--- title: Second Fermi Liquid Overview url: https://www.emergentmind.com/topics/second-fermi-liquid type: topic --- # Second Fermi Liquid Overview “Second Fermi liquid” is not a single universally standardized term. In the literature represented here, it denotes several distinct constructions that preserve some Fermi-liquid structures while departing from conventional Landau adiabatic continuity, the standard Luttinger count, or the usual charged-quasiparticle interpretation. These include the \(U(1)\)-Fermi liquid state that supports exclusion statistics [1610.08633], a gapless Mott-insulating phase with a neutral Fermi surface that is explicitly described as a “second Fermi liquid” [1206.6530], and a “small” Fermi liquid in a bilayer model whose Fermi-surface volume differs by exactly one half of the Brillouin zone per flavor from a conventional large Fermi liquid [2401.08753]. In a separate but related sense, “second-order Fermi-liquid” can refer to a low-energy expansion carried to quadratic order in excitation energy and distribution-function deviations, as in the single-impurity Anderson model [1409.3451], or to a second-order \(\lambda\)-expansion within the extremely correlated Fermi liquid formalism for the \(t\)-\(J\) model [1211.0594]. The shared theme is the persistence of Fermi-surface-based low-energy organization together with nonstandard quasiparticle content, gauge structure, or correlation effects. ## 1. Terminological range and conceptual core The most direct use of the phrase occurs in the projective construction of continuous transitions out of the composite Fermi liquid, where the gapless Mott insulator is described as a “second Fermi liquid” because “the physical electrons are localized (insulating), yet the neutral fermions \(f\) form a sharp Fermi surface coupled to a \(U(1)\) gauge field” [1206.6530]. A related but not identical usage appears in the \(U(1)\)-Fermi liquid paper, which states that the proposed state is “a second FL-like state” and “a new fixed point of the kinetic equation” [1610.08633]. In the bilayer deconfined transition study, the small-FL phase is also presented as a second, distinct Fermi liquid, though there the emphasis is on a Fermi-surface-volume jump without symmetry breaking or low-energy fractionalization [2401.08753]. A separate nomenclature concerns approximation order rather than a new thermodynamic phase. In the Anderson-model work, the “second Fermi-liquid” framework is the low-energy Fermi-liquid theory including the second-order parameters \(\alpha_2\) and \(\phi_2\), needed away from particle-hole symmetry [1409.3451]. In the ECFL treatment of the two-dimensional \(t\)-\(J\) model, “second order theory” refers to truncation at \(O(\lambda^2)\), with self-consistent solutions for an auxiliary Fermi-liquid-type propagator and an adaptive spectral weight [1211.0594]. This terminological heterogeneity suggests a useful unifying description: a “second Fermi liquid” is a Fermi-liquid-adjacent state or framework in which the low-energy organization remains controlled by a Fermi surface or by Fermi-liquid response theory, but the identification of quasiparticles, the counting of states, or the relation to bare electrons differs from the conventional Landau paradigm. ## 2. \(U(1)\)-Fermi liquid as a second FL-like state The \(U(1)\)-Fermi liquid theory extends Landau Fermi-liquid theory by introducing \(k\)-dependent gauge fields \(\phi_k(r,t)\) and \(a_k(r,t)\), with a Lagrangian \[ L[n,\phi,a] = L_G + L_1[n;\phi,a] + L_2[\phi,a], \] where \[ L_1[n;\phi,a] = \sum_k \int d^dr\, n_k(r,t)\,[ -\epsilon_k - \phi_k(r,t) + \dot r_k\cdot a_k(r,t) ] \] and \[ L_2[\phi,a] = \frac{1}{2V} \sum_{k,k'} [ \phi_k\,(f^{-1})_{k,k'}\,\phi_{k'} - a_k\cdot G^{-1}_{k,k'}\cdot a_{k'} ]. \] After eliminating \(\phi_k\) and \(a_k\), one obtains \[ H[n] = E_G + \sum_k \epsilon_k\,n_k + \frac{1}{2V} \sum_{k,k'} [ f_{k,k'}\,n_k\,n_{k'} + v_k\cdot G_{k,k'}\cdot v_{k'}\,n_k\,n_{k'} ]. \] The term \(v_k\cdot G_{k,k'}\cdot v_{k'}\,n_k\,n_{k'}\) is the additional \(k\)-dependent current-current interaction that distinguishes the theory from conventional Landau FL [1610.08633]. Its key physical ingredient is that the kinetic momentum depends on the occupation numbers: \[ \tilde k = k_0 + a_{k_0}[n], \qquad a_k[n] = \sum_{k'} g(k,k')\,\delta n_{k'}, \] with \(\delta n_{k'} \equiv n_{k'} - n^{(0)}_{k'}\). In the presence of an external electromagnetic vector potential \(A\), the total gauge field is \[ A_k(r) = a_k(r) + A(r), \qquad \tilde k = k_0 + A_k. \] Because \(k\to \tilde k\) is a phase-space coordinate transformation, the Jacobian modifies the local density of states through \[ z(k)=\det M(k), \qquad M_{ij}(k) \equiv \frac{\partial k_{0,i}}{\partial k_j} = \delta_{ij} - \frac{\partial A_{k,i}}{\partial k_j}. \] To linear order, \[ z(k)=1 + \nabla_k\cdot A_k + O(A^2), \] which reproduces Haldane’s exclusion-statistics Ansatz \[ z(k)=1-\sum_{k'}\alpha_{kk'}\,\delta n_{k'}, \qquad \alpha_{kk'} = -\nabla_k\cdot g(k,k'). \] The interaction-induced \(\nabla_k\cdot A_k\) term is therefore identified as the microscopic origin of fractional exclusion statistics in this Fermi liquid [1610.08633]. Thermodynamically, the state remains Fermi-liquid-like once the Landau function is replaced by \[ \tilde f_{kk'} = f_{kk'} + v_k\cdot G_{kk'}\cdot v_{k'}, \qquad \tilde F_l = N(0)\tilde f_l. \] The quoted low-\(T\) results include \[ \kappa = n^{-2}(\partial n/\partial \mu)=\frac{N(0)}{1+\tilde F_0}, \] \[ \gamma = (\pi^2/3)N^*(0), \] and \[ m^*/m = 1 + \tilde F_1/d. \] At the same time, the theory departs sharply from Landau FL in its quasiparticle interpretation. The quasiparticle charge becomes \[ e^*(k)=1 - \sum_{k'}D_0(k')\,v_{k'}\cdot g(k',k), \] and the Fermi-surface volume \[ V_{\rm FS} = \int_{\tilde \epsilon_k<0} d^dk = \int_{\epsilon_k<\mu} d^dk_0\,\det M \] need not equal the Luttinger value \((2\pi)^d n\). The paper states that the \(U(1)\)-FL “has Fermi-liquid like properties except that the quasi-particles are not adiabatically connected to bare fermions in the system and the state may not satisfy Luttinger theorem” [1610.08633]. That combination of conventional thermodynamics with non-adiabatic quasiparticles is one of the clearest meanings of a second FL-like state. ## 3. Gapless Mott insulator with a neutral Fermi surface In the projective treatment of transitions out of the composite Fermi liquid, the electron is decomposed as \[ c(r)=b(r)f(r), \qquad n_e=n_b=n_f. \] The gapless Mott insulator (GMI) is defined by bosons \(b\) in a trivial Mott insulator and neutral fermions \(f\) forming a Fermi surface. Its low-energy Lagrangian is \[ \mathcal{L}_{\rm GMI} = f^\dagger(i\partial_t+a_0-\mu_f)f -\frac{1}{2m_f}f^\dagger(\nabla-i\mathbf a)^2 f +\frac{1}{4g^2}f_{\mu\nu}f^{\mu\nu} +\cdots. \] Because the bosons are in a trivial Mott insulator, there is no Chern-Simons term for the emergent gauge field \(a_\mu\) [1206.6530]. The neutral Fermi surface is fixed by the electron density. For spinless fermions in two dimensions, \[ n_f = \frac{1}{(2\pi)^2}\,\pi k_F^2 \quad\Longrightarrow\quad k_F=\sqrt{4\pi n_e}, \] with \[ N_f(0)=\frac{m_f}{2\pi}, \qquad v_F=k_F/m_f. \] This is a Fermi surface without electric charge transport. The paper therefore characterizes the phase as a “genuine ‘second Fermi liquid’ without charge transport” [1206.6530]. The compressibility is controlled by the Ioffe-Larkin composition rule \[ \kappa_e^{-1}=\kappa_b^{-1}+\kappa_f^{-1}. \] At the boson critical point, with correlation length \(\xi\sim |\delta|^{-\nu}\), the boson compressibility behaves as \(\kappa_b\sim \xi\to\infty\), implying \[ \kappa_e \sim \frac{1}{\xi}\propto |\delta|^\nu \longrightarrow 0 \qquad (T=0,\;\delta\to0). \] At finite temperature in the quantum-critical fan, \[ \kappa_e(T)\sim T+O(T^2). \] Thus the critical point is incompressible at \(T=0\) even though both adjacent phases are compressible [1206.6530]. Transport and thermodynamics exhibit two crossover scales, \[ T^*\sim \frac{1}{\xi}\sim |\delta|^\nu, \qquad T^{**}\sim \frac{1}{k_0 c \xi^2}\sim |\delta|^{2\nu}. \] For \(T^{**}\alpha_{a,\psi}\), the interlayer pairing channel is marginally relevant, leading to \[ \Delta_{\rm sc}\sim \Lambda_\omega \exp\!\left[-\frac{\pi}{\sqrt{\alpha_{b,\psi}-\alpha_{a,\psi}}}\right]. \] The paper therefore predicts an interlayer-pairing dome cloaking the deconfined FL-to-FL transition at \(T=0\) unless an additional parameter is tuned to a tricritical point [2401.08753]. ## 5. Second-order Fermi-liquid theory in the Anderson model In the single-impurity Anderson model, “second Fermi-liquid” refers not to a distinct metallic phase but to a systematic low-energy expansion of the phase shift to second order in energy and distribution-function deviations. The elastic scattering phase shift is expanded as \[ \delta_\sigma(\varepsilon) = \delta_0+\alpha_1\varepsilon+\alpha_2\varepsilon^2 -\phi_1\Delta N_{\bar\sigma} -\phi_2\varepsilon\Delta N_{\bar\sigma} +\cdots, \] where \[ \Delta N_{\bar\sigma}\equiv \int d\varepsilon'\,\delta n_{\bar\sigma}(\varepsilon'). \] Here \(\alpha_1\) and \(\phi_1\) are the first-order Fermi-liquid parameters, while \(\alpha_2\) and \(\phi_2\) are the second-order Fermi-liquid parameters [1409.3451]. All four parameters can be expressed in terms of zero-temperature local charge and spin susceptibilities and their derivatives with respect to the level position: \[ \chi_c = -\frac{\partial n_d}{\partial \varepsilon_d}\Big|_{B=0}, \qquad \chi_s = \frac{\partial m_d}{\partial B}\Big|_{B=0}, \] \[ \chi_c'=\frac{\partial \chi_c}{\partial \varepsilon_d}, \qquad \chi_s'=\frac{\partial \chi_s}{\partial \varepsilon_d}. \] The explicit relations are \[ \frac{\alpha_1}{\pi}=\chi_s+\tfrac14\chi_c, \qquad \frac{\phi_1}{\pi}=\chi_s-\tfrac14\chi_c, \] \[ \frac{\alpha_2}{\pi}=-\tfrac34\chi_s'-\tfrac1{16}\chi_c', \qquad \frac{\phi_2}{\pi}=-\chi_s'+\tfrac14\chi_c'. \] At particle-hole symmetry, \(\chi_c'=\chi_s'=0\), so \(\alpha_2=\phi_2=0\) and the theory reduces to the Nozières Kondo FL [1409.3451]. The low-energy conductance through a symmetrically coupled quantum dot is written as \[ G(B,T,V)= G_0\Bigl[ 1-c_B(B/T_K)^2-c_T(\pi T/T_K)^2-c_V(eV/T_K)^2+\cdots \Bigr], \] with \[ G_0=\frac{2e^2}{h}\sin^2\delta_0, \qquad T_K\equiv E^*=\frac{\pi}{4\alpha_1}. \] The coefficients \(c_B\), \(c_T\), and \(c_V\) are exact functions of \(\alpha_1,\phi_1,\alpha_2,\phi_2,\delta_0\), and the paper states that they change sign across the Kondo to empty-orbital crossover. At particle-hole symmetry one finds \[ c_T^{\rm K}=\frac{\pi^4}{16}\approx 6.009,\quad c_V^{\rm K}=\frac{3\pi^2}{32}\approx 0.925,\quad c_B^{\rm K}=\frac{\pi^2}{16}\approx 0.617. \] The generalized Fano factor is \[ F=\frac{1}{2e}\lim_{V\to 0}\frac{\delta S}{\delta I}, \] with the universal Kondo value \(F=-5/3\) at particle-hole symmetry and the Poissonian value \(F=+1\) in the non-interacting limit [1409.3451]. This usage of “second Fermi-liquid” is therefore methodological rather than phase-defining. It designates a higher-order Fermi-liquid expansion capable of describing low-energy transport away from particle-hole symmetry. ## 6. Extremely correlated and quasi-Fermi-liquid contrasts The “second order theory” of extremely correlated Fermi liquids applies Shastry’s ECFL formalism to the two-dimensional \(t\)-\(J\) model. The physical Green’s function is factorized as \[ G(k,\omega)=g(k,\omega)\,\mu(k,\omega), \] where \(g(k,\omega)\) is an auxiliary Fermi-liquid-type propagator and \(\mu(k,\omega)\) is an adaptive spectral weight, or caparison factor. At \(O(\lambda^2)\), \[ g^{-1}(k,\omega)=\omega+\mu'-\overline\epsilon_k-\lambda^2\overline\Phi(k,\omega), \] \[ \mu(k,\omega)=1-\lambda \frac{n}{2}+\lambda^2\frac{n^2}{4}+\lambda^2\Psi(k,\omega). \] The formalism is argued to be quantitatively valid in the overdoped regime \(0