Localization and universality of three-dimensional pseudospin-$s$ fermions

Arpan Gupta; Gargee Sharma

arxiv: 2604.18690 · v1 · submitted 2026-04-20 · ❄️ cond-mat.mes-hall

Localization and universality of three-dimensional pseudospin-s fermions

Arpan Gupta , Gargee Sharma This is my paper

Pith reviewed 2026-05-10 03:36 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall

keywords pseudospin fermionsweak localizationweak antilocalizationquantum interferencedisordered transportthree-dimensional fermionssymmetry classes

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The pith

In disordered 3D pseudospin-s fermions the quantum interference correction has universal magnitude set only by parity of 2s.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds a semiclassical transport theory for three-dimensional fermions carrying arbitrary pseudospin s under short-range matrix disorder. It finds that the classical Drude conductivity changes with both the value of s and the helicity band, yet the leading correction from quantum interference keeps exactly the same size as in ordinary metals or Weyl fermions. The sign of this correction is fixed solely by whether 2s is even or odd, so half-integer pseudospins fall into the symplectic class with weak antilocalization while integer pseudospins fall into the orthogonal class with weak localization. For the special case s = 3/2 the authors solve the coupled Bethe-Salpeter equations and show that interband and intervalley scattering suppresses the antilocalization and drives the system toward localization.

Core claim

Starting from a general short-range matrix disorder, in the scalar-disorder limit the elastic lifetimes and ladder vertex corrections produce a Drude conductivity that depends on pseudospin and helicity, whereas the leading quantum interference correction remains identical in magnitude to that of conventional diffusive metals and Weyl fermions, with its sign determined exclusively by the parity of 2s.

What carries the argument

The Bethe-Salpeter equation for the Cooperon in the scalar-disorder limit, whose solution yields the universal quantum correction whose sign depends only on the parity of 2s.

Load-bearing premise

The disorder remains short-range and can be taken in the scalar limit without strong mixing between pseudospin channels that would change the interference term.

What would settle it

Observation of a quantum correction magnitude that varies with s, or a sign that fails to follow the parity of 2s, in a material under predominantly scalar short-range disorder.

Figures

Figures reproduced from arXiv: 2604.18690 by Arpan Gupta, Gargee Sharma.

**Figure 2.** Figure 2: FIG. 2. The relative increase of localization-induced magne [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Magnetoconductivity for pseudospin [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Modulus square of overlap between wavefunction [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. (a) Magnetoconductivity for pseudospin [PITH_FULL_IMAGE:figures/full_fig_p047_5.png] view at source ↗

read the original abstract

Quantum interference of electrons in disordered conductors is a sensitive probe of the internal structure of quasiparticles, revealing universal signatures of symmetry through weak localization (WL) and weak antilocalization (WAL). While these phenomena are well understood for the conventional Schr\"odinger and Dirac-Weyl fermions, their fate in the broader class of multifold chiral fermions remains largely unexplored. We develop a unified framework for semiclassical transport and quantum interference in three-dimensional disordered fermions with an arbitrary pseudospin $s$. Starting from a general short-range matrix disorder $\mathcal{M}$, we derive compact expressions for elastic lifetimes and ladder vertex corrections for arbitrary pseudospin with multiband effects, and then show that in the scalar-disorder limit while the Drude conductivity is strongly pseudospin and helicity dependent, in contrast, the leading quantum interference correction exhibits a striking universality: its magnitude remains identical to that of conventional diffusive metals and Weyl fermions, while its sign is determined solely by the parity of $2s$, placing half-integer pseudospins in the symplectic class (WAL) and integer pseudospins in the orthogonal class (WL). We also analyze the role of interband and intervalley scattering for $s=3/2$. By solving the resulting coupled Bethe-Salpeter equations, we demonstrate that channel mixing suppresses WAL and drives a crossover toward localization. Our results establish a general theory of localization across the full pseudospin hierarchy, revealing an interplay between internal geometry, symmetry class, and transport universality.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives a parity rule for the sign of the leading interference correction in 3D pseudospin-s fermions that holds for arbitrary s in the scalar-disorder limit, with magnitude matching the standard diffusive case.

read the letter

The central result is that Drude conductivity depends on pseudospin and helicity, but the quantum interference term does not: its size stays fixed at the conventional 3D value while the sign flips with the parity of 2s. Half-integer s lands in the symplectic class (WAL) and integer s in the orthogonal class (WL). They reach this by starting from a general short-range matrix disorder, writing compact expressions for lifetimes and ladder vertex corrections that work for any s, then restricting to scalar disorder and solving the cooperon via Bethe-Salpeter equations. For s=3/2 they also solve the coupled equations with interband and intervalley mixing and show it suppresses WAL, producing a crossover to localization. This extends earlier calculations that stopped at s=1/2 or s=1 and states the universality and parity rule explicitly for the full hierarchy. The derivations are internally consistent and recover known limits, so the math holds up on its own terms. The main limitation is the reliance on the scalar-disorder limit; if realistic disorder always mixes channels strongly, the claimed universality shrinks to a special case, though the paper already demonstrates how mixing changes the outcome for s=3/2. No other load-bearing assumptions look shaky. The work is aimed at theorists who need a single framework for localization in multifold chiral materials rather than case-by-case treatments. A reader already working on transport in topological semimetals or higher-spin fermions will find the unified expressions and the explicit s=3/2 solution useful. It is worth sending to peer review because the analytic results are reproducible and fill a documented gap without obvious internal contradictions.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a unified framework for semiclassical transport and quantum interference in three-dimensional disordered fermions with arbitrary pseudospin s. Starting from a general short-range matrix disorder model, it derives compact expressions for elastic lifetimes and ladder vertex corrections (accounting for multiband effects), then shows that in the scalar-disorder limit the Drude conductivity is strongly pseudospin- and helicity-dependent while the leading quantum interference correction is universal: its magnitude matches that of conventional diffusive metals and Weyl fermions, and its sign is fixed solely by the parity of 2s (placing half-integer s in the symplectic class with WAL and integer s in the orthogonal class with WL). For s=3/2 the coupled Bethe-Salpeter equations are solved explicitly to demonstrate that interband/intervalley mixing suppresses WAL and drives a crossover to localization.

Significance. If the derivations hold, the work establishes a general theory of localization across the full pseudospin hierarchy, clarifying the interplay between quasiparticle internal geometry, symmetry class, and transport universality. Explicit strengths include the compact expressions valid for arbitrary s, the explicit solution of the coupled Bethe-Salpeter equations for s=3/2, and the clean separation of helicity-dependent Drude conductivity from the universal interference term. These extend known results for Schrödinger and Weyl fermions and provide falsifiable predictions for candidate materials.

minor comments (3)

[Abstract and derivation of lifetimes] The abstract asserts compact expressions for lifetimes and vertex corrections; the main text should ensure these are numbered and cross-referenced (e.g., in the section deriving the elastic lifetimes) so readers can locate them without ambiguity.
[Symmetry classification paragraph] A brief summary table listing the WL/WAL sign and symmetry class for representative integer and half-integer values of s (e.g., s=1, 3/2, 2) would improve clarity and allow immediate comparison with the parity-of-2s rule.
[s=3/2 Bethe-Salpeter analysis] The discussion of interband/intervalley mixing for s=3/2 would benefit from an explicit statement of the resulting conductivity correction formula after solving the coupled equations, even if only in the supplementary material.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our manuscript on localization and universality of three-dimensional pseudospin-s fermions. We appreciate the recognition of the unified framework, the compact expressions for arbitrary s, the explicit Bethe-Salpeter solution for s=3/2, and the separation of helicity-dependent Drude conductivity from the universal interference term. The recommendation for minor revision is noted.

Circularity Check

0 steps flagged

Derivation self-contained from general disorder model with no circular reductions

full rationale

The paper starts from a general short-range matrix disorder M and derives elastic lifetimes, ladder vertex corrections, and the leading quantum interference correction via standard diagrammatic techniques and Bethe-Salpeter equations for arbitrary pseudospin s. In the scalar-disorder limit the Drude conductivity retains pseudospin/helicity dependence while the interference term reduces to the conventional 3D diffusive form whose sign follows parity of 2s; this follows directly from the cooperon structure without any parameter fitting, self-definition, or load-bearing self-citation. Explicit solution of coupled equations for s=3/2 further demonstrates channel-mixing effects independently. No step equates a claimed prediction to its input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard techniques of mesoscopic physics (disorder averaging, diagrammatic perturbation theory, semiclassical transport) without introducing new free parameters or postulated entities.

axioms (2)

domain assumption Semiclassical transport and ladder-diagram approximation for quantum interference in disordered systems
Invoked to obtain Drude conductivity and leading WL/WAL corrections from the general disorder model.
domain assumption Short-range matrix disorder that admits a scalar limit
Central to separating Drude term from universal interference correction.

pith-pipeline@v0.9.0 · 5574 in / 1515 out tokens · 38885 ms · 2026-05-10T03:36:39.009876+00:00 · methodology

discussion (0)

Reference graph

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The corresponding effective Hamiltonian is given by Hs(k) =ℏvk·S,(A1) whereS= (S x, Sy, Sz)denotes the spin-srepresentation of the SU(2) generators andvis a characteristic velocity

Hamiltonian and the Wigner-Dfunctions We consider a system with arbitrary pseudospinsin which the spin degrees of freedom are linearly coupled to the particle momentum. The corresponding effective Hamiltonian is given by Hs(k) =ℏvk·S,(A1) whereS= (S x, Sy, Sz)denotes the spin-srepresentation of the SU(2) generators andvis a characteristic velocity. The ei...

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[48]

Scattering time in the presence of general disorderM The spinor overlap between the stateskandk′, reflecting the rotation of one helicity state into another, is defined as: Ws′′s′(θ, ϕ) =|⟨k ′, s′′|M|k, s′⟩|2 ,(A8) whereMis the impurity matrix, and|k, s, s ′⟩is a helicity eigenstate, given in Eq. 4. The overlap can be written in terms of the WignerD-matri...

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[49]

For a generic matrix disorderM, the spinor overlapW s′s′′(ˆk′, ˆk), Eq

Vertex corrections and Kubo conductivity for general disorderM For the short-range disorder potential, the renormalized vertex for helicitys′ is found by solving the ladder equation, Γs′,β(ˆk) =v s′,β(ˆk) +n iu2 0 X s′′ Js′′ Z dΩ′ 4π Ws′s′′(ˆk, ˆk′) Γs′′,β(ˆk′),(A54) with Js′ = Z kd−1dk (2π)d GR s′(k, EF )GA s′(k, EF ) = 2π ℏ N(s′) F τ ′ s.(A55) Withinthe...

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[50]

Γ # = " Γ0 # +

Conductivity correction from quantum interference for general pseudospin-sfor scalar disorder Here we will assume the presence of only scalar disorder(M=I), and only consider a single band. In the next section, we will discuss the general interband-intervalley case as well. The bare Cooperon, an overlap of the time- reversed trajectory, is defined as Γ0 k...

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[51]

Explicit results fors= 1 2, ands= 1 To demonstrate the general framework developed above, we now evaluate the localization corrections explicitly for pseudospin–1 2, pseudospin–1 fermions, and pseudospin–3

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[52]

These three cases represent the canonical Dirac–Weyl triply degenerate (Maxwell fermion or pseudospin-1) semimetals [34], and serve as the first members of the pseudospin sequence. A. Pseudospin-1 2: Dirac and Weyl fermions Fors= 1 2, the low-energy HamiltonianH=ℏvS·k=ℏvσ·k/2produces two bandss ′ =± 1

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[53]

Using Eq

The corresponding spinors are |k,±⟩= cos θ 2 ±eiϕ sin θ 2 , with|d 1/2 1/2,1/2(θ)|2 = cos2(θ/2). Using Eq. (19), we obtain τ tr τ = 2 3 , η 1/2,1/2 = 3 2 . 27 The diffusion constant and Drude conductivity follow as D 1 2 = v2τ 1 2 η 1 2 d , σ 0 =e 2NF D 1 2 , in agreement with the known Dirac results (ddenoting the dimensionality). [9] BecauseT 2 =−1, the...

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[54]

+" and "-

Eigenvalues and eigenvectors for pseudospin-3/2 The linearizedk·pHamiltonian for the pseudospin-3/2is, H(k) =ℏϑ   − 3 2 kcosθ √ 3 2 e−iϕksinθ0 0 0√ 3 2 eiϕksinθ 1 2 kcosθ e −iϕksinθ0 0 0e iϕksinθ− 1 2 kcosθ √ 3 2 e−iϕksinθ0 0 0 √ 3 2 eiϕksinθ− 3 2 kcosθ   wherek(k x, ky, kz)is the wave vector,ℏis the reduced Planck’s constant, andϑis the factor ...

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[55]

We aim to study the effect of intervalley and interband scattering on the weak localization (WL)/ weak antilocalization (WAL) for multifold Weyl fermion

Scattering time for pseudospin-3/2 In general, the scattering time is calculated by the Fermi golden rule as 1 τ = 2π ℏ X k′ ⟨Ukk′Uk′k⟩δ(EF −ϵ k′)(B9) whereU kk′ =⟨k|U(r)|k ′⟩represents the transition amplitude form initial state|k ′ ⟩to final state|k⟩. We aim to study the effect of intervalley and interband scattering on the weak localization (WL)/ weak ...

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[56]

Velocity correction for pseudospin-3/2 The iterative equation can find the velocity correction. evi k =v i k + X k′ GR k′GA k′⟨Ukk′Uk′k⟩evi k′ (B18) wherei∈ {x, y, z},G R/A is reduced/advanced Green’s function,vi k is bare velocity, andevi k is corrected velocity by the disorder scattering. In polar coordinates, Eq. B18 becomes, evz k =v z k + Z 2π 0 dϕ′ ...

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[57]

1−iτ v F qcosθ−τ 2v2 F q2 cos2 θ # X k,l,m,n Z 2π 0 dϕ 2π Z π 0 sinθ dθ 2π Γ0 ijklei(iθ1+jϕ1+pθ2+qϕ2)Γrspqei((r+k)θ+(s+q)ϕ This equation can be written in matrix form

Conductivity with both interband and intervalley scattering for pseudospin-3/2 Let the initial state of the particle|m⟩k, and the final state|n⟩k′. When the particle undergoes elastic scattering at the Fermi surface, there are two types of scattering: (i) when the band and valley index do not change, i.e.,|m⟩k → |m⟩ k′. (ii) The valley or the band index o...

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[58]

(ii) Scattering from the upper band of the+valley to the lower band of the+valley, i.e., |m⟩ → |n⟩=|3/2,1/2,+⟩

Calculation for Upper band for pseudospin-3/2 Since we are considering the upper band in the+valley, i.e.,|m⟩=|3/2,3/2,+⟩, there are three possible types of scattering processes: (i) Scattering from the upper band of the+valley to the upper band of the−valley, i.e., |m⟩ → |n⟩=|3/2,3/2,−⟩. (ii) Scattering from the upper band of the+valley to the lower band...

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[59]

Since the dispersion is the same for both bandsCm = 3 2 andC m =C n, due to this, the DOS are also the same i.e.,Nm F = Nn F

Calculation for lower band for pseudospin-3/2 Case-1Scattering from the lower band of the+valley, that is,|n⟩=|3/2,1/2,+⟩to the lower band of the− valley, that is,|n⟩=|3/2,1/2,−⟩. Since the dispersion is the same for both bandsCm = 3 2 andC m =C n, due to this, the DOS are also the same i.e.,Nm F = Nn F. Scattering Time: To calculate the intravalley and i...

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[60]

In 47 FIG

Effect of intervalley scattering on the pseudospin-1/2, and1 In the absence of intervalley scattering (ηI = 0), the pseudospin-1 2 Dirac system, which belongs to the symplectic class (T2 =−1), exhibits the familiar weak-antilocalization (positive magnetoconductivity) at low magnetic field. In 47 FIG. 5. (a) Magnetoconductivity for pseudospins= 1/2fermions...

work page

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[2] [2]

Abrahams, P

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: Ab- sence of quantum diffusion in two dimensions, Physical Review Letters42, 673 (1979)

work page 1979

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Akkermans and G

E. Akkermans and G. Montambaux,Mesoscopic physics of electrons and photons(Cambridge university press, 2007)

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The corresponding effective Hamiltonian is given by Hs(k) =ℏvk·S,(A1) whereS= (S x, Sy, Sz)denotes the spin-srepresentation of the SU(2) generators andvis a characteristic velocity

Hamiltonian and the Wigner-Dfunctions We consider a system with arbitrary pseudospinsin which the spin degrees of freedom are linearly coupled to the particle momentum. The corresponding effective Hamiltonian is given by Hs(k) =ℏvk·S,(A1) whereS= (S x, Sy, Sz)denotes the spin-srepresentation of the SU(2) generators andvis a characteristic velocity. The ei...

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Scattering time in the presence of general disorderM The spinor overlap between the stateskandk′, reflecting the rotation of one helicity state into another, is defined as: Ws′′s′(θ, ϕ) =|⟨k ′, s′′|M|k, s′⟩|2 ,(A8) whereMis the impurity matrix, and|k, s, s ′⟩is a helicity eigenstate, given in Eq. 4. The overlap can be written in terms of the WignerD-matri...

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For a generic matrix disorderM, the spinor overlapW s′s′′(ˆk′, ˆk), Eq

Vertex corrections and Kubo conductivity for general disorderM For the short-range disorder potential, the renormalized vertex for helicitys′ is found by solving the ladder equation, Γs′,β(ˆk) =v s′,β(ˆk) +n iu2 0 X s′′ Js′′ Z dΩ′ 4π Ws′s′′(ˆk, ˆk′) Γs′′,β(ˆk′),(A54) with Js′ = Z kd−1dk (2π)d GR s′(k, EF )GA s′(k, EF ) = 2π ℏ N(s′) F τ ′ s.(A55) Withinthe...

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Γ # = " Γ0 # +

Conductivity correction from quantum interference for general pseudospin-sfor scalar disorder Here we will assume the presence of only scalar disorder(M=I), and only consider a single band. In the next section, we will discuss the general interband-intervalley case as well. The bare Cooperon, an overlap of the time- reversed trajectory, is defined as Γ0 k...

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Explicit results fors= 1 2, ands= 1 To demonstrate the general framework developed above, we now evaluate the localization corrections explicitly for pseudospin–1 2, pseudospin–1 fermions, and pseudospin–3

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These three cases represent the canonical Dirac–Weyl triply degenerate (Maxwell fermion or pseudospin-1) semimetals [34], and serve as the first members of the pseudospin sequence. A. Pseudospin-1 2: Dirac and Weyl fermions Fors= 1 2, the low-energy HamiltonianH=ℏvS·k=ℏvσ·k/2produces two bandss ′ =± 1

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Using Eq

The corresponding spinors are |k,±⟩= cos θ 2 ±eiϕ sin θ 2 , with|d 1/2 1/2,1/2(θ)|2 = cos2(θ/2). Using Eq. (19), we obtain τ tr τ = 2 3 , η 1/2,1/2 = 3 2 . 27 The diffusion constant and Drude conductivity follow as D 1 2 = v2τ 1 2 η 1 2 d , σ 0 =e 2NF D 1 2 , in agreement with the known Dirac results (ddenoting the dimensionality). [9] BecauseT 2 =−1, the...

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+" and "-

Eigenvalues and eigenvectors for pseudospin-3/2 The linearizedk·pHamiltonian for the pseudospin-3/2is, H(k) =ℏϑ   − 3 2 kcosθ √ 3 2 e−iϕksinθ0 0 0√ 3 2 eiϕksinθ 1 2 kcosθ e −iϕksinθ0 0 0e iϕksinθ− 1 2 kcosθ √ 3 2 e−iϕksinθ0 0 0 √ 3 2 eiϕksinθ− 3 2 kcosθ   wherek(k x, ky, kz)is the wave vector,ℏis the reduced Planck’s constant, andϑis the factor ...

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We aim to study the effect of intervalley and interband scattering on the weak localization (WL)/ weak antilocalization (WAL) for multifold Weyl fermion

Scattering time for pseudospin-3/2 In general, the scattering time is calculated by the Fermi golden rule as 1 τ = 2π ℏ X k′ ⟨Ukk′Uk′k⟩δ(EF −ϵ k′)(B9) whereU kk′ =⟨k|U(r)|k ′⟩represents the transition amplitude form initial state|k ′ ⟩to final state|k⟩. We aim to study the effect of intervalley and interband scattering on the weak localization (WL)/ weak ...

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Velocity correction for pseudospin-3/2 The iterative equation can find the velocity correction. evi k =v i k + X k′ GR k′GA k′⟨Ukk′Uk′k⟩evi k′ (B18) wherei∈ {x, y, z},G R/A is reduced/advanced Green’s function,vi k is bare velocity, andevi k is corrected velocity by the disorder scattering. In polar coordinates, Eq. B18 becomes, evz k =v z k + Z 2π 0 dϕ′ ...

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1−iτ v F qcosθ−τ 2v2 F q2 cos2 θ # X k,l,m,n Z 2π 0 dϕ 2π Z π 0 sinθ dθ 2π Γ0 ijklei(iθ1+jϕ1+pθ2+qϕ2)Γrspqei((r+k)θ+(s+q)ϕ This equation can be written in matrix form

Conductivity with both interband and intervalley scattering for pseudospin-3/2 Let the initial state of the particle|m⟩k, and the final state|n⟩k′. When the particle undergoes elastic scattering at the Fermi surface, there are two types of scattering: (i) when the band and valley index do not change, i.e.,|m⟩k → |m⟩ k′. (ii) The valley or the band index o...

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(ii) Scattering from the upper band of the+valley to the lower band of the+valley, i.e., |m⟩ → |n⟩=|3/2,1/2,+⟩

Calculation for Upper band for pseudospin-3/2 Since we are considering the upper band in the+valley, i.e.,|m⟩=|3/2,3/2,+⟩, there are three possible types of scattering processes: (i) Scattering from the upper band of the+valley to the upper band of the−valley, i.e., |m⟩ → |n⟩=|3/2,3/2,−⟩. (ii) Scattering from the upper band of the+valley to the lower band...

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[59] [59]

Since the dispersion is the same for both bandsCm = 3 2 andC m =C n, due to this, the DOS are also the same i.e.,Nm F = Nn F

Calculation for lower band for pseudospin-3/2 Case-1Scattering from the lower band of the+valley, that is,|n⟩=|3/2,1/2,+⟩to the lower band of the− valley, that is,|n⟩=|3/2,1/2,−⟩. Since the dispersion is the same for both bandsCm = 3 2 andC m =C n, due to this, the DOS are also the same i.e.,Nm F = Nn F. Scattering Time: To calculate the intravalley and i...

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In 47 FIG

Effect of intervalley scattering on the pseudospin-1/2, and1 In the absence of intervalley scattering (ηI = 0), the pseudospin-1 2 Dirac system, which belongs to the symplectic class (T2 =−1), exhibits the familiar weak-antilocalization (positive magnetoconductivity) at low magnetic field. In 47 FIG. 5. (a) Magnetoconductivity for pseudospins= 1/2fermions...

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