Second Fermi Liquid Overview

• Second Fermi Liquid refers to states that maintain a Fermi-surface structure while deviating from conventional Landau theory through nonstandard quasiparticle definitions and gauge interactions.
• These constructions include variants such as U(1)-Fermi liquids with exclusion statistics, gapless Mott insulators with neutral Fermi surfaces, and small Fermi liquids exhibiting Fermi-surface volume fractionalization.
• The framework spans both phase-defining states and systematic higher-order expansions, offering insights into unconventional transport, thermodynamics, and many-body correlation effects.

“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 (Ng, 2016), a gapless Mott-insulating phase with a neutral Fermi surface that is explicitly described as a “second Fermi liquid” (Barkeshli et al., 2012), 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 (Wu et al., 2024). 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 (Mora et al., 2014), or to a second-order $\lambda$-expansion within the extremely correlated Fermi liquid formalism for the $t$-$J$ model (Hansen et al., 2012). 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” (Barkeshli et al., 2012). 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” (Ng, 2016). 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 (Wu et al., 2024).

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 (Mora et al., 2014). In the ECFL treatment of the two-dimensional $t$-$\lambda$0 model, “second order theory” refers to truncation at $\lambda$1, with self-consistent solutions for an auxiliary Fermi-liquid-type propagator and an adaptive spectral weight (Hansen et al., 2012).

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. $\lambda$2-Fermi liquid as a second FL-like state

The $\lambda$3-Fermi liquid theory extends Landau Fermi-liquid theory by introducing $\lambda$4-dependent gauge fields $\lambda$5 and $\lambda$6, with a Lagrangian

$\lambda$7

where

$\lambda$8

and

$\lambda$9

After eliminating $t$0 and $t$1, one obtains

$t$2

The term $t$3 is the additional $t$4-dependent current-current interaction that distinguishes the theory from conventional Landau FL (Ng, 2016).

Its key physical ingredient is that the kinetic momentum depends on the occupation numbers: $t$5 with $t$6. In the presence of an external electromagnetic vector potential $t$7, the total gauge field is

$t$8

Because $t$9 is a phase-space coordinate transformation, the Jacobian modifies the local density of states through

$J$0

To linear order,

$J$1

which reproduces Haldane’s exclusion-statistics Ansatz

$J$2

The interaction-induced $J$3 term is therefore identified as the microscopic origin of fractional exclusion statistics in this Fermi liquid (Ng, 2016).

Thermodynamically, the state remains Fermi-liquid-like once the Landau function is replaced by

$J$4

The quoted low-$J$5 results include

$J$6

$J$7

and

$J$8

At the same time, the theory departs sharply from Landau FL in its quasiparticle interpretation. The quasiparticle charge becomes

$J$9

and the Fermi-surface volume

$f$0

need not equal the Luttinger value $f$1. The paper states that the $f$2-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” (Ng, 2016). 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

$f$3

The gapless Mott insulator (GMI) is defined by bosons $f$4 in a trivial Mott insulator and neutral fermions $f$5 forming a Fermi surface. Its low-energy Lagrangian is

$f$6

Because the bosons are in a trivial Mott insulator, there is no Chern-Simons term for the emergent gauge field $f$7 (Barkeshli et al., 2012).

The neutral Fermi surface is fixed by the electron density. For spinless fermions in two dimensions,

$f$8

with

$f$9

This is a Fermi surface without electric charge transport. The paper therefore characterizes the phase as a “genuine ‘second Fermi liquid’ without charge transport” (Barkeshli et al., 2012).

The compressibility is controlled by the Ioffe-Larkin composition rule

$U(1)$0

At the boson critical point, with correlation length $U(1)$1, the boson compressibility behaves as $U(1)$2, implying

$U(1)$3

At finite temperature in the quantum-critical fan,

$U(1)$4

Thus the critical point is incompressible at $U(1)$5 even though both adjacent phases are compressible (Barkeshli et al., 2012).

Transport and thermodynamics exhibit two crossover scales,

$U(1)$6

For $U(1)$7, the theory predicts “marginal Fermi-liquid” physics, including

$U(1)$8

For $U(1)$9, on the GMI side,

$U(1)$0

The phase also supports a $U(1)$1 singularity in density-density correlations. These properties sharply distinguish it from an ordinary Landau metal while preserving a well-defined neutral Fermi surface (Barkeshli et al., 2012).

A frequent misconception is that a Fermi liquid must be electrically conducting. The GMI construction provides a counterexample within the terminology of the cited work: the “second Fermi liquid” is insulating for charge transport yet retains Fermi-surface-controlled thermodynamics and neutral gapless fermions.

4. Small-FL, deconfined FL–FL transitions, and Fermi-surface-volume jumps

The 2024 bilayer study considers a single-orbital bilayer Hubbard model with electron density per layer $U(1)$2 and Hamiltonian

$U(1)$3

with $U(1)$4. Because each layer’s charge is separately conserved, the model admits two distinct symmetric, paramagnetic Fermi-liquid solutions consistent with Oshikawa’s theorem (Wu et al., 2024).

The two Fermi-surface volumes per flavor are stated as

$U(1)$5

for the conventional large-FL and

$U(1)$6

for the small-FL, where the minus sign indicates a hole-like Fermi surface centered at $U(1)$7. The two volumes differ by exactly one half of the Brillouin zone per flavor (Wu et al., 2024).

The continuum formulation fractionalizes the physical electron into fermionic partons $U(1)$8 and $U(1)$9, subject to on-site constraints enforced by compact $\alpha_2$0 gauge fields $\alpha_2$1 and $\alpha_2$2. The theory is organized by

$\alpha_2$3

with Higgs fields $\alpha_2$4 determining whether the gauge fields are Higgsed or deconfined. In the large-FL, $\alpha_2$5, both gauge fields are Higgsed, and the state is conventional. In the sFL, $\alpha_2$6 but an interlayer pairing $\alpha_2$7 develops; then $\alpha_2$8 is confined, $\alpha_2$9 is Higgsed to lock onto $\phi_2$0, and the remaining gapless $\phi_2$1 carries hole-like charge with small Fermi surface (Wu et al., 2024).

The paper emphasizes that the sFL has “neither symmetry breaking nor fractionalization” at low energies and identifies it with symmetric mass generation in a compressible setting. In both FL and sFL, “the parton gauge fields are either Higgsed or confined, so the low-energy excitations are ordinary electrons.” Yet the sFL “violates the naive Luttinger count of a weak-coupling FL yet is fully consistent with the exact (topological) version” (Wu et al., 2024). This makes the sFL conceptually close to the $\phi_2$2-FL: both preserve a Fermi-liquid endpoint while modifying the conventional relation between Fermi-surface volume and a weak-coupling electron count.

At the putative continuous transition, the theory becomes a deconfined $\phi_2$3 gauge theory coupled to Fermi-surface partons. The one-loop RG analysis yields fixed-point values

$\phi_2$4

and in the physical $\phi_2$5 limit the couplings are marginally irrelevant, with zero quasiparticle residue at criticality. The critical point is unstable to pairing; for the BCS coupling $\phi_2$6,

$\phi_2$7

and when $\phi_2$8, the interlayer pairing channel is marginally relevant, leading to

$\phi_2$9

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 (Wu et al., 2024).

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

$t$1

where

$t$2

Here $t$3 and $t$4 are the first-order Fermi-liquid parameters, while $t$5 and $t$6 are the second-order Fermi-liquid parameters (Mora et al., 2014).

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: $t$7

$t$8

The explicit relations are

$t$9

$\lambda$00

At particle-hole symmetry, $\lambda$01, so $\lambda$02 and the theory reduces to the Nozières Kondo FL (Mora et al., 2014).

The low-energy conductance through a symmetrically coupled quantum dot is written as

$\lambda$03

with

$\lambda$04

The coefficients $\lambda$05, $\lambda$06, and $\lambda$07 are exact functions of $\lambda$08, and the paper states that they change sign across the Kondo to empty-orbital crossover. At particle-hole symmetry one finds

$\lambda$09

The generalized Fano factor is

$\lambda$10

with the universal Kondo value $\lambda$11 at particle-hole symmetry and the Poissonian value $\lambda$12 in the non-interacting limit (Mora et al., 2014).

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 $\lambda$13-$\lambda$14 model. The physical Green’s function is factorized as

$\lambda$15

where $\lambda$16 is an auxiliary Fermi-liquid-type propagator and $\lambda$17 is an adaptive spectral weight, or caparison factor. At $\lambda$18,

$\lambda$19

$\lambda$20

The formalism is argued to be quantitatively valid in the overdoped regime $\lambda$21, with the estimate that for $\lambda$22 the $\lambda$23 theory remains accurate to $\lambda$24 error in the high-frequency weight while preserving the exact Luttinger-volume sum rule (Hansen et al., 2012).

The computed spectral function exhibits a narrow quasiparticle peak of weight $\lambda$25, a very broad incoherent background extending to $\lambda$26 several eV, and pronounced skew toward $\lambda$27. The results display the high energy kink, non-saturating bad-metal resistivity at high $\lambda$28, and $\lambda$29 at low $\lambda$30 crossing over to nearly linear-$\lambda$31 above $\lambda$32 K for $\lambda$33 (Hansen et al., 2012). Although this is not labeled a “second Fermi liquid” in the same sense as the GMI or $\lambda$34-FL, it is part of the broader family of noncanonical Fermi-liquid constructions in which quasiparticle phenomenology remains central but strongly modified by correlation effects.

A useful contrast is provided by the one-dimensional “quasi-Fermi liquid,” realized for spinless fermions with the usual marginal interaction tuned to zero and only an irrelevant coupling $\lambda$35. The state has a finite jump in $\lambda$36 at $\lambda$37, but no pole in the single-particle Green’s function, with $\lambda$38 (Rozhkov, 2014). This is explicitly described as dissimilar to both the Tomonaga-Luttinger and Fermi liquids. The comparison helps delimit the meaning of “second Fermi liquid”: not every nonstandard Fermi-surface state with partial FL features is classified that way in the literature, and distinct papers reserve different names for different departures from the Landau paradigm.

7. Common structural themes and open distinctions

Across these works, several recurring structures define what the phrase “second Fermi liquid” or closely related constructions are intended to capture.

Setting	Preserved low-energy structure	Nonstandard feature
$\lambda$39-FL (Ng, 2016)	Fermi-liquid-like thermodynamics	exclusion statistics, $\lambda$40, possible Luttinger-theorem violation
GMI / “second Fermi liquid” (Barkeshli et al., 2012)	neutral Fermi surface	insulating charge transport, emergent $\lambda$41 gauge field
sFL in bilayer model (Wu et al., 2024)	symmetric paramagnetic Fermi liquid	small Fermi surface differing by $\lambda$42 BZ per flavor
Anderson-model second FL (Mora et al., 2014)	low-energy FL expansion	second-order parameters $\lambda$43 away from PH symmetry
ECFL second-order theory (Hansen et al., 2012)	Fermi-surface-based Green’s-function framework	caparison factor, strong incoherence, small $\lambda$44

One common misconception is that all “second Fermi liquid” usages refer to the same phenomenon. The cited literature does not support that identification. Instead, the phrase spans at least three different conceptual domains: a new FL-like state with exclusion statistics (Ng, 2016); a neutral-Fermi-surface phase in a fractionalized Mott insulator (Barkeshli et al., 2012); and a small-FL endpoint of a deconfined metallic transition (Wu et al., 2024). Separately, “second-order Fermi-liquid” refers to expansion order in low-energy effective theory (Mora et al., 2014, Hansen et al., 2012).

Another common misconception is that a Fermi liquid must be adiabatically connected to bare electrons and must obey the conventional Luttinger theorem. The $\lambda$45-FL paper explicitly states the opposite possibility: quasiparticles “are not adiabatically connected to bare fermions” and the state “may not satisfy Luttinger theorem” (Ng, 2016). Likewise, the bilayer small-FL is “fully consistent with the exact (topological) version” of the Luttinger theorem while violating the naive weak-coupling count (Wu et al., 2024). The GMI goes further by separating charge transport from Fermi-surface structure altogether (Barkeshli et al., 2012).

Taken together, these works indicate that “second Fermi liquid” functions less as the name of a single universality class than as a label for Fermi-surface-based low-energy organization beyond conventional Landau assumptions. A plausible implication is that future usage of the term will remain context dependent unless a more unified taxonomy of FL-like but non-Landau phases is adopted.