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arXiv:cond-mat/9602142 · Physical review. B, Condensed matter

Specific heat and validity of the quasiparticle approximation in the half-filled Landau level

Yong Baek Kim, Patrick A. Lee

Published Jul 15, 1996• 15 citations
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Abstract

We calculate the specific heat of a composite fermion system in the half-filled Landau level. Two different methods are used to examine the validity of the quasiparticle approximation when the two-body interaction is given by v(q)=${\mathit{V}}_{0}$/${\mathit{q}}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$ (1\ensuremath{\le}\ensuremath{\eta}\ensuremath{\le}2). The singular part of the specific heat is calculated from the free energy of the gauge field, which is compared with the specific heat calculated from the quasiparticle approximation via the singular self-energy correction, due to the gauge field fluctuations. It turns out that two results are, in general, different and they coincide only for the case of the Coulomb interaction (\ensuremath{\eta}=1). This result supports the fact that the quasiparticle approximation is valid only for the case of the Coulomb interaction. It is emphasized that this result is obtained by looking at a gauge-invariant quantity namely, the specific heat. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.2715

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We calculate the specific heat of a composite fermion system in the half-filled Landau level. Two different methods are used to examine the validity of the quasiparticle approximation when the two-body interaction is given by v(q)=V0{\mathit{V}}_{0}V0​/q2\ensuremath−\ensuremathη{\mathit{q}}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}q2\ensuremath−\ensuremathη (1\ensuremath{\le}\ensuremath{\eta}\ensuremath{\le}2). The singular part of the specific heat is calculated from the free energy of the gauge field, which is compared with the specific heat calculated from the quasiparticle approximation via the singular self-energy correction, due to the gauge field fluctuations. It turns out that two results are, in general, different and they coincide only for the case of the Coulomb interaction (\ensuremath{\eta}=1). This result supports the fact that the quasiparticle approximation is valid only for the case of the Coulomb interaction. It is emphasized that this result is obtained by looking at a gauge-invariant quantity namely, the specific heat. \textcopyright{} 1996 The American Physical Society.
In this paper, we propose a gauge-invariant way to define and calculate the effective mass for quasiparticles in systems with gauge interactions, and apply it to a model closely related to the half-filled Landau level problem. Our model is equivalent to the Halperin-Lee-Read ν=1/2ν= 1/2ν=1/2 Hamiltonian with an ultraviolet cutoff ΛΛΛ for the gauge fields, and we expand our answer in powers of Λ/kFΛ/ k_FΛ/kF​, assuming it is small. In this definition the effective mass depends only on the gauge-invariant density and current response functions of the system. Within RPA, this definition yields a finite result for the effective mass in our model. We also comment on corrections to this effective mass formula when processes beyond RPA are included. Finally, we comment briefly on the observation that organizing the perturbation expansion in powers of Λ/kFΛ/ k_FΛ/kF​ is a way to systematically study the physics beyond RPA.
Fluctuations of the Chern-Simons gauge field in the composite Fermi liquid description of the half-filled Landau level are pair breaking in all angular momentum channels. For short-range electron-electron interactions these fluctuations are sufficiently strong to drive any T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 pairing transition first order. For Coulomb interactions these fluctuations are weaker and a continuous transition is possible.
We derive the quantum Boltzmann equation (QBE) of composite fermions at/near the \ensuremath{\nu}=1/2 state using the nonequilibrium Green's-function technique. The lowest-order perturbative correction to the self-energy due to the strong gauge-field fluctuations suggests that there is no well-defined Landau quasiparticle. Therefore, we cannot assume the existence of the Landau quasiparticles a priori in the derivation of the QBE. Using an alternative formulation, we derive the QBE for the generalized Fermi-surface displacement which corresponds to the local variation of the chemical potential in momentum space. From this QBE, one can understand in a unified fashion the Fermi-liquid behaviors of the density-density and the current-current correlation functions at \ensuremath{\nu}=1/2 (in the long-wavelength and the low-frequency limits) and the singular behavior of the energy gap obtained from the finite-temperature activation behavior of the compressibility near \ensuremath{\nu}=1/2. Implications of these results for recent experiments are also discussed.
The theory of composite fermion provides a simple and unified picture to understand a vast amount of phenomenology in the quantum Hall regime. However it has remained challenging to formulate this concept properly within a single Landau level, which provides the relevant degrees of freedom in the limit of strong magnetic field. Recently a low-energy noncommutative field theory for bosons at Landau-level filling factor $\ensuremath{\nu}=1$ has been formulated by Dong and Senthil [Z. Dong and T. Senthil, Phys. Rev. B 102, 205126 (2020)]. In the limit of long-wavelength and small-amplitude gauge fluctuation, they found it reduces to the celebrated Halperin-Lee-Read theory of a composite fermion liquid. In this work we consider a Bose-Fermi mixture at total filling factor $\ensuremath{\nu}=1$. Different from previous work, the number density of composite fermions in the mixture and corresponding Fermi momentum can be tuned by changing the filling factor of bosons, ${\ensuremath{\nu}}_{b}=1\ensuremath{-}{\ensuremath{\nu}}_{f}$. This tunability enables us to study the dilute limit ${\ensuremath{\nu}}_{b}\ensuremath{\ll}1$, which allows for a controlled and asymptotically exact calculation of the energy dispersion and effective mass of composite fermions. Furthermore, the approximation of the low-energy description by a commutative field theory is manifestly justified. Most importantly, we demonstrate that gauge fluctuations acquire a Higgs mass due to the presence of a composite boson condensate, as a result of which the system behaves like a genuine Landau- Fermi liquid. Combined with the irrelevance of four-fermion interaction in the dilute limit, we are able to obtain asymptotically exact properties of this composite fermion Fermi liquid. In the opposite limit of ${\ensuremath{\nu}}_{f}\ensuremath{\ll}1$, the Higgs mass goes to zero and we find crossover between Fermi liquid and non-Fermi liquid as temperature increases. Observing these properties either experimentally or numerically provides unambiguous evidence of not only the composite fermions and the Fermi surface they form, but also the presence of emergent gauge fields and their fluctuations due to strong correlation.
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.