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arXiv:1112.2709 · Physical Review B

Multiparticle pseudopotentials for multicomponent quantum Hall systems

Simon C. Davenport, Steven H. Simon

Published Feb 24, 2012• 18 citations
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Abstract

The Haldane pseudopotential construction has been an extremely powerful concept in quantum Hall physics---it not only gives a minimal description of the space of Hamiltonians but also suggests special model Hamiltonians (those where certain pseudopotential are set to zero) that may have exactly solvable ground states with interesting properties. The purpose of this paper is to generalize the pseudopotential construction to situations where interactions are $N$-body and where the particles may have internal degrees of freedom such as spin or valley index. Assuming a rotationally invariant Hamiltonian, the essence of the problem is to obtain a full basis of wave functions for $N$ particles with fixed relative angular momentum $L$. This basis decomposes into representations of SU$(n)$ with $n$ the number of internal degrees of freedom. We give special attention to the case where the internal degree of freedom has $n=2$ states, which encompasses the important cases of spin-1/2 particles and quantum Hall bilayers. We also discuss in some detail the cases of spin-1 particles ($n=3$) and graphene ($n=4$, including two spin and two valley degrees of freedom).

DOI
10.1103/physrevb.85.075430

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The Haldane pseudopotential construction has been an extremely powerful concept in quantum Hall physics---it not only gives a minimal description of the space of Hamiltonians but also suggests special model Hamiltonians (those where certain pseudopotential are set to zero) that may have exactly solvable ground states with interesting properties. The purpose of this paper is to generalize the pseudopotential construction to situations where interactions are NNN-body and where the particles may have internal degrees of freedom such as spin or valley index. Assuming a rotationally invariant Hamiltonian, the essence of the problem is to obtain a full basis of wave functions for NNN particles with fixed relative angular momentum LLL. This basis decomposes into representations of SU(n)(n)(n) with nnn the number of internal degrees of freedom. We give special attention to the case where the internal degree of freedom has n=2n=2n=2 states, which encompasses the important cases of spin-1/2 particles and quantum Hall bilayers. We also discuss in some detail the cases of spin-1 particles (n=3n=3n=3) and graphene (n=4n=4n=4, including two spin and two valley degrees of freedom).
In fractional quantum Hall physics, the Hilbert space is projected to a single Landau level and the entire Hamiltonian consists of just the projected interelectron interaction. Haldane's pseudopotential formalism has been an extremely useful tool both for understanding these interactions and for understanding the quantum Hall states that result. In the current paper, we consider the analog of this pseudopotential construction that results from general MMM-body interactions rather than the usual (Coulomb) two-body interaction.
We consider trial wave functions exhibiting SU(K)\text{SU}(K)SU(K) symmetry, which may be well suited to grasp the physics of the fractional quantum Hall effect with internal degrees of freedom. Systems of relevance may be either spin-unpolarized states (K=2)(K=2)(K=2), semiconductor bilayers (K=2,4)(K=2,4)(K=2,4), or graphene (K=4)(K=4)(K=4). We find that some introduced states are unstable, undergoing phase separation or phase transition. This allows us to strongly reduce the set of candidate wave functions eligible for a particular filling factor. The stability criteria are obtained with the help of Laughlin's plasma analogy, which we systematically generalize to the multicomponent SU(K)\text{SU}(K)SU(K) case. The validity of these criteria is corroborated by exact-diagonalization studies for SU(2) and SU(4). Furthermore, we study the pair-correlation functions of the ground state and elementary charged excitations within the multicomponent plasma picture.
A subtle relation between quantum Hall physics and the phenomenon of pairing is unveiled. By use of second quantization, we establish a connection between (i) a broad class of rotationally symmetric two-body interactions within the lowest Landau level and (ii) integrable hyperbolic Richardson-Gaudin-type Hamiltonians that arise in (px+ipy)({p}_{x}+i{p}_{y})(px​+ipy​) superconductivity. Specifically, we show that general Haldane pseudopotentials (and their sums) can be expressed as a sum of repulsive noncommuting (px+ipy)({p}_{x}+i{p}_{y})(px​+ipy​)-type pairing Hamiltonians. The determination of the spectrum and individual null spaces of each of these noncommuting Richardson-Gaudin-type Hamiltonians is nontrivial yet is Bethe ansatz solvable. For the Laughlin sequence, it is observed that this problem is frustration free and zero-energy ground states lie in the common null space of all of these noncommuting Hamiltonians. This property allows for the use of a new truncated basis of pairing configurations in which to express Laughlin states at general filling factors. We prove separability of arbitrary Haldane pseudopotentials, providing explicit expressions for their second quantized forms, and further show by explicit construction how to exploit the topological equivalence between different geometries (disk, cylinder, and sphere) sharing the same topological genus number, in the second quantized formalism, through similarity transformations. As an application of the second quantized approach, we establish a ``squeezing principle'' that applies to the zero modes of a general class of Hamiltonians, which includes but is not limited to Haldane pseudopotentials. We also show how one may establish (bounds on) ``incompressible filling factors'' for those Hamiltonians. By invoking properties of symmetric polynomials, we provide explicit second quantized quasihole generators; the generators that we find directly relate to bosonic chiral edge modes and further make aspects of dimensional reduction in the quantum Hall systems precise.
The discovery of the quantum Hall effect in 1980 opened to physics one of the simplest systems for studying many-body correlations. Numerical techniques and trial wavefunctions have proven useful for describing the novel collective behavior of the electrons, but have not fully explained all features of the fractional quantum Hall effect. For example, it is predicted that Landau level mixing should have a moderate effect on the system for all but the very strongest magnetic fields, but this effect has not been extensively studied. Among the tools most useful to modeling and describing the quantum Hall system is the Haldane pseudopotential, which is the energy of electron pairs as a function of their coupled total angular momentum. Because the pseudopotential uniquely determines the correlative behaviors in the quantum Hall system, the study of the pseudopotential is particularly important. In this dissertation I will first derive novel analytic expressions for the pair interactions and the pseudopotential using the techniques of angular momentum algebra. Then, I will use these results to numerically determine the effect of Landau level mixing on the pair pseudopotentials.
The quasiholes of the Read-Rezayi clustered quantum Hall states are considered, for any number of particles and quasiholes on a sphere, and for any degree kkk of clustering. A set of trial wavefunctions, that are zero-energy eigenstates of a k+1k+1k+1-body interaction, and so are symmetric polynomials that vanish when any k+1k+1k+1 particle coordinates are equal, is obtained explicitly and proved to be both complete and linearly independent. Formulas for the number of states are obtained, without the use of (but in agreement with) conformal field theory, and extended to give the number of states for each angular momentum. An interesting recursive structure emerges in the states that relates those for kkk to those for $k\ensuremath{-}1$. It is pointed out that the same numbers of zero-energy states can be proved to occur in certain one-dimensional models that have recently been obtained as limits of the two-dimensional k+1k+1k+1-body interaction Hamiltonians, using results from the combinatorial literature.
We consider a two-dimensional model of particles interacting in a Landau level. We work in a finite disk geometry and take the particles to interact with a linearly decreasing two-body Haldane pseudopotential. We show that the ground-state subspace of this model is spanned by the wave functions that can be written as polynomial conformal blocks (of an arbitrary conformal field theory) consistent with the filling fraction (scaling dimension). To remove degeneracies, we then add a quadratic perturbation to the Hamiltonian and show that (1) conformal blocks constructed using the Moore-Read construction (e.g., Laughlin, Pfaffian, and Read-Rezayi states) remain exact eigenstates of this model in the thermodynamic limit, and (2) by tuning an externally imposed single-body $\ensuremath{-}{L}_{z}^{2}$ potential we can enforce Moore-Read conformal blocks to become exact ground states of this model in the thermodynamic limit. We cannot rule out the possibility of residual degeneracies in this limit. This model has no filling dependence and is comprised only from two-body long-range interactions and external single-body potentials. Our results provide insight into how conformal block wave functions can emerge in a Landau level.