Fermi Surface Locking Mechanisms

• Locking of the Fermi surface is the process by which interaction and symmetry constraints replace conventional free-electron pole surfaces with constrained structures like Luttinger surfaces and spin-momentum locked textures.
• Different mechanisms such as interaction-driven reconfiguration, momentum-selective locking forming Fermi arcs, and interface-induced Fermi-level pinning highlight its role in both correlated and topological systems.
• The phenomenon is investigated using experimental techniques like ARPES, quantum oscillations, and NMR alongside theoretical models that incorporate strong correlations, symmetry breaking, and topological effects.

Searching arXiv for relevant papers on locking of the Fermi surface and closely related usages in correlated, topological, and spin-locked systems. Locking of the Fermi surface denotes a set of mechanisms by which the low-energy momentum-space structure at $E_F$ ceases to behave as the freely deformable pole surface of an ordinary Fermi liquid and instead becomes constrained by interaction-driven fixed points, commensurate symmetry breaking, spin-orbit-coupled texture, or interface electrostatics. In correlated systems, this language is used for the replacement of a Fermi surface of poles by a Luttinger surface of Green’s-function zeros; in topological and chiral systems, it refers to spin-momentum locking of the Fermi contour; and in strong-spin-orbit superconductors, it appears as a geometric constraint on residual spin response through a Fermi-surface average of the locking texture (Huang et al., 2021, Polyakov et al., 2020, Zhou, 16 May 2026). A complementary experimental usage concerns whether quantum oscillations reveal a single extremal branch or a more complex topology, as in the bag-shaped hole Fermi surface of p-type $\mathrm{Bi_2Se_3}$ (Piot et al., 2016).

1. Interaction-driven locking into a Luttinger surface

In the Mott literature, locking of the Fermi surface is formulated as the loss of an emergent local-in-momentum-space $\mathbb Z_2$ symmetry of the Fermi-liquid fixed point. At the Fermi surface, where the kinetic energy vanishes, one may perform the particle-hole-like transformation

$$c_{p\uparrow}\rightarrow c^\dagger_{p\uparrow},\qquad n_{p\uparrow}\rightarrow 1-n_{p\uparrow},$$

while leaving the other spin species untouched. In an ordinary Fermi liquid, all states below the chemical potential are doubly occupied. In the Mott regime, some of those states become singly occupied, the particle-hole equivalence between adding and removing an electron is lost, and the $\mathbb Z_2$ symmetry is broken. In this formulation, the low-energy structure is “locked” away from the ordinary pole surface and into a Luttinger surface of zeros of the single-particle Green function (Huang et al., 2021).

The microscopic exemplar is the Hatsugai–Kohmoto interaction

$$H_{\rm int}^{HK}=U\sum_{k}n_{k\uparrow}n_{k\downarrow},$$

which penalizes double occupancy in momentum space directly. In the Hatsugai–Kohmoto model, once $U>W$ the system becomes a Mott insulator with a hard gap, and the relevant microscopic effect is that some momentum states become singly occupied. The same paper argues that the Hubbard model flows to the same strongly coupled quartic fixed point because its Fourier decomposition contains the $q=0$, $k=p$ component with the same structure. This is the basis for the claim that the Hubbard and Hatsugai–Kohmoto models lie in the same high-temperature universality class (Huang et al., 2021).

The defining object of the locked phase is the Luttinger surface,

$\Omega := \{ \det G = 0\},$

rather than a Fermi surface of poles. Its stability is treated K-theoretically. If the Green function on a surrounding sphere $\mathrm{Bi_2Se_3}$0 defines a nontrivial element of $\mathrm{Bi_2Se_3}$1, the zero surface cannot be removed by small perturbations. Using Bott periodicity, the authors conclude that Luttinger surfaces of codimension $\mathrm{Bi_2Se_3}$2 are stable for $\mathrm{Bi_2Se_3}$3 odd and unstable for $\mathrm{Bi_2Se_3}$4 even. The experimental manifestation emphasized in this framework is particle-hole asymmetry, presented as the direct low-energy consequence of broken momentum-space $\mathrm{Bi_2Se_3}$5 symmetry (Huang et al., 2021).

2. Momentum-selective locking, Fermi arcs, and the Mott boundary

A more momentum-selective version of locking appears in the analytically solvable model for “Fermi and Luttinger arcs.” There the key object is a combined surface defined by

$\mathrm{Bi_2Se_3}$6

which can contain both pole segments, visible as Fermi-surface arcs, and zero segments, invisible to ARPES as Luttinger arcs. The central mechanism is that the electronic states on the Fermi arcs remain intact, while the part of the Fermi surface where the gap opens transforms into a Luttinger arc. In this picture, the Fermi surface does not literally terminate; its character changes from pole-like to zero-like under correlation effects (Worm et al., 2023).

The model couples $\mathrm{Bi_2Se_3}$7 and $\mathrm{Bi_2Se_3}$8, making the antiferromagnetic zone boundary the special locus for reconstruction. In the hole-doped case, the nodal or intrazonal segment remains intact because it couples to empty states, whereas the antinodal segment couples to filled states and is shifted away from the Fermi level, leaving a Luttinger surface. For electron doping, the pattern is reversed relative to the antiferromagnetic zone boundary. The same qualitative phenomenology is reported for the two-dimensional Hubbard model using dynamical vertex approximation with $\mathrm{Bi_2Se_3}$9, including nodal metallic segments in hole-doped cuprates and the complementary momentum-sector reconstruction in electron-doped cuprates (Worm et al., 2023).

At the metallic border of a Mott transition, locking need not imply loss of Fermi-surface volume. In NiS$\mathbb Z_2$0, high-pressure quantum oscillatory measurements at $\mathbb Z_2$1 GPa reveal a dominant frequency of $\mathbb Z_2$2 kT and $\mathbb Z_2$3 kT in two runs, matching the calculated $\mathbb Z_2$4 kT belly orbit of a large “Cube”-like Fermi-surface sheet. The extracted effective mass is $\mathbb Z_2$5, compared with the band-structure value $\mathbb Z_2$6. The result supports a large Fermi surface consistent with Luttinger’s theorem and a strongly enhanced carrier effective mass, rather than a shrinking Fermi volume. In Brinkman–Rice language, localization is approached through suppression of quasiparticle weight $\mathbb Z_2$7 and concomitant mass divergence, $\mathbb Z_2$8, so that electrons become effectively locked in place as coherence collapses (Friedemann et al., 2015).

3. Commensurability, symmetry breaking, and nesting

Locking also appears in reconstructed metals where translational symmetry breaking and pre-existing anisotropy jointly determine the surviving Fermi-surface topology. In the smectic-phase model for YBCO, a weak, period-4, unidirectional charge density wave reconstructs the large hole-like cuprate Fermi surface into a small electron pocket only when the underlying band already has sufficiently strong $\mathbb Z_2$9-symmetry breaking. That symmetry breaking can come from orthorhombic lattice anisotropy or an electron nematic tendency. The stripe wavevector is

$$c_{p\uparrow}\rightarrow c^\dagger_{p\uparrow},\qquad n_{p\uparrow}\rightarrow 1-n_{p\uparrow},$$0

and the reconstruction is described as folding the large Fermi surface into the reduced Brillouin zone, hybridizing the crossings through $$c_{p\uparrow}\rightarrow c^\dagger_{p\uparrow},\qquad n_{p\uparrow}\rightarrow 1-n_{p\uparrow},$$1 and $$c_{p\uparrow}\rightarrow c^\dagger_{p\uparrow},\qquad n_{p\uparrow}\rightarrow 1-n_{p\uparrow},$$2, opening gaps, and reconnecting the remaining segments (Yao et al., 2011).

In this context, “locking” means that the reconstructed topology becomes commensurately tied to the period-4 stripe potential together with the $$c_{p\uparrow}\rightarrow c^\dagger_{p\uparrow},\qquad n_{p\uparrow}\rightarrow 1-n_{p\uparrow},$$3-breaking geometry of the original Fermi contour. The pocket appears when the maximum spanning vector satisfies

$$c_{p\uparrow}\rightarrow c^\dagger_{p\uparrow},\qquad n_{p\uparrow}\rightarrow 1-n_{p\uparrow},$$4

For representative small stripe amplitudes,

$$c_{p\uparrow}\rightarrow c^\dagger_{p\uparrow},\qquad n_{p\uparrow}\rightarrow 1-n_{p\uparrow},$$5

the pocket area is of order $$c_{p\uparrow}\rightarrow c^\dagger_{p\uparrow},\qquad n_{p\uparrow}\rightarrow 1-n_{p\uparrow},$$6 of the unreconstructed Brillouin zone. The model yields an effective mass of order $$c_{p\uparrow}\rightarrow c^\dagger_{p\uparrow},\qquad n_{p\uparrow}\rightarrow 1-n_{p\uparrow},$$7 if $$c_{p\uparrow}\rightarrow c^\dagger_{p\uparrow},\qquad n_{p\uparrow}\rightarrow 1-n_{p\uparrow},$$8 eV, but the authors also identify a limitation: the total specific heat from all reconstructed segments is too large compared with experiment (Yao et al., 2011).

A related but distinct mechanism occurs on topological-insulator edges and surfaces under Fermi-surface nesting. Because of spin-momentum locking and time-reversal symmetry, the spin susceptibility at the nesting wavevector acquires a helical structure. On a two-dimensional topological-insulator edge, the nesting vector is $$c_{p\uparrow}\rightarrow c^\dagger_{p\uparrow},\qquad n_{p\uparrow}\rightarrow 1-n_{p\uparrow},$$9, and only the helicity-changing susceptibilities $\mathbb Z_2$0 and $\mathbb Z_2$1 are nonzero. On a three-dimensional surface in the Bi$\mathbb Z_2$2Se$\mathbb Z_2$3 class, hexagonal warping generates three strong nesting vectors $\mathbb Z_2$4, $\mathbb Z_2$5, and $\mathbb Z_2$6, and the dominant susceptibility eigenmode is nearly helical. The resulting ordered state is a helical spin-density wave or, for magnetically doped surfaces, a helical magnetic order (Jiang et al., 2010).

4. Spin-momentum locking of the Fermi contour

In topological and chiral materials, locking of the Fermi surface often denotes a momentum-dependent spin texture imposed by symmetry and topology. The Landau theory of helical Fermi liquids provides a systematic formulation for the surface of a three-dimensional time-reversal-invariant topological insulator. Before projection, interactions between quasiparticles are described by ten independent Landau parameters per angular momentum channel. After projection onto the single helical Fermi surface, the theory becomes effectively spinless, with a single projected Landau parameter per channel that still encodes spin-momentum locking and the nontrivial Berry phase of the surface state (Lundgren et al., 2015).

The projected theory makes the locking explicit: $\mathbb Z_2$7 Spin is therefore not an independent low-energy variable. A notable consequence is that projection can increase or lower the angular momentum of the quasiparticle interactions; the $\mathbb Z_2$8-th projected channel receives contributions from unprojected channels $\mathbb Z_2$9, $$H_{\rm int}^{HK}=U\sum_{k}n_{k\uparrow}n_{k\downarrow},$$0, and $$H_{\rm int}^{HK}=U\sum_{k}n_{k\uparrow}n_{k\downarrow},$$1. The helical Fermi surface is thus “locked” both in the ordinary spin-texture sense and in the Berry-phase sense that reshuffles the angular content of effective interactions (Lundgren et al., 2015).

For multifold fermions, the topological-quantum-chemistry formulation replaces direct spin expectation values by a spin reduced density matrix,

$$H_{\rm int}^{HK}=U\sum_{k}n_{k\uparrow}n_{k\downarrow},$$2

obtained after tracing out orbital and site degrees of freedom. The winding number of $$H_{\rm int}^{HK}=U\sum_{k}n_{k\uparrow}n_{k\downarrow},$$3 over a constant-energy surface can take the integer values $$H_{\rm int}^{HK}=U\sum_{k}n_{k\uparrow}n_{k\downarrow},$$4, $$H_{\rm int}^{HK}=U\sum_{k}n_{k\uparrow}n_{k\downarrow},$$5, $$H_{\rm int}^{HK}=U\sum_{k}n_{k\uparrow}n_{k\downarrow},$$6, $$H_{\rm int}^{HK}=U\sum_{k}n_{k\uparrow}n_{k\downarrow},$$7, and $$H_{\rm int}^{HK}=U\sum_{k}n_{k\uparrow}n_{k\downarrow},$$8, depending on on-site spin-orbit coupling, crystal-field splitting, and Wyckoff multiplicity. This establishes that spin-momentum locking is not fixed solely by the topological charge of a degeneracy; it also depends on orbital content and basis structure (Lin et al., 2022).

The experimentally most direct realization in the supplied corpus is PtGa. Spin- and angle-resolved photoelectron spectroscopy shows that near the projection of the bulk multifold fermion, the electron spin of the Fermi-arc states is orthogonal to the Fermi-surface contour. Because the contour is tangential to the projected bulk pocket, this geometry is consistent with purely parallel spin-momentum locking of the underlying multifold fermion. The inner pocket is described as practically spherical and showing parallel spin-momentum locking along all momentum directions, whereas the outer pocket is nearly spherical with small deviations due to slight rectangular distortion (Krieger et al., 2022).

5. Interface engineering, Fermi-level pinning, and relaxation of locking

Interfaces can both create and destroy locked Fermi-surface textures. At the Ta/Bi$$H_{\rm int}^{HK}=U\sum_{k}n_{k\uparrow}n_{k\downarrow},$$9Se$U>W$0 boundary, an interface reaction forms a single Se-Ta-Se triple layer of H-type TaSe$U>W$1, but the layer does not retain ideal $U>W$2 symmetry. Surface x-ray diffraction shows that the central Ta atom is vertically displaced from $U>W$3 to about $U>W$4, corresponding to a downward shift of roughly $U>W$5 pm. This lowers the point-group symmetry from $U>W$6 to $U>W$7, which allows the appearance of an in-plane component of the spin polarization. The resulting states at the Fermi surface acquire a helical Rashba-like spin texture on the $U>W$8-centered hole pocket and the “dog-bone” pocket near $U>W$9, coupled to the Dirac cone of the Bi$q=0$0Se$q=0$1 substrate (Polyakov et al., 2020).

This case is an engineered form of locking: the momentum-space contour itself acquires a fixed spin winding because strong spin-orbit coupling in the Ta $q=0$2 states is combined with symmetry lowering and proximity to the topological surface state. The observed Ta-derived spin texture is reported to be opposite in chirality to that of the Bi$q=0$3Se$q=0$4 topological surface state. The mechanism therefore links structural relaxation, point-group symmetry reduction, and Fermi-surface chirality in a single heterostructure (Polyakov et al., 2020).

A different interface effect is Fermi-level pinning at metal contacts to Bi$q=0$5Se$q=0$6. Large-scale ab initio calculations for Au, Ni, Pt, Pd, and graphene show that regardless of the contact, the Fermi level is located in the conduction band of Bi$q=0$7Se$q=0$8, producing an n-type Ohmic contact to the first quintuple layer. The band-bending potential can shift the topological-insulator bands by as much as $q=0$9 eV near the interface, and the conduction-band minimum at the surface lies several hundred meV below $k=p$0. Au shifts the near-surface Dirac cone downward by about $k=p$1 eV. There is no Schottky barrier for electron injection in the first and second quintuple layers, although graphene shows interface tunneling barriers because of its larger separation and weaker coupling (Spataru et al., 2014).

This electrostatic pinning does not preserve locking uniformly across contact materials. Au and graphene leave the spin-momentum locking mostly unaltered, whereas Ni, Pd, and Pt strongly hybridize with Bi$k=p$2Se$k=p$3 and relax the helical surface-state texture. The material dependence is traced to interface separation, density of states near $k=p$4, and the large electronegativity of the Se-terminated surface. Locking at the Fermi surface is therefore not only a bulk or surface-band property; it can be relaxed or obscured by charge-transfer physics at the contact (Spataru et al., 2014).

6. Response functions, strong-locking theorems, and spectroscopic tests

In noncentrosymmetric superconductors, strong locking yields a particularly sharp geometric statement. In the strong-locking regime, the residual $k=p$5 Knight shift is completely determined by the Fermi-surface projector

$k=p$6

through

$k=p$7

Because $k=p$8, the three principal Knight shifts at $k=p$9 lie on a two-dimensional simplex, termed the Knight-shift ellipsoid. The stated result is independent of pairing symmetry, gap magnitude, and Fermi-surface shape; it depends only on the Fermi-surface average of the local locking direction $\Omega := \{ \det G = 0\},$0 (Zhou, 16 May 2026).

The same work derives a spin Ferrell–Glover–Tinkham sum rule and a vanishing-projection theorem for $\Omega := \{ \det G = 0\},$1. Applied to $\Omega := \{ \det G = 0\},$2As NMR data on $\Omega := \{ \det G = 0\},$3, the observed ellipsoid sits at the oblate-axial vertex $\Omega := \{ \det G = 0\},$4 and saturates the trace bound. The decoupled-pocket spin-orbit-coupling baseline is excluded by $\Omega := \{ \det G = 0\},$5 in normalized units, requiring a common $\Omega := \{ \det G = 0\},$6-axis locking on all three pockets. The suppression of $\Omega := \{ \det G = 0\},$7 is identified as a fingerprint of finite-$\Omega := \{ \det G = 0\},$8 ferromagnetic spin-fluctuation gap formation (Zhou, 16 May 2026).

In the Kondo problem with arbitrary spin-momentum locking on a spherical Fermi surface, the locking texture affects the impurity scale only through the Fermi-surface-averaged spin

$\Omega := \{ \det G = 0\},$9

The Kondo temperature is

$\mathrm{Bi_2Se_3}$00

It is unchanged for $\mathrm{Bi_2Se_3}$01, decreases as $\mathrm{Bi_2Se_3}$02 increases, and vanishes for $\mathrm{Bi_2Se_3}$03. The detailed angular structure of the locking texture thus matters only through its Fermi-surface average spin polarization (Goto et al., 5 Jun 2025).

Quantum oscillations provide an orthogonal diagnostic by determining whether the relevant low-energy manifold is single-branched or topologically split. In p-type Ca-doped $\mathrm{Bi_2Se_3}$04, the angular dependence of the Shubnikov–de Haas and de Haas–van Alphen frequencies shows a downturn as $\mathrm{Bi_2Se_3}$05 approaches $\mathrm{Bi_2Se_3}$06, indicating a bag-shaped three-dimensional closed hole Fermi surface rather than a simple ellipsoid. A single dominant frequency is observed for all tilt angles, which rules out multiple extremal cross-sections and, in particular, rules out a camel-back valence band down to $\mathrm{Bi_2Se_3}$07 meV. The absence of frequency splitting is therefore a decisive negative test: it excludes a donut-like or dumbbell-like topology that would otherwise mimic a more complicated form of locking (Piot et al., 2016).

These results collectively show that locking of the Fermi surface is not a single mechanism but a family of constraints on low-energy momentum-space structure. Depending on context, the locked object may be a zero surface, a commensurately reconstructed pocket, a helical or parallel spin texture, a contact-pinned near-interface contour, or a superconducting response ellipsoid. The unifying theme is that the Fermi surface, or its correlated replacement, is no longer characterized only by volume and curvature; it is constrained by symmetry, topology, and many-body structure in ways that are directly measurable in transport, quantum oscillations, ARPES, spin-ARPES, NMR, and impurity response.