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arXiv:2104.05777 · Nuclear Physics B

Zk(r) -algebras, FQH ground states, and invariants of binary forms

Hamed Pakatchi

Published Oct 31, 2022• 1 citation
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Abstract

A prominent class of model FQH ground states is those realized as correlation function of Zk(r)-algebras. In this paper, we study the interplay between these algebras and their corresponding wavefunctions. In the hopes of realizing these wavefunctions as a unique densest zero energy state, we propose a generalization for the projection Hamiltonians. Finally, using techniques from invariants of binary forms, an ansatz for computation of correlations 〈ψ(z1)⋯ψ(z2k)〉∏i<j(zi−zj)2r/k is devised. We provide some evidence that, at least when r=2, our proposed Hamiltonian realizes Zk(2)-wavefunctions as a unique ground state.

DOI
10.1016/j.nuclphysb.2022.116010

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Model wavefunctions are essential for studying fractional quantum Hall (FQH) phases. They reveal the intricate connections between model FQH states, conformal field theory, Jack polynomials, and pseudopotential Hamiltonians. However, most model wavefunctions studied so far are in the continuum. In this work, we construct model wavefunctions and the corresponding exact parent Hamiltonians for the entire ZkZ_kZk​ Read--Rezayi series on a lattice. Our model wavefunctions obey a generalized clustering rule implemented on the lattice, which we find is encoded in the Macdonald polynomials providing a unified algebraic description of our lattice-specific model states. This ensures that the model states are exact zero-energy ground states of local density interactions and exhibit an infinitely large gap in the particle entanglement spectrum. Numerical studies of the Laughlin and Moore--Read series find the expected zero-mode and one-flux quasihole counting and positive energy gaps at the system sizes studied, and the available finite-size scaling suggests nonzero thermodynamic gaps. Tuning the model interaction range can drive a phase transition out of the FQH phase. Our work not only advances the theoretical understanding of the underlying mathematical structure of lattice model FQH states but also is potentially useful for designing exotic many-body phases in engineered quantum platforms.
The wave functions corresponding to the zero-energy eigenvalue of a one-dimensional quantum chain Hamiltonian can be written in a simple way using quadratic algebras.We describe here two applications.The non-hermitian Hamiltonians related to one-dimensional stochastic processes have always zero-energy eigenvalues corresponding to the stationary states.A class of quadratic algebras corresponding to diffusion processes was classified.As a second application, for a class of q-deformed O(N ) symmetric antiferromagnetic quantum chains, we give the zero-energy wavefunctions for periodic boundary conditions corresponding to momentum zero and for zone boundary states.We also consider free and various non-diagonal boundary conditions.All correlation lengths are derived.
The wavefunctions corresponding to the zero-energy eigenvalue of a one-dimensional quantum chain Hamiltonian can be written in a simple way using quadratic algebras. Hamiltonians describing stochastic processes have stationary states given by such wavefunctions and various quadratic algebras have been found and applied to several diffusion processes. We show that similar methods can also be applied for equilibrium processes. As an example, for a class of q -deformed O( N ) symmetric antiferromagnetic quantum chains, we give the zero-energy wavefunctions for periodic boundary conditions corresponding to momenta zero and π. We also consider free and various non-diagonal boundary conditions and give the corresponding wavefunctions. All correlation lengths are derived.
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