Modified Periodic Anderson Model (MPAM)

• MPAM is a modified version of the periodic Anderson model that introduces additional orbitals, altered hybridization, and nonstandard interactions to explore rich quantum phases.
• It exhibits a continuous zero-temperature Mott transition with quantum critical scaling, characterized by a vanishing quasiparticle weight and distinctive self-energy behavior.
• Various MPAM variants—including Anderson-Hubbard, staggered hybridization, and phonon-coupled models—demonstrate tailored balances between Kondo screening, RKKY exchange, and valence fluctuations.

Modified Periodic Anderson Model (MPAM) denotes a family of Anderson-lattice Hamiltonians obtained by altering the conventional periodic Anderson model (PAM) through additional orbitals, modified hybridization structure, nonstandard interaction terms, phonons, or spatial inhomogeneity. In the most specific recent usage, MPAM refers to a particle-hole-symmetric three-orbital extension containing a correlated localized $f$ orbital, a conduction orbital $c$, and a second conduction orbital $c_M$ coupled by $t_\perp$; this construction supports a continuous zero-temperature Mott transition in a paramagnetic phase and has therefore become a central setting for genuine Mott quantum criticality (Majumder et al., 19 Jun 2025, K. et al., 29 Aug 2025). In a broader usage, the same label covers periodic Anderson-Hubbard models, PAMs with selective $d$-$f$ interactions, momentum-local Hatsugai–Kohmoto interactions, degenerate conduction orbitals, staggered hybridization, and electron-phonon coupling, all of which modify the balance between Kondo screening, RKKY exchange, valence fluctuations, and Mott localization (Hagymasi et al., 2011).

1. Definition and model space

The reference PAM contains a dispersive conduction band, a localized $f$ level, on-site $c$-$f$ hybridization, and local $f$-electron repulsion. One standard form is

$c$0

to which MPAM constructions add new couplings or degrees of freedom (Hagymasi et al., 2011).

Across the literature represented here, “modified” does not denote a unique Hamiltonian. It denotes controlled departures from the standard PAM in the conduction sector, the interaction sector, the hybridization structure, or the lattice representation. Representative variants documented in (Majumder et al., 19 Jun 2025, K. et al., 29 Aug 2025, Hagymasi et al., 2011, Hagymasi et al., 2011, Zhong, 2022, Yang et al., 2019, Li et al., 2017), and (Jiang, 2020) are summarized below.

Variant	Defining modification	Reported consequence
Three-orbital MPAM	Add $c$1 and interconduction hopping $c$2	Continuous zero-$c$3 Mott transition
Periodic Anderson-Hubbard model	Add $c$4 on conduction sites	Shifted/widened Kondo regime; half-filled Mott transition
Selective $c$5-$c$6 interaction model	Add $c$7 only for simultaneous double occupancy	Sharpened valence transition
HK-based solvable MPAM	Replace local Hubbard term by momentum-local HK interaction	Exact solvability; correlated metal, HI, MI
Three-orbital degenerate-conduction PAM	Replace one conduction band by two degenerate orbitals	AF1, AF2, PM; Lifshitz transition
Staggered PAM	Alternate $c$8 and $c$9 by sublattice	Enhanced AF ordering tendency
PAM with Holstein phonons	Couple phonons to conduction density	Enhanced cerium volume collapse

The main conceptual distinction is therefore between the standard PAM, where only the $c_M$0 orbital is correlated and only one conduction channel hybridizes locally, and MPAM constructions, where either the bath itself becomes correlated, the hybridization environment is qualitatively restructured, or the interaction acquires a nonstandard form. That distinction is decisive for whether the low-energy problem is dominated by Kondo insulating behavior, valence fluctuations, Lifshitz reconstruction, or genuine Mott criticality.

2. Canonical three-orbital MPAM and the zero-temperature Mott transition

In the 2025 DMFT literature, the MPAM is a three-orbital lattice model with one correlated localized orbital and two itinerant orbitals. Its momentum-space Hamiltonian is

$c_M$1

The particle-hole-symmetric limit is

$c_M$2

with chemical potential set to zero; in (K. et al., 29 Aug 2025) the calculations use fixed hybridization $c_M$3 (K. et al., 29 Aug 2025).

A closely related real-space form is written as

$c_M$4

with nearest-neighbor hopping $c_M$5 for both conduction orbitals on a two-dimensional square lattice, local $c_M$6-$c_M$7 hybridization $c_M$8, and interorbital $c_M$9-$t_\perp$0 hopping $t_\perp$1 (Majumder et al., 19 Jun 2025). The model admits two equivalent viewpoints: a three-orbital lattice model and a bilayer model consisting of a conventional PAM layer coupled to a metallic layer. When $t_\perp$2, it reduces to the standard PAM, which at half filling is a particle-hole-symmetric Kondo insulator; when $t_\perp$3 and the $t_\perp$4 electrons are made dispersive, one recovers the single-band Hubbard-model limit (Majumder et al., 19 Jun 2025).

Its defining property is the existence, within DMFT, of a strictly zero-temperature quantum critical point separating a Fermi liquid from a Mott insulator. The transition is continuous, shows no hysteresis, and is not the finite-$t_\perp$5 endpoint structure familiar from the DMFT single-band Hubbard model (Majumder et al., 19 Jun 2025). In (K. et al., 29 Aug 2025) this point is generalized to a continuous surface of zero-temperature QCPs in the $t_\perp$6-$t_\perp$7 plane.

The low-energy metallic scale is

$t_\perp$8

and the $t_\perp$9-spectral function obeys

$d$0

in the Fermi-liquid regime (K. et al., 29 Aug 2025). Approaching the QCP, the quasiparticle weight vanishes continuously with exponent $d$1 on the metallic side, while the Mott gap closes with exponent $d$2 on the insulating side. At the critical point $d$3,

$d$4

equivalently with soft-gap exponent $d$5 (K. et al., 29 Aug 2025).

Analytical control is obtained in the two-site or linearized DMFT approximation. There the impurity bath is reduced to a single bath site, the self-energy is expanded as

$d$6

and the self-consistency condition becomes

$d$7

Using the two-site impurity solution,

$d$8

one obtains

$d$9

which makes the continuous vanishing of quasiparticle weight explicit (Majumder et al., 19 Jun 2025).

3. Quantum-critical scaling, local criticality, and entanglement diagnostics

The finite-temperature and real-frequency theory of the canonical three-orbital MPAM is formulated in DMFT with local $f$0-self-energy,

$f$1

and self-consistency

$f$2

For the semi-elliptic noninteracting density of states,

$f$3

the lattice Green’s function entering the DMFT loop is

$f$4

The impurity problem is solved with NRG because it is nonperturbative and works directly on the real-frequency axis, which is essential for low-energy spectra, finite-$f$5 transport, optical conductivity, and $f$6 scaling (K. et al., 29 Aug 2025).

A central result is the scaling collapse of the DC resistivity,

$f$7

with $f$8. At fixed $f$9, the reported exponents are

$f$0

These values are presented as lying in the same range as those reported for the single-band Hubbard model, where values around $f$1, $f$2, and $f$3 had been found depending on solver and Widom-line choice (K. et al., 29 Aug 2025).

The same work gives strong evidence for local quantum criticality through $f$4 collapse of both single- and two-particle quantities. At the QCP,

$f$5

collapse over multiple decades, and the local dynamical spin susceptibility satisfies a corresponding collapse of $f$6 (K. et al., 29 Aug 2025). Away from the QCP this scaling breaks down.

Optical transport yields an additional set of diagnostics. With vertex corrections absent in DMFT, the conductivity reduces to the bare bubble and displays a low-frequency Drude peak in the metal, a mid-infrared peak from interband excitations, and a high-energy charge-excitation peak. As the QCP is approached, the MIR peak redshifts, reaches a minimum at criticality, and disappears for $f$7; at the critical point, the conductivity follows a power law with exponent $f$8, close to $f$9. Two isosbestic points are identified, one before and one after the MIR peak; the first moves to zero and vanishes at the QCP, while the second disappears before the QCP is reached (K. et al., 29 Aug 2025).

A complementary diagnostic is the single-site von Neumann entanglement entropy. For a half-filled correlated site with double occupancy $c$0, the local density matrix gives

$c$1

Near criticality, the two-site impurity solution yields

$c$2

and therefore

$c$3

The metallic limit has $c$4, while the Mott insulating limit has $c$5 (Majumder et al., 19 Jun 2025). Cluster extensions within linearized cellular DMFT retain the qualitative nature of the transition but shift the critical behavior near $c$6, indicating that short-range spatial correlations matter most close to criticality (Majumder et al., 19 Jun 2025).

4. Interaction-based and exactly solvable MPAM variants

One important interaction-driven extension is the periodic Anderson-Hubbard model,

$c$7

which makes the conduction electrons correlated as well as the $c$8 electrons. Gutzwiller variational calculations and exact diagonalization on short one-dimensional chains show that $c$9 shifts and widens the heavy-fermion regime in $f$0, can move it above the bare conduction band, and, at half filling, drives a robust Kondo-insulator-to-Mott-insulator transition (Hagymasi et al., 2011). In the half-filled symmetric case,

$f$1

and the conduction-site double occupancy behaves as

$f$2

vanishing at

$f$3

At that point,

$f$4

so the Kondo effect is quenched and the system becomes a Mott insulator (Hagymasi et al., 2011).

A second interaction-based MPAM adds a selective local $f$5-$f$6 repulsion,

$f$7

which acts only when both orbitals are doubly occupied on the same site. Within a Gutzwiller treatment, this selective interaction sharpens the valence transition, narrows the intermediate-valence region $f$8, extends the heavy-fermion regime to smaller $f$9, and renormalizes the conduction band even though there is no bare $f$0-$f$1 repulsion (Hagymasi et al., 2011). For the parameters $f$2, $f$3, and comparison of $f$4 with $f$5, the paper reports narrowing in the region $f$6 but not in $f$7, reflecting the selectivity of the added term (Hagymasi et al., 2011).

A third line of development replaces the local Hubbard interaction by an infinite-range Hatsugai–Kohmoto interaction. The resulting MPAM is

$f$8

with each momentum sector independent,

$f$9

The many-body ground state factorizes as

$c$00

which is the exact solvability mechanism (Zhong, 2022). The one-dimensional case yields a non-Fermi-liquid-like correlated metal violating the Luttinger theorem, a hybridization insulator at $c$01, and a featureless Mott insulator at $c$02. The metal-insulator transitions are argued to belong to Lifshitz universality, with

$c$03

consistent with $c$04 in $c$05 (Zhong, 2022).

5. Orbital degeneracy, staggered hybridization, and magnetic Lifshitz reconstruction

Orbital multiplicity produces a distinct MPAM class in which one localized $c$06 orbital hybridizes with two degenerate conduction orbitals,

$c$07

A noninteracting middle band remains exactly $c$08, because it is orthogonal to the conduction combination coupled to $c$09 (Yang et al., 2019). At half filling and $c$10, the symmetric case $c$11 supports three phases: AF1, AF2, and PM. AF1 has a hole-type large Fermi surface around $c$12; AF2 has an electron-type small Fermi surface and, at half filling, a semimetallic state with Dirac-cone-like touching at $c$13 and midway between $c$14 and $c$15; the AF1–AF2 boundary is first order and identified as a Lifshitz transition (Yang et al., 2019). At the particle-hole-symmetric point $c$16, $c$17 independently of $c$18, while the on-site $c$19-$c$20 correlation tends to $c$21 at large $c$22, signaling strong singlet-like local correlations (Yang et al., 2019).

The same work reports a scaling transformation for the nonsymmetric case $c$23, $c$24: local moments, densities, and spin correlations collapse onto the symmetric-case curves after rescaling the hybridization strength to

$c$25

which the authors interpret as indicating that the effective control parameter is $c$26 rather than $c$27 and $c$28 separately (Yang et al., 2019).

A different structural modification is the staggered PAM with two inequivalent local-moment sites per unit cell and alternating hybridization,

$c$29

Determinant quantum Monte Carlo on a $c$30 square lattice at half filling shows a marked enhancement of the antiferromagnetic structure factor when one sublattice lies in the Kondo-singlet regime of the homogeneous PAM and the other lies in the AF-insulating regime (Jiang, 2020). The enhancement ratio

$c$31

can reach about an order of magnitude when $c$32 and $c$33. The commensurate $c$34 modulation is essential; a stripe-like $c$35 pattern does not produce the same effect (Jiang, 2020).

Magnetic Lifshitz reconstruction is also found in the finite-$c$36 square-lattice PAM treated by variational Monte Carlo. Near half filling, $c$37, the sequence PM $c$38 AF1 $c$39 AF2 occurs, with AF1–AF2 clearly first order and associated with a Fermi-surface topology change. Far from half filling, $c$40, the sequence PM $c$41 FM0 $c$42 FM1 $c$43 FM2 appears, with FM1 half-metallic and FM2 exhibiting a small-FS localized-$c$44 character (Kubo, 2015). The effective mass is extracted from

$c$45

and the large ordered-moment phases AF2 and FM2 are interpreted as itinerant-localized transitions of the $c$46 electrons (Kubo, 2015).

6. Rigorous antiferromagnetism in modified lattice representations

A mathematically distinct use of “modified” appears in rigorous work on the half-filled symmetric PAM. There the usual Anderson lattice,

$c$47

is specialized to the symmetric case

$c$48

and then rewritten as a generalized Hubbard model on a doubled bipartite lattice with fermions

$c$49

The resulting Hamiltonian is

$c$50

with intralayer hopping $c$51 and interlayer coupling $c$52 on a finite two-dimensional square lattice doubled to two layers (Enaroseha et al., 2013).

An auxiliary regularization

$c$53

is introduced and $c$54 is taken at the end of the proof. After a particle-hole transformation on one spin sector, reflection-positivity methods yield three main results. First, for distinct sites $c$55,

$c$56

Second, the transverse spin correlation satisfies

$c$57

Third, SU(2) symmetry extends this sign structure to the longitudinal correlator

$c$58

with

$c$59

The ground state is therefore a unique singlet with short-range antiferromagnetic correlations (Enaroseha et al., 2013).

The scope is explicitly limited. The result applies to the symmetric, half-filled PAM on a bipartite lattice after rewriting it as a doubled-lattice generalized Hubbard model. It is not a theorem for arbitrary asymmetry, doping, non-bipartite lattices, or arbitrary additional terms, and it proves short-range antiferromagnetic order rather than long-range order (Enaroseha et al., 2013).

7. Spectroscopic, thermodynamic, and materials-oriented extensions

Several MPAM-related extensions are motivated less by universal criticality than by specific observables or materials contexts. One example is the PAM with Holstein phonons coupled to the conduction density,

$c$60

with dispersionless Einstein phonons of frequency

$c$61

This model is used as a minimal framework for the cerium $c$62 volume-collapse problem, where the experimental transition is an isostructural collapse of about $c$63 at roughly $c$64 atmospheres (Li et al., 2017). Within DMFT using CT-QMC and maximum entropy, the paper finds that without phonons ($c$65) the calculated $c$66-$c$67 curve shows no kink, whereas with $c$68 a clear kink and first-order transition appear around $c$69; at $c$70 the volume collapse is about $c$71, with

$c$72

At small hybridization, increasing $c$73 opens a conduction-electron gap at $c$74, while at large $c$75 Kondo screening dominates and the conduction DOS remains metallic (Li et al., 2017).

A different diagnostic thread concerns the Knight-shift anomaly in heavy-fermion systems. Determinant quantum Monte Carlo on the half-filled square-lattice PAM resolves

$c$76

and shows that below a crossover temperature $c$77 the Knight shift no longer tracks the bulk susceptibility because of the onset of $c$78-$c$79 correlations (Jiang et al., 2014). The heavy-electron Knight-shift component follows

$c$80

and $c$81 increases with increasing hybridization $c$82 (Jiang et al., 2014). Although this work is on the standard PAM, it provides a microscopic reference point for MPAM studies in which the hybridization environment is itself modified.

Real-frequency self-energy structure supplies another benchmark. DMFT plus NRG calculations on the PAM show purely electronic kinks in $c$83, implying a crossover between a strongly renormalized Fermi-liquid regime and a less strongly renormalized intermediate-energy regime (Kainz et al., 2012). The paper extracts, for one representative fit, $c$84, $c$85, and $c$86, and reports that the kink scale follows an exponential dependence on $c$87, the same scaling as the Kondo temperature. It also emphasizes that PAM kinks are harder to resolve than Hubbard-model kinks because hybridization produces additional curvature and gap-related structure (Kainz et al., 2012). This suggests that MPAM variants retaining Anderson-lattice self-consistency should inherit kink physics, although its visibility will depend strongly on how the modification restructures the low-energy hybridization bath.

Taken together, these developments show that MPAM is best understood not as a single model but as a technically diverse Anderson-lattice framework. In one direction it furnishes a clean platform for zero-temperature Mott quantum criticality; in others it captures valence sharpening, conduction-band correlations, exact non-Fermi-liquid solvability, orbital-topological reconstruction, enhanced antiferromagnetism, rigorous short-range AF sign structure, phonon-assisted volume collapse, and spectroscopic anomalies tied to hybridization coherence. The unifying theme is that modest changes to the PAM conduction sector, interaction structure, or spatial patterning can qualitatively reorganize the infrared physics.