Doping-Driven Lifshitz Transition

• A doping-driven Lifshitz transition is an electronic topological change where carrier doping shifts the chemical potential through a band extremum, modifying the Fermi surface.
• Experimental techniques such as ARPES, Hall measurements, and quantum oscillations reveal signature changes in materials like cuprates, iron pnictides, and graphene.
• Correlation effects and multiband interactions determine whether the transition enhances or suppresses superconductivity and magnetic order in complex quantum materials.

Searching arXiv for the cited work and closely related papers on doping-driven Lifshitz transitions. arXiv search query: "doping-driven Lifshitz transition cuprate Hubbard iron pnictide TMD graphene nickelate" A doping-driven Lifshitz transition is an electronic topological transition in which carrier doping shifts the chemical potential through a band extremum or saddle point and thereby changes the topology of the Fermi surface. Depending on band structure and dimensionality, the transition may appear as a hole-like to electron-like conversion, the appearance or disappearance of a pocket, a neck-closing-and-reconnection event, or a one-valley to multi-valley crossover. Across cuprates, iron-based superconductors, transition-metal dichalcogenides, graphene, nickelates, heavy-fermion metals, and topological crystalline insulators, the transition is commonly discussed in connection with van Hove singularities, pseudogap phenomenology, magnetic reconstruction, and superconducting phase diagrams (Ghosh et al., 2016, Ding et al., 2019).

1. Formal definition and electronic-topological criteria

In its most general form, a Lifshitz transition occurs when a band edge crosses the chemical potential at a critical doping $x_c$: $\epsilon_n(k_c)=\mu(x_c)\,,$ so that the Fermi-surface topology changes at $x=x_c$ (Ghosh et al., 2016). In a pocket language, the equivalent criterion is

$\mu(x_c)=E_n^*\,,$

with $E_n^*$ the minimum of an electron pocket or the maximum of a hole pocket; at $x_c$ the corresponding pocket either appears or disappears (Ghosh et al., 2016). In a saddle-point language, the transition coincides with a van Hove singularity crossing the Fermi level.

For quasi-two-dimensional cuprates, the transition is often identified by the closing and reopening of an antinodal neck. In heavily overdoped Pb-Bi2201, the criterion is the change in Fermi-surface curvature and connectivity when the neck at $( -\pi,0 )$ closes and reopens with opposite curvature; equivalently, the saddle point at $M$ crosses $E_F$ (Ding et al., 2019). The paper expresses the critical condition either as

$$\left.\frac{\partial^2 E}{\partial k^2}\right|_{k=k_F,x_c}=0$$

or through the vanishing of the separation between the two antinodal Fermi crossings, $\epsilon_n(k_c)=\mu(x_c)\,,$0 (Ding et al., 2019).

In lattice models, the same notion appears in renormalized form. In the two-dimensional Hubbard model, the noninteracting saddle points at $\epsilon_n(k_c)=\mu(x_c)\,,$1 and $\epsilon_n(k_c)=\mu(x_c)\,,$2 yield $\epsilon_n(k_c)=\mu(x_c)\,,$3, but in the interacting case correlation effects shift and broaden the van Hove feature so that the critical filling occurs at finite hole doping: at $\epsilon_n(k_c)=\mu(x_c)\,,$4, $\epsilon_n(k_c)=\mu(x_c)\,,$5 for $\epsilon_n(k_c)=\mu(x_c)\,,$6, $\epsilon_n(k_c)=\mu(x_c)\,,$7 for $\epsilon_n(k_c)=\mu(x_c)\,,$8, and $\epsilon_n(k_c)=\mu(x_c)\,,$9 for $x=x_c$0 (Chen et al., 2012).

A central point is that the term refers to topology rather than symmetry breaking. The transition can occur with no change of broken symmetry, but in correlated materials it often coincides with other crossovers or phase boundaries, including pseudogap collapse, suppression of magnetic order, or the edge of a superconducting dome (Braganca et al., 2017, Ghosh et al., 2016).

2. Experimental identification and observables

Angle-resolved photoemission spectroscopy is the most direct probe because it resolves both Fermi-surface maps and the antinodal band geometry. In overdoped Pb-Bi2201, high-resolution ARPES with $x=x_c$1, measurements at $x=x_c$2 in the normal state, energy resolution $x=x_c$3–$x=x_c$4, and angular resolution $x=x_c$5 mapped the Fermi surface by integrating intensity within $x=x_c$6 of $x=x_c$7 (Ding et al., 2019). The decisive signature was the evolution from two antinodal Fermi crossings to a single crossing as doping increased, marking the reconnection from a hole-like barrel centered at $x=x_c$8 to an electron-like barrel enclosing $x=x_c$9 (Ding et al., 2019).

Transport can detect the same transition indirectly through changes in carrier type, mobility, and multiband compensation. In YBa$\mu(x_c)=E_n^*\,,$0Cu$\mu(x_c)=E_n^*\,,$1O$\mu(x_c)=E_n^*\,,$2, high-field Hall measurements showed that for $\mu(x_c)=E_n^*\,,$3 the normal-state Hall coefficient $\mu(x_c)=E_n^*\,,$4 changes sign from positive at high temperature to negative at low temperature, whereas for $\mu(x_c)=E_n^*\,,$5 it remains positive and rises monotonically as $\mu(x_c)=E_n^*\,,$6 (LeBoeuf et al., 2010). The disappearance of the high-mobility electron pocket at $\mu(x_c)=E_n^*\,,$7 coincides with a ten-fold drop in the low-temperature conductivity and with a jump in in-plane resistivity anisotropy, identifying a Lifshitz transition in the reconstructed Fermi surface (LeBoeuf et al., 2010).

Quantum oscillations provide a complementary criterion because extremal orbit areas track Fermi-surface topology. In underdoped cuprates, the disappearance of the $\mu(x_c)=E_n^*\,,$8 oscillation frequency and the logarithmic divergence of the cyclotron mass as $\mu(x_c)=E_n^*\,,$9 were interpreted as the merging of closed pockets into an open quasi-one-dimensional Fermi surface at $E_n^*$0 (Norman et al., 2010).

Raman spectroscopy is particularly sensitive when the critical states are antinodal or when a valley-selective phonon anomaly is involved. In overdoped Bi-2212, the integrated $E_n^*$1 Raman intensity changes slope above $E_n^*$2 and pins the collapse of the normal-state pseudogap to $E_n^*$3, the same doping at which the antibonding Fermi surface becomes electron-like (Benhabib et al., 2014). In doped single-layer MoS$E_n^*$4 and WS$E_n^*$5, the onset of a second valley produces a non-adiabatic red-shift and linewidth increase of Raman-active modes; for the $E_n^*$6 mode at $E_n^*$7, the reported non-adiabatic shift is $E_n^*$8 and the linewidth is $E_n^*$9–$x_c$0 above the transition (Novko, 2019).

Field-effect transport in multivalley semiconductors can reveal the transition through kinks in transconductance. In strained four-layer MoS$x_c$1, the densities $x_c$2 and $x_c$3 mark the occupations of the spin-orbit-split $x_c$4 and $x_c$5 valleys, and these densities coincide with experimentally observed kinks in $x_c$6 (Piatti et al., 2018).

3. Cuprates: topology change, pseudogap phenomenology, and superconductivity

The clearest cuprate example of a doping-driven Lifshitz transition directly tied to the end of superconductivity is heavily overdoped Pb-Bi2201. Partial Pb substitution suppresses the incommensurate Bi-O superstructure, and post-annealing controls the oxygen content and effective hole concentration. Using the Luttinger-volume relation

$x_c$7

the measured overdoped series spans approximately $x_c$8 to $x_c$9 (Ding et al., 2019). ARPES shows a large hole-like barrel at $( -\pi,0 )$0, progressive approach of the two antinodal sheets at $( -\pi,0 )$1 and $( -\pi,0 )$2, near-touching at $( -\pi,0 )$3, and an electron-like barrel at $( -\pi,0 )$4 (Ding et al., 2019). The Lifshitz transition occurs at $( -\pi,0 )$5, and this coincides with the change from superconducting to non-superconducting behavior: $( -\pi,0 )$6 decreases from $( -\pi,0 )$7 at $( -\pi,0 )$8 to $( -\pi,0 )$9 at $M$0 (Ding et al., 2019).

A distinct overdoped cuprate case is Bi$M$1Sr$M$2CaCu$M$3O$M$4, where the pseudogap and superconductivity decouple. Raman measurements on more than thirty crystals with $M$5 from $M$6 to $M$7 found that the normal-state pseudogap disappears at $M$8 (Benhabib et al., 2014). The same doping coincides with a Lifshitz transition of the antibonding Fermi surface from hole-like to electron-like, identified in a tight-binding model by the saddle-point condition $M$9 with $E_F$0 (Benhabib et al., 2014). However, the superconducting dome remains smooth through this point: $E_F$1 continues to decline with no visible kink or inflection at the Lifshitz transition (Benhabib et al., 2014). This is a direct counterexample to the idea that every topology change must produce a sharp anomaly in $E_F$2.

On the underdoped side, YBa$E_F$3Cu$E_F$4O$E_F$5 exhibits a different Lifshitz mechanism associated with density-wave reconstruction. High-field Hall data locate a critical doping $E_F$6 where a small electron pocket vanishes (LeBoeuf et al., 2010). A stripe-based model places the corresponding critical point near $E_F$7 and predicts that two symmetry-related electron pockets merge into an open quasi-one-dimensional Fermi surface, with a logarithmically divergent cyclotron mass

$E_F$8

in quantitative accord with the observed mass enhancement near the disappearance of quantum oscillations (Norman et al., 2010).

Taken together, the cuprate literature shows that the same topological event can correlate with distinct physical outcomes. In Pb-Bi2201 it coincides with the disappearance of superconductivity; in Bi-2212 it coincides with the collapse of the normal-state pseudogap while $E_F$9 remains smooth; in YBa$$\left.\frac{\partial^2 E}{\partial k^2}\right|_{k=k_F,x_c}=0$$0Cu$$\left.\frac{\partial^2 E}{\partial k^2}\right|_{k=k_F,x_c}=0$$1O$$\left.\frac{\partial^2 E}{\partial k^2}\right|_{k=k_F,x_c}=0$$2 it marks the loss of a reconstructed electron pocket and a large transport reorganization (Ding et al., 2019, Benhabib et al., 2014, LeBoeuf et al., 2010).

4. Iron-based superconductors: multiband Lifshitz points and superconducting domes

In Fe-based superconductors, doping-driven Lifshitz transitions are intrinsically multiband. The parent 122 compounds have three holelike pockets at $$\left.\frac{\partial^2 E}{\partial k^2}\right|_{k=k_F,x_c}=0$$3 and two electronlike pockets at $$\left.\frac{\partial^2 E}{\partial k^2}\right|_{k=k_F,x_c}=0$$4 and $$\left.\frac{\partial^2 E}{\partial k^2}\right|_{k=k_F,x_c}=0$$5, and doping shifts $$\left.\frac{\partial^2 E}{\partial k^2}\right|_{k=k_F,x_c}=0$$6 relative to the relevant band extrema (Ghosh et al., 2016). First-principles GGA-PBE+VCA calculations identified the following critical dopings:

Compound	$$\left.\frac{\partial^2 E}{\partial k^2}\right|_{k=k_F,x_c}=0$$7	Pocket undergoing topological change
Ba$$\left.\frac{\partial^2 E}{\partial k^2}\right|_{k=k_F,x_c}=0$$8K$$\left.\frac{\partial^2 E}{\partial k^2}\right|_{k=k_F,x_c}=0$$9Fe$\epsilon_n(k_c)=\mu(x_c)\,,$00As$\epsilon_n(k_c)=\mu(x_c)\,,$01	0.50	electron pocket at $\epsilon_n(k_c)=\mu(x_c)\,,$02
Ba$\epsilon_n(k_c)=\mu(x_c)\,,$03Na$\epsilon_n(k_c)=\mu(x_c)\,,$04Fe$\epsilon_n(k_c)=\mu(x_c)\,,$05As$\epsilon_n(k_c)=\mu(x_c)\,,$06	0.48–0.52	electron pocket at $\epsilon_n(k_c)=\mu(x_c)\,,$07
BaFe$\epsilon_n(k_c)=\mu(x_c)\,,$08Co$\epsilon_n(k_c)=\mu(x_c)\,,$09As$\epsilon_n(k_c)=\mu(x_c)\,,$10	0.11	inner hole pocket at $\epsilon_n(k_c)=\mu(x_c)\,,$11
BaFe$\epsilon_n(k_c)=\mu(x_c)\,,$12(As$\epsilon_n(k_c)=\mu(x_c)\,,$13P$\epsilon_n(k_c)=\mu(x_c)\,,$14)$\epsilon_n(k_c)=\mu(x_c)\,,$15	0.60	$\epsilon_n(k_c)=\mu(x_c)\,,$16-derived hole pocket
BaFe$\epsilon_n(k_c)=\mu(x_c)\,,$17Ru$\epsilon_n(k_c)=\mu(x_c)\,,$18As$\epsilon_n(k_c)=\mu(x_c)\,,$19	0.45–0.55	inner hole pocket at $\epsilon_n(k_c)=\mu(x_c)\,,$20

In these calculations, the Fermi-surface areas $\epsilon_n(k_c)=\mu(x_c)\,,$21 are more sensitive than band plots to the topological change: the critical sheet shows a kink or a jump in slope at $\epsilon_n(k_c)=\mu(x_c)\,,$22, and the corresponding quantum-oscillation frequencies $\epsilon_n(k_c)=\mu(x_c)\,,$23 should reflect this behavior (Ghosh et al., 2016). The same work reports that superconductivity is highest and magnetism disappears at or very near the Lifshitz concentration in each series (Ghosh et al., 2016). The minimal pairing estimate,

$\epsilon_n(k_c)=\mu(x_c)\,,$24

is used there to rationalize why a step or singularity in $\epsilon_n(k_c)=\mu(x_c)\,,$25 can enhance $\epsilon_n(k_c)=\mu(x_c)\,,$26 near the transition, whereas the loss of the critical sheet beyond $\epsilon_n(k_c)=\mu(x_c)\,,$27 cuts off interband scattering and suppresses superconductivity (Ghosh et al., 2016).

Strong-coupling formulations sharpen this multiband picture. In a two-orbital $\epsilon_n(k_c)=\mu(x_c)\,,$28–$\epsilon_n(k_c)=\mu(x_c)\,,$29 model for hole-doped iron pnictides, the Lifshitz point is where emergent electron bands at the commensurate-SDW momenta first cross the Fermi level; near this point an $\epsilon_n(k_c)=\mu(x_c)\,,$30-wave hole-pair ground state and a $\epsilon_n(k_c)=\mu(x_c)\,,$31-wave excited state become degenerate, yielding a soft $\epsilon_n(k_c)=\mu(x_c)\,,$32 collective mode whose resonant frequency collapses to zero at the quantum critical point in the low-density limit (Rodriguez, 2014). In heavily electron-doped iron selenides, an extended two-orbital Hubbard model with Eliashberg-type spin-fluctuation exchange yields a Lifshitz transition to electron and hole pockets centered at the corner of the two-iron Brillouin zone as on-site repulsion grows strong; after a rigid energy shift by sufficiently strong electron doping, only the two electron-type pockets remain, as in ARPES on iron selenide high-temperature superconductors (Rodriguez et al., 2018).

The iron-based literature therefore establishes a canonical multiband version of the phenomenon: the transition is pocket-selective, often tied to the suppression of spin-density-wave order, and its superconducting consequences depend on which pocket is removed or created and on how that pocket enters the interband pairing channel (Ghosh et al., 2016, Rodriguez, 2014).

5. Other material platforms and realizations

Doping-driven Lifshitz transitions are not confined to cuprates and pnictides. In monolayer MoS$\epsilon_n(k_c)=\mu(x_c)\,,$33 and WS$\epsilon_n(k_c)=\mu(x_c)\,,$34, adding electrons or holes shifts the chemical potential through successive valley extrema so that the Fermi surface changes from one-valley to multi-valley. Including non-local electron-electron interaction and spin-orbit coupling, the onset for electron doping is reported at $\epsilon_n(k_c)=\mu(x_c)\,,$35 in both MoS$\epsilon_n(k_c)=\mu(x_c)\,,$36 and WS$\epsilon_n(k_c)=\mu(x_c)\,,$37 (Novko, 2019). In four-layer strained MoS$\epsilon_n(k_c)=\mu(x_c)\,,$38, the corresponding multivalley thresholds are $\epsilon_n(k_c)=\mu(x_c)\,,$39 and $\epsilon_n(k_c)=\mu(x_c)\,,$40, directly mapped by transconductance kinks (Piatti et al., 2018).

In monolayer graphene, extreme chemical doping by an on-chip cesium diffusion architecture reaches $\epsilon_n(k_c)=\mu(x_c)\,,$41, close to the density range $\epsilon_n(k_c)=\mu(x_c)\,,$42–$\epsilon_n(k_c)=\mu(x_c)\,,$43 where tight-binding calculations place the saddle-point Lifshitz transition at the hyperbolic $\epsilon_n(k_c)=\mu(x_c)\,,$44 point (Aygar et al., 2024). Below the saddle, disconnected electron pockets surround $\epsilon_n(k_c)=\mu(x_c)\,,$45; above it, they merge into a hole-like pocket around $\epsilon_n(k_c)=\mu(x_c)\,,$46 (Aygar et al., 2024). The semiclassical cyclotron mass,

$\epsilon_n(k_c)=\mu(x_c)\,,$47

therefore diverges and changes sign across the transition, and Hall-coefficient inversion consistent with negative $\epsilon_n(k_c)=\mu(x_c)\,,$48 is reported in the most highly doped samples (Aygar et al., 2024).

Infinite-layer nickelates furnish an orbital-selective example. In Nd$\epsilon_n(k_c)=\mu(x_c)\,,$49Sr$\epsilon_n(k_c)=\mu(x_c)\,,$50NiO$\epsilon_n(k_c)=\mu(x_c)\,,$51, DFT+DMFT identifies a Lifshitz transition at $\epsilon_n(k_c)=\mu(x_c)\,,$52 when the Ni $\epsilon_n(k_c)=\mu(x_c)\,,$53-derived $\epsilon_n(k_c)=\mu(x_c)\,,$54 sheet is driven through the chemical potential and disappears (Leonov et al., 2020). Across this point, the Fermi surface changes from a three-sheet, partly three-dimensional topology to a more quasi-two-dimensional one, while magnetic correlations evolve from 3D $\epsilon_n(k_c)=\mu(x_c)\,,$55-type $\epsilon_n(k_c)=\mu(x_c)\,,$56 toward quasi-2D $\epsilon_n(k_c)=\mu(x_c)\,,$57-type $\epsilon_n(k_c)=\mu(x_c)\,,$58 behavior; the frustration ratio $\epsilon_n(k_c)=\mu(x_c)\,,$59 is maximal in the range $\epsilon_n(k_c)=\mu(x_c)\,,$60–$\epsilon_n(k_c)=\mu(x_c)\,,$61, the same range where superconductivity is observed (Leonov et al., 2020).

Topological crystalline insulators show a surface-state version of the same physics. In Pb$\epsilon_n(k_c)=\mu(x_c)\,,$62Sn$\epsilon_n(k_c)=\mu(x_c)\,,$63Se with $\epsilon_n(k_c)=\mu(x_c)\,,$64, ARPES on intrinsically $\epsilon_n(k_c)=\mu(x_c)\,,$65-doped samples resolves two eccentric Dirac cones, and alkali adsorption raises the surface chemical potential through two saddle energies

$\epsilon_n(k_c)=\mu(x_c)\,,$66

driving the system from hole pockets to an interlocked electron-hole configuration and then to purely electron pockets (Pletikosić et al., 2014).

Hydrogen-doped KCr$\epsilon_n(k_c)=\mu(x_c)\,,$67As$\epsilon_n(k_c)=\mu(x_c)\,,$68 provides a quasi-one-dimensional superconducting realization. Under the rigid-band approximation, the $\epsilon_n(k_c)=\mu(x_c)\,,$69-band Fermi surface changes from a connected three-dimensional sheet to two disconnected quasi-one-dimensional sheets at a critical hydrogen content $\epsilon_n(k_c)=\mu(x_c)\,,$70 in the tight-binding fit, while DFT places the transition near $\epsilon_n(k_c)=\mu(x_c)\,,$71–$\epsilon_n(k_c)=\mu(x_c)\,,$72 (Zhang et al., 2021). The triplet $\epsilon_n(k_c)=\mu(x_c)\,,$73 pairing channel remains dominant throughout the superconducting range $\epsilon_n(k_c)=\mu(x_c)\,,$74, and the calculated $\epsilon_n(k_c)=\mu(x_c)\,,$75 proxy peaks at the Lifshitz point even though the total density of states exhibits a dip there (Zhang et al., 2021).

6. Correlation effects, non-universality, and theoretical interpretation

A recurring interpretation is the van Hove scenario: tuning $\epsilon_n(k_c)=\mu(x_c)\,,$76 through a saddle point changes the density of states and pairing kernel. This idea is explicit in overdoped cuprates and Fe-based superconductors, where van Hove crossing is used to rationalize either enhanced or suppressed superconductivity depending on which sheet is involved (Ding et al., 2019, Ghosh et al., 2016). Yet the empirical record is not universal. In Pb-Bi2201, the Lifshitz transition at $\epsilon_n(k_c)=\mu(x_c)\,,$77 occurs where $\epsilon_n(k_c)=\mu(x_c)\,,$78 extrapolates to zero (Ding et al., 2019). In overdoped Bi-2212, by contrast, the normal-state pseudogap collapses at $\epsilon_n(k_c)=\mu(x_c)\,,$79 while $\epsilon_n(k_c)=\mu(x_c)\,,$80 remains smooth through the topology change (Benhabib et al., 2014). In hydrogen-doped KCr$\epsilon_n(k_c)=\mu(x_c)\,,$81As$\epsilon_n(k_c)=\mu(x_c)\,,$82, the calculated superconducting dome peaks at the Lifshitz point even though the total DOS dips, because the relevant type-II van Hove planes move to larger $\epsilon_n(k_c)=\mu(x_c)\,,$83 where the $\epsilon_n(k_c)=\mu(x_c)\,,$84 gap form factor is larger (Zhang et al., 2021). A plausible implication is that the superconducting consequence of a Lifshitz transition depends less on topology alone than on which orbital, momentum-space sector, and scattering channel are rendered critical.

Strong correlations can also change the order and symmetry of the transition. In large-scale dynamical-cluster QMC for the two-dimensional Hubbard model at $\epsilon_n(k_c)=\mu(x_c)\,,$85, the Lifshitz line is continuous at $\epsilon_n(k_c)=\mu(x_c)\,,$86, the van Hove singularity crosses the Fermi level at finite hole doping, and the bare $\epsilon_n(k_c)=\mu(x_c)\,,$87-wave pairing susceptibility near the Lifshitz points obeys

$\epsilon_n(k_c)=\mu(x_c)\,,$88

which is more singular than the standard Fermi-liquid or static-van-Hove expectations (Chen et al., 2012). In zero-temperature CDMFT, however, a different large-$\epsilon_n(k_c)=\mu(x_c)\,,$89 regime emerges: a pseudogap metal with a pole-like antinodal self-energy remains hole-like for all $\epsilon_n(k_c)=\mu(x_c)\,,$90, and the stable solution jumps at $\epsilon_n(k_c)=\mu(x_c)\,,$91 to a conventional metal that is already electron-like (Braganca et al., 2017). A later CDMFT study framed the distinction explicitly: at weak coupling the Lifshitz transition is continuous and approximately symmetric in $\epsilon_n(k_c)=\mu(x_c)\,,$92 and $\epsilon_n(k_c)=\mu(x_c)\,,$93 around $\epsilon_n(k_c)=\mu(x_c)\,,$94, while at strong coupling the pseudogap makes the evolution asymmetric and discontinuous because the Lifshitz point coincides with the pseudogap-to-Fermi-liquid transition (Aguiar et al., 11 Sep 2025). This directly contradicts the common simplification that every Lifshitz transition in correlated matter should look like a rigid-band saddle crossing.

Heavy-fermion metals make the failure of the rigid-band picture particularly explicit. In a periodic Anderson model with additional light bands, the shift of a heavy-band feature $\epsilon_n(k_c)=\mu(x_c)\,,$95 under carrier doping is controlled by a small parameter $\epsilon_n(k_c)=\mu(x_c)\,,$96, not by the noninteracting value $\epsilon_n(k_c)=\mu(x_c)\,,$97 (Benlagra et al., 2012). As a result, the critical doping required to drive a shallow pocket through the Fermi level is strongly enhanced relative to naive estimates based on the low-temperature specific heat, and doped carriers populate heavy and light bands in nearly equal numbers despite the dominant heavy-band DOS (Benlagra et al., 2012). This shows that in strongly renormalized multiband systems the experimental control parameter “doping” need not map linearly onto the band-topological control parameter “chemical-potential shift.”

The broad theoretical picture is therefore twofold. First, a doping-driven Lifshitz transition is a sharply defined change in Fermi-surface topology. Second, its physical meaning is material-dependent because correlations, orbital selectivity, density-wave reconstruction, and multiband carrier partitioning determine whether the transition is continuous or discontinuous, whether it coincides with pseudogap collapse or magnetic suppression, and whether it weakens, strengthens, or leaves largely unchanged the superconducting state (Braganca et al., 2017, Benlagra et al., 2012).