Probing Non-Fermi-Liquid Behaviour of Composite Fermi Liquid via Efficient Thermal Simulations

Abstract

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 quasiparticle of an electron with two magnetic flux quanta -- experience zero net magnetic field and form a Fermi sea, dubbed composite Fermi liquid (CFL). However, the seemingly simple appearance of CFL is a strongly correlated quantum many-body state in disguise, and to solve it in a controlled manner is extremely difficult, to the level that the thermodynamic properties of CFL is still largely unknown. In this work, we perform state-of-the-art thermal tensor network simulations on the $ν=1/2$ Landau level systems, 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 closed to $2/3$, clearly deviated from the ordinary linear-$T$ Fermi liquid behaviour, suggesting the coupling between the CFs and the dynamical emergent gauge field and therefore revealed the quantum many-body aspect of the CFL state. Relevance of our methodology to other quantum Hall settings and moiré systems is discussed.

Figures (6)

• Figure 1: The matrix product operator (MPO) setup in the lowest Landau level (LLL) basis for thermal density operator $\rho$. (a) In the Landau gauge with periodic boundary condition along the $y$-direction and open boundary condition along the $x$-direction. The LLL wavefunctions (labelled by $n \in[0,N-1]$) are extended along $y$- and localized along $x$-direction. (b) The matrix product operator of density operator $\rho$ is defined in the LLL basis, where the "local" degree of freedom of the $i$th tensor lives in the 2-dimensional physical space of either filling the $i$th LLL or leaving it empty.
• Figure 2: Benchmark and CBE optimization of tanTRG for the projected Coulomb-Yukawa interaction. We chose $N=16$ and compared to the exact diagonalization results. (a) Relative errors of free energy $\delta f \equiv |f_\mathrm{tanTRG} - f_\mathrm{ED}|/|f_\mathrm{ED}|$ calculated with different bond dimensions $D=400, 800, 1200$ are shown versus temperature $T$. (b) Relative errors of specific heat $\delta c_\mathrm{V}$ are shown versus temperature $T$. In the inset of both panels, the raw tanTRG data of the free energy $f$ and specific heat $c_\mathrm{V}$ is shown along with the ED data (the yellow curves). (c) Relative errors of specific heat $\delta c_\mathrm{V}$ for both 2-site (blue, the same as in (a)) and CBE (red) update scheme with various bond dimensions $D=400, 800, 1600$. (d) The CPU time (in the unit of hours) for both 2-site and CBE update scheme, shown as functions of bond dimension $D$.
• Figure 3: Thermodynamic properties of composite Fermi liquid. (a) Lower-temperature specific heat $c_\mathrm{V}$ behaviour for various number of orbitals $N=24, 36, 48$, exhibiting converged power-law scaling $T^{2/3}$ for $N=36$ and $48$. (b) The low-$T$ power $\alpha_\mathrm{Fit}$ from a linear fitting is shown versus system sizes $1/N$ for $\lambda=10$ and $2$.
• Figure 4: Formation of the composite Fermi surface in CFL. For the system with $L_y=12, N=48,\lambda=10$, (a) Two-dimensional momentum space view of the "guiding center" density-density correlations $\bar{D}(q) =\exp(q^2/2) \langle n_q n_{-q}\rangle_T$ is plotted as a function of 2D momenta $q=(q_x,q_y)$ at four different temperatures $T=10^1, 10^0, 10^{-1}, 10^{-2}$. It starts to exhibit a $2k_F$ circle for $T\lesssim10^{-1}$ suggesting the establishment of the composite Fermi surface. (b,c) $\bar{D}(q)$ is shown as functions of $q_x$ when $q_y=0, \frac{2\pi}{L_y}$ are fixed accordingly. The dashed grey line indicated the $2k_F$ points for the corresponding $q_y$. (d) $\bar{D}(q)$ at the three different $2k_F$ points of $q_y=m\frac{2\pi}{L_y}$ with $m=0,1,2$, which increase most quickly at $\sim10^{-1}$, suggesting the formation of the composite Fermi surface.
• Figure S1: Charge compressibility $\partial \langle n\rangle /\partial \mu$ is shown versus $T$, which quickly drops to zero at the low-$T$ regime (denoted by the grey area) suggesting the finite-size effect.