Marginal Metallic Phase in Condensed Matter

• Marginal metallic phase is a regime where a conventional metal borders a competing ordered state, showing anisotropic, critical transport features.
• It is characterized by operational diagnostics such as gap closure, scale-invariant conductance, and a blend of Fermi-liquid and marginal-Fermi-liquid behaviors.
• Research employs techniques like ADMR, DMRG, and scaling analyses to quantify transitions in systems ranging from overdoped cuprates to disordered topological insulators.

to=arxiv_search.search _植物百科通json 彩神争霸充值{"query":"\"marginal metallic phase\" OR \"marginal metal\" condensed matter", "max_results": 10} to=arxiv_search.search արսូនjson 天天买彩票 天天中彩票中奖ിയതി{"query":"(Kokalj et al., 2012) anisotropic marginal fermi liquid overdoped cuprate transport", "max_results": 5} to=arxiv_search.search 天天中彩票出票json 天天中彩票被ាតreplaced with to=arxiv_search.search 天天中彩票在哪ոնjson A marginal metallic phase is a metallic regime that, in the cited literature, appears at the boundary of another organizing tendency rather than as a conventional Landau Fermi liquid. Depending on context, that neighboring tendency is anisotropic marginal scattering in overdoped cuprates, a band-insulator–to–correlated-insulator interpolation, a superconducting state with disrupted phase coherence, or a disorder-driven localization transition with scale-invariant transport. The term therefore does not denote a single universal phase class; instead, it names a family of metallic states whose metallicity is retained while one or more defining properties of a conventional metal become borderline, critical, or strongly anisotropic (Kokalj et al., 2012).

1. Scope and defining diagnostics

Across recent condensed-matter usage, the phrase is attached to several distinct diagnostics. In overdoped cuprates, the metallic phase is described by a self-energy containing an isotropic Fermi-liquid contribution and an anisotropic marginal-Fermi-liquid contribution, with transport anomalies controlled by the latter (Kokalj et al., 2012). In the half-filled $t$-$t'$ ionic Hubbard chain, the metallic phase is identified by charge and spin gaps that extrapolate to zero between a band insulator and a correlated insulator (Rossini et al., 2023). In disordered two-dimensional altermagnets and semimagnetic topological insulators, the metallic phase is identified by size-independent conductance and normalized localization length, together with a vanishing beta function over a finite disorder window (Li et al., 14 Jul 2025). In two-dimensional superconducting films, the relevant metallic state has saturated longitudinal resistivity but vanishing Hall resistivity, and is interpreted as a “failed superconductor” retaining particle-hole symmetry (Breznay et al., 2017).

Setting	Control parameter	Metallic criterion
Overdoped Tl2201	Temperature and doping	Anisotropic marginal scattering in transport
$t$-$t'$ ionic Hubbard chain	$U$ at fixed $t',\Delta$	$\Delta_c \approx 0$, $\Delta_s \approx 0$
Disordered altermagnet	Disorder $W$	$\beta(g)=0$, $t'$0 size independent
Semimagnetic topological insulator	Disorder $t'$1	$t'$2, $t'$3 size independent
2D superconducting films	Magnetic field $t'$4	$t'$5 saturates, $t'$6

This diversity is itself a substantive feature of the concept. The common element is not a single microscopic mechanism, but a metallic state whose transport, spectroscopy, or scaling is pinned close to a crossover or transition that would ordinarily produce localization, gapping, or symmetry breaking. A plausible implication is that “marginal” is best read operationally: it identifies the specific quantity that sits at the border of its conventional metallic behavior.

2. Anisotropic marginal metallicity in overdoped cuprates

A particularly concrete formulation is given for overdoped $t'$7 through a phenomenological self-energy written as

$t'$8

The first term is isotropic over the Fermi surface, weakly doping dependent, and Fermi-liquid-like, with low-energy behavior quadratic in $t'$9 and $t$0. The second term is anisotropic, strongly doping dependent, and marginal-Fermi-liquid-like, with linear dependence on $t$1 (Kokalj et al., 2012).

The anisotropy is encoded by

$t$2

with $t$3 K. This makes the marginal contribution maximal in the antinodal directions and zero at the nodes, with the same angular structure as a $t$4 superconducting gap. The marginal channel therefore fades as overdoping reduces $t$5, while the isotropic Fermi-liquid channel remains present. In the associated 2011 formulation, the same two-component self-energy was used to reconcile the strongly doping dependent anomalous scattering rate observed in ADMR with the almost doping independent specific heat, and to give a consistent description of ADMR, specific heat, de Haas-van Alphen, and quasiparticle dispersion data (Kokalj et al., 2011).

Within transport, the consequences are sharply differentiated. The intra-layer resistivity, frequency-dependent optical conductivity, intra-layer magnetoresistance, and Hall coefficient of Tl2201 are reproduced quantitatively without introducing new parameters and while neglecting vertex corrections. The temperature dependence of magnetoresistance and Hall coefficient is especially sensitive to the anisotropy of the scattering rate and to the Fermi-surface shape, whereas the Hall angle is dominated by the Fermi-liquid contribution that controls the nodal scattering rate (Kokalj et al., 2012).

This construction gives the overdoped cuprate metallic phase a precise technical meaning. Quasiparticles remain well defined, yet the scattering is neither purely Fermi-liquid-like nor purely marginal, and the metallic state is governed by the strongly anisotropic superposition of the two. In that sense, the “marginal metallic phase” is not an incoherent strange metal, but a metal whose low-energy transport is marginal in selected regions of momentum space.

3. Interaction-driven metallic phases between insulating states

In one-dimensional correlated-electron systems, the phrase refers to a metal stabilized between two distinct insulators. For the half-filled $t$6-$t$7 ionic Hubbard chain with zero magnetization, an unusual metallic phase is proven to develop at intermediate repulsion $t$8 when second-neighbor hopping is tuned close to a Lifshitz transition. The model lies between a band insulator at low $t$9 and a correlated, Mott-like insulator at high $t'$0, and is analyzed by numerically exact DMRG together with mean field and bosonization (Rossini et al., 2023).

For representative parameters $t'$1, $t'$2, and $t'$3, the DMRG estimates place the metal between $t'$4 and $t'$5. In that interval, finite-size scaling is compatible with $t'$6 and with a spin gap that also tends to zero, so the phase has both charge and spin gapless excitations. Near the upper boundary, the metal supports spontaneous bond charge dimerization and antiferromagnetic correlations, with the BOW onset inside the metallic phase at $t'$7 (Rossini et al., 2023).

A separate but related correlated-electron usage appears in high-pressure $t'$8. At 300 K, a pressure-induced corundum-to-monoclinic transition occurs around $t'$9 GPa between two metallic phases. X-ray Raman scattering shows that the strong screening of the corundum phase becomes weakened at high pressure, and the theoretical analysis relates this to a decrease in coherent quasiparticle strength. The high-pressure monoclinic phase is therefore described as likely a critical correlated metal, on the verge of Mott-insulating behavior (Ding et al., 2013).

Taken together, these cases show a recurrent meaning of marginality: metallic transport survives, but only inside a narrow corridor in parameter space where the neighboring states are insulating and the coherence scale is suppressed. A plausible implication is that the phase is “marginal” because it is stabilized by competition rather than by a broad conventional metallic basin.

4. Scale-invariant critical metals under disorder

A second major usage is strictly scaling-theoretic. In this setting, the marginal metallic phase is not defined by $U$0-linear scattering or by proximity to Mottness, but by the fact that conductance and localization measures become scale invariant over a finite parameter window.

For transitions between obstructed atomic insulators protected by average magnetic crystalline symmetry, the intermediate metal can become a scale-invariant critical metal phase under disorder. The defining properties are that electronic states at the Fermi energy are delocalized, the conductance is independent of system size, and the normalized quasi-one-dimensional localization length

$U$1

approaches a finite, size-independent value rather than diverging. The proposed mechanism is a semiclassical percolation problem involving $U$2 domains, with average $U$3 symmetry enforcing $U$4 and the critical metal realized when $U$5 (Wang et al., 2023).

In two-dimensional disordered altermagnets, the metallic side is explicitly called an altermagnetic marginal metal. The quasi-one-dimensional localization length $U$6 defines

$U$7

and the conductance beta function is

$U$8

For a finite interval $U$9, $t',\Delta$0 is essentially independent of $t',\Delta$1, $t',\Delta$2 is almost independent of $t',\Delta$3, and $t',\Delta$4. On the insulating side, the correlation length follows the KT form

$t',\Delta$5

with $t',\Delta$6 and $t',\Delta$7 for $t',\Delta$8. Spectrally, the marginal metal has $t',\Delta$9, consistent with the Gaussian unitary ensemble, and a fractal dimension $\Delta_c \approx 0$0, close to the ideal two-dimensional metal value $\Delta_c \approx 0$1 (Li et al., 14 Jul 2025).

A closely related phase appears in disordered semimagnetic topological insulators. There the marginal metal lies between the weak-antilocalization half-quantized Hall metal and an Anderson insulator. The size dependence of the longitudinal conductivity is measured by

$\Delta_c \approx 0$2

and the marginal-metal regime is defined by $\Delta_c \approx 0$3, together with a normalized localization length $\Delta_c \approx 0$4 that collapses onto a size-independent curve. At $\Delta_c \approx 0$5 eV, the reported disorder interval is roughly $\Delta_c \approx 0$6; in that interval $\Delta_c \approx 0$7 is finite and size independent, while $\Delta_c \approx 0$8 is non-quantized and generally nonzero (Bi et al., 27 Aug 2025).

These works give the most literal implementation of the expression “marginal metal.” The metal is neither a conventional diffusive metal, for which the conductance would grow under scaling, nor an insulator, for which it would decay. Instead, it realizes a full phase with critical-like scale invariance.

5. Failed superconductivity and mixed-phase metallic states

In disordered two-dimensional superconducting films, the marginal metallic regime appears as a metallic state that preserves a superconducting hallmark while losing zero resistance. Transport studies on $\Delta_c \approx 0$9 and $\Delta_s \approx 0$0 show a field-tuned transition from a true superconductor to a metallic phase with saturated resistivity. The defining observation is that, over a wide field range above $\Delta_s \approx 0$1, the Hall resistivity remains zero within experimental resolution while the longitudinal resistivity is finite and saturates as $\Delta_s \approx 0$2. The vanishing $\Delta_s \approx 0$3 is interpreted as evidence that the metallic phase retains particle-hole symmetry from the disrupted superconducting state, leading to its identification as a “failed superconductor” (Breznay et al., 2017).

This state is distinguished from a trivial metal with superconducting inclusions. In the anomalous metallic regime, $\Delta_s \approx 0$4 deviates strongly from the normal-state Hall conductivity, while $\Delta_s \approx 0$5 persists between $\Delta_s \approx 0$6 and $\Delta_s \approx 0$7. Above $\Delta_s \approx 0$8, by contrast, $\Delta_s \approx 0$9 becomes finite and the transport crosses into a vortex-flow dominated metallic regime. The marginality here lies in the coexistence of finite dissipation with a transport tensor that still reflects superconducting particle-hole symmetry (Breznay et al., 2017).

A different boundary-metal picture is realized in epitaxial FeRh/$W$0 heterostructures. FeRh is metallic in both the low-$W$1 $W$2-type antiferromagnetic state and the high-$W$3 ferromagnetic state, and near the first-order AFM$W$4FM transition the free energies of the two metallic phases are nearly degenerate. The system is therefore treated as a mixed-phase metallic model system similar to phase-separated colossal magnetoresistance materials. The phase instability is quantified through a critical magnetic field $W$5 T at $W$6, corresponding to a characteristic energy $W$7 meV, and the heterostructure exhibits a $W$8 electroresistance modulation at $W$9 kV/cm near 376 K (Liu et al., 2017).

These superconducting and mixed-phase examples broaden the meaning of marginality. The metallic state is not defined by a special self-energy or by Anderson criticality, but by its location at the edge of a competing ordered phase, with transport controlled by residual symmetry, phase coexistence, or vortex dynamics.

6. Topological extensions and terminological boundaries

The term also has topological uses. In metallic single-wall carbon nanotubes, curvature and spin-orbit coupling open a tiny gap, $\beta(g)=0$0–$\beta(g)=0$1, and a magnetic field parallel to the tube shifts the circumferential momentum by an Aharonov–Bohm term

$\beta(g)=0$2

The gap closes when

$\beta(g)=0$3

at which point the winding number changes discontinuously and the number of edge states changes by bulk-edge correspondence. In that usage, the system is “almost metallic” yet still topologically classified, and the marginal metallic point is the gap-closing boundary where the invariant becomes ill defined (Okuyama et al., 2016).

A related boundary regime occurs in the two-dimensional periodic Anderson model away from half filling. There, a topological insulating state is hidden within a ferromagnetic metallic phase: the majority-spin sector develops a spin-selective gap with Chern number $\beta(g)=0$4, while the minority-spin sector remains metallic. At $\beta(g)=0$5, the spin-selective topological Kondo-insulator region extends roughly over $\beta(g)=0$6, even though the majority-spin gap is only of order $\beta(g)=0$7 (Yoshida et al., 2013). This is not named a marginal metallic phase in the formal sense of a zero-beta critical metal, but it is a metal whose topology is carried by only part of its spectrum.

At the same time, several papers use “marginal” in technically different ways. In the bosonic NRG treatment of impurity scattering in Luttinger liquids, “marginal spectral density” means

$\beta(g)=0$8

the case $\beta(g)=0$9 for which the usual NRG scale separation collapses because $t'$00 becomes independent of shell index $t'$01 (Freyn et al., 2011). In the $t'$02 chain on a two-dimensional metal, the metallic bath yields an Ohmic dissipative environment that is marginally relevant at the decoupled fixed point, whereas coupling to a semi-metal is irrelevant (Danu et al., 14 Sep 2025). These are important distinctions: “marginal” may refer to a phase, to a critical point, to a spectral density, or to an RG perturbation.

The main misconception therefore lies in treating the phrase as universally defined. The literature instead supports a more precise statement: a marginal metallic phase is any metallic regime for which the operative criterion of metallicity is itself borderline—anisotropic and marginal in self-energy, gapless between insulators, scale invariant under disorder, particle-hole symmetric despite finite resistance, or topological at the brink of gap closure.