Some topics on quantum transport Lingling CAO October 25, 2017 Lingling CAO (Cermics) Quantum transport October 25, 2017 1 / 23
Overview Overview 1 Junction of two 1-d embeded in 3d periodic systems 2 Briefing of other topics 3 Lingling CAO (Cermics) Quantum transport October 25, 2017 2 / 23
Resume of topics Study quantum transport within density functional theory. Junction of two 1-d embedded in 3d periodic systems. (A warming up problem) Quantum transport : i.e., conductivity etc. Coupling with phonons (Extension of Thomas- Fermi -von Weizs¨ acker model) Other interesting topics : Topological insulators, bulk-edge correspondence, Quantum Hall Effect, etc. Lingling CAO (Cermics) Quantum transport October 25, 2017 3 / 23
Prelimilary : Schatten Class H : a separable Hilbert space (usually used : L 2 ( R 3 ), H 1 ( R 3 )) with ( ψ i ) ∞ i =1 as orthogonal basis. L ( H ): bounded operator on H . For A ∈ L ( H ) which is positive , define its trace: ∞ � Tr ( A ) := ( ψ i , A ψ i ) . i =1 For Probabilists, please consider this as some form of expectation of some r.v. Schatten class S p ( H ) (Non-commutative L p space) : √ ⇒ Tr ( | A | p ) 1 / p < ∞ , A ∈ S p ( H ) ⇐ | A | = A ∗ A (1) A is in trace-class ⇐ ⇒ A ∈ S 1 ( H ), A is in Hilbert-Schmidt ⇐ ⇒ A ∈ S 2 ( H ). Lingling CAO (Cermics) Quantum transport October 25, 2017 4 / 23
Junction of two 1-d embeded in 3d periodic systems Motivation: study the junction of two 1-d embeded in 3d periodic systems with reduced Hartree-Fock model. Calculate its ground state → minimization of energy functional. Existence of ground state → existence of minimizer of energy functional. Lingling CAO (Cermics) Quantum transport October 25, 2017 5 / 23
reduced Hartree-Fock model For N nonrelativistic quantum electrons, reduced Hartree-Fock model is a mean-field model the state of N electrons described by one-body density matrix γ , where γ ∈ P N : � √ √ � � � P N = γ ∈ B ( L 2 ( R 3 )) | 0 ≤ γ ≤ 1 , Tr ( γ ) = N , Tr − ∆ γ − ∆ < ∞ i =1 H 1 ( R 3 ). N -body space of fermionic wavefunctions : ∧ N Hartree-Fock state : Φ := ψ 1 ∧ ψ 2 ∧ · · · ∧ ψ N ∈ ∧ N i =1 H 1 ( R 3 ) . γ = � N i =1 | ψ i � � ψ i | density matrix of Φ → diagonalizable in an orthogonal i =1 of L 2 ( R 3 ) : γ = � ∞ basis ( φ i ) ∞ i =1 n i | φ i � � φ i | , 0 ≤ n i ≤ 1 . Density associated with γ : ρ γ ( x ) = γ ( x , x ) = � ∞ i =1 n i φ 2 i ( x ) ≥ 0 . Lingling CAO (Cermics) Quantum transport October 25, 2017 6 / 23
reduced Hartree-Fock model Nuclei density of charge ρ nuc . reduced Hartree-Fock energy functional : � � − 1 + 1 E rHF ( γ ) = Tr 2 γ 2 D ( ρ γ − ρ nuc , ρ γ − ρ nuc ) . (2) � � ˆ f ( x ) g ( y ) f ( k )ˆ g ( k ) D ( f , g ) = | x − y | dxdy = 4 π dk . | k | 2 R 3 × R 3 R 3 The variational problem is : � E rHF ( γ ) , γ ∈ P N � I rHF = inf Theorem : for neutral or positively charged systems, the variational problem has a minimizer γ and ρ γ is unique . Lingling CAO (Cermics) Quantum transport October 25, 2017 7 / 23
1-d embeded in 3d periodic system Bloch decomposition for 1d embedded in 3d periodic infinite system: Unit cell: Γ := [ − 1 / 2 , 1 / 2) × R 2 . The first Brillouin zone (dual lattice): Γ ∗ := [ − π, π ) × { 0 } 2 ≡ [ − π, π ) Translation operator : τ k u ( x , r ) = u ( x − k , r ) , ∀ k ∈ R Density matrix of the electrons: γ , which is a self-adjoint operator acting on L 2 ( R 3 ) and 0 ≤ γ ≤ 1. Bloch decomposition: � � L 2 u ∈ L 2 loc ( R , L 2 ( R 2 )) | τ k u = e − ik ξ u , ∀ k ∈ Z ξ (Γ) = � γ = 1 γ ξ ∈ S ( L 2 Γ ∗ γ ξ d ξ, ξ (Γ)) 2 π Lingling CAO (Cermics) Quantum transport October 25, 2017 8 / 23
1-d embeded in 3d periodic system We can define a 1d embeded in 3d periodic rHF energy for γ ∈ P per : � � � E per ( γ ) = 1 − 1 d ξ + 1 Γ ∗ Tr L 2 2∆ γ ξ 2 D G ( ρ γ − µ per , ρ γ − µ per ) (3) ξ (Γ) 2 π The periodic rHF ground state energy (per unit cell) is given by � � � I per = inf E per ( γ ) , γ ∈ P per , ρ γ = Z (4) Γ � � D G ( f , g ) := G ( x − y ) f ( x ) g ( y ) d x d y Γ Γ ρ γ : density associate with γ . G ( · ): Green function Lingling CAO (Cermics) Quantum transport October 25, 2017 9 / 23
1-d embeded in 3d periodic system Theorem (Definition of the 1d periodic rHF minimizer) Let Z ∈ N \{ 0 } . The minimization problem (4) admits a unique minimizer γ per . Moreover, γ per satisfies the following self-consistent equation: � γ per = 1 ( −∞ ,ǫ F ] ( H per ) (5) H per := − 1 2 ∆ + ( ρ per − µ per ) ⋆ Γ G where ǫ F is a Lagrange multiplier called Fermi level (chemical potential). Lingling CAO (Cermics) Quantum transport October 25, 2017 10 / 23
Junction of two 1-d embeded in 3d periodic systems Difficulty: if not the same periodicity, there is breaking translation symmetry → Bloch decomposition cannot be applied → need to find a reference state. Periodic density operator corresponding to the left (right) system: γ per ,ℓ ( γ per , r ) solution of (5), with nuclei density µ per ,ℓ ( µ per , r ) and electronic density ρ per ,ℓ ( ρ per , r ). Density operator of junction system: γ s , with associated density ρ s . ˆ � f ( k )ˆ g ( k ) µ s = 1 x ≤ 0 · µ per ,ℓ + 1 x ≥ 0 · µ per , r , D ( f , g ) := 4 π dk . R 3 k 2 (Infinite) energy functional for the junction system is FORMALLY: � � − 1 + 1 E s ( γ s ) := Tr 2∆ γ s 2 D ( ρ s − µ s , ρ s − µ s ) (6) Objective: find a reference state γ r , and perturbative state Q , such that γ s = γ r + Q . Lingling CAO (Cermics) Quantum transport October 25, 2017 11 / 23
Choice of reference state Choice of reference state γ r : Need to be an orthogonal spectral projector of some well-chosen Hamiltonian, i.e., 0 ≤ γ r ≤ 1, γ 2 r = γ r and γ ∗ r = γ r . (If not we do not know yet how to treat its perturbation ...) Need to have enough regularity (Laplacian term ...) Need to approach the real state γ s such that the difference can be treated as perturbation (Very logic !) → should be something that is very similar to 1 x ≤ 0 · γ per ,ℓ + 1 x ≥ 0 · γ per , r . (Not this one, lack of regularity, the smooth version is not a spectral projector of some Hamiltonian ...) Lingling CAO (Cermics) Quantum transport October 25, 2017 12 / 23
Choice of reference state Introduce a smooth function χ ( x , y , z ): 1 if x ≤ − 1 / 2 χ ( x , · , · ) = 0 if x ≥ 1 / 2 (7) smooth elsewhere, bounded between 0 and 1 A regular potential V χ := χ 2 V per ,ℓ + (1 − χ 2 ) V per , r = χ 2 (( ρ per ,ℓ − µ per ,ℓ ) ⋆ Γ G ) + (1 − χ 2 ) (( ρ per , r − µ per , r ) ⋆ Γ G ). Define Hamiltonian associated with V χ writes: H χ := − 1 2∆ + V χ (8) Define a spectral projector γ r = γ χ := 1 ( −∞ ,ǫ F ] ( H χ ) we have [ γ χ , H χ ] = 0. Lingling CAO (Cermics) Quantum transport October 25, 2017 13 / 23
Choice of reference state � � ρ χ − µ χ := − 1 χ 2 ( ρ per ,ℓ − µ per ,ℓ ) + (1 − χ 2 )( ρ per , r − µ per , r ) 4 π ∆ V χ = + η χ . η χ is local term. ρ χ and is a priori unknown, is decided by ρ χ := 1 ( −∞ ,ǫ F ] ( H χ ). Perturbative energy formally � � − 1 + D ( ρ χ − µ χ , ρ Q ) + 1 ���� E s ( γ s ) − E r ( γ r ) = 2∆ Q 2 D ( ρ Q , ρ Q ) Tr − D ( ρ Q , ν χ ) − D ( ρ χ − µ χ , ν χ ) + 1 2 D ( ν χ , ν χ ) (9) where ν χ := µ s − µ χ = ( 1 x ≤ 0 − χ 2 ) µ per ,ℓ + ( 1 x ≥ 0 − (1 − χ 2 )) µ per , r (10) + ( χ 2 ρ per ,ℓ + (1 − χ 2 ) ρ per , r − ρ χ ) + η χ Objective: study the rigorous version of minimization problem (9). Lingling CAO (Cermics) Quantum transport October 25, 2017 14 / 23
Perturbative energy Proposition (Reference state density is exponentially close to the smoothed real density) Assume that Fermi level ǫ F < 0 (Fermi level is strictly negative), and a gap condition, we have χ 2 ρ per ,ℓ + (1 − χ 2 ) ρ per , r − ρ χ ∈ C � L 1 ( R 3 ) � L 2 ( R 3 ). So ν χ ∈ C � L 1 ( R 3 ) � L 2 ( R 3 ). Moreover, denote B ( Z ) a unit cube centred at Z ∈ Z , and w ( Z ) the characteristic function of unit cube B ( Z ), there exists positive constants c 1 , c 2 and m 1 , m 2 , and for α ∈ Z + and β ∈ Z , such that � � � χ 2 ρ per ,ℓ ( x , · , · ) + (1 − χ 2 ) ρ per , r ( x , · , · ) − ρ χ ( x , · , · ) w ( β ) dx | ≤ c 1 e − m 1 | β | , | R � � � χ 2 ρ per ,ℓ ( · , r , · ) + (1 − χ 2 ) ρ per , r ( · , r , · ) − ρ χ ( · , r , · ) w ( α ) dr | ≤ c 2 e − m 2 | α | . | R Proof. Write all in spectral projector form and use Cauchy formula representation, have norm estimations and by argument of duality to prove the result. Lingling CAO (Cermics) Quantum transport October 25, 2017 15 / 23
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