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arXiv:2407.11119 · Physical review. B./Physical review. B

Entanglement scaling and charge fluctuations in a Fermi liquid of composite fermions

Cristian Voinea, Songyang Pu, Ajit Coimbatore Balram, Zlatko Papić

Published Mar 6, 2025• 4 citations
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Abstract

The composite fermion Fermi liquid (CFL) state at ν = 1 / 2 filling of a Landau level is a paradigmatic non-Fermi liquid borne out purely by Coulomb interactions. But in what ways is this exotic state of matter different from a Fermi liquid? The CFL entanglement entropy was indeed found to exhibit a significant enhancement compared to free electrons [J. Shao , ], which was subsequently ruled out as a finite-size effect by the study of a lattice CFL analog [R. V. Mishmash and O. I. Motrunich, ]. Moreover, the enhancement was not observed in a quasi-one-dimensional limit of the Coulomb ground state at ν = 1 / 2 [S. D. Geraedts , ]. Here, we revisit the problem of entanglement scaling in the CFL state realized in a two-dimensional electron gas. Using Monte Carlo evaluation of the second Rényi entropy S 2 for the CFL variational wave function, we show that the entanglement enhancement is present not only at ν = 1 / 2 , but also at ν = 1 / 4 , as well as in bosonic CFL states at ν = 1 and ν = 1 / 3 fillings. In all cases, we find the scaling of S 2 with subsystem size to be enhanced compared to the noninteracting case, and insensitive to the choice of geometry and projection to the lowest Landau level. We also demonstrate that for CFL states, the variance of the particle number in a subsystem obeys area-law scaling with a universal subleading corner contribution, in stark contrast with free fermions. Our results establish the enhanced entanglement scaling and suppressed charge fluctuations as fingerprints of non-Fermi-liquid correlations in CFL states.

DOI
10.1103/physrevb.111.115119

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The composite fermion Fermi liquid (CFL) state at ν = 1 / 2 filling of a Landau level is a paradigmatic non-Fermi liquid borne out purely by Coulomb interactions. But in what ways is this exotic state of matter different from a Fermi liquid? The CFL entanglement entropy was indeed found to exhibit a significant enhancement compared to free electrons [J. Shao , ], which was subsequently ruled out as a finite-size effect by the study of a lattice CFL analog [R. V. Mishmash and O. I. Motrunich, ]. Moreover, the enhancement was not observed in a quasi-one-dimensional limit of the Coulomb ground state at ν = 1 / 2 [S. D. Geraedts , ]. Here, we revisit the problem of entanglement scaling in the CFL state realized in a two-dimensional electron gas. Using Monte Carlo evaluation of the second Rényi entropy S 2 for the CFL variational wave function, we show that the entanglement enhancement is present not only at ν = 1 / 2 , but also at ν = 1 / 4 , as well as in bosonic CFL states at ν = 1 and ν = 1 / 3 fillings. In all cases, we find the scaling of S 2 with subsystem size to be enhanced compared to the noninteracting case, and insensitive to the choice of geometry and projection to the lowest Landau level. We also demonstrate that for CFL states, the variance of the particle number in a subsystem obeys area-law scaling with a universal subleading corner contribution, in stark contrast with free fermions. Our results establish the enhanced entanglement scaling and suppressed charge fluctuations as fingerprints of non-Fermi-liquid correlations in CFL states.
The so-called "non-Fermi liquid" behavior is very common in strongly correlated systems. However, its operational definition in terms of "what it is not" is a major obstacle for the theoretical understanding of this fascinating correlated state. Recently there has been much interest in entanglement entropy as a theoretical tool to study non-Fermi liquids. So far explicit calculations have been limited to models without direct experimental realizations. Here we focus on a two-dimensional electron fluid under magnetic field and filling fraction ν=1/2, which is believed to be a non-Fermi liquid state. Using a composite fermion wave function which captures the ν=1/2 state very accurately, we compute the second Rényi entropy using the variational Monte Carlo technique. We find the entanglement entropy scales as LlogL with the length of the boundary L as it does for free fermions, but has a prefactor twice that of free fermions.
Quantum phases characterized by surfaces of gapless excitations are known to violate the otherwise ubiquitous boundary law of entanglement entropy in the form of a multiplicative log correction: $S\ensuremath{\sim}{L}^{d\ensuremath{-}1}logL$. Using variational Monte Carlo, we calculate the second R\'enyi entropy for a model wave function of the $\ensuremath{\nu}=1/2$ composite Fermi liquid (CFL) state defined on the two-dimensional triangular lattice. By carefully studying the scaling of the total R\'enyi entropy and, crucially, its contributions from the modulus and sign of the wave function on various finite-size geometries, we argue that the prefactor of the leading LlogLL\phantom{\rule{0.16em}{0ex}}logLLlogL term is equivalent to that in the analogous free fermion wave function. In contrast to the recent results of Shao et al. [Phys. Rev. Lett. 114, 206402 (2015)], we thus conclude that the ``Widom formula'' holds even in this non-Fermi liquid CFL state. More generally, our results further elucidate---and place on a more quantitative footing---the relationship between nontrivial wave function sign structure and $S\ensuremath{\sim}L\phantom{\rule{0.16em}{0ex}}logL$ entanglement scaling in such highly entangled gapless phases.
The physics of two-dimensional electron gas in a perpendicular magnetic field, i.e., the quantum Hall system, is remarkably rich. At half filling of the lowest Landau level, it has been predicted that "composite fermions"---emergent quasiparticles consisting of an electron attached to two magnetic flux quanta---experience zero net magnetic field and form a Fermi sea, dubbed composite Fermi liquid (CFL). However, despite its seemingly simple appearance, CFL is a strongly correlated quantum many-body state in disguise, and solving it in a controlled manner is extremely difficult, to the extent that the thermodynamic properties of CFL remain largely unknown. In this work, we perform state-of-the-art thermal tensor network simulations of the ν=1/2ν=1/2ν=1/2 Landau level system and observe low-temperature power-law behaviour of the specific heat, signaling the gapless nature of CFL. More importantly, the power is extracted to be close to 2/32/32/3, clearly deviating from the ordinary linear-TTT behaviour of Fermi liquid, suggesting coupling between the CFs and the dynamical emergent gauge field and thereby revealing the quantum many-body nature of the CFL state.
One of the most successful theories of a non-Fermi-liquid metallic state is the composite Fermi-liquid (CFL) theory of the half-filled Landau level. In this paper, we study continuous quantum phase transitions out of the CFL state and into a Landau Fermi liquid, in the limit of no disorder and fixed particle number. This transition can be induced by tuning the bandwidth of the Landau level relative to the interaction energy, for instance through an externally applied periodic potential. We find a transition to the Landau Fermi liquid through a gapless Mott insulator with a Fermi surface of neutral fermionic excitations. In the presence of spatial symmetries, we also find a direct continuous transition between the CFL and the Landau Fermi liquid. The transitions have a number of characteristic observable signatures, including the presence of two crossover temperature scales, resistivity jumps, and vanishing compressibility. When the composite fermions are paired instead, our results imply quantum critical points between various non-Abelian topological states, including the $\ensuremath{\nu}=1/2$ Moore-Read Pfaffian [Ising $\ifmmode\times\else\texttimes\fi{}$ U(1)U(1)U(1) topological order], a version of the Kitaev B phase (Ising topological order), and paired electronic superconductors. To study such transitions, we use a projective construction of the CFL, which goes beyond the conventional framework of flux attachment to include a broader set of quantum fluctuations. These considerations suggest a possible route to fractionalized Mott insulators by starting with fractional quantum Hall states and tuning the Landau-level bandwidth.
While an ordinary Fermi sea is perturbatively robust to interactions, the paradigmatic composite-fermion (CF) Fermi sea arises as a nonperturbative consequence of emergent gauge fields in a system where there was no Fermi sea to begin with. A mean-field picture suggests two Fermi seas, of composite fermions made from electrons or holes in the lowest Landau level, which occupy different areas away from half filling and thus appear to represent distinct states. Using the microscopic theory of composite fermions, which satisfies particle-hole symmetry in the lowest Landau level to an excellent approximation, we show that the Fermi wave vectors at filling factors ν and 1-ν are equal when expressed in units of the inverse magnetic length, and are generally consistent with the experimental findings of Kamburov et al. [Phys. Rev. Lett. 113, 196801 (2014)]. Our calculations suggest that the area of the CF Fermi sea may slightly violate the Luttinger area rule.
Composite Fermi liquids (CFLs) are compressible states that can occur for 2D interacting fermions confined in the lowest Landau level at certain Landau level fillings. They have been understood as Fermi seas formed by composite fermions which are bound states of electromagnetic fluxes and electrons as reported by Halperin, Lee, and Read [Phys. Rev. B 47, 7312 (1993)PRBMDO0163-182910.1103/PhysRevB.47.7312]. At half filling, an explicitly particle-hole symmetric theory based on Dirac fermions was proposed by Son [Phys. Rev. X 5, 031027 (2015)PRXHAE2160-330810.1103/PhysRevX.5.031027] as an alternative low energy description. In this work, we investigate the Berry curvature of CFL model wave functions at a filling fraction of one-quarter, and observe that it is uniformly distributed over the Fermi sea except at the center where an additional π phase was found. Motivated by this, we propose an effective theory which generalizes Son's half filling theory, by internal gauge flux attachment, to all filling fractions in which fermionic CFLs can occur. The numerical results support the idea of internal gauge flux attachment.