Adiabatic vs Sudden Flux Insertion and Nonlinear Electric Conduction Talk 2 by Masaki Oshikawa (ISSP , UTokyo) Condensed Matter Physics in All the Cities 2020 26 June 2020@Zoom 1
This presentation file is based on what was used in the actual talk at #CMPCity2020, but slightly revised and modified Talk 1 (Last week, Thursday 18 June) Applications of Adiabatic Flux Insertion ! to Quantum Many-Body Systems: A Pedagogical Introduction M. O. and T. Senthil, PRL 96 , 060601 (2006) Talk 2 (Today, Friday 26 June) Adiabatic vs Sudden Flux Insertion and Nonlinear Electric Conduction M. O. PRL 84 , 1535 (2000) / PRL 90 , 236401; 90 109901 (E) (2003) Haruki Watanabe and M.O., arXiv:2003.10390 Haruki Watanabe, Yankang Liu, and M. O., arXiv:2004.04561 2
Adiabatic Flux Insertion (i) Increase Aharonov-Bohm flux Φ adiabatically from 0 to Φ 0 (=2 ̟ ) | Ψ 0 � → | Ψ ′ 0 � Hamiltonian for the final state is Φ different from the original one, but we can (ii) eliminate the unit flux quantum by the large gauge transformation − 1 = H ( Φ = 0) U x H ( Φ = 2 π ) U x � � 2 π i � U x = exp xn � r L x � r | Ψ 0 � → | Ψ ′ 0 � → U x | Ψ ′ 0 � 3
Many Particles on Periodic Lattice For example, consider a many-particle system on the square lattice of L x × L y with periodic boundary conditions assume particle number conservation (U(1) symmetry) Φ assume that the system is gapped, and consider the adiabatic insertion of unit flux quantum through the “hole” M. O. 2000 4
Translation Invariance Translation invariance ⇒ Momentum Conservation P = − i � � ∇ Lattice model / periodic potential discrete (lattice) translation: T x = e iP x Let us now consider the adiabatic flux insertion A x = Φ 0 t TL x Hamiltonian is always translation invariant ⇒ momentum is exactly conserved! P (0) P (0) initial state: ⇒ final state: x x 5
Which Momentum? What is conserved exactly is the “canonical momentum” which is NOT gauge-invariant! P canonical = − i � � P kinetic = − i � � ∇ − � A ∇ kinetic momentum = covariant derivative (gauge invariant) After the insertion of the unit flux quantum, the system is equivalent to zero flux but in the different gauge! We must eliminate the vector potential by the large gauge transformation 6
Large Gauge Transformation Initial Groundstate Final State | Ψ 0 � | Ψ ′ 0 � = F x | Ψ 0 � T x | Ψ 0 � = e iP (0) 0 � = e iP (0) x | Ψ 0 � x | Ψ ′ T x | Ψ ′ 0 � groundstate of H (2 π ) groundstate of H (0) Large gauge transformation must be a groundstate of H (0) | ˜ Ψ ′ 0 � ≡ U x | Ψ ′ 0 � � � � � 2 π i 2 π i � − 1 T x U x = T x exp � U x n � U x = exp xn � r r L x L x � r � r 0 � = e i ( P (0) r ) | ˜ + 2 π T x | ˜ � r n � Ψ ′ Ψ ′ 0 � x � Lx 7
Momentum Shift total number of particles + 2 π P (0) → P (0) � n � r (conserved) x x L x � r We are usually interested in the thermodynamic limit for a fixed particle density (particle # / unit cell) ν ν = p Suppose and choose L y to be a coprime with q q p ∆ P x = 2 π L x L y ν = 2 π L y L x q Lattice momentum is defined modulo 2 ̟ momentum shifted if q ≠ 1 (fractional filling) The final state is different from the initial ground state ⇒ ground-state degeneracy! 8
“Lieb-Schultz-Mattis Theorem” General constraint on the spectrum of quantum many-body Hamiltonian on a periodic lattice Periodic (translation invariant) lattice ⇒ unit cell U(1) symmetry ⇒ conserved particle number ν : number of particle per unit cell (filling fraction) ν = p / q ⇒ “ingappability” - system is gapless OR - gapped with q-fold degenerate ground states gapped with unique ground state 9
History of the LSM Theorem 1961 LSM S =1/2 chain 1981 [ Haldane “conjecture”, dependence on 2 S mod 2] 1986 Affleck-Lieb LSM theorem for general S chain 1997 M.O.-Yamanaka-Affleck general magnetization 1997 Yamanaka-M.O.-Affleck electrons/particles 2000 M. O. “flux insertion” argument for d ≧ 2 2004 Hastings rigorous proof 2006 Nachtergale-Sims really rigorous proof … many recent extensions! (non-symmorphic crystal symmetry Parameswaran et al. 2013 etc.) 10
Gap Closing by AB Flux? the spectrum is identical! E E adiabatic evolution? gap gap Φ = 0 Φ = 2 π (= Φ 0 ) In principle, the ground state could evolve into an excited state, if there is a gap closing (level crossing with the “excited state”) at some value of Φ 11
Insulator vs Conductor Linear response theory: current induced by electric field Drude weight D =0 : insulator (Kohn, 1963) δ→ +0 D >0 : conductor In a realistic system, the Drude peak is broadened ( δ >0), but in an ideal model we can identify delta-function Drude peak as a signature of “perfect conductor” 12
Real-Time Formulation of D � t σ ( t − t ′ ) E x ( t ′ ) dt ′ j x ( t ) ∼ −∞ current induced by the electric field at t =0, t →∞ σ ( t ) = D lim that survives after an infinitely long time Initial condition at t =0: ground state | Ψ 0 � switch on an (infinitesimal) constant electric field for t >0 t E x = A x A x = A x adiabatic limit T →∞ T T j x ( t ) ∼ D A x M. O. 2003 T t Watanabe-M.O. 2020 13
Current vs Energy On the other hand, the current operator is V: volume � − 1 ∂ H j x = 1 ∂ H � dA x = 1 ˆ ( A x ) V ∂ A x V dt ∂ t j x ( t ) ∼ D A x T t For an adiabatic flux insertion � 2 T 2 � T � T � A x V ∆ E 0 = 1 1 � ∂ H ∂ t � dt = A x j x ( t ) dt ∼ D 2 V T T 0 0 A x = Φ 0 For the adiabatic insertion of unit flux quantum L x V 2 D ∆ E 0 ( Φ 0 ) = 2 Φ 0 G. S. energy increase in 2 L x the adiabatic flux insertion M. O. 2003 14
Gap Protection in Insulators ( d =2) E the spectrum is identical! E adiabatic evolution? gap gap V 2 D ∆ E 0 ( Φ 0 ) = 2 Φ 0 Φ = 0 2 L x Φ = 2 π (= Φ 0 ) O ( L d − 2 ) If this happens in d =2, energy gain ≧ gap ⇒ D >0 !! i.e. in an insulator, the groundstate must remain in the groundstate in the adiabatic flux insertion ⇒ LSM 15
Kohn Formula V 2 D ∆ E 0 ( Φ 0 ) = 2 Φ 0 2 L x A x = Φ 0 → 0 L x ∂ 2 E 0 � D = 1 � 2 ( A x ) � V ∂ A x � A x =0 Kohn’s formula for the Drude weight 16
Non-Linear Conductivities Non-linear electric conduction: AC E -field topic of current interest e.g. “shift current” application to photovoltaics DC current n -th order conductivity � t � t � t ∞ 1 � σ ( n ) ( t − t 1 , t − t 2 , . . . , t − t n ) j x ( t ) ∼ . . . n ! −∞ −∞ −∞ n =1 E x ( t 1 ) E x ( t 2 ) . . . E x ( t n ) dt 1 dt 2 . . . dt n Nonlinear Drude weights ∆ t 1 , ∆ t 2 ,..., ∆ t n →∞ σ ( n ) ( ∆ t 1 , ∆ t 2 , . . . , ∆ t n ) = D ( n ) lim 17
Non-Linear “Kohn Formula” Consider the same adiabatic flux insertion and include the non-linear Drude weights � n � A x x ( t ) ∼ 1 t j ( n ) n ! D ( n ) t n A x = A x T T � T � T 1 = 1 ∂ H ∂ t dt = A x V ∆ E ( n +1) j ( n ) x ( t ) dt 0 V T 0 0 � n +1 T n +1 � A x ∼ 1 1 n +1 n ! D ( n ) ( n + 1)! D ( n ) A x n + 1 = T ∂ n +1 E 0 � D ( n ) = 1 Watanabe-M.O. 2020 � n +1 ( A x ) � V ∂ A x � Watanabe-Liu-M.O. 2020 A x =0 18
Sudden Flux Insertion t E x = A x A x = A x T T T → 0: sudden insertion delta-function electric field pulse In this limit, quantum state (wavefunction) does not change but again we are in a different gauge, so need to apply the large gauge transformation to go back to the original gauge | Ψ 0 � → | Ψ 0 � → U x | Ψ 0 � 19
Energy Gain in Sudden Flux Insertion � T � T T → 0 V ∆ E = 1 1 � ∂ H ∂ t � dt = A x j x ( t ) dt V T 0 0 ∼ σ ( n ) (0 , 0 , . . . , 0) � n � A x 1 2 n n + 1 T � t � t � n ∞ � A x 1 � σ ( n ) ( t − t 1 , . . . , t − t n ) j x ( t ) ∼ . . . dt 1 dt 2 . . . dt n n ! T 0 0 n =1 � n ∼ σ ( n ) (0 , 0 , . . . , 0) � A x t n 2 n T 1 � � † H (0) U x | Ψ 0 � − � Ψ 0 |H (0) | Ψ 0 � cf.) LSM � Ψ 0 | U x V variational � � = 1 H ( A x = Φ 0 V � Ψ 0 | ) − H (0) | Ψ 0 � energy L x 20
Non-linear f -Sum Rules Comparing both sides, we obtain the identity instantaneous response in real-time σ ( n ) (0 , 0 , . . . , 0) = � Ψ 0 | ∂ n +1 H ( A x ) � � | Ψ 0 � � n +1 2 n ∂ A x � A x =0 � ∞ � ∞ � ∞ d ω 1 d ω 2 d ω n 2 π σ ( n ) ( ω 1 , ω 2 , . . . , ω n ) = 2 π . . . 2 π −∞ −∞ −∞ [frequency space representation] Watanabe-M.O. / Watanabe-Liu-M.O. 2020 cf.) Shimizu 2010, Shimizu-Yuge 2011 21
Example: Tight-Binding Model 22
Numerical Check 23
Summary Two general formulas for non-linear conductivity f -sum rules (instantaneous response = ω -integral) σ ( n ) (0 , 0 , . . . , 0) = � Ψ 0 | ∂ n +1 H ( A x ) � � | Ψ 0 � � n +1 2 n ∂ A x � A x =0 energy gain by sudden flux insertion “Kohn formulas” for non-linear Drude weights (long-time response = 1/ ω pole) ∂ n +1 E 0 � D ( n ) = 1 � n +1 ( A x ) � V ∂ A x � A x =0 energy gain by adiabatic flux insertion more general results are given in arXiv:2003.10390 & arXiv:2004.04561 24
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