,Noting the relations m1 M m3 M,k k k m2 m3 m4 m3 m1 m2. z22 2 z1 z2 cos Z14 z1 z2 z3 0,2 m1 M m4 M,k m1 m3 m4 m4 m1 m2. sin j j z21 z22 2 z1 zk2 sin Z23 z1 z2 z3 0,k 0 k 2 m2 M m3 M. P x 1 z21 z22 2 P z m1 m3 m4 m3 m1 m2,d z3 3 Z24 z1 z2 z3 0. we can show that Y z are homogeneous polynomials, of order in z1 z2 z3 The numerical problem is solved independently in each. The projection Eq B2 is applied using the trans sector p1 p2 p3 p4 and checked for consistency with the. formation matrix O g similar sector p4 p3 p2 p1 In the following as an example. we solve the sector 1342 which is limited by Z34 Z13 and. PA Y z g Y O g z B4 Z24 coincidence planes We isolate z1 in the above. G g G equations and introduce nonstandard spherical coordi. nates with z1 cos z2 sin cos and z3, This projection will be zero unless is in the spectrum sin sin so that cos 0 in sector 1342 Then. given by Eq 6c in the main text Note that it may also be substituting spherical coordinates into the coincidence. zero for any particular but there must be as many linearly plane equations and dividing by z1 cos 0 we find. independent polynomials that also solve the spherical. Laplacian as there are solutions for n1 and n2 for a given tan cos 0. in Eq 6c For H 3 the first time there are multiple. solutions is when 45 Explicit calculation for 0 1 a tan cos b tan sin 0. 15 confirms Eq 5 from the main text for the ground state 1 c tan cos d tan sin 0 C2. of H 3 and we perform the same calculations for the other. N 4 Coxeter groups with,APPENDIX C NUMERICAL METHOD m1 m2 m1 m4 Mm1. FOR SOLVING SPHERICAL TRIANGLE a b,m3 m4 m3 m2 m3 m4 m2. Here we present the numerical method to calculate the s s. energy spectrum for N 4 particles with arbitrary masses m1 m2 m2 m3 Mm2. which is an extension of the method found in Ref 31 to m3 m4 m1 m4 m3 m4 m1. spherical triangles The general strategy is outlined in the. main text here we provide details about the coordinates Introducing new coordinates u tan cos and. the flattening and the exact diagonalization we use to v tan sin the boundary conditions. construct Fig 4, After separation of variables we must solve for the u 0. hyperangular wave function within a sector p This,function must satisfy 1 au bv 0. 1 C1 1 cu dv 0, with Dirichlet boundary conditions on the three bounding are simple and describe a triangle in flat space However. coincidence planes Zp1 p2 Zp2 p3 and Zp3 p4 To solve the the differential operator has become more complicated. N L HARSHMAN et al PHYS REV X 7 041001 2017, 2 2 excited state energies then one needs to increase N max in. 1 u v 1 u2 2 1 v2 2, order to get a better precision at the higher end of the. 2uv 2u 2v C3 Additionally numerical results are compared to the exact. u v u v algebraic results for the integrable Coxeter sector for. several mass families in A3 C3 and H3 to confirm the. Next we introduce a final coordinate transformation uncertainty estimates And as we describe in the main text. we compare the level density of the spectrum to the. s au bv prediction of Weyl s law in order to establish that all. b d b d eigenstates are found by this method, This can be inverted as 1 S Albeverio F Gesztesy R Hoegh Krohn and H Holden. Solvable Models in Quantum Mechanics Springer Verlag. b d Berlin 1988, 2 ad bc 2 B Sutherland Beautiful Models 70 Years of Exactly Solved. 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