Disordered fermions in two dimensions: is Anderson insulating phase - PowerPoint PPT Presentation

Disordered fermions in two dimensions: is Anderson insulating phase the only possibility? Luca Dell’Anna Department of Physics and Astronomy ”Galileo Galilei”, University of Padova Conference on Frontiers in Two-Dimensional Quantum Systems ICTP Trieste, 13 November 2017

Outline ◮ Basic concepts on the Anderson localization ◮ QFT approach: derivation of the non linear σ -model and symmetry classifications ◮ (Anti)-localization effects for all symmetry classes ◮ Combined effects of disorder and interactions: not-universal behaviors and enhacement of critical temperatures.

Anderson localization In the presence of strong enough disorder in D > 2 or for any amount of disorder in D ≤ 2 a metal can turn into an insulator. Interference effect ( λ DB ≃ ℓ ) ⇒ localization of the wavefunctions The probability to find the particle at point C is: | a 1 | 2 + | a 2 | 2 + 2 Re ( a 1 a ∗ 2 ) = 4 | a 1 | ⇒ enhancement of probability to find a particle at C ⇒ reduction of probability to find it at B (conductivity ց ) Probability of self-intersection  δσ ∝ − ( 1 1 ℓ − L ϕ ), d = 3 � τ ϕ v λ d − 1 dt δσ   δσ ∝ − log( L ϕ σ ∼ − ⇒ ℓ ), d = 2 ( Dt ) d / 2 τ  δσ ∝ − ( L ϕ − ℓ ), d = 1  � D τ ϕ and τ ϕ ∼ T − 1 with L ϕ =

Scaling theory of localization (Thouless, Phy.Rep. (1974); Abrahams, Anderson, Licciardello and Ramakrishnan PRL (1979); Gor’kov, Larkin, and Khmel’nitskii, JETP (1979)) Thouless idea: sample (2 L ) d made of cubes L d ⇒ an eigenstate for (2 L ) d is a mixture of e.s. of L d depending on overlap integrals and energy differences (as in perturbation theory) ◮ energy differences ∼ level spacing δ W = ( ν 0 L d ) − 1 ◮ overlap ∼ bandwidth δ E (if localized e.s. δ E exp. small, otherwise ∼ � D / L 2 ) δ W related to the conductance G (units of e 2 / � ) δ E One parameter: ◮ small disorder: G( L ) = σ L d − 2 ◮ strong disorder: G( L ) ∼ exp( − L /ξ )

Scaling theory of localization ◮ strong disorder : G( L ) ∼ exp ( − L /ξ ) β (G) = d log G d log L = log G . G c ◮ small disorder : G ( L ) ∼ σ L d − 2 , expanding in 1 / G β (G) = ( d − 2) − a G  ( 1 ℓ − 1 L ) d=3 (metal)   log( L ⇒ σ ( L ) − σ 0 ∝ − ℓ ) d=2 (insulator)  ( L − ℓ ) d=1 (insulator) 

Diagrammatics Hamiltonian with some random potential H = H 0 + V Disorder variance V ( r ) V ( r ′ ) = w 0 δ rr ′ = Bare Green function G 0 = In Born approximation, Σ = = i / 2 τ ( τ mean free time) Green function G ± ( E , p ) = ( E − H 0 ( p ) ± i / 2 τ ) Kubo formula for conductivity (paramagnetic part) σ ( ω ) = e 2 ≃ e 2 ν v 2 � d ε∂ n ε τ v G + v ( G + ε − G − � � F ∂ε Tr ˆ ε + ω ˆ ε ) 2 π d 1 + i ωτ σ 0 = σ (0) = e 2 ν v 2 F τ (Drude conductivity) d

Diagrammatics The dc electrical conductivity can be written in terms of current-current or density-density correlation functions 1 ω ω K ij ( 0 , ω ) δ ij = i lim q 2 K 00 ( q , ω ) σ = i lim ω → 0 lim ω → 0 q → 0 Ladder summation (diffuson) 1 1 D ( q , ω ) = = D q 2 − i ω 2 πντ 2 with D = v F ℓ/ d = v 2 F τ/ d (diffusion coefficient) D q 2 K 00 ( q , ω ) = = − e 2 ν + D q 2 − i ω from which σ = σ 0 = e 2 ν D .

Diagrammatics: Weak Localization Inclusion of crossing diagrams Ladder summation in the particle-particle channel: cooperon 1 1 C ( q , ω ) = = D q 2 − i ω 2 πντ 2 Since now q = p + p ′ , the contribution to the current-current correlator = i ωσ 0 � 1 δ K ii ( 0 , ω ) = d q D q 2 − i ω νπ The correction to the dc conductivity is ( 1 ℓ − 1  L ) d = 3  δσ = − σ 0 � 1  log( L d q D q 2 − i ω ∝ − ℓ ) d = 2 νπ  ( L − ℓ ) d = 1 

Anderson insulator ◮ 1D - 2D: Weak localization is IR-divergent in 1D and 2D: δσ ∼ σ 0 at a scale ξ ∼ πν D , for 1D ξ ∼ ℓ exp ( π 2 ν D ) , for 2D ◮ 3D: Localization only above a critical value of the disorder localization lenght at criticality ξ ∼ ( σ 0 − σ c ) − ν In the localized phase D ( q , ω ) = C ( q , ω ) becomes massive D ( r , ω ) ∼ exp ( − r /ξ )

Field theory approach: non-linear σ -model (Wegner, ZPB (1979); Efetov, Larkin, Khmel’nitsky, JETP (1980)) ◮ Write G ± in terms of Grassmann variables with action � ¯ S = Ψ( E − H 0 − V ± i ω )Ψ ◮ Average over disorder V by replica method � � ¯ (¯ ΨΨ) 2 S eff = Ψ( E − H 0 ± i ω )Ψ+ w 0 ◮ Hubbard Stratonovich transformation (auxiliary field Q ) ◮ Integrating over fermionic fields ⇒ S ( Q ) δ S ◮ Saddle point: δ Q = 0 ⇒ Q sp ◮ Fluctuations around saddle point ◮ Gradient expansion ⇒ N.L. σ M.

Hubbard Stratonovich transformation Integration over disorder ⇒ a quartic term in the action e − S eff = e − ( S 0 + S imp ) By Hubbard-Stratonovich decoupling, � ( ΨΨ ) � 2 2 w 0 Tr [ QQ † ] − iTr [ Ψ Q Ψ ] 1 � e − S imp = e w 0 = dQ e For bipartite lattices the auxiliary field is not hermitian Q j = Q 0 j + i ( − 1) j Q 3 j (smooth and staggered components) Integrating over Ψ 1 − 1 � � � Q † Q S ( Q ) = Tr 2 Tr ln ( − H + iQ ) 2 w 0 δ S → Q sp = Σ ∝ τ − 1 Saddle point: δ Q = 0 − the self-energy at the Born level, in the diagrammatics!

Transverse modes and symmetry classification Quantum fluctuations around Q sp that leave H invariant Q = U − 1 Q sp U U ∈ G and [ U , Q sp ] � = 0 If H subgroup of G such that h ∈ H, [ h , Q sp ] = 0 ⇒ U ∈ G / H (Coset) Hamiltonian Class RMT SU(2) NL σ -model manifolds T Wigner-Dyson classes A GUE − ± U(2 n )/U( n ) × U( n ) AI GOE + + Sp(4 n )/Sp(2 n ) × Sp(2 n ) AII GSE + − O(2 n )/O( n ) × O( n ) Chiral classes AIII chGUE U( n ) − ± BDI chGOE + + U(4 n )/Sp(2 n ) CII chGSE + − U( n )/O( n ) Bogoliubov-de Gennes C − + Sp(2 n )/U(2 n ) CI + + Sp(2 n ) D − − O(2 n )/U( n ) DIII + − O( n )

Non linear σ -model From the real part of S ( Q ) � − H − iQ † � Tr ln ( − H + iQ ) + Tr ln = H 2 + Q 2 � � = − Tr ln − Tr ln (1 + G 0 U ) , sp � − 1 and H 2 + Q 2 � where G 0 = sp U RR ′ = iQ † R H RR ′ − iH RR ′ Q R ′ ≃ − � J · � ∇ Q the current operator appears J = − iH RR ′ ( R − R ′ ) Expanding in U RR ′ , the second term reads ∂ Q † ∂ Q � � Tr ( G 0 UG 0 U ) ≃ ( JG 0 JG 0 ) Tr the factor ( JG 0 JG 0 ) is the Kubo formula for the conductivity!

Effective action (NLSM) The final effective action in long wavelength limit S [ Q ] = π � � ∇ Q ∇ Q † � 8 σ dR Tr − 4 ν Tr (ˆ ω Q ) the bare σ = e 2 ν D is the Drude conductivity! Quantum corrections from Renormalization Group (RG) procedure: Gaussian propagators = diffuson and cooperon in diagrammatics � d 2 q 1 1 < QQ > = q 2 − i ω ≡ g log( s ) 4 π 2 2 πσ where the effective coupling constant which controls the 1 perturbative expansion is given by g = 2 π 2 σ (the resistivity) d g β ( g ) = ( s energy scaling factor ) d log s g is the running coupling constant.

RG of NLSMs (Wigner-Dyson classes) in (2 + ǫ ) d Beta-functions by ǫ -expansion ◮ Class A (unitary symmetry class, broken T ) β ( g ) = − ǫ g + g 3 / 2 + 3 g 5 / 8 + O ( g 7 ) ◮ Class AI (ortogonal symmetry class, preserved T and SU(2)) β ( g ) = − ǫ g + g 2 + 3 ζ (3) g 5 / 4 + O ( g 6 ) ◮ Class AII (simplettic symmetry class, preserved T , no SU(2)) β ( g ) = − ǫ g − g 2 + 3 ζ (3) g 5 / 4 + O ( g 6 ) (+ g 2 ⇒ weak localization, − g 2 ⇒ weak anti-localization ) Anderson transitions ( β ( g c ) = 0 ) ◮ 3D ( ǫ = 1). Example: class AI critical point: g c = ǫ − 3 ζ (3) ǫ 4 / 4 + O ( ǫ 5 ) localization lenght exponent: ν = − 1 /β ′ ( g c ) = ≃ 1 . 7 (in good agreement with numerics, ν ≃ 1 . 57) ◮ 2D for class AII critical point: g c = (4 / 3 ζ (3)) 1 / 3 ≃ 1 Metal-Insulator transition in 2D

Two-subattice models (Chiral classes) (Gade, Wegner, NPB (1991)) The Hamiltonian is defined on a bipartite lattice ✟ � t ij e i φ ij c † � µ c † ✟✟✟✟✟ H = − i σ c j σ − i σ c i σ i ,σ � ij � σ ◮ t ij = t ji random hopping, ◮ φ ij = − φ ji , if � = 0, breaks time reversal symmetry ( T ), ◮ µ � = 0 breaks sublattice symmetry ( S ) The effective action π � � ∇ Q ∇ Q † � S [ Q ] = 16 σ dR Tr − 4 ν Tr (ˆ ω Q ) − π � �� 2 � � Q † ( R ) � 8 Π dR Tr ∇ Q ( R ) (for µ � = 0 ⇒ Π = 0)

Results with and without sublattice symmetry in 2D Coset space Symm. class β ( g ) g 2 µ � = 0, φ ij = 0 Sp(4 n )/Sp(2 n ) × Sp(2 n ) AI O ( g 3 ) µ � = 0, φ ij � = 0 U(4 n )/U(2 n ) × U(2 n ) A µ = 0, φ ij = 0 U(8 n )/Sp(4 n ) BDI 0 µ = 0, φ ij � = 0 U(4 n ) × U(4 n )/U(4 n ) AIII 0 ◮ without sublattice symmetry ( µ � = 0): 1 σ = σ 0 − 2 π 2 log( τ ϕ /τ ) (insulator, like for the on-site disorder) ◮ with sublattice symmetry ( µ = 0): σ = σ 0 (conductor, Gade-Wegner criticality) at any order in g β ( g ) = 0 also for CII ( Fabrizio, Dell’Anna, Castellani, PRL (2002))