Polarized electrons in the lowest Landau level

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Numerical studies of the FQHE

In this Chapter we set up the numerical (exact) diagonalization of the FQH problem in finite geometries. Throughout this thesis, by exact diagonalization (ED) we refer to the procedure where the Hamiltonian of the system, subject to particular boundary conditions, is represented in a convenient (finite) basis and its energy spectrum numerically evaluated using the computer. This method offers the possibility to study the quantum system in an unbiased manner and it was used since the early days of the FQHE, for example in working out the physical properties of the Laughlin wave function. [41] By now it has assumed the status of a standard tool in FQH studies where perturbative approaches are not guaranteed to work and physical properties expected in the thermodynamic limit can be identified already in very small systems. Although the energy spectrum itself offers substantial information about the physical system, it is furthermore possible to make direct comparison between the numerically-obtained eigenvector and the given trial wave function believed to describe the system’s state. This amounts to evaluating a simple scalar product between the two vectors and we refer to it as the overlap calculation. Knowledge of the eigenvectors also allows one to calculate various operator mean values, propagators etc. The virtue of exact diagonalization is its versatility because it can be applied to any system. However, in practice the computational effort exponentially increases with the size of the system and therefore ED is usually restricted to small systems. Despite this, ED remains a powerful tool of choice for the theoretical studies of the FQHE.
Popular choices of the geometry in the FQH literature have been the disk [42], the sphere [127] and the rectangle with periodic boundary conditions (torus) [128]. In this thesis we focus on the latter two geometries which do not have open boundaries and therefore are particularly useful in examining the bulk properties of incompressible FQH liquids. This Chapter will largely present an overview of the pioneering papers by Haldane [42, 127, 129] and their extensions. [1, 18, 21, 130]

Exact diagonalization: Sphere

The original insight of Haldane [127] was that the two-dimensional sheet containing electrons can be wrapped around the surface of a sphere, while the perpendicular (radial) magnetic field is generated by a fictitious Dirac magnetic monopole at the sphere’s center. The single particle problem had previously been solved by Wu and Yang. [131, 132] We will quote the main results without derivation. A recent pedagogical summary containing some of the derivations is given in [1].
The magnetic flux through the surface of the sphere (measured in units of Φ0), is quantized to be an integer and denoted by Nφ = 2Q. The position of an electron is given by the spherical coordinates R = √QlB , θ, φ. Because of the rotational symmetry enforced by the choice of the geometry, the orbital angular momentum and its z-component are good quantum numbers, denoted by l and m, respectively. Due to the presence of the monopole, their allowed values are l = |Q|, |Q| + 1, . . . and m = −l, −l + 1, . . . , l. Different angular momentum shells correspond to Landau levels in the spherical geometry. The degeneracy of each Landau level is equal to the total number of m values, i.e. 2l + 1, it is obviously finite and increasing by two units for each successive Landau level. The single particle eigenstates are the monopole harmonics, denoted by YQlm, which are a generalization of the familiar spherical harmonics for Q = 0. In the lowest Landau level, they reduce to YQQm = 2Q+1 2Q (−1)Q−mvQ−muQ+m, (2.1) 2Q − m where u, v are the spinor variables [127] u = cos θ/2eiφ/2, v = sin θ/2e−iφ/2. (2.2)
The spinor variables are not independent because |u|2 + |v|2 = 1 and we can transform to a single z complex coordinate by stereographic mapping z = 2Rv/u, R being the radius of the sphere. The basis states are given by zm/(1 + |z|2/4R2)1+Nφ /2, where the Lz momentum quantum number is Lz = Nφ/2 − m. Note that if Nφ, R → ∞ (with Nφ/R 2 fixed), the sphere becomes effectively flat and the single particle states reduce to those on the disk, zme−|z|2 /4.
For N electrons, the filling factor corresponding to the thermodynamic limit (1.8) is given by ν = lim N , (2.3) N→∞2Q but in case of the finite system the relation (2.3) requires a slight modification:
Nφ = 1 νN −δ, (2.4)
which defines the so-called shift, δ. The shift is the topological number that characterizes each FQH state on the spherical surface. It is of order unity (e.g. the Laughlin state at ν = 1/3 has δ = 3) and it is required to specify the system in addition to the filling factor ν. In the thermodynamic limit of an infinite plane, the shift plays no role, but for a finite sphere it is a crucial aspect of the ED technique [42] as it can lead to an “aliasing” problem: at a fixed choice of (Nφ, N ), more than one quantum Hall state (having different ν, δ and, therefore, different physical properties) may be realized.
We are now in a position to set up the Hilbert space which contains the Slater determinants | m , m , . . . , m N = c† . . . c† | 0 built from the single-particle states labelled by m . It is 1 2 m1 mN i often advantageous to exploit symmetries in order to reduce the dimension of the Hamiltonian matrix because we only consider transtionally/rotationally invariant Hamiltonians. An obvious symmetry which is used in the construction of Hilbert space is the conservation of Lz = i mi component. In principle it is possible to demand full rotational symmetry and diagonalize the Hamiltonian in the invariant subspaces of the L2 operator, but the numerical implementation of this symmetry is difficult. One can also use the symmetry under the discrete parity transforma-tion Lz → −Lz in diagonalizing within Lz = 0 sector of the Hilbert space.
The general interacting Hamiltonian in the basis of states |m1, m2, . . . , mN quantized form, is given by H = 12 m1 ,m2 ,m3 ,m4 =−Q m1, m2|V |m3m4 c†m1 c†m2 cm4 cm3. , expressed in second- (2.5)
The interaction V usually is such that it depends solely on the relative coordinate and not the center of mass. This is the case for Coulomb (1.9) and any other physical interaction. Furthermore, if the interaction involves only two particles at a time, it is convenient to express the matrix element m1, m2|V |m3, m4 in the following form 2QL m1, m2|V |m3, m4 = Qm1, Qm2|L, M V2Q−L L, M |Qm3, Qm4 (2.6) L=0 m=−L where the numbers V2Q−L are called the Haldane pseudopotentials and can be directly evaluated (a useful formula for the Coulomb interaction in the LLL is given in [130]). Evaluated on the sphere, V2Q−L represents the energy of two particles in a state with relative angular momentum L in the plane. Apart from geometrical Clebsch-Gordan coefficients L, M |Qmi, Qmj , the set of {VL |L = 0, . . . , 2Q} uniquely specifies the interacting Hamiltonian in any one Landau level. If the particles are fermions, the only physically relevant pseudopotentials are those with odd 2Q − L because of the exchange antisymmetry. Using the algebra of the harmonic oscillator raising operators, it can be shown that the Fourier transform of the effective interaction Veff of the nth LL projected to the LLL is simply related to the Fourier transform of the LLL interaction V : Veff (k) = Ln( k22 ) 2 V (k). Therefore, as long as we remain in a single Landau level, the problem is uniquely specified by a simple sequence of numbers VL. Another important insight also due to Haldane [42] was that specifying a few non-zero VL’s defines special short-range Hamiltonians that uniquely produce certain FQH trial states as their zero-energy modes. The simplest example is the Laughlin state at ν = 1/3 which is the unique, highest density zero mode of the following pseudopotential Hamiltonian: V1>0, V3=V5=…=0 (2.7)
This is the hard-core interaction that is represented by (1.22) in the real space. We can numer-ically obtain the ground state of the interaction (2.7), inserting (2.7) into the Hamiltonian (2.5) and choosing the proper shift Nφ = 3N − 3. We will recover a unique zero-energy state with a gap controlled by the value of V1 and a magneto-roton branch in the excitation spectrum, [45] the known facts from the physics of the Laughlin state (Sec. 2.2.1).

Example: Effect of finite thickness on the Laughlin ν = 1/3 state

To illustrate the principle of numerical investigation, let us try to tackle the following simple problem: what is the effect of finite layer width on the ν = 1/3 state. We can model the finite-width FQH system using the effective Zhang-Das Sarma interaction, (1.17). We evaluate the Haldane pseudopotentials for the ZDS interaction, numerically diagonalize the Hamiltonian (2.5) for several values of the width parameter d and obtain the exact eigenvectors |Ψexact . On the other hand, the ground state of the interaction (2.7) is just the Laughlin state |ΨL . By evaluating the scalar product between the two vectors, ΨL|Ψexact , we can monitor how well the Laughlin wave function describes the exact ground state. We know that for d = 0 the overlap is very large (nearly 1), which means that the Laughlin wave function is an excellent representative of the ground state. More details on the numerical calculations are given in the Appendix. The results for several values of d are given in Fig. 2.1. We see that the overlap decreases with the thickness of the sample, which is consistent with the experiments that see the weakening of the Laughlin state in the samples of very wide width. In the inset to Fig. 2.1 we repeated the calculation for the first excited LL, using the effective interaction Veff as we explained above. Here we may note that the Laughlin state is a much worse description of the exact state for small values of d. Indeed, for N = 5 particles we even obtain zero overlap. Small overlap usually means that we have a wrong candidate wave function. Zero overlap, on the other hand, means that we are testing wave functions of different symmetry, e.g. an incompressible L = 0 state (such as the Laughlin) against the compressible L > 0 state. However, for large d we see that the Laughlin state is stabilized. Thus, we can argue that finite width has the opposite effect on the Laughlin state in the two Landau levels: while it destabilizes the ν = 1/3, it enhances (and may even be necessary for the appearance of) ν = 7/3 state.
We would like to stress that in order to draw conclusions from the overlap calculations, one needs to assure that a range of system sizes has been studied that would allow at least a minimal finite-size scaling to be performed. The results should also be robust to the slight variation of the interaction parameters in order to be able to identify the phase of the system. Still, examples are known [82] where two fundamentally different trial states both have high overlaps with the exact state and it is not obvious which one is the wrong candidate. In such cases, the study should be complemented with the analysis of excitations, different choice of boundary conditions (Sec. 2.2) and/or alternative tools like the entanglement spectrum.

Entanglement spectrum on the sphere

The idea of entanglement, borrowed from the field of quantum information [134, 135, 136], provides a useful tool to study FQH states on the sphere [137] and torus [138]. Imagine that the sphere is divided at a line of latitude into two regions, A and B, so that 2Q + 1 orbitals are partitioned into NorbA around the north pole and NorbB around the south pole. A general many-body state |ψ can be decomposed on the product basis HA ⊗ HB involving a sum over the basis of subsystem A and and a sum over the basis of subsystem B. We can alternatively perform a Schmidt decomposition (equivalent to the singular value decomposition of a matrix) of |ψ |ψ = ie− 1 ξi |ψAi ⊗ | ψBi (2.8 where exp(− 1 ξi) ≥ 0, |ψAi ∈ HA, |ψBi ∈ HB , and ψAi|ψAj = ψBi |ψBj = δij , giving exp (− 1 ξi) as the singular values and |ψAi and |ψBi the singular vectors. If the state is normalized, i exp (−ξi) = 1. The ξi’s can be thought of as the “energy levels” of a system with ther-modynamic entropy at “temperature” T = 1. The entanglement entropy, S = iξi exp (−ξi) i.e. the von Neumann entropy of the subsystem A, has been shown to contain information on the topological properties of the many-body state. [139] The full structure of the “entanglement spectrum” of levels ξi contains much more information than the entanglement entropy S, a single number. This is analogous to the extra information about a condensed matter system given by its low-energy excitation spectrum rather than just by its ground state energy. For model states, such as Laughlin or Moore-Read, the low-lying part of the entanglement spectrum displays the structure of the Virasoro levels of the corresponding conformal field theory, up to some limit set by the size of the spherical surface. The counting can be used as a fingerprint of topological order also in the generic states, such as Coulomb, which have much more complicated Slater decomposition than the model states. We illustrate this with the example of the entanglement spectrum for the exact ground state of the Coulomb interaction at ν = 5/2, Fig. 2.2. A very large system of N = 20 is split in two parts with the same number of particles and number of orbitals. Because the FQH ground state is translationally and rotationally invariant (with quantum number Ltot = 0 on the sphere), and the partitioning of Landau-level orbitals conserves both gauge symmetry and rotational symmetry along the z-direction, in either block A or B both the electron number (NeA and NeB ) and the total z-angular momentum (LAz and LBz ) are good quantum numbers constrained by NeA + NeB = Ne, LAz + LBz = 0. The entanglement spec-trum splits into distinct sectors labeled by NeA and LAz. The low-lying part of the entanglement spectrum (with the “banana”-like shape) displays the counting 1, 1, 3, 5, . . . which is the same as that of the Moore-Read state (we show the entanglement spectrum of the Moore-Read state in Chapter 5, Fig. 5.6). This result contributes to the belief that the generic ν = 5/2 state is indeed described by the Pfaffian wave function. Furthermore, by changing the location of the cut, one can establish that there are exactly three nonequivalent countings that one may obtain and these correspond to the three topological sectors of the Ising CFT we mentioned in Sec. 1.3.2 (see also Sec. 2.2.1).
Figure 2.2: Entanglement spectrum for N = 20 electrons on the sphere, Coulomb interaction at the filling factor ν = 5/2. The block A consists of NA = 10 electrons and lA = 19 orbitals, corresponding to the cut [0|0] in Ref. [137].

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Example: Multicomponent states in the ν = 1/4 bilayer

We consider now a second example to illustrate the extension of ED formalism to the case when there is an internal degree of freedom. This degree of freedom can be represented by an ordinary SU(2) spin or the layer/subband index in bilayer systems or wide quantum wells. The basis states are now specified by pairs of indices |j1, σ1; j2, σ2; . . . jNe , σNe , where σ =↑, ↓ is the eigenvalue of the Sz projection of spin. If we are dealing with the physical spin, the Hamiltonian is additionally contrained to be an eigenstate of S2 operator of the total spin which acts as a Casimir operator for the SU(2) group. In the application to bilayer systems where the SU(2) symmetry is broken for any nonzero value of d (distance between the layers), S2 no longer commutes with the Hamiltonian and the maximal symmetry classification is provided by Sz . In addition, while working in the Sz = 0 subspace, it is possible to use the discrete symmetry Sz → −Sz to further reduce the dimension of the Hilbert space. Haldane’s pseudopotential formalism is straightforwardly extended to the spinful/bilayer case which yields two sets of interaction coefficients, VLintra and VLinter, for the intra and inter-layer interaction, respectively(we assume the layers are indentical). We consider the filling factor ν = 1/4 in the LLL, which was the subject of recent experiments [118], and for illustration purpose we analyze to what extent this state can be described by the multicomponent wave functions, Ψ5 5 3 ≡ 553, Ψ7 7 1 ≡ 771, introduced in Sec. 1.4.1. The experiment where ν = 1/4 state was discovered was performed in a single wide quantum well which we will study within a more complete model in Chapter 5. We assume the system is effectively a bilayer, described by the interaction (1.49), and calculate overlap between the exact state and each of the trial states as a function of d, Fig. 2.3. The trial states are defined as the zero mode of the following potentials V553intra = {0, 1, 0, 1, 0, . . .}, V553inter = {1, 1, 1, 0, 0, . . .} and V771intra = {0, 1, 0, 1, 0, 1, 0 . . .}, V771inter = {1, 0, 0, . . . }. 553 displays a familiar maximum [108] in the overlap for intermediate distance between the layers. The 771 overlap starts to increase with larger d, consistent with the large difference in the correlation exponents for intra and inter components. Since our numerics is performed in the spherical geometry, 771 appears as the usual incompressible state, but in the experiment this state would closely compete with the two coupled Wigner crystals. We cannot address this competition in the spherical diagonalization because this geometry is not adapted to assess states with broken translational symmetry. Note also that we cannot make a direct comparison between the Coulomb ground state overlap with each of the trial states, 553 and 771, because the latter are characterized by different shifts (2.4), δ = 5 and δ = 7 respectively. This examples illustrates the fundamental problem of the spherical geometry in describing the competition between different phases realized at the same filling factor, but characterized by different shifts.

Example: Abelian vs. non-Abelian states on the torus

We mentioned above that an incompressible state on the torus is characterized by k = 0 pseudo-momentum. In Fig. 2.4 we show an example of the low-lying energy spectrum for N = 10 electrons on the torus at the filling factor ν = 1/3 and interacting with Coulomb and hard-core V1 interaction (2.7). The latter interaction produces a unique zero-energy ground state with k = 0, which is described by the Laughlin wave function, along with a characteristic magneto-roton branch of excitations. The high overlap between the ground states of the two interactions, as well as the general similarity of the low-lying spectra, is a convincing evidence that the Coulomb interacting state at ν = 1/3 is the Laughlin state.

Example: Torus degeneracy of the 331 multicomponent state

Analysis of translation symmetry can be immediately generalized to the multicomponent states. We can construct multicomponent Halperin states (1.47) from the Chern-Simons theory using the so-called K-matrix [143] which turns out to be nothing else but a set of wave function exponents m1, …, mK ; {nij }. The torus degeneracy of such a state is given by det K = qN ′ (2.20) where q is the denominator of the filling factor that describes the overall center-of-mass degen-eracy, while N ′ is an integer that accounts for the different translations of the centers of mass of the different components. The points which have parity invariance are k = (0, 0), (0, N ′/2), (N ′/2, 0), (N ′/2, N ′/2) and therefore we can express N ′ as the sum of the number of k = 0 states (N0) and the number of states that lie on the zone boundary NB . For homogeneous liquids, we have N ′ = N0 + 3NB . [144] For the 331 state, we have q = 2 and the associated K-matrix is given by which gives det K = 8. The explicit form of the 331 wave function in the torus geometry is a straightforward generalization of (2.18) and can be found e.g. in Ref. [144]. Therefore, the multicomponent degeneracy that distinguishes the 331 state is 4 and the degenerate ground states are expected to have momenta k = (0, 0), (0, N ′/2), (N ′/2, 0), (N ′/2, N ′/2). We can verify this fact to numerical precision using the pseudopotential interaction that produces the 331 state as the unique and densest zero mode. For Coulomb interaction, we expect the multiplet of states to be only approximately degenerate. Let us consider the Coulomb bilayer system at the filling factor ν = 1/2 as a function of distance d. We assume that the tunneling between the layers is negligible and the layers are in balance, so Sz = 0 is a good quantum number. The Hamiltonian for each d is diagonalized in all the different k sectors and the energies are plotted relative to the ground state at a given d, Fig. 2.7. We consider N = 8 particles and set the aspect ratio to r = 0.99, to avoid further degeneracies due to the special symmetry of the Brillouin zone. To track the evolution of the ground state, we highlight the states belonging to the momentum sectors of interest. We see that the tentative quadruplet of states for the 331 begins to form only beyond d ≥ lB . The multiplet structure is robust to variation of the aspect ratio of the torus, which eliminates the CFL as the other competing phase at the same filling factor (furthermore, CFL in the same-size system has additional degeneracies which are not seen in the spectrum of Fig. 2.7). This is consistent with the result in the spherical geometry which shows maximum overlap between the Coulomb state and the 331 for this range of d. However, on the sphere the overlap is not very low for smaller d’s and even for d = 0. The torus spectra clearly prove that the overlap on the sphere gives an incomplete picture because we also must be in the region of d which gives us the correct ground-state degeneracy. The two geometries, of course, should not give inconsistent results if a range of system sizes is considered and all the competing phases are properly taken into account.

Table of contents :

1 Introduction 
1.1 Fractional quantum Hall effect
1.2 Polarized electrons in the lowest Landau level
1.2.1 Laughlin’s wave function
1.2.2 Effective field theories
1.2.3 Composite fermions
1.2.4 Compressible state ν = 1/2
1.3 Second Landau level
1.3.1 The ν = 5/2 state
1.3.2 Conformal field theory approach
1.4 Multicomponent quantum Hall systems
1.4.1 Wide quantum wells
1.4.2 Quantum Hall bilayer
1.5 Multicomponent systems studied in this thesis
1.5.1 Quantum Hall bilayer at ν = 1
1.5.2 ν = 1/2
1.5.3 ν = 2/5
1.5.4 ν = 1/4
1.5.5 Graphene
2 Numerical studies of the FQHE 
2.1 Exact diagonalization: Sphere
2.1.1 Example: Effect of finite thickness on the Laughlin ν = 1/3 state
2.1.2 Entanglement spectrum on the sphere
2.1.3 Example: Multicomponent states in the ν = 1/4 bilayer
2.2 Exact diagonalization: Torus
2.2.1 Example: Abelian vs. non-Abelian states on the torus
2.2.2 Example: Torus degeneracy of the 331 multicomponent state
2.3 Summary
3 Quantum disordering of the quantum Hall bilayer at ν = 1 
3.1 Chern-Simons theory for the Halperin 111 state
3.2 Trial wave functions for the quantum Hall bilayer
3.3 Basic response of trial wave functions
3.4 Chern-Simons theory for the mixed states
3.4.1 Case 1
3.4.2 Case 2
3.4.3 Generalized states
3.5 Possibility for a paired intermediate phase in the bilayer
3.5.1 First-order corrections to the 111 state
3.5.2 Discussion
3.5.3 Numerical results
3.6 Conclusion
4 Transitions between two-component and non-Abelian states in bilayers with tunneling 
4.1 Transition between 331 Halperin state and the Moore-Read Pfaffian
4.1.1 BCS model for half-filled Landau level
4.1.2 Exact diagonalization
4.1.3 Pfaffian signatures for intermediate tunneling and a proposal for the phase diagram
4.1.4 Generalized tunneling constraint
4.2 Transition from 332 Halperin to Jain’s state at ν = 2/5
4.2.1 The system under consideration
4.2.2 Exact diagonalizations
4.2.3 Intepretation of the results within an effective bosonic model
4.3 Conclusions
5 Wide quantum wells 
5.1 Finite thickness and phase transitions between compressible and incompressible states
5.2 Two-subband model of the quantum well
5.2.1 Connection between the quantum-well model and the bilayer
5.3 ν = 1/2 in a quantum well
5.4 ν = 1/4 in a quantum well
5.5 Conclusion
6 Graphene as a multicomponent FQH system 
6.1 Interaction model for graphene in a strong magnetic field
6.1.1 SU(4) symmetry
6.1.2 Effective interaction potential and pseudopotentials
6.2 Multicomponent trial wave functions for graphene
6.2.1 [m;m,m] wave functions
6.2.2 νG = 1/3 state in graphene
6.2.3 [m;m − 1,m] wave functions
6.2.4 [m;m − 1,m − 1] wave functions
6.3 Conclusions
7 Outlook 

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