Spatially Random Yukawa Interaction in 2+1D Metals

• Spatially random Yukawa-type interaction is defined by a fermion–boson coupling with spatially quenched randomness that breaks momentum conservation.
• The large-N Schwinger–Dyson framework reveals a marginal Fermi liquid self-energy, leading to T-linear resistivity and T log(1/T) specific heat in 2+1 dimensions.
• This approach uniquely classifies Yukawa scalar couplings as the sole route to linear-in-T resistivity, with lattice models further supporting non-Boltzmann magnetotransport.

Searching arXiv for the cited work and closely related papers on spatially random Yukawa interactions. arxiv_search(query="Spatially random Yukawa interaction strange metals Patel (Patel et al., 2022) linear resistivity Yukawa uniqueness (Sin et al., 13 Jul 2025) vector coupling (Wang et al., 2024)", max_results=10) Spatially random Yukawa-type interaction denotes a coupling between fermions and bosonic modes in which the Yukawa vertex is a quenched random field in space, rather than a uniform translationally invariant constant. In the formulation studied most extensively, the fermions form a metal with a Fermi surface and the bosons represent quantum critical order parameters or emergent gauge-sector degrees of freedom. The central result of this framework is that, in $2+1$ dimensions and at large $N$, spatial randomness in the Yukawa coupling produces momentum relaxation directly at the interaction vertex, yielding a marginal–Fermi–liquid self-energy, a transport scattering rate linear in temperature, $\rho(T)\sim T$, and a specific heat $C_v(T)\sim T\ln(1/T)$ (Patel et al., 2022). Subsequent classification results argue that among scalar interactions of the form $(\psi^\dagger\psi)^n\phi^m$, the Yukawa case $(n,m)=(1,1)$ in $2+1$ dimensions is the unique scalar coupling within this disordered large-$N$ framework that gives linear-in-$T$ resistivity, while a spatially random vector coupling provides a parallel non-scalar route to the same transport scaling (Sin et al., 13 Jul 2025, Wang et al., 2024).

1. Definition and field-theoretic setup

The basic Euclidean action for the scalar version consists of an $N$-component fermion $N$0, an $N$1-component real scalar $N$2, and a Yukawa coupling

$N$3

with nonzero spatial average $N$4 and random fluctuations $N$5 (Patel et al., 2022). The action is

$N$6

The random part is taken Gaussian with correlator

$N$7

(Patel et al., 2022).

A closely related formulation uses zero-mean random Yukawa coupling from the outset,

$N$8

with

$N$9

(Sin et al., 13 Jul 2025). In large-$\rho(T)\sim T$0 generalizations, one may also consider matrix-valued random couplings $\rho(T)\sim T$1 with disorder variance proportional to $\rho(T)\sim T$2 (Wang et al., 2024).

The defining structural feature is locality of the disorder correlator in real space. Because the Yukawa vertex is $\rho(T)\sim T$3-correlated in space, the disorder-averaged self-energies become momentum independent in the purely spatially random version, a point emphasized both in continuum analyses and in later lattice formulations (Wang et al., 2024, Valentinis et al., 2 Nov 2025). This locality is the technical origin of the non-Boltzmann character of the resulting transport theory.

2. Large-$\rho(T)\sim T$4 solution and saddle-point equations

After averaging over disorder and introducing bilocal Green’s functions and self-energies, the theory is solved at leading order in a large-$\rho(T)\sim T$5 expansion by Dyson or Schwinger–Dyson equations (Patel et al., 2022). In momentum-frequency space these take the form

$\rho(T)\sim T$6

$\rho(T)\sim T$7

(Patel et al., 2022).

In the spatially random limit, the $\rho(T)\sim T$8-function at each disorder-averaged vertex collapses momentum integrals, and one obtains momentum-independent self-energies,

$\rho(T)\sim T$9

(Wang et al., 2024). This simplification is one of the distinctive features of the construction. It differs sharply from clean translationally invariant Yukawa theories, where $C_v(T)\sim T\ln(1/T)$0 retains nontrivial momentum structure and transport is strongly constrained by momentum conservation.

In the $C_v(T)\sim T\ln(1/T)$1-dimensional strange-metal solution, a scaling ansatz in imaginary time,

$C_v(T)\sim T\ln(1/T)$2

gives

$C_v(T)\sim T\ln(1/T)$3

by matching the time-domain Schwinger–Dyson equations (Wang et al., 2024). This scaling solution is consistent with a marginal fermionic self-energy at low frequency.

The same large-$C_v(T)\sim T\ln(1/T)$4 logic extends to generalized “SYK-rised” interactions involving $C_v(T)\sim T\ln(1/T)$5, normalized as

$C_v(T)\sim T\ln(1/T)$6

with replica disorder averaging leading again to a $C_v(T)\sim T\ln(1/T)$7–$C_v(T)\sim T\ln(1/T)$8 saddle-point description (Sin et al., 13 Jul 2025). That broader construction provides the basis for the later uniqueness argument for Yukawa coupling.

3. Marginal self-energy and thermodynamics

For fermions on the Fermi surface, the leading self-energy in the original continuum treatment contains an elastic term and two logarithmic dynamical terms,

$C_v(T)\sim T\ln(1/T)$9

(Patel et al., 2022). The last term, proportional to $(\psi^\dagger\psi)^n\phi^m$0, is identified as the crucial new contribution generated by spatial fluctuations of the Yukawa coupling (Patel et al., 2022).

This logarithmic dependence is the characteristic marginal–Fermi–liquid structure. In the alternative notation of the later analysis, the retarded self-energy behaves as

$(\psi^\dagger\psi)^n\phi^m$1

and at finite temperature

$(\psi^\dagger\psi)^n\phi^m$2

(Wang et al., 2024). The agreement between these formulations indicates that the disorder-induced Yukawa mechanism consistently produces marginal dissipation in $(\psi^\dagger\psi)^n\phi^m$3 dimensions.

The same self-energy controls the low-temperature specific heat. The disorder-averaged free energy with a self-energy of the form $(\psi^\dagger\psi)^n\phi^m$4 yields

$(\psi^\dagger\psi)^n\phi^m$5

up to non-universal subleading terms (Patel et al., 2022). The combination of $(\psi^\dagger\psi)^n\phi^m$6 and $(\psi^\dagger\psi)^n\phi^m$7 is one of the defining signatures of the strange-metal regime in this approach.

A plausible implication is that the thermodynamic and transport anomalies arise from the same local critical scattering process rather than from unrelated sectors. That implication is suggested directly by the shared origin of both observables in the marginal fermion self-energy.

4. Transport mechanism and linear resistivity

The transport result is obtained through the Kubo formula. In the original large-$(\psi^\dagger\psi)^n\phi^m$8 calculation, the dc conductivity receives three contributions: the elastic Drude term, $(\psi^\dagger\psi)^n\phi^m$9 self-energy and vertex corrections that cancel, and a $(n,m)=(1,1)$0 self-energy correction that does not cancel (Patel et al., 2022). For $(n,m)=(1,1)$1,

$(n,m)=(1,1)$2

which implies

$(n,m)=(1,1)$3

(Patel et al., 2022). In the dc limit $(n,m)=(1,1)$4,

$(n,m)=(1,1)$5

so that the transport scattering rate obeys

$(n,m)=(1,1)$6

(Patel et al., 2022).

The later continuum treatment recasts the same mechanism more directly: with spatial randomness, the usual Maki–Thompson and Aslamazov–Larkin vertex corrections vanish, so a single Yukawa self-energy insertion into the current-current bubble produces

$(n,m)=(1,1)$7

(Wang et al., 2024). The disappearance of the standard cancellations is therefore not incidental; it is tied to the fact that momentum conservation is broken locally by the random Yukawa vertex.

This mechanism also clarifies the contrast with several more familiar settings. In a clean critical metal, full vertex corrections restore the Drude weight and can lead to vanishing resistivity despite singular self-energies. With scalar potential disorder added to an otherwise uniform boson coupling, one obtains different boson dynamics and $(n,m)=(1,1)$8 in $(n,m)=(1,1)$9. Spatial randomness in the Yukawa coupling itself changes the problem qualitatively by breaking momentum conservation at each vertex and exposing the marginal self-energy directly in transport (Wang et al., 2024).

5. Dimensional restriction and uniqueness of the Yukawa case

A major later development was the systematic classification of scalar random couplings of the form $2+1$0 in arbitrary spatial dimension $2+1$1 (Sin et al., 13 Jul 2025). Tree-level engineering dimensions give

$2+1$2

so that marginality requires $2+1$3 (Sin et al., 13 Jul 2025). For $2+1$4, this condition gives $2+1$5, namely the usual $2+1$6-dimensional Yukawa coupling.

Beyond tree level, the generalized low-frequency self-energies behave as

$2+1$7

$2+1$8

(Sin et al., 13 Jul 2025). Writing the fermion self-energy exponent as

$2+1$9

the conductivity correction scales as $N$0, and hence

$N$1

(Sin et al., 13 Jul 2025). Asking for linear-in-$N$2 resistivity sets $N$3.

In the regime relevant to the random Yukawa strange metal, this reduces to the condition

$N$4

whose unique positive integer solution is $N$5, $N$6 (Sin et al., 13 Jul 2025). The same work states that no other scalar coupling and no higher dimension reproduces the hallmark $N$7, and concludes that only the scalar Yukawa coupling in $N$8 dimensions yields linear resistivity within this scalar disordered class (Sin et al., 13 Jul 2025).

The comparison with the vector case is also explicit. A companion analysis shows that a random vector coupling can likewise give $N$9, and the two papers together conclude that spatially random Yukawa and QED-type interactions exhaust the class of random couplings capable of producing linear resistivity in this framework (Sin et al., 13 Jul 2025, Wang et al., 2024).

6. Extensions, lattice realization, and magnetotransport

A lattice realization of the same basic idea appears in the two-dimensional spatially disordered Yukawa–Sachdev–Ye–Kitaev model on a square lattice, or 2D-YSYK model (Valentinis et al., 2 Nov 2025). Its Hamiltonian contains fermion hopping on a square lattice, bosonic modes with a lattice stiffness term, and a local random Yukawa interaction

$T$0

(Valentinis et al., 2 Nov 2025). The couplings are Gaussian-distributed, zero mean, and delta-correlated in real space, with a parameter $T$1 controlling Cooper-pair breaking versus time-reversal-symmetric real couplings (Valentinis et al., 2 Nov 2025).

In the $T$2 limit, the model again yields momentum-independent self-energies,

$T$3

$T$4

(Valentinis et al., 2 Nov 2025). The momentum independence is interpreted as a non-Boltzmann regime because the self-energy is purely local and frequency dependent rather than weak and momentum resolved (Valentinis et al., 2 Nov 2025).

The corresponding Kubo formulas for the DC conductivities at linear order in perpendicular magnetic field are

$T$5

$T$6

(Valentinis et al., 2 Nov 2025). In the low-$T$7 marginal-Fermi-liquid regime, $T$8, so

$T$9

(Valentinis et al., 2 Nov 2025). Above that regime, the interplay between the YSYK interaction, square-lattice transport functions, and frequency-dependent broadening produces a crossover in which $N$0 decreases roughly as $N$1 while $N$2 remains proportional to $N$3, leading to

$N$4

with numerics reporting values up to $N$5 (Valentinis et al., 2 Nov 2025).

This lattice extension does not alter the core mechanism of the continuum theory. Rather, it shows how spatially random Yukawa interactions, once embedded in a concrete band structure, can generate linear longitudinal resistivity together with distinct Hall-sector scaling, because the local random interaction “breaks” the standard assumption that a single quasiparticle scattering time controls both channels (Valentinis et al., 2 Nov 2025).

7. Interpretation, scope, and common points of confusion

The central physical interpretation is that spatial randomness in the Yukawa interaction supplies a direct momentum-relaxing channel while preserving the critical boson-mediated singularity in the fermionic self-energy. The resulting strange metal is therefore neither an ordinary impurity-dominated Fermi liquid nor a clean quantum critical metal. It is a disordered critical metal whose randomness is located in the interaction vertex itself (Patel et al., 2022, Wang et al., 2024).

One recurring point of confusion concerns the role of disorder. The framework does not identify arbitrary disorder with strange-metal transport. The random scalar potential contribution produces an elastic Drude term and sets scales such as $N$6, but the linear-in-$N$7 contribution is attributed specifically to spatial fluctuations of the Yukawa coupling, through the term proportional to $N$8 and the associated uncancelled Kubo correction (Patel et al., 2022). This distinction is essential.

A second point concerns dimensionality. The later classification and the vector generalization both state that the mechanism for linear-$N$9 resistivity works only in $N$00 dimensions and not in higher dimensions, regardless of whether the interaction is scalar or vector (Sin et al., 13 Jul 2025, Wang et al., 2024). This is not presented merely as a perturbative accident; it follows from the scaling structure of the Schwinger–Dyson equations and from the corresponding resistivity exponents.

A third issue is the status of the so-called Planckian bound. Defining the transport scattering time by a Drude form,

$N$01

the original theory gives

$N$02

with

$N$03

(Patel et al., 2022). For sufficiently large $N$04, the logarithms cancel and $N$05, realizing the “Planckian” bound $N$06; for smaller $N$07, $N$08 (Patel et al., 2022). The theory therefore presents Planckian behavior as emergent and parameter dependent, not as an independent postulate.

Taken together, these works define a controlled large-$N$09 paradigm in which spatially random Yukawa-type interactions provide a universal route to several canonical strange-metal signatures in $N$10 dimensions: a marginal fermion self-energy, $N$11-linear resistivity, $N$12 specific heat, and, under suitable conditions, an emergent Planckian transport rate (Patel et al., 2022). Subsequent analysis sharpens this result by arguing that, among scalar couplings of the form $N$13, the Yukawa vertex is uniquely capable of producing the linear resistivity, while lattice generalizations show how the same local random interaction can generate non-Boltzmann magnetotransport beyond the longitudinal channel (Sin et al., 13 Jul 2025, Valentinis et al., 2 Nov 2025).