Non-Hermitian Quantum Mechanics & Topology of Finite-Lifetime - PowerPoint PPT Presentation

Non-Hermitian Quantum Mechanics & Topology of Finite-Lifetime Quasiparticles Liang Fu Kyoto workshop, 11/09/2017

Quantum Theory of Solids: Two Foundations Band Theory Quasiparticle • • Bloch waves Landau quasiparticles • • Energy bands in k-space mass & lifetime particle-wave duality

Modern Developments Band Theory Topological Band Theory • • Top. crystalline insulator global property of Bloch • Top. Kondo insulator eigenstates 𝜔 𝑙 in k-space • Weyl/Dirac semimetals Thouless et al (1982), Haldane (1988)… diverse phenomena, unified framework

Bewildering Behaviors of Correlated Electron Systems • Fermi arc: topological integrity of Fermi surface violated

Bewildering Behaviors of Correlated Electron Systems Sebastian et al, Science (2015); Li et al, Science (2014) • SmB 6 & YbB 12 : quantum oscillation in heavy fermion insulators

Bulk Fermi arc & Insulator’s Fermi surface • shaking the fundamental of solid-state theory? • special or generic phenomena? • theoretical framework?

Quasiparticles vs. Noninteracting Electrons Fundamental distinction: • Quasiparticles have finite lifetime resulting from e-e and e-phonon interaction at 𝑈 ≠ 0 , and impurity scattering at all 𝑈 . • Non-interacting electrons last forever. ImΣ 𝑙 𝐺 , 𝜕 = 0 ∝ 𝑈 2 < 𝑙 𝐶 𝑈 Fermi liquid: Electron-phonon, quantum ImΣ 𝜕~0 ~ 𝑙 𝐶 𝑈 critical & chaotic systems:

Damping + Dispersion 𝐻 𝑆 𝒍, 𝜕 = −1 𝜕 − 𝐼 𝒍, 𝜕 Green’s function: 𝐼 𝒍, 𝜕 ≡ 𝐼 0 𝒍 + Σ(𝒍, 𝜕) Quasiparticle Hamiltonian: (non-Hermitian) Bloch Hamiltonian self-energy Finite lifetime means Σ is non-Hermitian: Σ = Σ ′ + 𝑗Σ′′ Complex spectrum of 𝐼 𝒍, 𝜕~0 determines quasiparticle properties Re( 𝐹 𝑙 ) : quasiparticle dispersion Im( 𝐹 𝑙 ) : inverse lifetime single-band system: 𝐹 𝑙 = 𝜗 k − 𝑗𝛿 k

Damping Reshapes Dispersion 𝐻 𝑆 𝒍, 𝜕 = −1 𝜕 − 𝐼 𝒍, 𝜕 Green’s function: 𝐼 𝒍, 𝜕 ≡ 𝐼 0 𝒍 + Σ(𝒍, 𝜕) Quasiparticle Hamiltonian: (non-Hermitian) Bloch Hamiltonian self-energy Finite lifetime means Σ is non-Hermitian: Σ = Σ ′ + 𝑗Σ′′ Complex spectrum of 𝐼 𝒍, 𝜕~0 determines quasiparticle properties Re( 𝐹 𝑙 ) : quasiparticle dispersion Im( 𝐹 𝑙 ) : inverse lifetime multi-orbital systems: 𝐼 0 & Σ are matrices and generally do not commute • imaginary part of self-energy resulting from qp decay can have dramatic feedback effect on qp dispersion in zero/small-gap systems. Kozii & LF, arXiv:1708.05841

Damping Reshapes Dispersion 𝐻 𝑆 𝒍, 𝜕 = −1 𝜕 − 𝐼 𝒍, 𝜕 Green’s function: 𝐼 𝒍, 𝜕 ≡ 𝐼 0 𝒍 + Σ(𝒍, 𝜕) Quasiparticle Hamiltonian: (non-Hermitian) Bloch Hamiltonian self-energy Finite lifetime means Σ is non-Hermitian: Σ = Σ ′ + 𝑗Σ′′ Complex spectrum of 𝐼 𝒍, 𝜕~0 determines quasiparticle properties Re( 𝐹 𝑙 ) : quasiparticle dispersion yin and yang of quasiparticles Im( 𝐹 𝑙 ) : inverse lifetime multi-orbital systems: 𝐼 0 & Σ are matrices and generally do not commute • imaginary part of self-energy Σ′′ resulting from qp decay can have dramatic feedback effect on qp dispersion in zero/small-gap systems. The whole is more than the sum of its parts !

Quasiparticles in Zero/Small Gap Systems Bloch Hamiltonian + self-energy with two lifetimes near hybridization ∼ nodes light band − heavy band + Two orbitals unrelated by symmetry generally have different lifetimes. Example: d- and f-orbitals in heavy fermion systems. Kozii & LF, arXiv:1708.05841

Microscopic Origins of Two Lifetimes Electron-phonon interaction: Kozii & LF, 1708.05841 • two orbitals with different e-ph coupling constants 𝜇 1,2 . Electron-electron interaction: • periodic Anderson model Yang Qi, Kozii & LF, to appear + 𝑒 𝑙 + 𝜗 𝑔 𝑙 𝑔 + 𝑔 + 𝑔 𝐼 = ෍ 𝜗 𝑒 (𝑙)𝑒 𝑙 𝑙 + 𝑊 𝑙 𝑒 𝑙 𝑙 + ℎ. 𝑑. + ෍ 𝑉𝑜 𝑔↑,𝑗 𝑜 𝑔↓,𝑗 𝑙 𝑙 𝑗 interaction on f-orbital only

Asymmetric Damping Reshapes Dispersion Quasiparticle Hamiltonian Quasiparticle dispersion Re( E k ) : Γ 1 ≠ Γ 2 • Γ 1 − Γ 2 ≡ 2𝛿 ≠ 0 ∶ two bands stick together to form a bulk Fermi arc, terminating at 𝑙 𝑦 = 0, 𝑙 𝑧 = ±𝛿/𝑤 𝑧 • new fermiology: constant dos, strong anisotropy

Asymmetric Damping Reshapes Dispersion Quasiparticle Hamiltonian Quasiparticle dispersion Re( E k ) : Prediction: bulk Fermi arc in heavy fermion systems Γ 1 ≠ Γ 2 • Γ 1 − Γ 2 ≡ 2𝛿 ≠ 0 ∶ two bands stick together to form a bulk Fermi arc, terminating at 𝑙 𝑦 = 0, 𝑙 𝑧 = ±𝛿/𝑤 𝑧 • new fermiology: constant dos, strong anisotropy

Interplay of Damping and Coherence Non-Hermitian quasiparticle Hamiltonian: 𝛿 ≡ (Γ 1 −Γ 2 )/2, Γ ≡ (Γ 1 + Γ 2 )/2. Complex-energy spectrum: 𝜗 1𝒍 − 𝜗 2𝒍 − 𝑗𝛿 2 + |Δ 𝒍 2 | − 𝑗Γ 𝐹 ± 𝒍 = 𝜗 𝒍 ± without hybridization, Fermi surface at band crossing 𝜗 1𝒍 = 𝜗 2𝒍 Re( E ) with hybridization and asymmetric damping, • for |Δ 𝒍 | > 𝛿 : Re( E + ) ≠ - Re( E - ) (gap opens) k y • for |Δ 𝒍 | < 𝛿 : Re( E + ) = Re( E - ) = 0 (gap closes)

Spectral Function 1 1 𝐵 𝒍, 𝜕 = −Im Tr𝐻 𝑆 𝒍, 𝜕 = −Im( 𝜕−𝐹 + 𝒍 + 𝜕−𝐹 − 𝒍 ) constant-energy contour linecuts 𝜕 < 0 𝜕 = 0 𝜕 > 0 𝜕 k y k y k x k x • Dirac point spreads into an arc • Asymmetry due to two lifetimes

Topological Stability of Bulk Fermi Arc 𝐼(𝒍) = (𝑤 𝑦 𝑙 𝑦 − 𝑗 𝛿)𝜏 𝑨 + 𝑤 𝑧 𝑙 𝑧 𝜏 𝑦 → 𝛿(−𝑗 𝜏 𝑨 ± 𝜏 𝑦 ) at ±𝒍 𝟏 ≡ (0, ±𝛿/𝑤 𝑧 ) Re( E ) 𝑤 𝑦 𝑙 𝑦 − 𝑗𝛿 2 + 𝑤 𝑧 2 𝑙 𝑧 2 𝐹 ± 𝑙 = ± k y • at two ends of Fermi arc 𝒍 = ±𝒍 𝟏 , matrix H is non-diagonalizable and has only one eigenstate! • eigenvalue coalescence is unique to non-Hermitian operators.

Exceptional Points 2004 2015 1966 1971

Exceptional Points 2004 2015 1966 • In open systems, non-Hermiticity results from coupling with external bath. • In interacting many-body systems, microscopic Hamiltonian is Hermitian, while one-body quasiparticle Hamiltonian is non-Hermitian due to damping. 1971 Kozii & LF, arXiv:1708.05841

Topology of Finite-Lifetime Quasiparticles k-space: Non-Hermitian Hamiltonian: 𝐼 𝒍, 𝜕 ≡ 𝐼 0 𝒍 + Σ(𝒍, 𝜕) Topology of non-Hermitian quasiparticle Hamiltonian: the generalization of topological band theory to interacting electron systems. *quasiparticles can be electron, magnon, exciton... Shen, Zhen & LF, arXiv:1706.07435

arXiv:1706.07435 generic Hamiltonian 𝐼(𝑙 𝑦 , 𝑙 𝑧 ) near exceptional point (EP): 𝐼 = 𝜗𝜏 + + 𝜇 𝑗𝑘 𝑙 𝑗 𝜏 𝐹 ± 𝑙 = ± 𝜗(𝑤 𝑦 𝑙 𝑦 + 𝑤 𝑧 𝑙 𝑧 ) 𝑘 ( 𝑤 𝑦 , 𝑤 𝑧 , 𝑤 𝑧 /𝑤 𝑦 are complex) due to double-valuedness of square root, encircling an EP in k-space swaps the pair of complex eigenvalues: 𝐹 + → 𝐹 − , 𝐹 − → 𝐹 +

arXiv:1706.07435 generic Hamiltonian 𝐼(𝑙 𝑦 , 𝑙 𝑧 ) near exceptional point (EP): 𝐼 = 𝜗𝜏 + + 𝜇 𝑗𝑘 𝑙 𝑗 𝜏 𝐹 ± 𝑙 = ± 𝜗(𝑤 𝑦 𝑙 𝑦 + 𝑤 𝑧 𝑙 𝑧 ) 𝑘 ( 𝑤 𝑦 , 𝑤 𝑧 , 𝑤 𝑧 /𝑤 𝑦 are complex) topological index: vorticity of complex energy “gap” in k-space topological charge of exceptional point: 𝜉 = ± 1 2 topology guarantees a line of real gap closing (= Fermi arc) emanates from EP.

Exceptional Points: Ubiquitous in 𝑒 ≥ 2 How many parameters must be tuned to hit a degeneracy? • Hermitian: 3 𝑏 ⋅ Ԧ Ԧ 𝜏 is degenerate when Ԧ 𝑏 = 0 => topological Weyl points in 3D • non-Hermitian: 2 ( Ԧ 𝑏 + 𝑗𝑐) ⋅ Ԧ 𝜏 is defective when Ԧ 𝑏 ⋅ 𝑐 = 0 and Ԧ 𝑏 = |𝑐| => topological exceptional points in 2D and exceptional loops in 3D.

Exceptional Points in 2D Introducing generic damping to 2D Dirac fermion: arXiv:1706.07435 𝐼(𝒍) = (𝑙 𝑦 − 𝑗𝜆 1 )𝜏 𝑨 + (𝑙 𝑧 − 𝑗𝜆 2 )𝜏 𝑦 + (𝑛 − 𝑗𝜀 )𝜏 𝑧 • imaginary vector potential: Dirac point turns into Fermi arc ending at a pair of EPs. • imaginary Dirac mass Dirac point turns into “Fermi disk” ending at a ring of EPs

Exceptional Points in 2D Introducing generic damping to 2D Dirac fermion: arXiv:1706.07435 𝐼(𝒍) = (𝑙 𝑦 − 𝑗𝜆 1 )𝜏 𝑨 + (𝑙 𝑧 − 𝑗𝜆 2 )𝜏 𝑦 + (𝑛 − 𝑗𝜀 )𝜏 𝑧 | Ԧ 𝜆| 𝑙 ⊥ 𝑙 ∥ 𝑛 Direct quantum Hall transition is replaced by intermediate phase with a pair of exceptional points at momenta:

arXiv:1706.07435 Exceptional points cannot be created or removed alone. Merging a pair of exceptional points: 1 2 + 1 • 2 = 1 : results in a topological “vortex point”. 1 2 − 1 • 2 = 0 : results in a “hybrid point”, which can be gapped.

Fermi Arc in Heavy Fermion Systems 𝜍(𝜕) DMFT shows temperature-dependent bulk Yuki Nagai Fermi arc in periodic Anderson model with (JAEA & MIT) d-wave hybridization.