Density Matrix Renormalization Group Analysisof Spin Chains and Quantum Hall Systems

Emil Johansson Bergholtz

Master's Thesis

Department of PhysicsStockholm University

13th December 2002

Abstract

There is often a need for simpli�cation and approximations in condensed mat-ter physics. A very powerful method for these purposes, the Density MatrixRenormalization Group, is studied in this thesis. The DMRG method has hadtremendous success when it has been applied to one-dimensional systems suchas the antiferromagnetic Heisenberg spin chain, the Ising model and the XY-model. In this thesis, various previously known properties of the antiferromag-netic Heisenberg chain are presented as well as a new application to a QuantumHall system. The QH system that is examined consists of a two-dimensionalelectron gas on a cylinder with a strong magnetic �eld perpendicular to the elec-tron layer. For a thin cylinder, a homogenous ground state is found at �llingfactor ν = 1

2 and examined in some detail. Furthermore, we observe particleas well as hole excitations in the ν = 1

2 ± ε ground states. There is reasonto believe that relatively small adjustments would make it possible to studythe fractionally charged Quantum Hall states corresponding to �lling factorsν = 1

2m+1 .

Acknowledgements

First of all I would like to thank my supervisor Anders Karlhede for introducingme to this subject and for being very helpful during all stages of my work onthis thesis. During this work on I have learned a lot of physics, programmingand how to deal with a large project. It has also opened a new and interesting�eld of physics that I knew almost nothing about prior to this work.

I would also like to thank my brother Olle for helping me with all kinds ofgraphics related problems and all the people in the Field and Particle Theorygroup that has been helping me in various ways.

Contents

1 Introduction and Outline 2

2 Density Matrix Renormalization Group 42.1 The 1-D standard numerical renormalization group . . . . . . . . 42.2 Optimal states from density matrices . . . . . . . . . . . . . . . . 62.3 In�nite systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . 82.3.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Spin Chains and Model Hamiltonians 133.1 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 The Heisenberg, XY and Ising models . . . . . . . . . . . . . . . 14

3.2.1 The Heisenberg model . . . . . . . . . . . . . . . . . . . . 153.2.2 The XY model . . . . . . . . . . . . . . . . . . . . . . . . 163.2.3 The Ising model . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 A DMRG example: The Heisenberg model . . . . . . . . . . . . . 173.3.1 Comments on the algorithms . . . . . . . . . . . . . . . . 173.3.2 The �rst DMRG steps for the Heisenberg chain . . . . . . 173.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Application to Quantum Hall Systems 244.1 Quantum mechanics in strong B-�elds . . . . . . . . . . . . . . . 254.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2.1 Instead of the Coulomb interaction . . . . . . . . . . . . . 274.2.2 Second quantized form of the Hamiltonian . . . . . . . . . 284.2.3 Number representation of fermionic states . . . . . . . . . 30

4.3 DMRG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.3.1 Ground states for any given density . . . . . . . . . . . . 324.3.2 Comments on the algorithm . . . . . . . . . . . . . . . . . 334.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5 Conclusions and Outlook 42

1

Chapter 1

Introduction and Outline

Strongly correlated systems are among the most active areas of theoretical con-densed matter physics today. The models describe systems where a large numberof particles interact strongly with each other and therefore cannot be analyzedstarting from properties of single particles. Even though there exists quantummany body systems that are relatively easy to understand many of them are ex-tremely complicated. This can easily be seen by studying a rather simple manyparticle system of N free charged particles that interact according to Coulomb'slaw in vacuum. In this case the Schrödinger equation reads

[− h2

2

N∑

i=1

∇2i

mi+

14πε0

∑

i<j

qiqj

|~ri − ~rj | ]Ψ({~ri}, t) = ih∂ Ψ({~ri}, t)

∂t; (1.1)

which is clearly far to hard to solve, even numerically, for systems of large N .These systems often need a description based on collective phenomena when ahuge amount of atoms or particles interact. In order to �nd such a descriptionone must look for a simpli�ed model or a so called e�ective theory. When sucha theory is constructed it may still be very hard or even impossible to calculatethe properties that one is interested in analytically. Thus one must also �ndgood numerical prescriptions to deal with that problem.

There are many di�erent ways to simplify the calculations for strongly cor-related systems and one of the most successful ones is a method called DensityMatrix Renormalization Group theory (DMRG), invented by White [1] in 1992.The method obeys the variational principle because it does not introduce un-physical states. It only gives a method of �nding the most relevant ones. Thisparticular method has had tremendous success over the last decade when it hasbeen applied to one-dimensional systems such as the ferro- and antiferromag-netic Heisenberg spin chains, the Ising model and the XY- model.

Another hot area of theoretical condensed matter physics is the quantum Halle�ect. This area deals with interacting electrons moving in two spatial dimen-sions in a very strong perpendicular magnetic �eld. This is also an exampleof a strongly correlated system where there is a need for e�ective theories andnumerical methods.

2

Several phenomena observed in QH systems, such as the fractional QH e�ect1and Skyrmions2, are based on collective rather than on single electron proper-ties. Despite this Laughlin [2] was able to give a microscopic understanding ofthe fractional quantum Hall systems by providing a very good trial wave func-tion for the many particle ground state.

In this thesis we �rst investigate properties of DMRG for some previously stud-ied systems and then we explore the possibility of applying the successful DMRGmethods to a quantum Hall system.

The basic concepts of DMRG are presented in chapter two and these are imple-mented on the antiferromagnetic Heisenberg chain in chapter three. The latterchapter also deals with the description of the Heisenberg, XY and Ising Hamil-tonians and contains some of the results that we have achieved with our DMRGprogram.

Chapter four deals with a quantum Hall system which consists of a two-dimensionalelectron gas on a cylinder with a strong magnetic �eld perpendicular to the elec-tron layer. The Hamiltonian is analyzed and some approximations are made inorder to make it possible to apply the DMRG method. These methods are newand we have been able to extract interesting results for the ν = 1/2 ground stateon a thin cylinder where we �nd a homogenous state. The results are presentedas well as the strengths and weaknesses of the approach. Conclusions and anoutlook are contained in the �fth chapter.

1Particle excitations may have fractional charges (q = e/3, e/5, . . .) at low temperatures inthe fractional QH e�ect.

2Skyrmions are particles (or topological objects) formed by the collective motion of severalelectron. They have the same charge as the particle excitations but the spin is higher.

3

Chapter 2

Density MatrixRenormalization Group

Renormalization is used in many di�erent areas of theoretical physics such asquantum �eld theory and the theory of critical phenomena. However we willonly consider the Density Matrix Renormalization Group [1] and its precursor,the one-dimensional standard numerical Renormalization Group (RG) [3], inthis chapter.

The objective of these renormalization methods is to �nd e�ective descriptionsof large systems. This is done by starting with a small system that is well knownand then increase the size gradually. At the same time as much irrelevant infor-mation as possible is discarded. This procedure is motivated by the fact that thenumber of quantum states of a system increases exponentially with the numberof particles and therefore the computers of today are only able to treat verysmall systems by use of exact methods. However there are many examples ofmacroscopic systems1 where quantum e�ects are crucial. In the following weprovide a systematic way of dealing with some of these systems.

2.1 The 1-D standard numerical renormalizationgroup

Let us consider a chain that one �rst breaks into a set of identical blocks A,as shown in �gure 2.1. From two neighboring blocks A, the correspondingHamiltonians HA and the interactions between the blocks, one forms new andlarger blocks AA and Hamiltonians HAA.

Figure 2.1: A chain divided into identical blocks A.1Two famous examples are the quantum Hall e�ect and the Bose-Einstein condensation.

4

In order to reduce the number of states kept one diagonalizes the Hamilto-nian matrix HAA and uses the m lowest lying eigenstates (if one wants to studythe ground state) to form a new simpler Hamiltonian HA′ which now representsa block twice as large, as in �gure 2.2. In doing this, one assumes that only thelowest lying block eigenstates play a dominant role in forming larger blocks atlater iterations.

Figure 2.2: Two identical blocks A that are used to form A′.

The procedure above represents one iteration and it is repeated until thedesired size is reached. In a given iteration one must represent the block AAby m2 states since each of the blocks A is represented by m states. A slightlymore general case is if one connects two neighboring blocks A and B that arerepresented by m and n states respectively. In this case the block AB is clearlyrepresented by m× n states.

Figure 2.3: Two blocks A and B that are used to form A′.

The result for the blocks A and B will be important later since it makesit possible to keep many states m in every iteration as long as the number ofstates n of the added block B is su�ciently small.

To form the e�ective Hamiltonian we consider the approach used by Wilson[3], involving a change of basis to the new set of eigenstates. This truncationprocedure can be written as

HA′ = OHABO† (2.1)

5

where O is an m×d matrix when d is the dimension of HAB and m is the num-ber of states kept. In the standard numerical renormalization group procedurethe rows of O are the m lowest lying eigenstates of HAB .

It is helpful to notice that no unphysical states are introduced during the calcu-lations and therefore a study of the ground state obeys the variational principlewhich states that

Evariational ≥ E0, (2.2)where E0 is the exact ground state energy. Thus the calculated energy gives anupper bound for the ground state energy.

2.2 Optimal states from density matricesThe lowest energy eigenstates of HAB seem to be the natural states to keepbut as we shall see they are not optimal simply because they are not taking theinteractions with the surrounding blocks into account. An alternative approach,proposed by White and Noack [4], is the superblock method. In this method,one diagonalizes a larger system, the superblock, composed of at least three2blocks including the two blocks A and B which are used to form A′.

Figure 2.4: A scheme of how an iteration of the superblock method is carried out.

The states of the superblock are projected onto AB and these projected3states are kept. However, for a many-particle state, the projection can be donein many di�erent ways. This raises the question of which states are the mostimportant ones. The density matrix provides a good answer according to White[1, 5]. As it turns out the eigenstates with the largest eigenvalues of the densitymatrix4 for AB, produced from the ground state5 of the superblock, are the

2We use four blocks because it will make it easier, as the reader will see later, to takeadvantage of certain symmetries.

3The projection have the same role as the truncation in the standard RG.4If you are unfamiliar with density matrices you should review Feynman's [7] or Sakurai's

[6] introductions to the subject.5Or any other state one wants to study.

6

optimal states to keep in a certain well de�ned sense.

Assume that we have diagonalized the superblock and extracted a particularstate |Ψ 〉, usually the ground state. Let | i 〉, i = 1, . . . , n, be a complete set ofstates for the system AB and let | j 〉, j = 1, . . . , J , be the states of the rest ofthe superblock, i.e. the surroundings. We can then write6

|Ψ 〉 =∑

ij

ψij | i 〉| j 〉. (2.3)

However we do not always want to keep all the states | i 〉. If the number ofstates n required to describe |Ψ 〉 exactly, is greater than the number of states mkept in each iteration we cannot represent |Ψ 〉 exactly with only m states. Hencewe want to construct a set of orthogonal states |uα 〉 =

∑i uα

i | i 〉, α = 1, . . . ,mwhich can be used to accurately represent |Ψ 〉 as follows

|Ψ 〉 ≈ |ψ 〉 =∑

jα

ajα|uα 〉| j 〉. (2.4)

One way of getting an accurate expansion is to demand that

S = ||Ψ 〉 − |ψ 〉|2 (2.5)will be minimized by varying over all ajα and |uα 〉. It can be shown [5] fromlinear algebra that the optimal states |uα 〉 are the eigenstates of the densitymatrix ρ of the system AB which are associated to the m largest eigenvaluesωα of ρ. This can also be explained, or at least hinted at, intuitively. Given thestate (or the ensemble) of the superblock ABCD it is possible to construct theso called density operator ρ for the system AB

ρ =∑

i

ωi|ui 〉〈ui |, (2.6)

where ωi is the probability of AB to be in the state |ui 〉. It is now easy tocalculate the probability of AB to be in an arbitrary state | s 〉

P (| s 〉) = 〈 s |ρ| s 〉 =∑

i

ωi〈 s |ui 〉〈ui | s 〉 =∑

i

ωi|〈 s |ui 〉|2. (2.7)

It is at least plausible that the expression in (2.7) is maximized by | s 〉 = |uα 〉where ωα = max{ωi}. But the states |ui 〉 happen to be the eigenstates ofthe density matrix ρ associated with the eigenvalues ωi. Given this, it is clearthat the best state to keep is the state corresponding to the largest eigenvalue,the second best7 is the state corresponding to the next largest eigenvalue andso on. Thus the density matrix tells us which states are the most important ones.

Suppose that the superblock is in a pure state8 (2.3). In this case9 the reduceddensity matrix for the block AB can be written as

6Note that | i 〉| j 〉 ≡ | i 〉 ⊗ | j 〉.7In the case when only orthogonal states are considered. It is of course just a matter of

taste which m states one projects on as long as they span the same subspace as {|uα 〉}.8I.e. the superblock is certain to be in a quantum state |Ψ 〉.9This is actually assumed to be the case in all applications in this thesis.

7

ρij =∑

k

ψ∗ikψjk, (2.8)

where ψik are the coe�cients in (2.3). A more general expression for the densitymatrix, valid when the superblock is not in a pure state, is

ρij =∑

δ

wδ

∑

k

ψδ∗ik ψδ

jk; (2.9)

where wδ is the probability that the superblock is in the state |Ψδ 〉.

After ρ has been diagonalized it is easy to form the truncation matrices O.Just as in the case of the standard method the truncation procedure can bewritten as

HA′ = OHABO†. (2.10)However, this time the rows of O are the states |uα 〉 corresponding to the mlargest eigenvalues of ρ. This approach does of course also obey the variationalprinciple

EDMRG ≥ E0. (2.11)

2.3 In�nite systemsThere are two di�erent ways to implement the DMRG algorithm. The in�nitemethod in which the system size is increased in every iteration and the �nitemethod where the size of the system is held �xed and the iterations are per-formed to increase the precision of the calculations. Only the in�nite method isdiscussed and used in this thesis.

2.3.1 The algorithmThe algorithm is in general quite abstract and therefore an example of themethod is provided in section 3.3 to make the ideas more transparent. In thefollowing we provide the main steps in a fairly general algorithm for a one-dimensional system with re�ection symmetry with respect to the center of thesystem.

Step 1Construct four initial blocks A, B, C and D and set up matrices representing theblock Hamiltonians and other operators. At this stage the superblock consistsof the four initial blocks and one usually chooses each initial block to representa single site just as in �gure 2.5.

8

Figure 2.5: An initial con�guration of the superblock consisting of only four sites.

Step 2Form all relevant operators (such as Hamiltonians and spin operators etc) of thecomposite blocks AB and CD and the Hamiltonian matrix for the superblockABCD. The operators of the blocks AB and CD are formed from the operatorsof A, B and C, D respectively and from these one can form the superblockHamiltonian HABCD. Note that for systems with re�ection symmetry the blockCD is just the re�ection of AB. In particular when only one site is added ineach iteration we get a very simple con�guration as shown in �gure 2.6.

Figure 2.6: The con�guration of the superblock in the case of systems with re�ectionsymmetry and where only one site is added in each iteration.

Step 3Diagonalize the superblock Hamiltonian to �nd a target state Ψi1i2i3i4 , where i1numbers the states of block A and i2 the states of B etc. In most applicationsΨ is the ground state. At this stage it is possible to measure expectation valuesof various operators expressed in the superblock basis using

〈Q 〉 ≡ 〈Ψ |Q|Ψ 〉 =∑

i1i2i3i4i′1

Ψ∗i1i2i3i4 [Q]i1i′1Ψi′1i2i3i4 , (2.12)

for an operator Q acting on block A. The formulas for the expectation valuesof operators on the other blocks are of course similar to (2.12).

Step 4Form the density matrix for the two-block system AB using

ρi1i2i′1i′2 =∑

i3i4

Ψ∗i1i2i3i4Ψi′1i′2i3i4 . (2.13)

At this stage it is possible to measure expectation values of operators Q actingon the system AB from the relation

〈Q 〉 = TrρQ. (2.14)

9

Step 5Diagonalize ρ and discard all eigenvectors |uα 〉 but the ones associated with them largest eigenvalues ωα, α = 1, . . . , m. At this stage it is possible to estimatethe truncation error because it corresponds to the deviation of

Pm =∑α

wα (2.15)

from unity. If Pm happens to equal unity then the truncation error is zero.

Step 6Form a new block A′ by changing basis and truncating to m states according to

QA′ = OQABO† (2.16)

etc, where the rows of O are the chosen density matrix eigenvectors |uα 〉, for allthe operators Q of the system AB. These are the e�ective matrix representationsof the operators for the new block A′.

Step 7Replace the old block A with the new block A′ and the new block D with there�ection of A′. This is shown in �gure 2.7.

Figure 2.7: Iteration procedure for systems with re�ection symmetry and where onlyone site is added in every iteration.

Step 8Return to step 2.

2.3.2 MeasurementsAs we have already seen it is fairly easy to measure expectation values of arbi-trary, rather complicated, operators on the system or the superblock accordingto

10

〈Q 〉 = TrρQ (2.17)and

〈P 〉 =∑

i1i2i3i4i′1

Ψ∗i1i2i3i4 [P ]i1i′1Ψi′1i2i3i4 (2.18)

respectively. However it can be costly because all the matrices of the operatorsthat are to be examined have to be stored and updated in a new truncated basisat every iteration. This is of course no problem if one only wants to study afew operators at a time, but if many operators are stored the iterations becomeslower and one may run out of computer memory. The obvious solution is thatone only studies a few operators at a time and that the program is customizedfor these. When other operators are to be studied one simply rewrites someparts of the DMRG program. One may argue that there are not that many dif-ferent operators that it would cause problems for most simple systems such asthe Heisenberg or Ising chains. But it turns out that it is far more cumbersometo calculate correlation functions.

It is natural to expect that one can calculate expectation values etc of anyoperator in the system or superblock if one has access to matrices for all pos-sible single-site-operators (such as spin-operators for each site). However thisis not the case. If one tries to measure correlation functions such as 〈QiP j 〉,where i and j denote di�erent sites of the same block10 in a spin chain, onemust be careful since:

〈QiP j 〉 6=∑

i1i2i3i4i′1i′′1

Ψ∗i1i2i3i4 [Qi]i1i′1 [P

j ]i′1i′′1 Ψi′′1 i2i3i4 . (2.19)

However if j and k are sites on di�erent blocks11 there is no problem because

〈QjP k 〉 =∑

i1i2i3i4i′1i′4

Ψ∗i1i2i3i4 [Qj ]i1i′1 [P

k]i4i′4Ψi′1i2i3i′4 . (2.20)

The reason for this is that the sum over i′1 in (2.19) should, but does not, runover a complete set of states. To be able to measure quantities as the one in(2.19), one has to store (and update at each iteration according to (2.16)) thematrix representations [QiP j ]i1i′1 throughout the calculation. By doing this itis possible to calculate the expectation value according to

〈QiP j 〉 =∑

i1i2i3i4i′1

Ψ∗i1i2i3i4 [QiP j ]i1i′1Ψi′1i2i3i4 . (2.21)

An alternative way of looking at this is to consider matrix elements. In factmatrix elements of di�erent operators, expressed in di�erent bases, are all thatwe have access to during the DMRG calculations. Let |A,B 〉 ≡ |A 〉 ⊗ |B 〉 bea state of the block AB. If Qi operates only on A and P k operates only on Bit is clear that we can write

〈A′, B′ |QiP k|A, B 〉 = 〈A′ |Qi|A 〉〈B′ |P k|B 〉. (2.22)10In equation (2.19) we assume that the block is A.11In equation (2.20) we assume that the blocks are A and D.

11

From (2.22) it is obvious that we only need to have access to the matrix rep-resentations of the single-site-operators in this case. However if both Qi andP k operate on the same site, say A, we need a matrix representation for thecorrelation function since

〈A′, B′ |QiP k|A,B 〉 = 〈A′ |QiP k|A 〉〈B′ |B 〉 = δBB′〈A′ |QiP k|A 〉. (2.23)

This fact turns out to severely limit the possibilities of applying DMRG tolong range interactions because the Hamiltonians will generally contain termsconsisting of such correlation functions discussed above. As we shall see later,in chapter four, these terms do not only exist, but the number of them becomesenormous as the length of the interaction is increased.

12

Chapter 3

Spin Chains and ModelHamiltonians

As mentioned previously there is a need for simpli�cation in many areas ofphysics and therefore one must try to �nd e�ective theories which are repre-sented by model Hamiltonians. Three such models - the Heisenberg, XY andIsing models - are presented and discussed brie�y in this chapter. Furthermorea description of how to apply DMRG to these systems and some results of ourown DMRG calculations are included. However, �rst of all we need a descriptionof the quantum mechanical concept of spin.

3.1 SpinIn classical mechanics a rigid body is often said to have two kinds of angularmomentum1. Orbital angular momentum (−→L = −→r × −→p ) associated with themotion of the center of mass, and spin (−→S = I · −→ω ) associated with motionabout the center of mass. However this distinction is just a matter of conve-nience because −→S is nothing but the sum of all orbital angular momenta of allparts of the body that circle around the center of mass.

In quantum mechanics2 there is a fundamental distinction between −→S and −→L .The orbital angular momentum has a similar structure3 as in classical mechan-ics. The spin however has no analog in the classical theory since it cannot bethought of as being made up by the angular momentum of constituent parts. Afundamental di�erence, between −→L and −→S , that arises in quantum mechanics(QM) is that the spin can have both integer and half integer values while onlyinteger values of the orbital angular momentum are allowed.

The quantum mechanical operator4 ~Si = (Sxi , Sy

i , Szi ) operates on site i and

1This is nicely described in chapter three of Scheck's book [8].2See Sakurai [6] for a more detailed description of spin and angular momentum in QM.3Just let (x1, x2, x3) → (x1, x2, x3), (p1, p2, p3) → (p1, p2, p3) and impose the canonical

commutation relations [xi, pj ] = ihδij for the components. The quantummechanical operatorsare then obtained as Li =

∑jk

εijkxj pk.4The operator hats are not explicitly written out when there is no risk for confusion.

13

the components are Hermitian5 operators which obey the SU(2) algebra

[Ski , Sl

j ] = ihδij

∑m

εklmSmj . (3.1)

It is convenient to specify eigenkets of ~S2 and Sz

~S2| s, m 〉 = h2s(s + 1)| s,m 〉 (3.2)Sz| s,m 〉 = hm| s,m 〉. (3.3)

With use of these de�nitions it is now possible to construct ladder operatorsS± = Sx ± iSy and it is not hard to �gure out how they act on the basisvectors6:

S±| s,m 〉 = h√

s(s + 1)−m(m± 1) | s,m± 1 〉. (3.4)Observations in nature tell us that every elementary and composite particle

has a speci�c value of spin, s. π-mesons have spin 0; electrons have spin 1/2;photons have spin 1; ∆-particles have spin 3/2; gravitons have spin 2 and so on.There is a remarkable connection between the spin of a particle and the statis-tics it obeys. The half integer (s = 1

2 , 32 , . . .) spin particles are fermions and

they obey the Pauli exclusion principle7 and Fermi-Dirac statistics. The integer(s = 0, 1, . . .) spin particles obey a fundamentally di�erent kind of statistics,Bose-Einstein statistics, where the particles favor to be in the same state. Innon-relativistic QM this has to be taken as an empirical postulate. However, inthe framework of relativistic quantum mechanics it can be proved8 that the halfinteger spin particles are fermions and that the integer spin particles are bosons.

We are now in a position to list the possible values of the spin s and the z-component m of the spin:

s = 0,12, 1,

32, . . . ; m = −s,−s + 1, . . . , s− 1, s. (3.5)

In many applications there is no confusion about the spin s of the particlesand therefore that index is dropped. For a spin 1/2 particle we write the twopossible states simply as | 1

2 〉 and | − 12 〉 or as | ↑ 〉 and | ↓ 〉. From this we

can easily write down kets of more than one particle by just specifying their mvalues. Such a ket may for example be written like | ↓↓↑↓↑ 〉 for a system of �vespin 1/2 particles.

3.2 The Heisenberg, XY and Ising modelsIn this section we treat three closely related models, which deal with interactingspin systems. We begin with the most complex one, the Heisenberg model, and

5A quantum mechanical operator Q is said to be Hermitian if Q = Q†. † means thatQ†ij = Q∗ji if the operator is expressed in matrix form.

6The only assumption needed to prove this is that the eigenkets are properly normalizedi.e. 〈 s, m′ | s, m 〉 = δm′m.

7The Pauli exclusion principle states that two fermions cannot occupy the same quantummechanical state at the same time.

8This is shown in [9] for the important special cases of s = 0, 12and 1.

14

Figure 3.1: A qualitative picture of the structure of ferromagnets (left) and antiferro-magnets (right).

then impose constraints which lead to the other two models.

3.2.1 The Heisenberg modelThe Heisenberg model is a rather realistic model which describes magnetic sys-tems as interacting spin systems. It was introduced in 1926 by Werner Heisen-berg [10] and it leads to parallel or antiparallel spin ordering at low tempera-tures9. At higher temperatures the spin system will of course be in a disorderedphase.

One can motivate the form of the Hamiltonian by generalizing the results froma simple system of just two spins with

H12 = −J ~S1 · ~S2 (3.6)where J is the exchange coupling constant. A negative J Hamiltonian is calledantiferromagnetic (AFM), since it favors antiparallel spins, while a positive JHamiltonian is called ferromagnetic (FM) since its ground state has parallelspins. If one only takes the interaction between nearest neighbors into accountthe straightforward extension of (3.6) to an N -particle system is

H = −J∑

〈 i,j 〉

~Si · ~Sj (3.7)

which is the so called Heisenberg Hamiltonian10, where 〈 i, j 〉 means that thesummation is to be taken over pairs of nearest neighbors only. In this thesis wepresent results for the one-dimensional version of (3.7). It becomes

H = −J

N−1∑

i=1

~Si · ~Si+1. (3.8)

To be able to do calculations it is useful to rewrite the scalar product using

~Si · ~Sj = Szi Sz

j +12(S+

i S−j + S−i S+j ). (3.9)

The Heisenberg model may seem simple but the solution of the antiferromagneticchain is extremely complicated and it is only solved for the s=1/2 case in one-dimension. The exact 1-D solution11 is obtained by use of the Bethe ansatz [11]

9The one-dimensional AFM Heisenberg chain is disordered at �nite temperatures.10To study the system in a magnetic �eld just add the term

∑i

~B · ~Si.11See [13] for the solution. A nice and pedagogical introduction to the solution of the

problem is given in [14].

15

and the ground state energy12 is found to be

EAFM = NJh2(−14

+ ln 2). (3.10)

The ferromagnetic Heisenberg chain is much simpler and it has a solution, validin any dimension, for which the ground state have all the spins parallel. Theenergy becomes

EFM = −Nh2s2zJ/2, (3.11)where z is the number of nearest neighbors of a spin.

Of course a number of conditions, like that the next nearest neighbor inter-actions can be ignored, has to be ful�lled for (3.7) to be reasonable. Never-theless there are some physical systems13 that are accurately described by theHeisenberg Hamiltonian.

3.2.2 The XY modelThe XY model is obtained by eliminating the z components of the spin interac-tions from the Heisenberg Hamiltonian. This means that the Hamiltonian canbe written as

H = −J∑

〈 i,j 〉(Sx

i Sxj + Sy

i Syj ) (3.12)

or more conveniently as

H = −J

2

∑

〈 i,j 〉(S+

i S−j + S−i S+j ). (3.13)

The advantage of (3.13) compared to (3.12) is that the same algorithms as theone used for the Heisenberg model can be used14 and that S+

i = (S−i )†.

The XY model is completely solvable in one-dimension for spin 1/2 and thesolution is much simpler than the Bethe ansatz solution for the Heisenbergchain. The model arises when one deals with super�uidity or superconductivity[17].

3.2.3 The Ising modelBy further simpli�cation of the spin chain one can obtain the Ising model15[12]; in which only one of the spin components is taken into account. TheHamiltonian can then be written as

H = −J∑

〈 i,j 〉Sz

i Szj . (3.14)

12The �rst one to compute this energy was Lamek Hultén [15], who worked in Stockholm.13The model describes ferromagnetism of magnetic insulators e.g. EuS or EuO [16].14This is done by imposing Sz

i = 0 for all i.15The Ising model was in fact not proposed by Ising himself, but he was the �rst to solve

the 1-D model.

16

This model seems very simple and it is also classical because all the operatorsSz commute. Despite its relative simplicity the model is not (yet) solved in threedimensions even though Onsager [18] found the 2-D solution as early as 1944.

3.3 A DMRG example: The Heisenberg modelThe excellent results for the one-dimensional Heisenberg chain were one of theearly triumphs of the Density Matrix Renormalization Group algorithm. Theground state of the antiferromagnetic S = 1/2 Heisenberg chain is highly non-trivial16, hence it constitutes a good test for the algorithm. Another advantagewith this system is that the actual code can be made fairly compact and trans-parent even for readers that are not used to similar calculations.

3.3.1 Comments on the algorithmsThe in�nite system method is the one used for the calculations. The startingpoint is a small system of only four sites representing four spins according to therecipe given in section 2.3.1. This system is the superblock in the �rst iterationand all the matrix elements of all relevant operators (such as the Hamiltonianand the spin operators) can easily be computed by hand once a convenient basisis chosen. However it is very helpful to think of how all these matrix elementscan be computed from tensor products of the one-site-operators and while doingthis one has to keep track of what the basis looks like. Once this small systemis fully understood it is not hard (in principle) to construct matrix representa-tions of operators for chains of arbitrary size. If the basis is chosen carefully itis possible to simplify the programs to large extent when the re�ection symme-try of the system is used. An example where this is used is given is section 3.3.2.

It is also helpful to consider what the individual matrix elements of di�erentoperators for a larger block look like. Consider two neighboring blocks A andB of an antiferromagnetic (J = −1) Heisenberg chain. The two blocks interactand we �nd that

[HAB ]i1i2i′1i′2 = [HA]i1i′1δi2i′2 + δi1i′1 [HB ]i2i′2 +

+[Szr ]i1i′1 [S

z` ]i2i′2 +

12[S+

r ]i1i′1 [S−` ]i2i′2 +

12[S−r ]i1i′1 [S

+` ]i2i′2 (3.15)

and[Sz

r ]i1i2i′1i′2 = [Szr ]i1i′1δi2i′2 (3.16)

etc. The new tensors, with four indices, contain the matrix elements of theoperators for the composite block AB. In the next section we go on to showhow this is carried out in practice for the antiferromagnetic Heisenberg chain.

3.3.2 The �rst DMRG steps for the Heisenberg chainIn this section we provide an example of how the calculations for the Heisen-berg S = 1/2 chain are performed. We start our calculations by constructing

16The exact solution, obtained by the Bethe ansatz [11, 15], is very complicated to use, asthe reader can check by studying [14].

17

an initial superblock consisting of four sites as in �gure 3.2.

Figure 3.2: The initial con�guration of the superblock of four sites representingABCD.

Now we turn our attention to the formation of relevant operators. In orderto do this we start by choosing a single particle basis:

| 1 〉 ≡ | ↑ 〉 =(

10

); | 2 〉 ≡ | ↓ 〉 =

(01

). (3.17)

We can now express the needed single particle operators in matrix form as

Sz =(

12 00 − 1

2

); S+ =

(0 10 0

); S− =

(0 01 0

), (3.18)

where we have put h = 1. To be able to form the operators of the block AB,which consists of two sites, we must pick a basis for this system. The naturalchoice is the direct products of the single site states:

| 1 〉 ≡ | ↑ 〉 ⊗ | ↑ 〉 = | ↑↑ 〉, | 2 〉 ≡ | ↑ 〉 ⊗ | ↓ 〉 = | ↑↓ 〉,| 3 〉 ≡ | ↓ 〉 ⊗ | ↑ 〉 = | ↓↑ 〉, | 4 〉 ≡ | ↓ 〉 ⊗ | ↓ 〉 = | ↓↓ 〉. (3.19)

We can now compute the matrices of the relevant operators acting on AB.This is done with the help of tensor products. From equation (3.6) we obtainthe Hamiltonian17 for the two-spin system AB:

H12 = Sz1 ⊗ Sz

2 +12(S+

1 ⊗ S−2 + S−1 ⊗ S+2 ). (3.20)

In matrix form we get

H12 =

14 0 0 00 − 1

4 0 00 0 − 1

4 00 0 0 1

4

+

12

0 0 0 00 0 1 00 0 0 00 0 0 0

+

+12

0 0 0 00 0 0 00 1 0 00 0 0 0

=

14 0 0 00 − 1

412 0

0 12 − 1

4 00 0 0 1

4

(3.21)

Because of the interaction of the block AB with the block CD we must alsoform the spin matrices of the rightmost site18 in the basis of AB. This is doneaccording to

17Here we study the antiferromagnetic chain and we set J=-1.18In this case site B.

18

[Sz2 ]AB = [δ1]A ⊗ [Sz

2 ]B (3.22)etc. We are now in a position to write down the Hamiltonian for the superblock:

HABCD = H12 ⊗ δ34 + δ12 ⊗H34 + [Sz2 ]AB ⊗ [Sz

3 ]CD +

+12([S+

2 ]AB ⊗ [S−3 ]CD + [S−2 ]AB ⊗ [S+3 ]CD). (3.23)

Taking advantage of the fact that the Heisenberg spin chain has a re�ectionsymmetry19 we �nd that we can use the same matrices for H34 as for H12 and[Sα

2 ]AB as for [Sα3 ]CD where α = z, +,−. We can also take advantage of the fact

that S+i = (S−i )†.

The superblock Hamiltonian, which is a 16×16 matrix, is now diagonalized andwe obtain the ground state |Ψ 〉 and the corresponding energy E0 ≈ −1, 616 . . ..From |Ψ 〉 we form the density matrix for AB using (2.13):

ρ ≈

0.0223 0 0 00 0.4777 −0.4553 00 −0.4553 0.4777 00 0 0 0.0223

. (3.24)

In this example we keep m = 3 states so we diagonalize the density ma-trix ρ and keep the three eigenvectors |uα 〉 corresponding to the three largesteigenvalues ωα. These eigenvectors are

|u1 〉 ≈

0.0000−0.70710.70710.0000

, |u2 〉 ≈

−0.54990.39400.3940−0.6222

, |u3 〉 ≈

0.80760.11640.1164−0.5663

(3.25)

and the eigenvalues are

ω1 ≈ 0.9330, ω2 ≈ 0.0223, ω3 ≈ 0.0223. (3.26)

The fourth (discarded) eigenvector |u4 〉 happened to have almost the sameeigenvalue, ω4 ≈ 0.0223, as two of the kept eigenvectors. So our choice of statesto keep could (almost) equally well have been any other set as long as |u1 〉 iskept20.

We now form the truncation matrix O, where the rows of O are the threeeigenstates |uα 〉:

O ≈

0 −0.7071 0.7071 0−0.5499 0.3940 0.3940 −0.62220.8076 0.1164 0.1164 −0.5663

. (3.27)

19The block CD is the same as the re�ection of AB with respect to the center of ABCD.20The di�erence between ω4 and ω2,ω3 is only 1.0×10−11 and 2.5×10−11 respectively. This

di�erence is of course negligible because the error made by discarding one of the eigenstatesis larger - by nine orders of magnitude!

19

The next step is to form the new e�ective matrices for the relevant operatorson AB and then rename the block AB as A′; which is then replacing block Aand the next iteration can be performed. This procedure is illustrated in �gure3.3.

Figure 3.3: The procedure carried out in the �rst iteration of our example.

The truncation of all the operators21 are of the form of (2.10) and, as anexample, we get the following e�ective Hamiltonian:

HA′ = OHABO† ≈−0, 7500 0 0

0 0, 2500 00 0 0, 2500

. (3.28)

The observant reader may recognize that we seem to have recovered22 thesinglet state and two of the triplet states. This is expected because in the earlyiterations the surroundings CD is small23. This will however not be the caselater in the calculation because CD grows just as fast as AB.

3.3.3 ResultsAs mentioned earlier the results of DMRG calculations on Heisenberg chainsare excellent. In �gure 3.4 below, we present the ground state energies as wellas the truncation errors of a Heisenberg chain calculation keeping only fourstates. Even though only four states are kept and the calculation time is just amatter of seconds on an ordinary workstation, the above example shows goodprecision. As seen in �gure 3.4 the ground state energy seems to converge asthe length of the system goes to in�nity. As a matter of fact an extrapolationof the curve to in�nity shows that it approaches a value that di�ers by about2.8 × 10−3 from the exact result E0 = 1

4 − ln 2 = −0.443147 . . . obtained byuse of the Bethe ansatz [11]. It is also worth noting that this di�erence is at

21To be able to continue the calculations one has to keep, as a minimum, the HamiltonianHAB as well as the spin operators Sz

r and one of S+r or S−r , where r denotes the rightmost

site of AB.22If more digits were to be presented in (3.28) it would be clear that it is only a good

approximation and not the exact result.23In the limit of no surroundings the DMRG will be equivalent to standard RG and the

exact eigenstates of HAB will be kept.

20

the order of the typical truncation error at each iteration. As the number ofstates kept, m, is increased the results are naturally improved. In table 3.1 wepresent some of these results for the in�nite S = 1/2 Heisenberg chain obtainedusing our DMRG program, and we illustrate how these results are obtained in�gure 3.5. The results are obtained from calculations for N = 4000 sites byextrapolating 1

N → 0. Furthermore the early iterations have been excluded inthese calculations since the surroundings are so small that the calculations maynot be reliable.

40 60 80 100 120 140 160 180 200−0.45

−0.445

−0.44

−0.435

−0.43

Ene

rgy

per

site

Exact energy for the infinite chain

Calculated energy for the finite chain

40 60 80 100 120 140 160 180 2000

0.2

0.4

0.6

0.8

1

1.2

1.4

x 10−3

Iteration

Tru

ncat

ion

erro

r

Figure 3.4: A DMRG calculation of the average ground state energy per site for theantiferromagnetic spin 1/2 Heisenberg chain keeping only four states (m = 4). Theenergy per site (above) is computed from the superblock con�guration as a function ofthe number of iterations and is given in units of h = 1. The superblock includes 2i + 2

sites at the ith iteration. The truncation error (below) at each iteration is given by1− Pm = 1−∑m

i=1ωi.

m EDMRG0 − Eexact

0

4 2.8× 10−3

8 3.0× 10−4

12 1.5× 10−4

16 5.9× 10−5

20 2.6× 10−5

Table 3.1: DMRG results for the in�nite AFM spin 1/2 Heisenberg chain.As seen in table 3.1 the error is falling rapidly with increasing m. It is ofcourse just a matter of computer time to get even better results. However, oneshould keep in mind that one has to diagonalize a 4m2 × 4m2 matrix for thesuperblock in every iteration, so the iterations become very time consuming forlarge m.

21

We have also done calculations on the XY and Ising models, but these resultsare not presented because we want to keep this thesis short and also becausethese results are easier24 to obtain25 using other methods such as the standardRenormalization Group or Monte Carlo simulations.

200 400 600 800 1000 1200 1400 1600 1800 2000−0.45

−0.445

−0.44

−0.435

−0.43

Ene

rgy

per

site

Iteration

Exact energy for the infinite chain

Calculated energy for the finite chain

0 0.2 0.4 0.6 0.8 1 1.2 1.4

x 10−3

3

3.5

4

4.5

5

5.5x 10

−4

Dev

iatio

n

1/N

Figure 3.5: An illustration of how the results in table 3.1 are obtained. The full linecorrespond to calculated deviations from the exact in�nite chain ground state energyand the dashed line represents the extrapolated curve. We kept eight states in eachiteration in this particular example.

24This is particulary true for the Ising model.25Compared to the Heisenberg results.

22

40 60 80 100 120 140 160 180 200−0.45

−0.445

−0.44

−0.435

−0.43

Exact energy for the infinite chain

Calculated energy for the finite chain

40 60 80 100 120 140 160 180 2000

0.5

1

1.5

2

2.5

3

3.5

x 10−5

Iteration

Tru

ncat

ion

erro

r

Figure 3.6: A DMRG calculation of the ground state energy for the antiferromagneticspin 1/2 Heisenberg chain keeping twelve states. Compared to the results in �gure 3.4,the truncation error is much smaller and the energy is closer to the exact value for thein�nite chain.

40 60 80 100 120 140 160 180 200−0.45

−0.445

−0.44

−0.435

−0.43

Ene

rgy

per

site

Exact energy for the infinite chain

Calculated energy for the finite chain

40 60 80 100 120 140 160 180 2000

1

2

3

4

5

x 10−6

Iteration

Tru

ncat

ion

erro

r

Figure 3.7: A DMRG calculation of the groundstate energy for the AFM spin 1/2Heisenberg chain keeping twenty states. The results are now very accurate.

23

Chapter 4

Application to Quantum HallSystems

In this chapter we apply the ideas of DMRG to a two-dimensional system withlonger ranged interactions. This system is a quantum Hall system1 consistingof a two-dimensional electron gas on a cylinder with a strong magnetic �eldperpendicular to the electron layer.

B

y

x

L/

Figure 4.1: A cylinder with a magnetic �eld ~B perpendicular to the electron layer.

Quantum Hall systems are of course more complicated than the one-dimensionalsystems studied in chapter three. Hence we need to generalize the methods weused for the spin chains. There are two big di�erences that have to be takencare of: 1) the system is now two-dimensional and 2) the interaction is longrange instead of only between nearest neighbors.

The �rst problem is handled by mapping the two-dimensional system onto a1See Girvin [19] for a good review.

24

set of one-dimensional harmonic oscillators. This is done in section 4.1. Thelonger range generalization is discussed in section 4.2 and in section 4.3 we im-plement these ideas in our DMRG calculations.

In order to accomplish our task we �rst study the single particle states andthereafter we take the interaction into account. From the interaction and thesingle particle states we �nd a second quantized form of the Hamiltonian. TheHamiltonian is then analyzed and the ground states are approximated by useof the DMRG method.

4.1 Quantum mechanics in strong B-�eldsIn this section we study the quantum states of a single electrically charged par-ticle in a two-dimensional system with a perpendicular magnetic �eld. Sucha system is illustrated by �gure 4.1 and the states we seek correspond to thestates where just one particle is present on the surface of the cylinder.

In order to �nd the single particle Hamiltonian we use the concept of mini-mal coupling and choose an appropriate gauge for the vector potential. A goodchoice is the so-called Landau gauge:

~A = Bxy (4.1)

which gives a magnetic �eld in the direction of z because ∇× ~A = Bz. We cannow write the Hamiltonian as

H =1

2m(p2

x + (py +eB

cx)2) (4.2)

using minimal coupling. As one can see from this expression the Hamiltonianis not invariant under translation along the x-axis, even though the physics is(B-�eld only in the z-direction).

Let us now consider this system on a cylinder as shown in �gure 4.1. We takeadvantage of the translation symmetry in the y-direction and write the wavefunction as

ψ(x, y) = eikyξ(x) (4.3)where we have used separation of variables as an ansatz. Written in this formit is an eigenstate of py and hence we can make the substitution py → hk in theHamiltonian. The system also gives us a constraint on k. Imposing periodicboundary conditions

ψk(x, y) = ψk(x, y + L) (4.4)gives

k =2πp

L, p = 0,±1, . . . (4.5)

where L is the circumference of the cylinder. We then arrive at an e�ectiveone-dimensional Schrödinger equation

hkξ(x) = εkξ(x) (4.6)

25

where

hk ≡ 12m

(p2x + (hk +

eB

cx)2). (4.7)

This is a familiar equation, namely the one-dimensional harmonic oscillator.We can therefore use our knowledge from elementary quantum mechanics [6].The eigenvalues are

εn = (n + 1/2)hωc, n = 0, 1, . . . (4.8)where ωc ≡ eB

mc is the cyclotron frequency. The corresponding eigenfunctions2are

ψnk(~r) =1√

π1/22nn!LeikyHn(x + k`2)e−

12`2

(x+k`2)2 , (4.9)

where Hn is the nth Hermite polynomial and ` ≡√

hceB is the so-called magnetic

length. In this thesis we are only interested in the lowest Landau level (n = 0)with the following eigenfunctions:

ψk(~r) =1√

π1/2L`eikye−

12`2

(x+k`2)2 . (4.10)

As we shall se later these eigenfunctions are well suited for the DMRG method.

Equation (4.8) tells us that the number n gives the energy of a single parti-cle state. Since n is an integer it means that there are discrete energy levels,these are the so called Landau levels [20].

From �gure 4.2 we see that the area per state in the lowest Landau level is2π`2, thus the density of states is ns = 1

2π`2 . Given the number of electrons ne

per unit area in the system we can de�ne the �lling fraction ν as

ν ≡ ne

ns= 2π`2ne =

2πhc

eBne. (4.11)

Figure 4.2: An illustration on how the quantum single particle states are distributedon the cylinder. The dotted lines represent the lines Xk = −k`2 around which thestates ψk are centered.

2Note that the eigenfunctions are centered at x = Xk = −k`2, i.e. the single particle wavefunctions are centered at di�erent x-coordinates depending on the momentum of the state.

26

4.2 The HamiltonianIn analogy to the case of the spin chains we need to �nd a matrix formulation ofthe Hamiltonian to be able to apply the DMRG methods. We also need to makesome simpli�cations to get a system that the computers of today can handle.

4.2.1 Instead of the Coulomb interactionThat the Coulomb interaction has in�nite range means that there will be prob-lems for the algorithms when large systems are studied. In fact the problemswill be enormous because when the size of the system goes to in�nity so doesthe number of matrices that one has to keep in every iteration and therefore thesize of the code also diverges. Unfortunately the Coulomb interaction

V (~r) =e2

4πε|~r| (4.12)

does not vanish fast enough for our purposes. Therefore a more local interactionis introduced. Trugman and Kivelson [21] proposed that the interaction can beexpanded as

V (~r) =∑

j

cjb2j∇2jδ2(~r), (4.13)

where b is the range of the interaction. In the limit of short range, b → 0, theexpression (4.13) reduces to

V (~r) = c1b2∇2δ2(~r) (4.14)

since 〈 δ2(~r) 〉 = 0 for any antisymmetric state. The form of this interaction canbe motivated [21] by the fact that the physics of quantum Hall systems doesnot change signi�cantly if the interaction is made more local. Furthermore theLaughlin states [2] can in fact be shown to be exact eigenstates of this particularinteraction. Expressed in complex coordinates z = x + iy the Laughlin wavefunction is

Ψ 12m+1

(z1, z2, . . . , zN ) =∏

i<j

(zi − zj)2m+1∏

i

e−14 |zi|2/`2 . (4.15)

This describes the quantized Hall states at �lling fraction ν = 12m+1 in a circular

geometry. At ν = 12m the Fermi liquid like states are observed. They are

described by wave functions obtained from ν = 12m Laughlin states. One forms

an antisymmetric state while doing as little damage as possible to the Laughlinstate. This is done by multiplying with a determinant3. One gets

Ψ 12m

(z1, z2, . . . , zN ) = PLLLDet[eiki·rj ]∏

i<j

(zi − zj)2m[∏

i

e−14 |zi|2/`2 ], (4.16)

which is the so called Rezayi-Read [23] wave function where PLLL is a projectionoperator which projects the state onto the lowest Landau level.

3See [22] for a more complete discussion of the properties of these states.

27

4.2.2 Second quantized form of the HamiltonianInspired by previous attempts with momentum space methods for DMRG [24,25] we look for a suitable form of the Hamiltonian. Let us introduce creationand annihilation operators4 c† and c respectively and impose anti-commutation5relations:

{c†i , cj} ≡ c†i cj + cjc†i = δij (4.17)

and{ci, cj} = {c†i , c†j} = 0. (4.18)

We then get the following expression for the second quantized Hamiltonian:

H =12

∑

klmn

Vklmnc†kc†l cmcn (4.19)

whereVklmn =

∫ ∫ψ†k(~r)ψ†l (~r

′)V (~r − ~r′)ψm(~r′)ψn(~r)d2rd2r′ (4.20)

with the eigenfunctions ψk from (4.10). We can now compute the matrix ele-ments (4.20) using the interaction (4.14) and the eigenfunctions in (4.10). Sincewe are dealing with in�nite systems in this thesis we simply let the height ofthe cylinder in �gure 4.1 go to in�nity. We set6 ` = 1 and �nd that

Vklmn =∫ ∫

d2rd2r′[ψ†k(~r)ψ†l (~r′)[c1b

2∇2r−r′δ

2(~r − ~r′)]ψm(~r′)ψn(~r)]

=c1b

2

πL2

∫ L/2

−L/2

∫ ∞

−∞

∫ L/2

−L/2

∫ ∞

−∞dx dy dx′ dy′[(∂2

x−x′+∂2y−y′)δ(x− x′)δ(y− y′)

×ei(n−k)y− 12 ((x+k)2+(x+n)2)ei(m−l)y′− 1

2 ((x′+l)2+(x′+m)2)]

=c1b

2

√2πL

δk+l,m+n((n− l)2 − (n− k)2 − 1)ekl− 12 (m2+n2). (4.21)

From now on we set c1 = 2√

2πLb2 in order to get a repulsive interaction and

units that are easy to work with. We now note that if k = l (or m = n) wewill get no contribution. Furthermore we note that only the parts of Vklmn

that are antisymmetric in k ←→ l and m ←→ n survive in the sum due to theanti-commutation relations (4.18). Given this we can write our Hamiltonian as

H =∑

klmn

(1− δkl)δk+l,m+n((n− l)2 − (n− k)2)ekl− 12 (m2+n2)c†kc†l cmcn. (4.22)

We can easily see from (4.22) that the physics is invariant under translation inmomentum space7. This means that we only have to calculate a few matrixelements and use them throughout the calculations because we can re-numberthe sites at each iteration. Furthermore, we note that the matrix elements just

4c†kcreates a fermion with momentum k and cl annihilate a fermion with momentum l.

5The anti-commutation relations assure that the fermion wave functions are antisymmetricunder particle exchange.

6I.e. we work in units where the magnetic length is the unit length.7To see this just add a constant α to k, l, m, and n, and use the constraint k + l = m + n.

28

depend on two indices and some index-manipulation leads us to the followingform of the Hamiltonian

H =∑

n

∑

k>l

(k2 − l2)e−12 (k2+l2)c†l+nc†k+ncl+k+ncn. (4.23)

Written in the form (4.23) it is easier to see how the interaction falls with dis-tance and the translation invariance is also made more transparent. However,the expression (4.22) is more convenient when we implement the DMRG algo-rithm.

Even though a very local interaction is used it is obvious that the interactionnever vanishes completely. The solution to this problem is that we introducea cuto� distance d, in terms of the number of interacting neighbors8, for theinteraction. This means that for distances larger than d the interaction is con-sidered so weak that we just neglect it.

There are a great number of symmetries that can be used to improve the algo-rithms, but it would not be meaningful to discuss all of them here. However,there are two nice properties of the Hamiltonian that are worth mentioning. Byuse of the anti-commutation relations (4.17) and (4.18) we can easily derive

Vklmnc†kc†l cmcn + Vlknmc†l c†kcncm

+Vklnmc†kc†l cncm + Vlkmnc†l c†kcmcn

= (Vklmn + Vlknm − Vklnm − Vlkmn)c†kc†l cmcn (4.24)

which is valid for all non-vanishing operators appearing in the Hamiltonian(4.19). The expression (4.24) shows that one does not have to calculate asmany operators as matrix elements of V . This is of course good news becauseit is a lot more demanding to compute matrix representations of the operatorsthan it is to compute the matrix elements, since these are just numbers.

In order to further reduce the number of computed operators we use that

(c†kcmcnc†l )† = (cnc†l )

†(c†kcm)†

= (c†l )†c†nc†m(c†k)† = clc

†nc†mck. (4.25)

Since (4.25) is valid for all strings of operators we have reduced the number ofneeded operators even further.

A special case of (4.24) can be used to study which terms are the importantones in our calculations. We simply gather the matrix elements (4.21) and �ndthat

(Vklmn + Vlknm − Vklnm − Vlkmn) = −4(k − l)(m− n)ekl− 12 (m2+n2). (4.26)

In order to analyze this result we note that the terms where k − l = m − nare the largest ones9. Let us look at the coe�cient corresponding to interaction

8For d = 1 we only consider nearest neighbor interactions and for d = 2 we also includenext nearest neighbor interactions etc.

9This is of course only valid when terms corresponding to the same range are compared.

29

between sites with momentum 0 and k. We conclude that

(Vk0k0 + V0k0k − Vk00k − V0kk0) ∼ k2e−k2/2. (4.27)

We can now see clearly that our interaction, that was constructed in order tobe very local, is in fact not maximal for small k. The maximum magnitude isin fact at k =

√2 and if we consider the expression (4.5) it is clear that we can

only handle systems with small L since we cannot handle interactions betweensites separated by long distances. In fact we need to specify the cuto� distanced in terms of how many neighbors a given site interacts with, to the left or tothe right of the site itself. We can interpret the length of the interaction inmomentum space as

∆k =2πd

L. (4.28)

Hence we can only treat systems with relatively small circumference L. Typi-cally we are forced to use L ∼ 2π in order to get reliable results.

4.2.3 Number representation of fermionic statesWe need to construct a many particle basis for the Hilbert space in which wecan perform our calculations. It is helpful to note that the momentum quantumnumber of a particle has the same meaning as the number/position of the sitehad in our calculation for the spin chains. This is natural because, as we sawearlier, the single particle wave functions are centered at positions governed bythe momentum. With this in mind we now adopt the name site for a state witha given momentum.

A good basis is the number representation where we start out by de�ning thevacuum state

ci| 0 〉 = 0, i = 1, 2, . . . , Ns (4.29)where Ns is the maximal possible number of particles in the lowest Landau level.We now act with the creation operators on the vacuum state in order to createparticles with a speci�ed momentum

c†k| 0 〉 = | 01, . . . , 1k, . . . , 0Ns 〉, k = 1, 2, . . . , Ns. (4.30)

A more general state is of course given by

|n1, . . . , nk, . . . , nNs 〉 =Ns∏

i=1

(c†i )ni | 0 〉. (4.31)

These are the basis states which span our 2Ns dimensional Hilbert space10.Hence any state can be written as a linear combination of the states (4.31). Sincewe are dealing with fermions all the occupation numbers ni, i = 1, 2, . . . , Ns,are either zero or one.

10A nice property of these states is that if the vacuum is normalized (〈 0 | 0 〉 = 1), so are allthe other states.

30

It is nice to introduce number operators, which indeed justify the name of therepresentation. We form the operator that counts the number of particles ofmomentum k as

Nk ≡ c†kck (4.32)and an operator that counts the total number of particles:

Ntotal ≡Ns∑

i=1

Ni. (4.33)

It is not hard to see that the number representation provides a set of eigenketsof these operators. We get

Nk|n1, . . . , nk, . . . , nNs 〉 = nk|n1, . . . , nk, . . . , nNs 〉, (4.34)and

Ntotal|n1, . . . , nk, . . . , nNs 〉 = (Ns∑

i=1

ni)|n1, . . . , nk, . . . , nNs 〉. (4.35)

In order to be able to use these results we introduce a matrix representation forthe basis states. Just as in the case of the spin chain take

| 1 〉 =(

10

); | 0 〉 =

(01

), (4.36)

where | 1 〉 and | 0 〉 represent the existence and non-existence of a particle with agiven momentum, respectively. A many particle state is now straight forwardlygiven by direct products as

|n1, . . . , nk, . . . , nNs 〉 =(

n1

1−n1

)⊗. . .⊗

(nk

1−nk

)⊗. . .⊗

(nN

1−nNs

). (4.37)

We now produce a representation of the creation and annihilation operatorsusing this basis. It is easy to see that these matrices do the job for a single site:

c† =(

0 10 0

); c =

(0 01 0

). (4.38)

We can of course also construct representations of arbitrary composite operatorsby tensor products of the single site operators in the same manner as we formedthe spin operators earlier. An alternative way of getting states as in (4.37) is to�rst construct single site creation operators, expressed in the many site basis,and then use equation (4.31) to obtain the states from the vacuum state11. Afurther discussion on how relevant operators are formed is given in section 4.3.2.

11Which is a simple special case of (4.37) with nk = 0 for all k.

31

4.3 DMRGSince the Hamiltonian we investigate in this thesis has a longer interaction rangethan the nearest neighbor interaction of the spin chains, we are forced to writelonger and more intricate computer programs. But we are not facing any newfundamental di�culties.

4.3.1 Ground states for any given densityBecause the interactions between the particles are repulsive one naturally ex-pects that the ground state should be the empty state and the highest lyingenergy state should correspond to a �lled state. However it is more interestingto investigate states with a de�nite number of particles i.e. a de�nite density.This is motivated by the fact that the physical system as a hole should be elec-trically neutral, i.e. the electrons are compensated by a positive backgroundcharge.

In order to target a state of a speci�c �lling fraction (density), ν , we addan extra term to the Hamiltonian that makes it energetically favorable to bein a state with that density. We also want to �nd a state with no spread ofthe density i.e. the mean density of the system should be ν and the standarddeviation should be zero. This is natural to demand because we want to targeta state of de�nite density and not a linear combination of other states that hap-pens to give ν when we compute the expectation value of the density. In orderto achieve this we introduce a new operator

n ≡ 1N

N∑

i=1

c†i ci (4.39)

which gives the average density of the system. Now we add an extra term tothe Hamiltonian:

H → H ′ = H + ∆H = H + γN2(n− ν)2 (4.40)where ν ∈ [0, 1]. This means that we can express ∆H in matrix representationas

∆Hij = γN2∑

k

(nik − νδik)(nkj − νδkj). (4.41)

Let us go on to see what this means. Using the de�nition of the uncertainty ofan operator ∆n and the linearity of expectation values we �nd

〈Ψ |∆H|Ψ 〉 = 〈∆H 〉 = 〈 γN2(n− ν)2 〉= γN2〈 n2 − 2νn + ν2 〉

= γN2(〈 n2 〉 − 〈 n 〉2︸ ︷︷ ︸≡(∆n)2

+〈 n 〉2 − 2ν〈 n 〉+ ν2)

= γN2(∆n)2 + γN2(〈 n 〉 − ν)2. (4.42)

32

When γ is large enough the ground state of the new Hamiltonian H ′ will notbe the empty state, but a state which minimizes the contribution from ∆H.However, this contribution is clearly positive semi-de�nite:

〈 a |∆H| a 〉 ≥ 0 (4.43)for all states | a 〉. This means that the ground state, |Ψ 〉, of H ′ will be suchthat it satis�es

〈Ψ |∆H|Ψ 〉 = 0. (4.44)This clearly �xes the average density to ν and minimizes the spread in thisvariable. So we have found a term that accomplishes our task when it is addedto the Hamiltonian. There is of course a huge number of possible states thatful�lls this condition but the state we �nd when we minimize 〈Ψ |H ′|Ψ 〉 withrespect to |Ψ 〉 is naturally also the state that has the lowest energy because ofthe �rst term in the expression (4.40) for H ′.

4.3.2 Comments on the algorithmThe algorithm for the QH systems is equivalent to the spin chain methods inmost parts. However, there are some vital di�erences due to the fact that theinteraction is no longer only between nearest neighbors. The Hamiltonian isbuilt up by operators of the form

c†kc†l cmcn (4.45)where k, l,m and n represent either two or four di�erent sites. In the case oftwo distinct sites we have simple interactions corresponding to operators like

c†kckc†l cl ≡ NkNl. (4.46)The above expression (4.46) just gives the (repulsive) interaction between theparticles of momentum k and l and it is easily treated just in the same manneras the spin interactions were treated earlier. The only di�erence, in this case,is that one has to keep12 more operators Nk = c†kck because of the increasedlength of interaction.

In the case when all of k, l, m and n are di�erent we end up with slightly morecomplicated operators. These are formed in steps by iterative use of tensorproducts of the single site operators. The following example will probably makethe idea more transparent:

Example 1Suppose that we want to construct the operator

c†i c†i+4ci+1ci+3 = c†i ci+1ci+3c

†i+4. (4.47)

Starting from the matrix representation of c†i when i is the rightmost site of theblock A we add a single site i + 1 representing B. We get

12And of course update to new basis sets.

33

[c†i ci+1]AB = [c†i ]A ⊗ [ci+1]B . (4.48)This operator is now truncated using

[c†i ci+1]A′ = O[c†i ci+1]ABO† (4.49)

and A′ is renamed A′ → A. We now add another site i+2 but this time there isno creation or annihilation operator acting on the site in the expression (4.47).We insert a unit matrix instead:

[c†i ci+1]AB = [c†i ci+1]A ⊗ [δi+2]B . (4.50)

This is now truncated and a new representation is obtained;

O : [c†i ci+1]AB 7→ [c†i ci+1]A′ . (4.51)

Now A′ is renamed A′ → A and the site i + 3 is added. The new operator

[c†i ci+1ci+3]AB = [c†i ci+1]A ⊗ [ci+3]B (4.52)

is formed. Just as in the previous steps the operator is truncated and in thelast step we form

[c†i ci+1ci+3c†i+4]AB = [c†i ci+1ci+3]A ⊗ [c†i+4]B . (4.53)

We have now arrived at a representation of one of the operators13 that form theQH Hamiltonian (4.22).

4.3.3 ResultsAs already mentioned we are forced to study the quantum states on thin cylin-ders. Although we have tried to vary the circumference L of the cylinder wedecided to only present the results obtained for L = 2π. Hence the physicalsystem can be interpreted as a very local interaction on a thin but very longcylinder. Furthermore we present results obtained with d ranging from four tosix. There are no observed qualitative di�erences when we use these di�erentinteraction lengths.

The results were quite surprising. We expected, or hoped, to �nd some ofthe fractional quantum Hall states [2] for the lowest Landau level correspondingto

ν =1

2m + 1, m ∈ N (4.54)

where m is a positive integer. However these states were not found but in-stead we seem to have found another interesting state - a homogenous stateat ν = 1/2. There are two more states that were easily found. The emptyν = 0 state and the �lled ν = 1 state. The structure of these two states is ofcourse trivial but the fact that they are found indicates that our calculationsare making sense. Another nice thing that we observed was that the states weretotally symmetric in particles ←→ holes when ν ←→ 1 − ν for ν ≤ 1/2. In

13Namely the operator (4.47).

34

50 100 150 200 250 300 350 400

0.3

0.4

0.5

0.6

0.7

0.8

Ave

rage

ene

rgy

per

site

50 100 150 200 250 300 350 4000.4995

0.5

0.5005

Number of sites

Ave

rage

den

sity

Figure 4.3: The ν = 1/2 ground state energy per site (above) and the average densityof the system (below) obtained with four states kept. The average density is almostperfectly constant and the energy seems to converge smoothly. That the energy increasesis due to the fact that the edge e�ects become less important when the system size isincreased.

�gure 4.3 we present results obtained for the ν = 1/2 state. The results seemto be reasonable and a further investigation shows that the found state has theproperty that the density is perfectly uniform throughout the system, see �gure4.4. A calculation shows that the mean density is 1/2 up to the precision of atleast 16 digits and the standard deviation of the single site density is as smallas 10−12 for large systems. Even though the density is not as perfectly uniformas in �gure 4.4 when we let the interaction range d be �ve or six the results arestill strikingly uniform14. Another fact that leads us to believe that we havefound a very special state is that the energy seems to be very low compared toother states15. Thus the obtained state has used the o� diagonal elements ofthe Hamiltonian to reduce the energy in a very e�cient way.

The results we obtained when we targeted ν = 1/3 were quite discouragingcompared to the ν = 1/2 results. As shown in �gure 4.5 the results are notlooking good and we cannot observe any convergence. Furthermore the cal-culated ν = 1/3 state does not have a uniform density distribution as shownin �gure 4.6. As a matter of fact we were able to �nd another ν = 1/3 statewith considerably lower energy than the one found in �gure 4.5 and, since theDMRG method obeys the variational principle, it is clear that we have not foundthe true ground state. This lower energy state is treated in �gure 4.7 and itwas surprisingly obtained by our algorithm when we kept only m = 2 states

14As an example, the standard deviation of the single site density is at the order of 10−7

for d = 615As we shall see soon it has even lower energy per site, and therefore also per particle, than

ground states with lower densities ν.

35

in each iteration and it qualitatively has the structure hole-hole-particle-hole -hole-particle etc. The probable cause to the problems when we keep more statesis that the DMRG method is not robust if the number of nonzero eigenvalues ofthe density matrix is smaller than the number of states m kept in each iteration.

Inspired by the results for the ν = 1/2 state we investigate densities near 1/2.As shown in �gure 4.8 the state looks like the ν = 1/2 state in the beginningand then it suddenly jumps. This is then repeated periodically. In order to gainfurther knowledge of what is going on we compute the expectation value of thedensity at di�erent sites of a block of density ν = 0.499. The results are shownin �gure 4.9, 4.10 and 4.12. Furthermore, we present results with a targeteddensity ν = 0.501 in �gure 4.11.

We can clearly see that we recover the ν = 1/2 state even though ∆H in(4.41) makes it energetically unfavourable to be in a state of that density. Butwhen we continue the iterations something dramatic happens, as we were leadto believe by studying �gure 4.8.

In order to analyze what is happening we recapitulate how we perform thecalculations. In each iteration we add two sites at the center of the superblockwhich is the block which is used to �nd a target state |Ψ 〉. This state is obtainedby diagonalizing H ′ = H + ∆H = H + γN2(n− ν)2 and extracting the groundstate |Ψ 〉 of H ′. What we �nd is that the ν = 1/2 ± ε ground state is alwaysthe ν = 1/2 state unless it is more favourable to place a hole or an extra particleat a localized position. No intermediate states are formed, i.e. in each iterationwe add two sites to the superblock and we always �nd that we add zero, one ortwo particles to the system depending on the situation. An illustration of this isprovided in �gure 4.9 and �gure 4.10 where we show the density distribution atthe center of the superblock at two consecutive iterations. We observe a suddentransition and the net e�ect is that in the last iteration no particles were addedto the superblock even though two more sites were added16. The exact shapeof the hole or particle excitation happens to depend on the number of statesm kept and the strength of the coupling γ but the number of holes or particlesadded is independent of these parameters. This is illustrated by the resultsdisplayed in �gure 4.12, where we observe a more localized hole excitation. Theresults are also found to be practically independent of the range of interactionwhen d = 4, 5 and 6.

We can only speculate on how much our obtained ν = 1/2 ground states has todo with known results for quantum Hall systems such as (4.16). It is possiblethat the restriction of the system to small circumferences L rules out these statesbut it is not certain. Hence, it would be very nice to be able to compare ourDMRG results with the analytical ones corresponding to (4.15) and (4.16) inthe cylinder geometry. However this is not manageable because we would be re-stricted to compare small systems with these results for practical reasons. Sincewe su�er from large edge-e�ects for these small systems this kind of comparisonwould not be meaningful.

16Adding more sites have the physical interpretation of increasing the magnetic �ow per-pendicular to the electron layer.

36

0 5 10 15 20 25 300.4995

0.4996

0.4997

0.4998

0.4999

0.5

0.5001

0.5002

0.5003

0.5004

0.5005

Center

Den

sity

dis

trib

utio

n

Figure 4.4: A plot of the single site density 〈 c†i ci 〉 for a few sites close to the centerof the superblock (at the right edge of the system). The calculations are from the samesystem and for the same number of states kept as in �gure 4.3 and we have usedd = 4. The distribution seems to be perfectly uniform and a calculation shows that themean density is 1/2 up to the precision of at least 16 digits. Furthermore the standarddeviation of the single site density is as small as 10−12.

50 100 150 200 250 300 350 400

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Ave

rage

ene

rgy

per

site

50 100 150 200 250 300 350 400

0.3

0.32

0.34

0.36

0.38

Number of sites

Ave

rage

den

sity

Figure 4.5: Results from a DMRG calculation with the same conditions as in �gure4.3 except that the targeted density is ν = 1/3. No smooth convergence is seen.

37

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Center

Den

sity

dis

trib

utio

n

Figure 4.6: The ν = 1/3 density distribution close to the center of the block in thelast iteration performed in �gure 4.5.

0 20 40 60 80 100 120 140 160 180 200

0.04

0.06

0.08

0.1

0.12

0.14

Ave

rage

ene

rgy

per

site

0 20 40 60 80 100 120 140 160 180 200

0.3

0.32

0.34

0.36

0.38

Iteration

Ave

rage

den

sity

0 5 10 15 20 25 30 350

0.5

1

Center

Den

sity

dis

trib

utio

n

Figure 4.7: An example of a state with considerably lower energy than the state in�gure 4.5 even though the average density is approximately 1/3. Note that this statealso has a non uniform density distribution.

38

50 100 150 200 250 300 350 400

0.3

0.4

0.5

0.6

0.7

Ave

rage

ene

rgy

per

site

50 100 150 200 250 300 350 4000.44

0.46

0.48

0.5

0.52

0.54

Number of sites

Ave

rage

den

sity

Figure 4.8: Results from a DMRG calculation with the same conditions as in �gure4.3 except that the targeted density is ν = 0.49. We observe a rather stable processwith periodic jumps.

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Center

Den

sity

Figure 4.9: This shows the expectation values of the single site density operators〈 c†i ci 〉. This plot is identical to a plot of the densities in the ν = 1/2 system althoughthe results are obtained from the superblock con�guration in a DMRG calculation withthe targeted density ν = 0.499 and 498 sites.

39

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Center

Den

sity

dis

trib

utio

n

Figure 4.10: This is the ground state obtained for the same system as in �gure 4.9when we performed one extra iteration. Hence the results are obtained from the su-perblock con�guration in a DMRG calculation with the targeted density ν = 0.499 and500 sites. This plot indicates the possibility of a hole excitation in the ν = 1/2 system.The results di�er dramatically from those in �gure 4.9 and it is obvious that somethinghas happened in the last iteration. This localized object (or hole) has exactly the samenumber of particles as the ν = 1/2 state with two sites less. I.e. the density jumpsfrom a state with ν = 1/2 to a state with ν = 0.498. The center of the plot correspondsto the center of the superblock.

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Center

Den

sity

dis

trib

utio

n

Figure 4.11: Here we see evidence for a particle excitation. The results are obtainedfrom a DMRG calculation with the targeted density ν = 0.501 and 502 sites.

40

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Center

Den

sity

dis

trib

utio

n

Figure 4.12: A clear indication of holes. Note that two of the sites are practicallyempty (〈 c†i ci 〉 ≈ 10−12). The typical length of the excitation is one magnetic length`. The results are obtained by letting γ = 20 be much smaller than in the previouscalculation shown in �gure 4.10. The circumstances are equivalent of those in �gure4.10 in all other respects.

41

Chapter 5

Conclusions and Outlook

In this thesis we have been able to reproduce some previously known resultsfor one-dimensional nearest neighbor interacting systems using our own DMRGprograms. These systems are the FM and AFM spin 1/2 Heisenberg, XY andIsing chains. The most striking results of the DMRG method, that we repro-duced, are the highly nontrivial ground state results for the antiferromagneticspin 1/2 Heisenberg chain.

Furthermore, we have mapped a two-dimensional quantum Hall system onto aone dimensional system and we have been able to handle interactions of longerrange. We have also developed a method that makes it possible to study groundstate properties at di�erent densities, ν.

We looked for homogeneous QH states, since the Laughlin states have uniformdensity distribution, and found a very promising ν = 1/2 state. Furthermore, weexamined our ν = 1/2 results and found some interesting properties such as verylow energy, perfectly constant density distribution and localized particle as wellas hole excitations. The excitations were observed when we targeted ν = 1/2±ε.

When we targeted other states than the ν = 1/2 state we could not observeany smooth convergence and the states had nonuniform density distributions.However one observable e�ect was that the states had a symmetry in particles←→ holes when ν ←→ 1− ν for ν ≤ 1/2.

Despite our relative success, we have not been able to study the whole classof systems that we would like to have done. For practical reasons we were re-stricted to study the states on cylinders with relatively small circumferences L.Furthermore we were not able to compare our results with analytical groundstate results because our systems are too large and the smaller systems that westudied su�er from disturbances from edge e�ects.

Our work has so far answered a few questions and raised many new. A naturalway to continue this project would be to implement the �nite DMRG methodfor a new, closely related, system. This new system is a system with periodicboundary conditions in both spatial directions, i.e. a torus. This would have anumber of advantages. For example we would be able to treat smaller systems

42

with high accuracy. Therefore we would be able to compare our results withanalytical results for this system [26, 27]. Moreover we would have no problemsrelated to the edge e�ects and we would probably even be able to resolve theproblem that forced us to study systems with small circumferences L. The onlyobvious problem with this approach is that the number of operators that weneed to store grows with the size of the system. However this problem is notsevere and a study of a closely related system, the quantum Hall e�ect in higherLandau levels, has in fact been performed by Shibata and Yoshioka [25].

As a matter of fact we have already dealt with many of the problems related tothe �nite size method when we have constructed our in�nite method programs.The largest hurdle at the moment seems to be that we need to change program-ming language. It might also be cumbersome to �nd a matrix representation ofthe Hamiltonian since the single particle states on the torus are elliptical thetafunctions.

43

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[13] D.C. Mattis, The theory of Magnetism I: statistics and dynamics (Springer-Verlag, Berlin Heidelberg New York, 1981)

[14] M. Karbach and G. Müller, Introduction to the Bethe ansatz I, cond-mat/9809162 (1998)

[15] L. Hultén, Ark. Met. Astron. Fysik 26A, Na. 11 (1938)

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[17] J. Cardy, Scaling and Renormalization in Statistical Physics (CambridgeUniversity Press, Cambridge, 1996)

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[21] S.A. Trugman and S. Kivelson, Exact results for the fractional quantumHall e�ect with general interactions, Phys. Rev. B31, 5280 (1985)

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45