Quantum world on a sphere Using symmetry to obtain the energy spectra of polyhedron configurations of hopping and antiferromagnetic spin-1/2 particles

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Quantum world on a sphere

Using symmetry to obtain the energy spectra of polyhedron configurations of hopping and antiferromagnetic spin-1/2 particles

Rik Albers (10195343) July 2, 2014

Report Bachelor Project Physics and Astronomy, size 15 EC, conducted between April 1, 2014 and July 2, 2014

Institute for Theoretical Physics FNWI, Universiteit van Amsterdam

Supervisor: Prof. dr. K. Schoutens Second assessor: Dr. P.R. Corboz

Abstract

Symmetry operators are used to reduce the dimensionality of a number of three-dimensional polyhedron hopping and antiferromagnetic spin-1/2 problems with N = 4, 6, 8, 12 particles. An attempt is made to optimize and generalize this method. The energy spectra are obtained as a function of total spin. For the cube, results are com-pared to the 8 particle chain and 8 particle fully interacting system. The efficiency of the symmetry operators is analysed: up to N ' 10, spatial symmetry is efficient, while for larger systems its effect grows slower than the dimensionality of the problems.

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Popular scientific summary (Dutch)

In de natuurkunde wordt veel onderzoek gedaan naar hoe materialen zich gedragen wanneer ze bijvoorbeeld in een magneetveld worden neergezet, of wanneer de temperatuur verandert. Een interessant soort materiaal is bijvoorbeeld de antiferromagneet. Dit is een materiaal dat niet vanzelf magnetisch is, en dat zich pas als een magneet gaat gedragen nadat de temperatuur heel hoog wordt of nadat het in een heel sterk magnetisch veld wordt neergezet.

Dat sommige materialen antiferromagneet zijn heeft te maken met spin. De spin van een deeltje is voor te stellen als de as waar het deeltje om draait. Voor de deeltjes in dit onderzoek kan de spin omhoog (+1₂) of omlaag (−1₂) staan (dus niet naar de zijkant!).

Nu wordt er op elk hoekpunt van een vorm (bijvoorbeeld een kubus, of een driehoekige piramide) zo een deeltje neergezet met spin. De deeltjes hebben zelf een spin, maar voelen ook de spin van hun buren. Als twee deeltjes naast elkaar de spin dezelfde kant op hebben staan, kost dit meer energie dan wanneer ze de tegenovergestelde kant op staan. De deeltjes willen zo weinig mogelijk energie hebben, dus ze zullen zoveel mogelijk tegenovergesteld gaan staan van hun buren. Nu kunnen we uitrekenen hoe de deeltjes gaan staan en wat dan hun energie is. Dit is belangrijk om te weten als je de eigenschappen van het materiaal wilt onderzoeken. Stel je wilt dit uitrekenen voor een vorm met N deeltjes. Het aantal mogelijkheden om de deeltjes op verschillende manieren omhoog en omlaag wijzend naast elkaar te zetten is dan 2N_{. Voor vier deeltjes (de driehoekige piramide) zijn er dus 16 toestanden. Om de energie¨}_en uit te rekenen moet je dan een matrix maken van 16 × 16 getallen (elk getal vertelt hoe de ene toestand met de andere te maken heeft). Die kun je in een computer stoppen en die rekent dan de energie¨en uit. Voor zo een kleine matrix kan de computer dat makkelijk, maar voor grotere matrices duurt dit heel snel veel te lang. Voor de kubus bijvoorbeeld wordt de matrix veel te groot (256 × 256 !). -- -- -- --1 0 1 2 3 4 5 S -6 -4 -2 0 2 4 E Om dit probleem op te lossen wordt symmetrie gebruikt. Als je een kubus een kwartslag draait ziet hij er nog precies het-zelfde uit. Dit kun je ge-bruiken om de toestanden in groepjes bij elkaar te vegen. Als je dit trucje een paar keer doet, houd je opeens geen matrix van 256 × 256 over, maar een-tje van 12 × 12. Het re-sultaat zie je hiernaast (de energie als functie van de totale spin).

Het doel van dit project

was om de symmetrie te gebruiken om grote problemen zo klein mogelijk te maken. Dit is aardig gelukt, maar voor hele grote problemen (die van 12 deeltjes bijvoorbeeld) is de sym-metrie nog niet genoeg. Als je een te grote matrix overhoudt, duurt het nog steeds uren om de energie¨en uit te rekenen. Voor die systemen hebben we dus nog handigere manieren nodig.

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1 Introduction

To study properties of (anti)ferromagnetic quantum spin systems, it is often necessary to obtain the energy levels of these systems. The solutions of a spin system can be of use to a number of applications, including studies of magnetization and temperature processes and attempts to find approximations and bounding values of the energies of (other) three-dimensional systems, as was done by Schmidt et al. (2001), to find approximate solutions for much larger systems. An example that has been of interest recently is the C60 molecule, Buckminsterfullerene, which has icosahedral symmetry (Johnson et al. 1990).

For 2D systems there are methods available that provide analytical solutions for large systems. A widely used method is the Bethe Ansatz (Karbach & Muller 1998). This method makes it possible to solve an N -particle system without diagonalising an entire Hamiltonian. For 3D systems, however, such methods are not available. Therefore, methods are developed to efficiently solve these problems.

Recent studies have successfully obtained energy spectra of the cube, icosahedron and the dodecahedron using symmetry properties of the systems (Moustanis & Thanos 1994) (Konstantinidis 2005). This method has been shown to be successful and provides the ability to greatly reduce the dimensionality of the problems, without reduction of exactness.

The method described relies on operators that are in the symmetry group of the system of interest. The eigenstates ψ of a Hamiltonian H are those satisfying the equation

Hψ = Eψ, (1)

where E is the energy eigenvalue. If an operator O commutes with a system’s Hamiltonian, one can show that all states Oψ are also eigenstates of the Hamiltonian with the same energy eigenvalues (Bhagavantam & Venkatarayudu 1969), because

H(Oψ) = O(Hψ) = O(Eψ) = E(Oψ). (2) This property can be exploited to reduce a problem’s degeneracy by finding the eigenstates of a set of operators that commute with both the Hamiltonian and all other used operators. This paper will aim to optimise this method. An attempt will be made to provide read-ability and simplicity in the notation of states and Hamiltonians, while optimising the speed with which the solutions of a problem can be found. It will attempt to optimally use spacial symmetry to reduce the numerical calculations needed. To what extend can spacial symmetry reduce the degeneracy of (anti)ferromagnetic spin-1/2 problems?

For some of the systems analysed, this paper will also solve the hopping problem. These problems describe one particle that can jump to different positions in a system. The use of this is that when hopping and spin-1/2 models are combined, it will start to resemble the Hubbard model and its extensions such as the t−J model (Spa lek 2007). The Hubbard model describes a half filled lattice, where the particles have the ability to move around according to the hopping model. If two particles are in the same position, they will have an extra energy U . The t − J model extends the Hubbard model by adding spin interactions. An example of such an application is shown in section 4.3.

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2 Applying symmetry

2.1 Permutation subsystems

A system of N spin-1/2 particles has a total of 2N _{possible states. The basis can be written} as the combination of all permutations of N − n↓ spin-up and n↓ spin-down particles, where n↓ = 0, 1, .., N . As an example, the basis of a system of 4 spin-1/2 particles can be written as

(↑4) = {|1111i}

(↑3↓1) = {|1110i , |0111i , |1011i , |1101i}

(↑2↓2) = {|1100i , |0110i , |0011i , |1001i , |1010i , |0101i} (↑3↓1) = {|1000i , |0100i , |0010i , |0001i}

(↓4) = {|0000i},

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where a 1 or 0 represent a particle in the spin-up or spin-down state respectively. The order of (↑N −n↓n) is the number of states within the subsystem and is equal to

O(↑N −n↓n) = N n↓ = N ! (N − n↓)!n↓! , (4) note that N X n↑=0 O(↑N −n↓n) = N X n=0 N ! (N − n↓)!n↓! = 2N (5)

gives the total number of states in the system.

2.2 Rotation subsystems

For an operator RM that performs a cyclic permutation of particles, one can find groups of states {p0, p1, .., pM} that are affected by the operator according to the cyclic permutation

RMpn=

pn+1 if n 6= M p0 if n = M

(6) where n = 0, 1, .., M − 1 and pnis the m-th cyclic permutation in the rotation subsystem. The group can be denoted as the RM rotation group of p0, where p0 will be called the basis state of the group.

A system’s groups can be found in appendix A, where all basis states’ configurations are denoted by a diagram such as figure 1. In this figure, a spin-up particle is represented by an open circle, while a spin-down particle is represented by a filled circle. The first number and letter are a notation for the group, which, for the cube system, is explained in section 4.5. This notation differs for each system, as a generalized notation would not be optimally compact for the systems analysed in this paper. The second number, between brackets, is the amount of states within this rotation group.

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• • ◦ ◦ ◦ ◦ ◦ ◦

2a(4)

Figure 1: Example diagram of the group of states (11000000), consisting of |11000000i and its cyclic permutations under the operator R4.

2.3 Cyclic permutation operators RM

The rotation operator of interest is an operator RM that is in the symmetry group of some (sub)system of M states. To find the eigenvalues of RM, the characteristic equation

Det(RM − λI) = 0 (7)

can be used, where I is the identity matrix and solutions for λ are the eigenvalues of RM. This equation yields

(−1)M +1(1 − λM) = 0 (8)

λn= e−2nπi/M. (9) The eigenstates of RM are found by solving

RMψn= λnψn, (10)

= λn(x1, x2, .., xM) (11) = (xM, x1, .., xM −1) (12) where ψn are eigenstates of RM, with solutions

xk+1 = eikθxk. (13)

where θ is the rotation that RM causes to the (sub)system. For a rotation operator that causes a cyclic permutation as in equation 6, θ = 2π/M . It follows that the eigenstates of RM are given by (s1s2s3s4..sN)nθ = 1 √ M M −1 X k=0 einkθpk. (14) 2.3.1 Reflection operator F

An operator F that acts on a state φ as

F φ = f φ (15)

is equivalent to an operator R2 that rotates the states by an angle π. Therefore, the eigenstates of the operator F are, in the subsystem of φ and φ,

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φ±= 1 √ 2 φ ± φ (16) with eigenvalues ±1. 2.4 Degenerate operators

In case RM is not a cyclic permutation of all states, one can attempt write RM as a diagonal combination of rotation operators, in a form that resembles

RM =   Rk Rl Rm  . (17)

where k, l and m are the order of the subsystems. Each subsystem of order l will account for l eigenstates and eigenvalues of RM. These eigenstates and -values are those of Rl in accordance to equations 14 and 9 respectively.

In this form, RM has degenerate eigenvalues, and one can form subsystems Φp = {φp1, φ p 2, .., φ

p M} of eigenstates with eigenvalues op. The eigenstates of the Hamiltonian are linear combinations of states in Φp.

3 The Hamiltonian

3.1 Hopping

In the hopping problems, a system will be analysed with one particle that can jump to positions that are connected to its current position. The hopping Hamiltonian Hhop of a system solely describes the interactions in that system. For all possible particle positions i that are connected to another position j,

Hij = Hji= −t (18)

where t is a parameter describing the strength of the interactions, which is equal for all interactions in this model. For this reason, the hopping problems will be normalized by setting the value to

t ≡ 1. (19)

For a more compact description of the hopping Hamiltonian, diagrams will be used showing the interactions of different particles in a system. For example, the hopping Hamiltonian of a 4 particle chain system (equation 20) is described by figure 2.

Hhop = −     0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0     . (20)

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1 2

3 4

,

Figure 2: Diagram describing the interactions in the 4 particle chain system.

3.2 Spin-1/2 systems

The anti-ferromagnetic Hamiltonian for a system of spin-1/2 particles can be written, accord-ing to the Heisenberg model (Heisenberg 1928), as

Hspin= J X <ij> Si· Sj (21) = J X <ij> S_izS_jz+ S_ixS_jx+ S_iyS_jy, (22) where J is the spin coupling strength. The spin ladder operators are defined by the equations

S±= Sx± iSy, (23)

S±|1/2±1/2i = ¯h |1/2∓1/2i , (24) S±|1/2∓1/2i = 0, (25) in the notation |smi, where s is the spin value and m is the spin projection (Griffiths 2005). For spin-1/2 particles, equation 22 can be rewritten as

H = J X <ij> S_izS_jz+1 2(S + i S − j + S − i S + j ). (26)

In this equation, the operator Sz gives the spin projection to the z-axis, giving the diagonal terms of the Hamiltonian. The spin ladder operators account for the off-diagonal terms. For a system in which all particles are interacting with all other particles, the Hamiltonian is a sum over all particles. For a state of M interactions of which m are negative (up-down interaction) and M − m are positive (up-up or down-down interaction), the diagonal terms are the product of spin projections, given by

Hii= hψi| H |ψii = J X <ij> hψi| SkzSlz|ψii = ¯h2J 1 4(M − m) − 1 4m = ¯h 2_J 4 (M − 2m). (27)

For a system in which all particles are interacting, this term may also be written in the total number of particles N and the number of particles in the spin-up state n↑, according to the equation

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Hii= ¯ h2J 4   n↑ X k=1 (n↑− k) + N −n↑ X k=1 (N − n↑− k) − n↑(N − n↑)   = ¯h 2_J 4 1 2n↑(n↑− 1) + 1 2(N − n↑)(N − n↑− 1) − n↑(N − n↑) = ¯h 2_J 8 (N (N − 1) − 4n↑(N − n↑)) . (28)

The off-diagonal terms are given by

Hij = hψi| H |ψji = hψi| 1 2(S + kS − l + S − kS + l ) |ψji . (29) As the combination of S+ and S− is a transition from one state in (↑n↓m) to another one in the same permutation subsystem, this can be rewritten as

Hij = _¯_h2_J

2 if states differ in no more than one particle’s spin state

0 else . (30)

To simplify equations and explanations, the values of both J and ¯h will be set to

J ≡ 1, ¯ h ≡ 1

with the exception of section 4.3, which variates both the hopping strength t and J .

3.3 Representation (interaction tables)

The Hamiltonians for different problems will be represented by interaction tables in appendix B. This representation was chosen to provide a direct link to the origin of the interactions between different groups of states, while also representing multiple Hamiltonain matrices in a compact manner.

As an example, the interactions of the states in 2a of the cube spin-1/2 problem are analysed. First of all, the diagonal term can be derived from equation 27, which gives Hii= 1 for M = 12 and m = 4. From the representation in section A.2, one can show that that 2a should have 4 interactions (each of the two spin-down particles is connected to two spin-up particles). Particle 1 can switch spin states with particle 4, giving the first rotated state of 2a1 _(R

4 applied to the basis state in 2a1 - the one shown as a diagram). Particle 1 switched with particle 5 gives the basis state of 2b1. Furthermore, the state interacts with the basis state of 2a1 and the basis of 2b1.

The interactions are summarized in the table below. The subscripts represent the in-teractions, while the number between square brackets represents the diagonal term. Notice that, because all off-diagonal terms get a factor1_/₂_{, the diagonal element is shown as [2H}_ii_]. The number at the upper left corner of the table represents the number of particles in the spin-up state for this particular subsystem. The complete table for this subsystem ((↑6↓2) of the cube) is shown in table 11.

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2 a a1(2) b1 a [2] 0,1 0,0

,

Table 1: Interactions of the group 2a in the cube spin-1/2 problem.

To find the matrices from the interaction table, one should apply the Θ function to the interactions (thus only the subscripts), defined by the equation

Θθ(x1, x2, x3) = 1 2

eiθx1_{+ e}iθx2 _{± e}iθx3_, ₍₃₁₎

where θ is the angle associated with the rotation operator used for this system and ± is the F eigenvalue f . Applying this function to (↑6↓2) of the cube, one obtains the matrix as a function of R4 and F eigenvalues as is shown below. To obtain the correct matrix for a given θ and f , one should only take the rows and columns of the groups that contribute to those values. H_nπ/2± (n↑ = 2) = 1 2         a a1(2) b0 b1 b2 a 2 1 + in 0 1 ± 1 0 a1 1 + in+ i2n+ i3n 0 0 0 1 + i2n b0 0 0 2 1 ± 1 + i3n± i3n 0 b1 1 ± 1 0 1 + in 0 1 + i3n b2 0 1 ± 1 0 1 + in± i2n± i3n 0         (32)

4 Problems

4.1 Tetrahedron hopping

The tetrahedron is a fully interacting system of 4 positions with 6 connections. Its hopping Hamiltonian is defined by Hhop = −     0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0     . (33)

which can also be described by the following diagram:

The cyclic permutation (1234), described by the rotation operator R4, will be used to help find this system’s eigenstates. Note that (R4)4 = 1. Using this operator is allowed, because the tetrahedron problem can be simplified as a fully connected 2D square (it is displayed as such in figure 3).

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(a) Figure showing the tetrahedron

1 2

4 3

.

(b) Diagram describing the interactions in the tetrahedron system.

Figure 3: The tetrahedron (a) and its interactions (b).

(1000)0 = 1 2 3 X k=0 pk = 1

2(|1000i + |0100i + |0010i + |0001i) , (1000)π/2 = 1 2 3 X k=0 eikπ/2pk = 1

2(|1000i + i |0100i − |0010i − i |0001i) , (1000)π = 1 2 3 X k=0 eikπpk, = 1

2(|1000i − |0100i + |0010i − |0001i) , (1000)_3π/2= 1 2 3 X k=0 ei3kπ/2pk = 1

2(|1000i − i |0100i − |0010i + i |0001i) .

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The energy of (1000)0 is found by letting the Hamiltonian act on it:

Hhop|(1000)0i = E |(1000)0i = −3 |(1000)0i . (35) Similarly, the energies of the other eigenstates are

E[(1000)_π/2] = E[(1000)π] = E[(1000)2π/3] = −(i − 1 − i) = 1. (36)

4.2 Tetrahedron spin-1/2

From 4 spin-1/2 particles, one expects the following multiplets:

24 = (1 ⊕ 3) ⊕ (1 ⊕ 3) ⊕ (3 ⊕ 5). (37) The states in this system can be ordered as

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With N = 4, equation 28 becomes

Hii= 1

2(3 − n↑) (39)

Table 2 shows the system’s subsystems.

Order R4 subs. (↑4) 1 (1111) (↑3↓1) 4 (1110) (↑2↓2) 6 (1100), (1010) (↑1↓3) 4 (1000) (↓4) 1 (0000)

Table 2: Subsystems and R4 subsystems for spin-1/2 states on a tetrahedron

The R4 subsystems (1111) and (0000) give the eigenstates of the Hamiltonian (1111)0 = |1111i and (0000)0 = |0000i respectively. Their energies are given by the diagonal terms in the Hamiltonian,

E[(1111)0] = E[(0000)0] = 3/2. (40) The subsystem (1110) gives

(1110)0= 1 2 3 X k=0 pk (41) = 1

2(|1110i + |0111i + |1011i + |1101i) , (42) (1110)π/2= 1 2 3 X k=0 eikπ/2pk (43) = 1

2(|1110i + i |0111i − |1011i − i |1101i) , (44) (1110)π = 1 2 3 X k=0 eikπpk, (45) (1110)3π/2= 1 2 3 X k=0 ei3kπ/2pk, (46)

which are all eigenstates of the Hamiltonian. For the state (1110)0, all off-diagonal matrix elements add up, while the diagonal term is zero. Thus the energy of this state is

E[(1110)0] = 3/2, (47)

while for the other states two off-diagonal terms cancel out and one is multiplied by −1, giving

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Equivalently, the eigenstates from the subsystem (1000) and their energies are given by

E[(1000)0] = 3/2, (49)

E[(1000)π/2] = E[(1000)π] = E[(1000)3π/2] = −1/2. (50) 4.2.1 Degenerate subsystem (↑2↓2)

The subsystems (1100) and (1010) give states (1100)0, (1100)π/2, (1100)π, (1100)3π/2, (1010)0, (1010)π, but as the subsystems overlap, not all off these states are eigenstates of the Hamil-tonian. The states found by applying R4 can be rewritten as

|a₀, nπ/2i = (1100)_nπ/2, |a1, nπi = (1010)nπ,

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θ 0 π/2 π 3π/2 deg. 2 1 2 1

Table 3: Degeneracy in the states of (↑2↓2) after applying the operator R4.

To find the eigenstates in rotation groups with θ = 0 and θ = π, the interaction table for a0 and a1 is constructed, which gives table 4. The Hamiltonians that follow from this table are shown in equations 52 and 53.

2 a0 a1(2) a0 [−1]/1,3 0,1 a1(2) 0,1,2,3 [−1]

Table 4: Tetrahedron spin-1/2 (↑2↓2) interaction table

H0 = 1 2 a0 a1 a0 1 2 a1 4 −1 ! (52) Hπ = 1 2 a0 a1 a0 −3 0 a1 0 −1 ! (53) The resulting eigenstates and their energy eigenvalues are

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To find the multiplets in this problem, one can make use of the operators S− and S+. By applying the operator S− to different eigenstates, it is found, for example, that

S−(1111)0= S−|1111i = |1110i + |0111i + |1011i + |1101i = 2(1110)0, (55) showing that these two eigenstates belong to the same multiplet. Using this method, the multiplets in this system are found to consist of all states with the same eigenvalues of R4, so that

(1111)0, (1110)0, (1100)0+ (1010)0, (1000)0 and (0000)0 (56) forms a quintuplet with energy eigenvalue E = 3/2,

(1110)π/2, (1100)π/2, (1000)π/2, (57) (1110)π, (1010)π, (1000)π, (58) (1110)_3π/2, (1100)_3π/2, (1000)_3π/2, (59) form triplets with energy eigenvalues E = −1/2 and

(1100)π (60)

(1100)0− 2(1010)0 (61)

form singlets with energy eigenvalues E = −3/2.

The resulting energies as a function of their spin are shown in figure 4.

--

-0 1, 2, 3 2, 0 -1 0 1 2 3 S -6 -4 -2 0 2 4 E

Figure 4: Energy levels for spin-1/2 states on a tetrahedron as a function of their spin. The number of stripes represents the size of the multiplet. The values of n in θ = nπ/2 that share a particular energy eigenvalue are also shown, where an overline n means that f = −1. States with equal spin, energy, θ and f are shown once.

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4.3 Tetrahedron spin-1/2 hopping (2 particles)

To combine the problems of hopping and spin-1/2, two spin-1/2 particles are analysed on a tetrahedron with the combined Hamiltonian

Hspin+hop = Hhop+ Hspin (62) where Hspin sums over all interacting spin particles and Hhop sums over all interactions between an empty position and a spin particle. The total amount of states is equal to

4 × 4!

2!2! = 24, (63)

which are divided up into the subsystems shown in appendix A.1. It is expected that 6 singlets and 6 triplets are found. By applying the operator R4, one obtains the states shown in table 5. The notation was adopted from section 4.5.

states θ 0a nπ/2 0a1 nπ 1a0 0 nπ/2 1a1₀ nπ/2 1a1 nπ/2

Table 5: Subsystems and R4subsystems for spin-1/2 hopping states on a tetrahedron

The degeneracy left in this problem is shown in figure 5. The Hamiltonians of these degenerate subsystems are Hnπ/2(↑↑ / ↓↓) = J 2 a0 a1 a0 1/2 0 a1 0 1/2 + t a0 a1 a0 in+ i3n 1 + in a1 1 + in+ i2n+ i3n 0 , (64) Hnπ/2(↑↓) = J 2   a0 a10 a1 a0 −1/2 1 0 a1₀ 1 −1/2 0 a1 0 0 i2n−1/2  +t   a0 a10 a1 a0 0 in+ i3n 1 + i3n a1₀ in+ i3n 0 in+ i2n a1 in+ i3n i2n+ i3n 0  . (65)

If one sets the values of J = 1 and t = 0, one finds six triplets with E = 1/4 and six singlets with E = −3/4. This is equivalent to six equal solutions for the two spin-1/2 particle system. If the values are set to J = 0 and t = 1, one finds a solution as would be expected in a regular hopping problem; the ground state has θ = 0 and is thus the simplest linear combination of all possible positions.

The problem can also be seen as a spin problem with a constant J = 1 and a variable hopping perturbation t. The result of this approach can be seen in figure 6. At t = J/2, the ground state shifts from the one in the spin-1/2 problem (t = 0) to the one in the hopping problem (J = 0). The degeneracy at this value of t corresponds to the super symmetry of the t − J model at J = 2t (Foerster & Karowski 1992).

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0 Π2 Π 3Π2 0.0 0.5 1.0 1.5 2.0 2.5 3.0

_¯¯

¯

Figure 5: Degeneracy in the two spin-1/2 particle hopping problem on the tetrahedron.

Figure 6: The energy levels as a function of the hopping strength t in the two spin-1/2 particle hopping problem on the tetrahedron where J = 1. Each line is labelled with the values of θ of the associated eigenstates.

4.4 Cube hopping

The cube is an 8 particle system with 12 interactions. Its hopping Hamiltonian can be described by figure 7. The operators used to solve this problem are R4, which is defined as the cyclic perturbation of particles (1234) and (5678) (so that (R4)4 = 1), and F , which switches the positions of particles according to the permutation (15)(26)(37)(48) (so that F2 = 1).

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(a) Figure showing the cube

1 2 3 4

5 6 7 8 F

OO

R4ll // // //

(b) Diagram describing the interactions in the cube system. The operators R4 and T and their effects are also shown. Figure 7: The cube (a) and its interactions (b).

The notation of the cube spin-1/2 (↑7↓1) subsystem, found in appendix A.2, will be used. By applying R4, one finds 4 solutions for both chains:

|a, nπ/2i = (1000)_nπ/2(0000) (66) |a, nπ/2i = (0000)(1000)_nπ/2 (67) for the chain of particles 1−4 and 5−8, respectively. These equations, where n = 0, 1, 2, 3, each describe four solutions of one separate chain connected to the other (empty) chain. For example, the 4 particle chain solution

(1000)π = |1000i − |0100i + |0010i − |0001i (68) is combined with an empty chain (0000) to form the state

(1000)π(0000) = |10000000i − |01000000i + |00100000i − |00010000i . (69) These states are no eigenstates of the Hamiltonian, as they overlap with the other chain, but the operator F can be used to form the eight eigenstates

|a, nπ/2, ±i = |a, nπ/2i ± |a, nπ/2i , (70) so that there is no degeneracy left in the eigenvalues of F and R4.

The energies of the eigenstates are given by

E[|a, 0, +i] = −(2 + 1) = −3

E[|a, 0, −i] = −(2 − 1) = −1

E[|a, π/2, ±i] = E[|a, 3π/2, ±i] = −(i − i ± 1) = ∓1 E[|a, π, +i] = −(−1 − 1 + 1) = 1 E[|a, π, −i] = −(−1 − 1 − 1) = 3

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4.5 Cube spin-1/2

From 8 spin-1/2 particles, one expects the following multiplets:

28 = 114⊕ 328⊕ 520⊕ 77⊕ 9, (72) where the subscripts denote the amount of multiplets of that particular size.

Table 6 shows the subsystems in this problem. The operators R4 and F , also used in the cube hopping problem, were used to reduce the degeneracy in this problem.

Order (↑8) 1 (↑7↓1) 8 (↑6↓2) 28 (↑5↓3) 56 (↑4↓4) 70 (↑3↓5) 56 (↑2↓6) 28 (↑1↓7) 8 (↓8) 1

Table 6: Subsystems for spin-1/2 particles on a cube.

The resulting states and their R4 and F eigenvalues are shown in table 7. For the R4 eigenvalues, only the associated angles, nθ, are shown, where n = 0, 1, 2, .., (2π_θ − 1). Diagrams showing the configurations of the groups’ basis states can be found in appendix A.2. Because of the equivalence of (↑m↓n) and (↑n↓m) groups, the latter are not explicitly shown. The notation of the states represents the following:

the number represents the amount of spin-down particles in the state and thus the subsys-tem it belongs to;

the letter represents the amount of spin-down particles in the particles 5 − 8, where a = 0 particles;

a subscript represents the amount of cyclic permutations applied to the particles 5 − 8 (the notation gn−m represents all the groups gn, gn+1, .., gm);

a superscript gx,y represents the amount of spin-down particles that are placed between the spin-up particles on the row of particles 1 − 4 (x) or 5 − 8 (y);

an overline represents a group’s counterpart for the operator F , where F g = f g (their diagrams are not always explicitly shown).

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states θ f 0a 0 + 1a nπ/2 ± 2a0 nπ/2 ± 2a1₀ nπ ± 2b0 nπ/2 + 2b1 nπ/2 ± 2b2 nπ/2 (−)n 3a nπ/2 ± 3b0−3 nπ/2 ± 3b1₀₋₁ nπ/2 ± 4a 0 ± 4b0−3 nπ/2 ± 4c0 nπ/2 + 4c1 nπ/2 ± 4c2 nπ/2 (−)n 4c0,1₀₋₁ nπ/2 ± 4c1,1₀ nπ + 4c1,1₁ nπ (−)n

Table 7: States found by applying R4 and F for spin-1/2 states on a cube and their eigenvalues for these operators.

The degeneracy left in this problem is shown in figure 8. The operators R4 and F have reduced the largest subsystem of order 70 to 8 problems of orders 8 to 12. The 8 problems are strongly connected, as they can all be represented by the interaction table shown in appendix B.1. The interaction tables shown in that appendix provide a total of 24 matrices, which can be diagonalized to find the energy eigenvalues in this problem for (↑6↓2), (↑5↓3) and (↑4↓4). For the subsystem (↑7↓1), the energy eigenvalues, as a function of nπ/2 and f , are given by

1 2 3 + i

n_{+ i}3n_{+ f}_, ₍₇₃₎ and (↑8) has one eigenvalue with (E, n, f ) = (3, 0, +1).

+1 -1

Figure 8: Degeneracy left in the cube spin-1/2 problem after applying the operators R4 and F . The left and right sides account for the groups with f = +1 and f = −1, respectively.

By connecting each energy eigenvalue with the group’s associated eigenvalues for F and R4 , it is possible to find the multiplets by collecting groups of with equal (E, n, f ) values from different subsystems. A value (E, n, f ) that is found in all Ng groups between (↑k↓m) and (↑m↓k), represents an Ng-plet. The multiplets’ energy eigenvalues as a function of their spin are shown in figure 9.

By counting the number of unique (E, n, f ) values, one finds the amount of multiplets expected from equation 72. For S = 1, only 27 values are counted, while 28 are expected. This is due to a duplicate (0, 2, −1) triplet. The ground state is a singlet of (−4.82, 0, 1), while the first 3 × 7 excited states are triplets with all values of n and f except (E, 0, 1).

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The energy values for states (E, 1, f ) and (E, 3, f ) are equal, as all factors in the Hamil-tonians of n↑ = 3 are simply conjugates of those in n↑ = 1 matrices. The average degeneracy, the average number of multiplets with equal spin and energy values, is equal to 2.41.

--

-0 0, 1, 3 1, 2, 3 2 1, 1, 2, 2, 3, 3 0, 2 0, 1, 1, 2, 3, 3 0 0 0, 2 0, 2 1, 2, 3 0, 2 2 0, 1, 3 1, 2, 3 0, 1, 3 1, 2, 3 0, 1, 3 0, 1, 3 0, 1, 3 0 1, 2, 3 1, 2, 3 0, 2 0, 2 0 0 0 -1 0 1 2 3 4 5 S -6 -4 -2 0 2 4 E

Figure 9: Energy levels for spin-1/2 states on a cube as a function of their spin. The values of n in θ = nπ/2 that share a particular energy eigenvalue are also shown, where and overline n means that f = −1. States with equal spin, energy, θ and f are shown once.

4.6 8 Particle chain spin-1/2

The 8 particle chain system is describe by figure 10. The expected multplets are the same as those expected for the cube, see equation 72.

1 2 3 4 5 6 7 8

R8 nn // // // // // // //

Figure 10: Diagram describing the interactions in the 8 particle chain system. The operator R8 and its effect are also shown.

Applying the operator R8 yields the the groups of states shown in appendix A.3. The notation differs from the one used for the cube states. The particles are viewed from the chain of spin-down particles starting at position 1. All spin-down particles not connected to this chain, thus separated by a spin-up particle, will be referred to as loose particles. In this notation:

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a subscript represents the number of spin-up particles placed between the chain’s rightmost particle and the rightmost loose particle;

the letter represents a permutation for all (loose) middle spin-down particles, which are all spin-down particles placed between the chain and the rightmost loose particle (a means |↓_x↑_y↓↑_wi, b means |↓_x−1↑↓↑_y−1↓↑_wi, etc.).

The operator eigenvalues will not be written down for each state, as was done for the cube. The θ in their R8 eigenvalues are equal to

nθ = 2nπ

M , (74)

where M is the number of states in the group and n = 0, 1, 2, .., M − 1. The degeneracy left in this problem is shown in figure 11.

0 Π4 Π2 3Π4 Π 5Π4 3Π2 7Π4 0 2 4 6 8 10

8

¯

8

7

¯

1

¯

7

6

¯

2

¯

6

5

¯

3

¯

5

4

¯

4 Figure 11: Degeneracy left in the chain spin-1/2 problem after applying the operator R8.

By building and diagonalizing the Hamtiltonians derived from the interaction tables shown in appendix B.2, one yields the energy eigenvalues in this problem. The multiplets are found by collecting groups of equal (E, n) with the method explained for the cube system. The energies of the resulting multiplets as a function of their total spin are shown in figure 12. The Hamiltonians of all states with n↑ = 1 and n↑ = 7 are conjugates of eachother, as are those for n↑ = 2 and n↑ = 6 and those for n↑ = 3 and n↑ = 5. For this reason their energy values are equal, as can be seen in the figure.

Compared to the energy diagram of the cube, the average degeneracy has decreased to 1.56. This can be linked to chain system’s property of being less connected than the cube (each particle has 2 connections compared to 3 in the cube). Their ground states share the value of n = 0, while the cube’s ground state energy is lower than that of the chain system (E = −3.65).

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-- -- -- -0 1, 7 2, 6 3, 5 4 1, 7 2, 6 0 3, 5 4 1, 7 3, 5 4 2, 6 3, 5 0 1, 7 2, 6 0 3, 5 4 7, 1 2, 6 3, 5 0 1, 7 2, 6 7, 1 5, 3 0 4 2, 6 3, 5 1, 7 4 4 2, 6 0 7, 1 0 4 2, 6 3, 5 4 0 -1 0 1 2 3 4 5 S -6 -4 -2 0 2 4 E

Figure 12: Energy levels for spin-1/2 states on an 8 particle chain as a function of their spin, similar to figure 9. States with equal spin, energy and θ are shown once.

4.7 8 Particle fully interacting system spin-1/2

The 8 particle fully interacting system is a system equivalent to the 8 particle chain system, with the modification that all particles are interacting with eachother. This system is also solved by using the R8 operator. Its states after using this operator are the same as those found in the chain problem, as well as the R8 eigenvalues and degeneracy.

By using the interaction tables shown in B.3 and finding the eigenvalues of the resulting Hamiltonians, one finds the multiplets shown in figure 13. As this system is fully connected, this problem has a full degeneracy for all states with equal spin, as is the case in the tetrahe-dron problem. The average degeneracy is therefore equal to 14. The ground state energy is higher than that of the other 8 particle systems and has an 8-fold degeneracy.

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--

-0 1, 2, 3, 4, 5, 6, 7 0, 1, 2, 3, 4, 5, 6, 7 0, 1, 2, 3, 4, 5, 6, 7 0, 1, 2, 3, 4, 5, 6, 7 -1 0 1 2 3 4 5 S -6 -4 -2 0 2 4 6 8 E

Figure 13: Energy levels for spin-1/2 states as a function of their spin on a fully interacting system of 8 particles, similar to figure 9. States with equal spin, energy and θ are shown once.

4.8 Octahedron spin-1/2

The octahedron is a 6 particle system with, in equivalence with the cube, 12 interactions. Its connections are described by figure 14. The operators R4 and F , which act as the cyclic permutations (2345) and (16) respectively, will be used to reduce the degeneracy in this problem. The subsystems in this problem are shown in table 8.

(a) Figure showing the octahedron

1 F_OO

2 3 4 5 6 R4 kk // // //

(b) Diagram describing the interactions in the octahedron system. The effects of the operators R4 and F are also shown. Figure 14: The octahedron (a) and its interactions (b).

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Order (↑6) 1 (↑5↓1) 6 (↑4↓2) 15 (↑3↓3) 20 (↑2↓4) 15 (↑1↓5) 6 (↓6) 1

Table 8: Subsystems for spin-1/2 particles on a octahedron.

Applying the operators gives the groups of states shown in table 9. Diagrams showing the configurations of each group’s basis states can be found in appendix A.4. The notation is defined by the following:

the number represents the number of spin-down particles in the state and thus the subsys-tem that it belongs to;

the letter represents the lowest row of particles in which a spin-down particle can be found (a represents the upper, b the middle and c the lowest row);

a subscript represents a permutation on the middle row.

states θ f 0a 0 ± 1a 0 ± 1b nπ/2 + 2a nπ/2 ± 2b0 nπ/2 ± 2b1 nπ/2 + 2c 0 + 3a0 _nπ/2 _± 3a1 nπ ± 3b nπ/2 + 3c nπ/2 +

Table 9: States found by applying R4 and F for spin-1/2 states on an octahedron and their eigenvalues for these operators.

The degeneracy left in this problem is shown in figure 15. Using the method explained for the cube problem, one can find multiplets of equal (E, n, f ) values. The energy eigenvalues as a function of total spin are shown in figure 16.

The average level of degeneracy in the energy levels, 2.22, is comparable to that of the cube. Unlike in the 8 particle systems, the ground state has n = 2.

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+1 -1

Figure 15: Degeneracy left in the octahedron spin-1/2 problem after applying the operators R4 and F . The left and right sides account for the groups with f = +1 and f = −1, respectively.

--

-0 0, 1, 3 0, 2 1, 2, 3 1, 2, 3 0, 2 2 0, 1, 3 2 -1 0 1 2 3 4 S -6 -4 -2 0 2 4 6 8 E

Figure 16: Energy levels for spin-1/2 states on a octahedron as a function of their spin, similar to figure 9.

5 Discussion

5.1 Correctness of results

The correctness of the results in this paper can be checked in a number of ways. First of all, the energy eigenvalues for S > 0 originate from 2S + 1 subsystems with S + 1 different and independent Hamiltonians. This means that, should an energy value in one of these multiplets be incorrect, all 2S + 1 Hamiltonians contain errors that lead to the exact same (wrong) answers. Although incorrect results in S > 0 cannot be ruled out entirely, the chances

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of it do decrease.

Secondly, for the cube spin-1/2 problem, there have been numerous attempts at obtaining the eigenvalues, as was done in this paper. A study by Moustanis & Thanos (1994), which uses ”the method of hierarchy of algebras of spin systems”, is, for a large part, in agreement with this study. Some eigenvalues, however, do not agree with the results obtained in this paper. Another study by Kawabata & Suzuki (1969) does not explicitly present the numerical eigenvalues in the paper, but from the energy spectrum one can see that it contradicts the study done by Moustanis and seems to agree with this paper. A third study (Dresselhaus 1962) presents the exact results and agrees completely with this paper.

A program that applies a simplified version of the method (using only one symmetry operator) was used to find the energies in the icosahedron’s (↑6↓6) subsystem for θ = 0. As the ground states of all 8-particles systems had the property θ = 0, it was expected to find the system’s ground state. The lowest energy found was E = −6.18789, which is in agreement with the ground state energy found in a study by Konstantinidis (2005). The operator F does not commute with the icosahedron’s Hamiltonian, but a similar operator could be applied to reduce the need for numerical calculations.

5.2 Efficiency of spacial symmetry

Naively, one could expect an operator RM to reduce a problem’s degeneracy by a factor M . This is not the case, as there is a number of groups of states that are, to some extend, symmetric, which results in a number of groups that do not contribute to all values of (n, f ). This causes some values of (n, f ) to retain a higher level of degeneracy than expected, while other groups’ degeneracies decrease more than expected.

In the cube problem, for example, the operators R4 and F were applied, which reduces the degeneracy by an average of 4 × 2. In reality, however, the degeneracy shown in figure 8 is not constant across all (n, f ). The table below shows the maximum degeneracy left in each of the problems analysed and in the icosahedron problem.

Operators used tetrahedron octahedron cube icosahedron

None 6 20 70 924

RN 2 6 20 188

F − 4 12 ∼ 100

Factor 3 5 5.83 ∼ 9

Table 10: Efficiency of the operators used to reduce dimensionality in each problem. The operator F has not been applied to the icosahedron (its approximation is shown) and the tetrahedron. The factor by which the degeneracy was reduced is also shown.

5.3 Degeneracy across spin

In the energy spectra for the cube, 8 particle chain and octahedron systems, there is a degen-eracy in certain energy levels across multiple values of total spin. This degendegen-eracy is shown in figure 17. The cube system involves the largest amount of this type of degeneracy. This is not trivial, since the chain system has a lower degeneracy within each spin level and thus has a larger amount of unique energy levels. It would be expected that, if this degeneracy were the results of a pure coincidence, it would be found more often in the chain’s energy spec-trum. Furthermore, the cross-spin degenerate energy levels in the cube system are sometimes

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degenerate in the values of n and f as well. This means that there are, for example, triplets and quintets with equal (E, n, f ) values.

For two eigenstates from a subsystem (↑k↓m), this implies that it is not possible to deter-mine which eigenstate (or which linear combination of the degenerate eigenstates) belongs to which multiplet without measuring the total spin directly with an S2 operator or indirectly with an S± operator.

The (tetrahedron and 8 particle) fully interacting systems, having maximum degeneracy within each spin level, do not have this type of degeneracy.

1 0 8 Particle chain 1 0 -1 0.414214 -2.41421 Cube 0 -2 Octahedron 0 2 4 6 8 10 12

Figure 17: Barchart for energy degeneracy across multiple values for total spin. The height displays the amount of different spin values the degeneracy covers for the specific energy value shown in each bar.

6 Conclusion

In general, using symmetry operators greatly increases the efficiency with which spin-1/2 problems can be solved. However, as the dimensionality of a problem is proportional to 2N, while the factor by which it is reduced using spatial symmetry operators is at most equal to N , the effectiveness of spatial symmetry reduces significantly for larger problems. For the systems analysed in this paper, the degeneracy left after applying spacial operators is of an acceptably small order. For larger systems, however, one or two spatial operators, however effective they may be, will not reduce dimensionality to an acceptable level. To analyse these problems, it will be necessary to find other, non-spatial, operators.

Future research could focus on finding and generalising a number of extra (symmetry) operators to reduce dimensionality further. This could allow for efficient solving of larger systems, such as the icosahedron and the dodecahedron (a 20 particle system). Examples of operators that could be optimized for efficient application are the spin ladder operators S±. These operators were used in the tetrahedron problem to find the multiplets, but were not efficient enough to be used in the other problems. If these operators could be generalized so that they can be used efficiently in larger problems, they could aid in solving these systems. An ideal operator would be one for which the dimensionality reduction factor grows vastly as N increases.

Secondly, the methods used in this paper could be applied to different, more realistic, models, such as the Hubbard and t − J models mentioned before. This could provide more physically interesting and realistic results.

Another application of the method shown could be a program that numerically solves a system by applying multiple symmetry operators, thus reducing its dimensionality, and finds the energy spectrum. As explained in the discussion section, a program of this form was written, but, being a simple version, it only allows for the use of a single symmetry operator.

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The effectiveness of such a program could be improved by providing the ability to use multiple symmetry operators to maximally reduce a problem’s dimensionality.

References

Bhagavantam, S. & Venkatarayudu, T. 1969, Theory of Groups and its Application to Physical Problems (Academic Press), 183–184

Dresselhaus, G. 1962, Phys. Rev., 126, 1664

Foerster, A. & Karowski, M. 1992, Phys. Rev. B, 46, 9234

Griffiths, D. J. 2005, Introduction to Quantum Mechanics, 2nd edn. (Pearson Education) Heisenberg, W. 1928, Zeitschrift fr Physik, 49, 619

Johnson, R. D., Meijer, G., & Bethune, D. S. 1990, Journal of the American Chemical Society, 112, 8983

Karbach, M. & Muller, G. 1998, eprint arXiv:cond-mat/9809162 Kawabata, C. & Suzuki, M. 1969, Physics Letters A, 29, 264 Konstantinidis, N. P. 2005, Phys. Rev. B, 72, 064453

Moustanis, P. & Thanos, S. 1994, Physica B: Condensed Matter, 202, 65

Schmidt, H.-J., Schnack, J., & Luban, M. 2001, EPL (Europhysics Letters), 55, 105 Spa lek, J. 2007, Acta Physica Polonica A, 111, 409

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Appendix A

Groups of states

A.1 Tetrahedron ◦ ◦ × × 0a(4) ◦ × × ◦ 0a1(4) ◦ • × × 1a0(4) • ◦ × × 1a1(4) ◦ × × • 1a1 0(4) A.2 Cube ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 0a(1) • ◦ ◦ ◦ ◦ ◦ ◦ ◦ 1a(4) ◦ ◦ ◦ ◦ • ◦ ◦ ◦ 1a(4) • • ◦ ◦ ◦ ◦ ◦ ◦ 2a(4) ◦ ◦ ◦ ◦ • • ◦ ◦ 2a(4) • ◦ • ◦ ◦ ◦ ◦ ◦ 2a1(2) ◦ ◦ ◦ ◦ • ◦ • ◦ 2a1₍₂₎ • ◦ ◦ ◦ • ◦ ◦ ◦ 2b0(4) • ◦ ◦ ◦ ◦ • ◦ ◦ 2b1(4) ◦ • ◦ ◦ • ◦ ◦ ◦ 2b1(4) • ◦ ◦ ◦ ◦ ◦ • ◦ 2b2(4) • • • ◦ ◦ ◦ ◦ ◦ 3a(4), 3a(4) • • ◦ ◦ • ◦ ◦ ◦ 3b0(4), 3b0(4) • • ◦ ◦ ◦ • ◦ ◦ 3b1(4), 3b1(4) • • ◦ ◦ ◦ ◦ • ◦ 3b2(4), 3b2(4) • • ◦ ◦ ◦ ◦ ◦ • 3b3(4), 3b3(4) • ◦ • ◦ • ◦ ◦ ◦ 3b1 0(4), 3b10(4) • ◦ • ◦ ◦ • ◦ ◦ 3b1 1(4), 3b11(4)

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• • • • ◦ ◦ ◦ ◦ 4a(1), 4a(1) • • • ◦ • ◦ ◦ ◦ 4b0(4), 4b0(4) • • • ◦ ◦ • ◦ ◦ 4b1(4), 4b1(4) • • • ◦ ◦ ◦ • ◦ 4b2(4), 4b2(4) • • • ◦ ◦ ◦ ◦ • 4b3(4), 4b3(4) • • ◦ ◦ • • ◦ ◦ 4c0(4) • • ◦ ◦ ◦ • • ◦ 4c1(4), 4c1(4) • • ◦ ◦ ◦ ◦ • • 4c2(4) • • ◦ ◦ • ◦ • ◦ 4c0,1₀ (4), 4c0,1₀ (4) • • ◦ ◦ ◦ • ◦ • 4c0,1₁ (4), 4c0,1₁ (4) • ◦ • ◦ • ◦ • ◦ 4c1,1₀ (2) • ◦ • ◦ ◦ • ◦ • 4c1,1₁ (2)

A.3 8 particle chain & fully connected system

◦◦◦◦◦◦◦◦ 0a(1) •◦◦◦◦◦◦◦ 1a(8) ••◦◦◦◦◦◦ 2a0(8) •◦•◦◦◦◦◦ 2a1(8) •◦◦•◦◦◦◦ 2a2(8) •◦◦◦•◦◦◦ 2a3(4) •••◦◦◦◦◦ 3a0(8) ••◦•◦◦◦◦ 3a1(8) ••◦◦•◦◦◦ 3a2(8) ••◦◦◦•◦◦ 3a3(8) ••◦◦◦◦•◦ 3a4(8) •◦•◦•◦◦◦ 3b2(8) •◦•◦◦•◦◦ 3b3(8) ••••◦◦◦◦ 4a0(8) •••◦•◦◦◦ 4a1(8) •••◦◦•◦◦ 4a2(8) •••◦◦◦•◦ 4a3(8) ••◦••◦◦◦ 4b1(8) ••◦•◦•◦◦ 4b2(8) ••◦•◦◦•◦ 4b3(8) ••◦◦••◦◦ 4c2(4) ••◦◦•◦•◦ 4c3(8) •◦•◦•◦•◦ 4d3(2) A.4 Octahedron ◦ ◦ ◦ ◦ ◦ ◦ 0a(1) • ◦ ◦ ◦ ◦ ◦ 1a(1), 1a(1) ◦ • ◦ ◦ ◦ ◦ 1b(4)

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• • ◦ ◦ ◦ ◦ 2a(4), 2a(4) ◦ • • ◦ ◦ ◦ 2b0(4) ◦ • ◦ • ◦ ◦ 2b1(2) • ◦ ◦ ◦ ◦ • 2c(1) • • • ◦ ◦ ◦ 3a0_{(4), 3a}0₍₄₎ • • ◦ • ◦ ◦ 3a1_{(2), 3a}1₍₂₎ ◦ • • • ◦ ◦ 3b(4) • • ◦ ◦ ◦ • 3c(4)

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Appendix B

Interaction tables

B.1 Cube 2 a a1 _b 0 b1 b2 a [2] 0,1 0,0 a1 0,1,2,3 [0] 1,3 b0 [2] 0,0,3,3 b1 0,0 0,1 [0] 0,3 b2 0,0 0,1,2,3 [0]

Table 11: cube spin-1/2 (↑6↓2) interaction table

3 a b0 b1 b2 b3 b10 b11 a [1]_/ 1,3 0 1 0 b0 [1]/0 0 0 0 3 b1 0 [1]/0 0 1 0 b2 0 0 [−1] 0,1 2 1,0 b3 3 0 0,3 [−1] 3 2,3 b1 0 0 3 2 1 [−1]/0 0,2 b1₁ 0 1 0 3,0 2,1 0,2 [−3]

Table 12: cube spin-1/2 (↑5↓3) interaction table

4 a(1) b0 b1 b2 b3 c0 c1 c2 c0,10 c 0,1 1 c 1,1 0 (2) c 1,1 1 (2) a [2] 0,1,2,3 b0 [0] 0,3 0,1 0,0 b1 0,1 [0] 0,3 0,0 b2 0,1 [0] 0,3 1,1 b3 0 0,3 0,1 [−2] 0,1 0 c0 [2] 0,0 0,0 c1 0,0 [0] 0,1 0,1 c2 0,3,1,2 [−2] 0,2 0,2 c0,1₀ _0,0 0 0,3 0 [−2] 0 1 c0,1₁ _3,3 0 0,3 0 [−2] 1 0 c1,1₀ (2) _0,0,2,2 _1,1,3,3 [−2] c1,1₁ (2) _0,2,1,3 _1,3,0,2 _0,2,1,3 [−6]

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B.2 8 particle chain 2 a0 a1 a2 a3(4) a0 [2] 0,7 a1 0,1 [0] 0,7 a2 0,1 [0] 0,7 a3(4) 0,1,4,5 [0]

Table 14: 8 particle chain spin-1/2 (↑6↓2) interaction table

3 a0 a1 a2 a3 a4 b2 b3 a0 [2] 0 1 a1 0 [0] 0 2 7 a2 0 [0] 0 0 7 a3 0 [0] 0 5 0 a4 7 6 0 [0] 6 b2 1 0 3 2 [−2] 0,2 b3 1 0 0,6 [−2]/3,5

4 a0 a1 a2 a3 b1 b2 b3 c2(4) c3 d3(2) a0 [2] 0 1 a1 0 [0] 0 0 1 a2 0 [0] 0 0 1 a3 7 0 [0] 6 0 b1 0 2 [0] 0 3 b2 0 0 [−2] 0,2 0 1 b3 7 0 0,6 [−2] 0,2 c2(4) 0,4 [0] 0,4 c3 7 5 0,6 0 [−2] 0 d3(2) 1,3,5,7 0,2,4,6 [−4]

B.3 8 particle fully interacting system

2 a0 a1 a2 a3(4) a0 [2]/1,7 0,1,6,7 0,1,5,6 0,1 a1 0,1,2,7 [2]/2,6 0,2,5,7 0,2 a2 0,2,3,7 0,1,3,6 [2] 3,5 0,3 a3(4) 0,3,4,7 0,2,4,6 0,1,4,5 [2]

(34)

3 a0 a1 a2 a3 a4 b2 b3 a0 [−1]/7,1 0,1,7 0,1 0,1 0,1,2 0,6 0 a1 0,7,1 [−1] 0,7 0,3 0,2,3 1,7 1,3,6 a2 0,7 0,1 [−1]/4 0,3,4,7 0,3 0,4 4,7 a3 0,7 0,5 0,4,1,5 [−1]/4 0,7 1,5 0,3 a4 0,7,6 0,5,6 0,5 0,1 [−1] 4,6 1,4,6 b2 0,2 1,7 0,4 3,7 2,4 [−1]/2,4,6 0,2 b3 0 2,5,7 1,4 0,5 7,2,4 0,6 [−1]/3,5

Table 18: 8 particle fully interacting system spin-1/2 (↑5↓3) interaction table

4 a0 a1 a2 a3 b1 b2 b3 c2(4) c3 d3(2) a0 [−2]/7,1 0,1,7 0,1 0,1,2 0,7 0 0,2 2 a1 0,7,1 [−2] 0,7 0,2 0,1 1,7 1 0 0,4 0 a2 0,7 0,1 [−2] 0,7 1,5 0,5 2,7 0,1 1,4 a3 0,7,6 0,6 0,1 [−2] 5,6 1,5 0 1 0,2 0 b1 0,1 0,7 3,7 3,2 [−2] 0,3 0,3 0,3 0,3 b2 0 1,7 0,3 3,7 0,5 [−2] 0,2,5 0 5,7 1 b3 0,6 7 1,6 0 0,5 0,3,6 [−2]/3,5 0,2,5 c2(4) 0,4 0,3,4,7 3,7 0,1,4,5 0,4 [−2] 0,4 c3 6 0,4 4,7 0,6 0,5 1,3 0,3,6 0 [−2] 0 d3(2) 0,2,4,6 0,2,4,6 1,3,5,7 0,2,4,6 [−2]

Table 19: 8 particle fully interacting system spin-1/2 (↑4↓4) interaction table

B.4 Octahedron

1 a(1) b

a(1) [2] 0,1,2,3 b 0, 0 [2]_/

1,3

Table 20: Octahedron spin-1/2 (↑5↓1) interaction table

2 a b0 b1(2) c(1) a [0]_/ 1,3 0,3 0 0 b0 0,0,1,1 [0] 0,2 b1(2) 0,0,2,2 0,1,2,3 [−2] c(1) _{0,0,1,1,2,2,3,3} [−2]

Table 21: Octahedron spin-1/2 (↑4↓2) interaction table

3 a0 _a1₍₂₎ _b _c a0 [0] 0,1 0,3 0,1 a1₍₂₎ 0,1,2,3 [−2] 0,2 0,2 b _0,0,1,1 _0,0 [−2]/1,3 c _0,0,3,3 _0,0 [−2]/1,3

Quantum world on a sphere Using symmetry to obtain the energy spectra of polyhedron configurations of hopping and antiferromagnetic spin-1/2 particles

Quantum world on a sphere

Contents

Popular scientific summary (Dutch)

1

Introduction

2

Applying symmetry

3

The Hamiltonian

4

Problems

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