An exactly solvable quantum four-body problem associated with the symmetries of an octacube
aa r X i v : . [ m a t h - ph ] A ug An exactly solvable quantum four-body problem associated with the symmetries of anoctacube
Maxim Olshanii ∗ Department of Physics, University of Massachusetts Boston, Boston Massachusetts 02125, USA
Steven G. Jackson
Department of Mathematics, University of Massachusetts Boston, Boston Massachusetts 02125, USA (Dated: September 17, 2018)In this article, we show that eigenenergies and eigenstates of a system consisting of four one-dimensional hard-core particles with masses 6 m , 2 m , m , and 3 m in a hard-wall box can be foundexactly using Bethe Ansatz. The Ansatz is based on the exceptional affine reflection group ˜ F associated with the symmetries and tiling properties of an octacube —a Platonic solid unique tofour dimensions, with no three-dimensional analogues. We also uncover the Liouville integrabilitystructure of our problem: the four integrals of motion in involution are identified as invariantpolynomials of the finite reflection group F , taken as functions of the components of momenta . Introduction .– The relationship between exactly solv-able quantum one-dimensional multi-particle problemsand kaleidoscopes has been long appreciated [1]. A kalei-doscope is a system of mirrors where none of the multi-ple images of the original object is broken at the mirrorjunctions. A viewer situated inside a kaleidoscopic cav-ity has no means of distinguishing the images of objectsfrom the original. For a broad class of boundary condi-tions, the eigenmodes of a kaleidoscopic cavity are repre-sented by finite superpositions of plane waves; they canbe found exactly , using the method of images [2]. Whena many-body problem reduces to a solvable kaleidoscope,the resulting solution is known as the (coordinate) BetheAnsatz solution [3, 4]. The list of particle systems solv-able using Bethe Ansatz is so far exhausted by: equalmass hard-core bosons, on an open line, on a circle orin between walls [5]; δ -interacting, equal-mass, generallydistinguishable particles on a line and on a circle [6–8];this includes bosons [9], which could also be in the pres-ence of one or two walls [1]; systems of two hard-coreparticles with masses m and 3 m interacting with a wall,for both wall- m -3 m and wall-3 m - m spatial orders werebriefly commented on in [6], but discarded as inferior tothe problems with finite strength interactions.It has been long conjectured that no exceptional—specific to a given number of spatial dimensions—kaleidoscopes lead to solvable particle problems with re-alistic interactions [1, 3]. The search for physical real-izations was limited to systems of particles of the samemass , with a possible addition of immobile walls. In thisLetter, we show that the exceptional closed kaleidoscope˜ F induces a novel quantum integrable system: four one-dimensional quantum hard-core particles with different masses, 6 m , 2 m , m , and 3 m , in a hard-wall box. The so-lution utilizes the symmetries of an octacube, a Platonicsolid unique to four dimensions. Identifying the particle problem generated by the Cox-eter diagram ˜ F .— Consider six hard-core (i.e. impene-trable) particles on a line, with masses m , m , m , m , x =- L x = x x = x x = x x =0
12 36
60… 60… 45… 60… y =03 y = y z = z + z + z z =0 z = z z = z z + z =1 ! y = 2 y ! y = 3 y ! y =-1 !
12 36 12 36 12 36 12 36 L L - L - L - Lm m m m m m m m m m m m m m m m m m m m FIG. 1. The ˜ F kaleidoscope and its particle realization.From top to bottom: the affine Coxeter diagram ˜ F ; the an-gles between the mirrors; the four particles with masses 6 m ,2 m , m , and 3 m between two hard walls at x = − L and x = 0; the particle-particle and particle-wall hyperplanes ofcontact in all three coordinate systems used in the main text. m , and m . Their coordinates will be denoted as x , x , x , x , x , and x respectively. The natural coordinatetransformation x i = r M m i y i , for i = 0 , , . . . , M (to be fixed later) whose motion is constrainedby the following set of five inequalities: y i +1 / √ m i +1 >y i / √ m i , for i = 0 , , . . . , F [10] de-picted in the top line of Fig. 1. It encodes the geometry ofa particular simplex-shaped 4-dimensional kaleidoscope.The vertices label its five mirrors: the 3-faces (repre-sented by 3-dimensional simplexes) of the 4-dimensionalsimplex. The edges—the links—of the diagram (andtheir absences) encode the angles between the corre-sponding mirrors, i.e. the angles between their normals.When two vertices are not linked by an edge, the respec-tive mirrors are orthogonal to each other. An edge withno numbers above it corresponds to an angle of π/
3. Anumber n above an edge would produce an angle of π/n between the corresponding mirrors. The meaning of themark inside the leftmost vertex will be explained later.In our case, the non-right angles between the mirrors are π/ π/ π/
4, and π/
3, in the order of their appearanceon the Coxeter diagram, left to right.Let us try to identify the five consecutive vertices ofthe Coxeter diagram with the five hyperplanes of con-tact between neighboring particles: y i +1 √ m i +1 = y i √ m i for i = 0 , , . . . , i , i + 1, and i + 2, theangle between the i vs i + 1 and i + 1 vs i + 2 contacthyperplanes is given by θ i ( i +1) ( i +2) = arctan s m i +1 ( m i + m i +1 + m i +2 ) m i m i +2 , (see e.g. [6]). Notice that these angles do not dependon the overall mass scale. Therefore, any constraints onthese angles can only lead to relationships between thefive mass ratios, m i +1 /m i for i = 0 , , , , F kaleidoscope, theangles between the hyperplanes of contact must satisfyfour equations only, θ = π θ = π θ = π θ = π , thus being seemingly underdetermined. Nevertheless,once the non-negativity of the masses ( m i ≥ i = 0 , , , , ,
5) is invoked, a single solution (up toan overall scale m ) survives: m = + ∞ ; m = 6 m ; m = 2 m ; m = m ; m = 3 m ; m = + ∞ . In retrospect, that both the leftmost and the rightmostmass should be infinite could have been predicted fromthe onset: this is probably the only way to map six par-ticles on an open line to a single 4-dimensional particlein a closed cavity.Without loss of generality, we can place the infinitelymassive particles at x = − L ; x = 0 , where L is the size of the resulting hard-wall box (seethe hyperplanes of contact in terms of the original x -coordinates in Fig. 1). Now observe that m = m and L appear to be convenient mass and length scales, and ~ anatural scale of action. From now on, we will be usingthe system of units where m = L = ~ = 1 . Also, it will become clear from what follows that thechoice M = 2 m allows us to respect the existing con-ventions on the size of the octacube.Figure 1 also shows the corresponding formulae for thecontact hyperplanes expressed through the transformedcoordinates y , y , y , and y . These hyperplanes formthe boundaries of the ˜ F kaleidoscope we have in mind: − < √ y ; y < √ y ; y < √ y ; √ y < y ; y < z = ˆ T z ← y · y , with therotation-inversion matrix ˆ T z ← y given byˆ T z ← y = 12 √ −√ −√ − −√ −√ √ √ − √ √
30 0 − √ , brings the domain of our ˜ F kaleidoscope to the conven-tional form [2, 3, 11–15]: D ˆ e ≡ n z such that α · z > − α · α = − α , , , · z > o , (1)where y ≡ ( y , y , y , y ), and z ≡ ( z , z , z , z ).According to this convention, the kaleidoscope (1)is a 4-dimensional simplex with five 3-faces (mirrors)defined by five inward normals η j = α j / | α j | for j = 0 , , , ,
4, with α = ( − , − , , α =(0 , +1 , − , α = (0 , , +1 , − α = (0 , , , +1), α = (+ , − , − , − ) being the so-called minimal(or negative of maximal) root ( j = 0) and the simpleroots ( j = 1 , , ,
4) of the corresponding finite reflectiongroup F [16]. The mirrors from 1 to 4 pass through theorigin; the 0th mirror (marked by a dot in the Coxeter di-agram of Fig. 1) passes through the point (1 / , / , , e in K ˆ e has the meaning of the identity element of the finitegroup F , a convention whose meaning will become clearfrom what follows. Note that the “natural coordinates” z can be expressed through the particle coordinates x as z = ˆ T z ← x · x , withˆ T z ← x = 112 − − − − − − − . Periodic tiling by consecuitive reflections .— The keyto Bethe Ansatz solvability of the closed kaleidoscopicbilliards lies in their ability to periodically tile the fullspace—with neither holes nor overlaps—via consecutivereflections about their faces [2, 17]. Consider the ˜ F kalei-doscope (1). A union of the figures produced by sequen-tial reflections of (1) about its 3-faces from the 1-st tothe 4-th (“simple root” mirrors), C ≡ ∪ ˆ g ∈ F D ˆ g = n z such that (ˆ g α ) · z > −
12 (ˆ g α ) · (ˆ g α ) = − , for all ˆ g ∈ F o , (2)with D ˆ g ≡ n z such that (ˆ g α ) · z > −
12 (ˆ g α ) · (ˆ g α ) = − g α , , , ) · z > o , (3)leads to the so-called octacube, otherwise known as the24-cell [10]; it is the only 4-dimensional Platonic solidthat does not have any 3-dimensional analogues. Its24 vertices lie at points given by all coordinate per-mutations and sign choices applied to ( ± , , ,
0) and( ± , ± , ± , ± ). Its 24 octahedron-shaped 3-faces arecentered at all coordinate permutations and sign choicesof ( ± , ± , , F kaleidoscope form the sym-metry group of the octacube, the reflection group F .(The same label is used to mark both an open kaleido-scope, where the 0th, “minimal root” mirror is absentand the corresponding simple, “non-affine” Coxeter di-agram.) The union in (2) runs over all elements ˆ g ofthe group F . The “physical” domain (1) is the identitymember of the set (3).The 4-dimensional space can be periodically tiledby identical octacubes; the centers of the octacu-bical cells lie on all points with integer coordi-nates whose coordinates sum to an even number: { ( z , z , z , z ) : z + z + z + z = even } . The verysame tiling can be obtained by a consecutive reflectionof the (1) simplex about all five of its mirrors, includingthe “minimal root” one. Transformations generated bythese reflections form the symmetry group of the octacu-bical tiling— the affine reflection group ˜ F , homonymousto the corresponding closed kaleidoscope.In order to facilitate the possible classification of en-ergy levels by symmetry, we choose the lattice vectors a , , , to be proportional to the four simple roots of thereflection group D [16], closely associated with F : a =(1 , − , , a = (0 , , − , a = (0 , , , − a = (0 , , ,
1) . The reciprocal lattice vec-tors are thus κ = (2 π, , , κ = (2 π, π, , κ = ( π, π, π, − π ), and κ = ( π, π, π, π ). Finding the eigenenergies and eigenstates of theproblem .— In general, finding the eigenenergies andeigenstates of a D -dimensional kaleidoscopic billiardamounts to solving a system of D nonlinear algebraicequations—the so-called Bethe Ansatz equations [2, 3,11–15]. However, in the case of hard-core interactions, the problem greatly simplifies (see Corollary 3.2 in [13]).The eigenstates acquire the form ψ ( z ) = 1 p V D | G | X ˆ g ∈ G ( − P (ˆ g ) exp[ i (ˆ g k ) z ] , (4)where ˆ g are the members of the finite reflection group G (= F in our case) generated by sequential reflectionsabout the “simple root” mirrors of the kaleidscope inquestion; P (ˆ g ) is the parity of the number of the ele-mentary reflections needed to reach ˆ g from the identitytransformation; V D = V C / | G | (= 1 /
576 in our case) isthe kaleidoscope volume; V C = | det[( a , a , a , . . . )] | is the unit cell volume; | G | (= 1152 for F ) is theorder, i.e. the number of elements, of the group G .The conjecture contained in Theorem 1 in [15] indi-cates that the wavefunction (4) is normalized to unity: R D ˆ e dz dz dz . . . | ψ ( z ) | = 1 when integrated over thecorresponding kaleidoscope D ˆ e ((1) in the case of ˜ F ).This conjecture covers not only the case of the hard-wallboundary conditions but also a much more general classof boundary conditions, associated with finite strengthinteractions. The wavevectors k are simply drawn fromthe reciprocal lattice of the corresponding tiling [18]. Inorder to prevent both double counting and the formal ap-pearance of the eigenstates that are identically zero, thewavevectors k must be further restricted to those lyinginside the open kaleidoscope associated with the kalei-doscope in question, obtained by removing the “minimalroot” mirror: η j · k > j = 1 , , , . . . (see the endof Sec. 3 of [13]).For the case of the hard-wall boundary conditions, thenormalization formula used above can also be provendirectly without resorting to the conjecture [15]. Ob-serve that the octacube (2) is a unit cell of the lat-tice spanned by the lattice vectors a , , , and that the“seed” wavevector k , along with all its images ˆ g k belongto the respective reciprocal lattice. In that case, any twodistinct plane waves with wave vectors ˆ g k and ˆ g ′ k willbe orthogonal to each other if integrated over the wholecell: h ˆ g ′ k | ˆ g k i C = δ ˆ g ′ , ˆ g V C , with h k | k i A ≡ Z A d d z exp[ i ( k − k ) z ] , where A is an area of space. The normalization integralin question can now be evaluated as Z D ˆ e d d z | X ˆ g ∈ G ( − P (ˆ g ) exp[ i (ˆ g k ) z ] | = X ˆ g ′ X ˆ g ( − P (ˆ g ′ )+ P (ˆ g ) h ˆ g ′ k | ˆ g k i D ˆ e ˆ g ′′ ≡ ˆ g − ˆ g ′ = X ˆ g ′′ X ˆ g ( − P (ˆ g ˆ g ′′ )+ P (ˆ g ) h ˆ g ˆ g ′′ k | ˆ g k i D ˆ e = X ˆ g ′′ ( − P (ˆ g ′′ ) X ˆ g h ˆ g ˆ g ′′ k | ˆ g k i D ˆ e ˆ g ′′′ ≡ ˆ g − = X ˆ g ′′ ( − P (ˆ g ′′ ) X ˆ g ′′′ h ˆ g ′′ k | k i D ˆ g ′′′ = X ˆ g ′′ ( − P (ˆ g ′′ ) h ˆ g ′′ k | k i C = X ˆ g ′′ ( − P (ˆ g ′′ ) δ ˆ g ′ , ˆ e V C = V C = V D | G | ;this result justifies the choice of the normalization fac-tor in the expression (4). Above, we used ( − P (ˆ g ˆ g ′′ ) =( − P (ˆ g ) ( − P (ˆ g ′′ ) and ( − P (ˆ g ) = 1.We are now in the position to write down the spec-trum of our system explicitly. Starting from this pointwe abandon the m = L = ~ = 1 system of units.The eigenenergies, E = ( ~ / M ) k , and the eigen-states (4), with k = P j =1 , , , n j κ j , n j =1 , , , ∈ Z , and η j =1 , , , · k >
0, are then given by E n n n n = π ~ m L (2 n ( n + n + n + n )+ (5) n + n + n + n n + n n + n n (cid:1) Ψ n n n n ( x ) (6)= 1 √ L X ˆ g ( − P (ˆ g ) exp[ i (ˆ g k n n n n ) · ˆ T z ← x · x ]for n ≥ n > n > n ≥ , where k n n n n = n + 2 n + n + n n + n + n n + n − n + n πL ;the sum in (6) runs over all 1152 members ˆ g ofthe reflection group F [19] (transformations gen-erated by consecutive reflections about the four“simple root” mirrors of the ˜ F kaleidoscope);Ψ n n n n ( x ) ≡ q | det[ ˆ T z ← x ] | ψ n n n n ( ˆ T z ← x · x )are the eigenstates of the system expressed throughthe particle coordinates x , x , x , x , normalized as R x = − L R x = x R x = x R x = x d x | Ψ n n n n ( x ) | = 1; (cid:0) D z D x (cid:1) is the Jacobian matrix of the z ← x transforma-tion. FIG. 2. (color online). Exact energy spectrum for four parti-cles of mass m = 6 m , m = 2 m , m = m , and m = 3 m in ahard-wall box of length L (red solid line). Weyl’s law predic-tion for the spectrum of this system (blue dashed line).Theenergies up to 2 × ~ /m L are shown. At high energies, the spectrum converges to the Weyllaw prediction for the number of states with energies be-low a given energy E : N ( E ) = W ( E )(2 π ~ ) D =4 = m L E π ~ ,where W ( E ) is the classical phase-space volume occu-pied by points with energies below E , and D is numberof spatial dimensions (see Fig. 2).The ground state energy is E ground state = E n =3 , n =1 , n =1 , n =2 = 13 π ~ m L . (7)The ground state wave function can be obtained fromEq. (6), using k ground state = k n =3 , n =1 , n =1 , n =2 = πL . (8)Fig. 3 shows a particular section of the ground state den-sity distribution within the ˜ F simplex, along with itsspace-tiling mirror images. Integrals of motion .— The Bethe Ansatz integrabilityof our system can be also reinterpreted in terms of theLiouville integrability. To construct the three additionalintegrals of motion in involution with the Hamiltonianand each other, we suggest invoking the invariant poly-nomials of the group in question [20]: finite polynomi-als of coordinates w ( z , z , z , . . . ) that remain invariantunder the group action, w (ˆ g z ) = w ( z ) . (9)Consider now an operator ˆ I that is constructed by takingan invariant polynomial w as a function of the momenta associated with the z coordinates:ˆ I ≡ w ( − i ∇ z ) . FIG. 3. (color online). The ground state of our four-bodysystem. The plot is performed using the z coordinates, rele-vant to the tiling. The image is produced by first intersectingthe 4-dimensional density distribution by the 3-plane and the3-sphere indicated in the upper left and upper right corners,respectively (the formulas, however, are given in x coordi-nates, in which the 3-sphere becomes a 3-ellipsoid). The re-sult is a 2-sphere centered at the origin. Considering now this2-sphere as living in the usual 3D space, we pick, in this 3Dspace, a (2D) plane that does not intersect the 2-sphere, andproject onto it the 2-sphere’s hemisphere closest to the plane;this projection is what is shown in the image. The circleson the 2-sphere are the sections of the 3D hyperplanes of theoriginal 4D space; the great circles come from the hyperplanesthat pass through the origin, and the other circles come fromthe hyperplanes that do not. The lighter triangle in the mid-dle corresponds to a section of the “physical” 4-dimensionalsimplex to which the particle coordinates are bounded; the re-mainder of the sphere is a section of the entire 4-dimensionalspace when this space is octacubically tiled with the mirrorimages of the original “physical” simplex. The existence ofthis tiling ensures the applicability of the Bethe Ansatz [2]. Let us show that any eigenstate (4) of the kaleidoscopeassociated with the group in question is at the same timean eigenstate of ˆ I . Indeed,ˆ Iψ ( z ) = w ( − i ∇ z ) X ˆ g ( − P (ˆ g ) exp[ i (ˆ g k ) z ] = X ˆ g ( − P (ˆ g ) w (ˆ g k ) exp[ i (ˆ g k ) z ] = w ( k ) X ˆ g ( − P (ˆ g ) exp[ i (ˆ g k ) z ] , Q.E.D. For the reflection group F , the four lowest power functionally independent invariant polynomials read [21]: w ( F ) M ( z ) = ( z − z ) l M + ( z + z ) l M + ( z − z ) l M +( z + z ) l M + ( z − z ) l M + ( z + z ) l M +( z − z ) l M + ( z + z ) l M + ( z − z ) l M +( z + z ) l M + ( z − z ) l M + ( z + z ) l M (10) l M =1 = 2; l M =2 = 6; l M =3 = 8; l M =4 = 12 . Accordingly, the four fundamental integrals of motion forour system readˆ I ( F ) M = w ( F ) M ((( ˆ T z ← x ) − ) ⊤ · ˆ p x ) (11) M = 1 , , , , with ˆ p x ≡ ( − i ~ ∂∂x , − i ~ ∂∂x , − i ~ ∂∂x , − i ~ ∂∂x ) (12)being the particle momenta, corresponding to the particlecoordinates x , x , x , x . Above, ( . . . ) ⊤ stands for thetranspose of a matrix. Note that the first integral ofmotion is proportional to the Hamiltonian of the system:ˆ I ( F )1 /m = 144 ˆ H . Summary and outlook .– In this Letter, we obtain—using Bethe Ansatz—an exact expression for the eigenen-ergies and eigenfunctions of four hard-core particles withmass ratios 6 : 2 : 1 : 3 in a hard-wall box. The Ansatzis induced by a hidden symmetry of the system relatedto the symmetries of the tiling of a 4-dimensional spaceby octacubes. The exact spectrum stands in good agree-ment with the approximate Weyl’s law prediction.The following observation may serve as a seed for alonger research program. The procedure, outlined in thisLetter, for identifying a few-body problem relevant to aparticular Coxeter diagram [19] is not unique to ˜ F . Anydiagram, affine or not, that does not have bifurcations can potentially be used to generate a solvable few-bodyhard-core problem. The ˜ A N − ( N identical hard-cores ona circle) and ˜ C N ( N identical hard-cores in a hard-wallbox) diagrams have been already successfully introducedin the first years of many-body Bethe Ansatz, in [5] and[1], respectively. The I ( n ) diagrams were explored inRef. [22], albeit classically, still awaiting a quantum treat-ment: for each integer n a continuous one-parametricfamily of solvable three-body mass ratios (on a line withno boundary conditions) was obtained. A system of twohard-core particles with mass ratio 3:1 in a box (whoseone-wall version was explored in Ref. [6]) is expected toexhibit an exact solution associated with the ˜ G Coxeterdiagram. It can be conjectured that H and H Coxeterdiagrams lead to solvable four- and five-body problemson a line, respectively.The true challenge is posed by the Coxeter diagrams(affine or otherwise) with bifurcations , most notably the˜ E , , series. At the moment, it is not clear if thereare any realistic many-body problems that can be solvedusing these symmetries. (Some examples of “unphysical”many-body realizations of Coxeter diagrams—e.g. thosewhere a given particle can interact with an empty pointin between the other two—are given in Ref. [3].)The potential empirical context for this and furtherplanned explorations is one-dimensional cold gas mix-tures in optical lattices where the dynamics is governedby an effective (rather than physical) mass, controlled atwill [23]. Acknowledgments .– We are profoundly indebted tolate Marvin Girardeau for guidance and inspiration.We are thanking Jean-S´ebastien Caux for reading themanuscript and for the comments that followed. Theimpact of the numerous in-depth discussions of the sub-ject with Dominik Schneble can not be overestimated.This work was supported by grants from National Sci-ence Foundation (
PHY-1402249 ) and the Office of NavalResearch (
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