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arXiv:1306.3268 · Physical Review B

Repulsive interactions in quantum Hall systems as a pairing problem

Gerardo Ortíz, Zohar Nussinov, J. Dukelsky, Alexander Seidel

Published Oct 8, 2013• 43 citations
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Abstract

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 $({p}_{x}+i{p}_{y})$ superconductivity. Specifically, we show that general Haldane pseudopotentials (and their sums) can be expressed as a sum of repulsive noncommuting $({p}_{x}+i{p}_{y})$-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.

DOI
10.1103/physrevb.88.165303

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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 study of model Hamiltonians, whose exact ground states are well known quantum Hall states, have been investigated solely in the literature from the point of view of first quantized approach in the last few decades. However, a second quantized approach to such Hamiltonians has not been taken in the same measure. In this dissertation, we study Haldane pseudopotential, which is the parent Hamiltonian of 1/M Laughlin state, and Trugman-Kivelson pseudopotential, which is the parent Hamiltonian of 2/5 unprojected Jain state. We find that the study of these Hamiltonians in their second quantized forms not only reproduces all properties of their zero modes already known in first quantized approach, but also sheds lights on general studies of such structures of these Hamiltonians as frustration free lattice Hamiltonians.
Nearly fifty years after its experimental debut, the quantum hall effect has produced perhaps more unique descriptions than any other subject of condensed matter physics. From topological band theory to non-commutative geometry, topological and conformal field theories, random matrix models and more, each approach captures a subset of the essential features, and compliments the others to produce a beautiful mosaic description. In this thesis, we revisit the fractional quantum hall (FQH) effect from its oldest vantage point, that of the trial wavefunctions and the parent Hamiltonians which stabilize them. Laughlin's observation of the importance of the theory of holomorphic functions to describing incompressible states in a partially filled Landau level has produced a coherent paradigm for realizing FQH parent Hamiltonians as frustration-free exactly-solvable one-dimensional models through dimensional reduction. Trial wavefunctions are then classified in terms of a small number of product states which generate all other states in the orbital-occupation expansion of the full state by a sequence of angular-momentum-preserving squeezing operations. These unentangled states are called the DNA of the FQH state, and they can be described in terms of simple local rules called generalized Pauli principles (GPP) of the form "at most r electrons can occupy k consecutive orbitals". It was only recently shown that an analysis of this form can be carried out in second-quantization for states with Landau level mixing, such as the Jain composite-fermion and parton series. In this case, the GPPs are generalized once further to entangled Pauli principles (EPP) for which the DNA may be entangled with respect to the extra Landau level index. In the first part of this thesis, we prove a theorem that allows for a rigorous computation of EPPs given a frustration free k-body parent Hamiltonian by examining only k-particle zero-modes, and give an example of its utility by demonstrating the exact-solvability of a correlated pair-hopping model that can be derived as a special limit of a FQH parent Hamiltonian. In the second part, taking advantage of the second quantized EPP analysis from the continuum, we derive a class of bosonic lattice models for states in multiple Landau levels which, in the presence of a contact interaction, are shown to stabilize non-Abelian ground states. Furthermore, we show how the EPP can be used to compute the degeneracies of quasihole excitations, and therefore the topological order through the bulk-boundary correspondence, which we verify with simple numerics. To the best of our knowledge, this work represents the first exponentially-local exactly-solvable fractional Chern insulator model for non-Abelian states with only two-body interactions.
The family of ``Jack states'' related to antisymmetric Jack polynomials are the exact zero-energy ground states of particular model short-range many-body repulsive interactions, defined by a few nonvanishing leading pseudopotentials. Some Jack states are known or anticipated to accurately describe many-electron incompressible ground states emergent from the two-body Coulomb repulsion in the fractional quantum Hall effect. By extensive numerical diagonalization, we demonstrate the emergence of Jack states from suitable pair interactions. We find empirically a simple formula for the optimal two-body pseudopotentials for the series of most prominent Jack states generated by contact many-body repulsion. Furthermore, we seek a realization of arbitrary Jack states in realistic quantum Hall systems with Coulomb interaction, i.e., in the partially filled lowest and excited Landau levels in quasi-two-dimensional layers of conventional semiconductors such as GaAs or in graphene.
Recently, interest has increased in the hyperbolic family of integrable Richardson-Gaudin (RG) models. It was pointed out that a particular linear combination of the integrals of motion of the hyperbolic RG model leads to a Hamiltonian that describes ppp-wave pairing in a two-dimensional system. Such an interaction is found to be present in fermionic superfluids (3He{}^{3}\text{He}3He), ultracold atomic gases, and ppp-wave superconductivity. Furthermore the phase diagram is intriguing, with the presence of the Moore-Read and Read-Green lines. At the Read-Green line a rare third-order quantum phase transition occurs. The present paper makes a connection between collective bosonic states and the exact solutions of the px+ipy{p}_{x}+i{p}_{y}px​+ipy​ pairing Hamiltonian. This makes it possible to investigate the effects of the Pauli principle on the energy spectrum, by gradually reintroducing the Pauli principle. It also introduces an efficient and stable numerical method to probe all the eigenstates of this class of Hamiltonians. We extend the phase diagram to repulsive interactions, an area that was not previously explored due to the lack of a proper mean-field solution in this region. We found a connection between the point in the phase diagram where the ground state connects to the bosonic state with the highest collectivity, and the Moore-Read line where all the Richardson-Gaudin (RG) variables collapse to zero. In contrast with the reduced BCS case, the overlap between the ground state and the highest collective state at the Moore-Read line is not the largest. In fact it shows a minimum when most other bosonic states show a maximum of the overlap. We found remnants of the Read-Green line for finite systems, by investigating the total spectrum. A symmetry was found between the Hamiltonian with and without single-particle part. When the interaction was repulsive we found four different classes of trajectories of the RG variables.
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).
We report on our systematic attempts at finding local interactions for which the lowest-Landau-level projected composite-fermion wave functions are the unique zero energy ground states. For this purpose, we study in detail the simplest nontrivial system beyond the Laughlin states, namely, bosons at filling factor $\ensuremath{\nu}=2/3$, and identify local constraints among clusters of particles in the ground state. By explicit calculation, we show that no Hamiltonian up to (and including) four-particle interactions produces this state as the exact ground state and speculate that this remains true even when interaction terms involving a greater number of particles are included. Surprisingly, we can identify an interaction, which imposes an energetic penalty for a specific entangled configuration of four particles with relative angular momentum of $6\ensuremath{\hbar}$, that produces a unique zero energy solution (as we have confirmed for up to 12 particles). This state, referred to as the $\ensuremath{\lambda}$ state, is not identical to the projected composite-fermion state, but the following facts suggest that the two might be topologically equivalent: the two states have a high overlap, they have the same root partition, the quantum numbers for their neutral excitations are identical, and the quantum numbers for the quasiparticle excitations also match. On the quasihole side, we find that even though the quantum numbers of the lowest energy states agree with the prediction from the composite-fermion theory, these states are not separated from the others by a clearly identifiable gap. This prevents us from making a conclusive claim regarding the topological equivalence of the $\ensuremath{\lambda}$ state and the composite-fermion state. Our study illustrates how new candidate states can be identified from constraining selected many-particle configurations and it would be interesting to pursue their topological classification.