# Composite Fermi liquid

Fig-1: Main numerical results of Ref [1]; the main theoretical result is illustrated in Fig-2.

The Berry curvature distribution on the emergent Fermi sea at 1/4 filled Landau level. The strong peak at the Fermi sea center strongly suggest the emergence of relativistic Dirac particle. Motivated by this numerical observation, I proposed an effective theory generalizing the Son-Dirac theory to all other filling fractions that composite Fermi liquids can occur. See Ref. [1] for details.

During the graduate school at Princeton, where I mainly worked with my advisor Duncan Haldane, I worked on the emergent phenomena in a partially filled Landau level. Much of my Ph.D. work is about the *Composite Fermi Liquid** *phase, a strongly correlated gapless phase occurring in an even denominator (v=1/2m, such as 1/2, 1/4, ...) filled lowest Landau level.

My work [1,2,3] contribute to the fundamental understandings about the nature of composite fermion in this phase.

[1] **Jie Wang** (2019). Dirac Fermion Hierarchy of Composite Fermi Liquids. PhysRevLett.122.257203

[2] **Jie Wang**, S. Geraedts, E.H. Rezayi and F.D.M. Haldane (2019). Lattice Monte Carlo for Quantum Hall States on a Torus. PhysRevB.99.125123

[3] S. Geraedts, **Jie Wang**, E.H. Rezayi and F.D.M. Haldane (2018). Berry phase and model wavefunctions in the half-filled Landau level. PhysRevLett.121.147202 (Editor's suggestion)

## Background

The composite Fermi liquids (CFL) was initially understood by B.I. Halperin, Patrick Lee, Nicholas Read in 1993 [PRB.47.7312] as Fermi seas of composite fermions, which are bound states of one electron and 2m flux quanta at filling fraction v=1/(2m). A long-standing open problem of Halperin-Lee-Read theory is the lack of particle-hole symmetry at half filling (m=1). This motivated Dam Thanh Son in 2015 to propose an alternative low energy effective theory [PRX.5.031027]. According to Son, the composite fermions are instead relativistic Dirac particles.

## My Contributions

Important contributions of our works include the definition and the numerical detection of the many-body Berry phase associated to the adiabatic transportation of the composite fermion around the Fermi surface. The Berry phase at half filling was found to have a Z2 character [3], supporting Sonâ€™s theory.

The Z2 phase was further confirmed on much larger system sizes by a fast *Lattice Monte Carlo technique* developed by us, based on our exact discretized lattice formulation of the continuum torus quantum Hall problems [2].

In an independent study [1], I generalized Son-Dirac theory from half filling to all other filling fractions motivated by the Berry phase studied in a one-quarter filled Landau level [Fig-1]. According to my theory, composite fermions at generic filling fractions are flux-attached Dirac fermions [Fig-2]. Same effective theory is proposed at the same time by H. Goldman and E. Fradkin [PRB.98.165137] from a different but complementary point of view.

Fig-2: Main theoretical results of Ref [1]: illustration of my Dirac effective theory for composite Fermi liquids at all filling fractions.

My theory generalizes Son-Dirac theory (only for half filling) by internal gauge flux-attachment represented by 'a' in the effective theory. 'eta' and 'm' determines the filling fraction.