Non-equilibrium almost-stationary states for interacting electrons - PowerPoint PPT Presentation

Non-equilibrium almost-stationary states for interacting electrons on a lattice Stefan Teufel, Universit¨ at T¨ ubingen Quantissima II, Venice, 2017. Based on joint work with Domenico Monaco .

1. Example and setup As a microscopic model for a quantum Hall system consider a system of interacting fermions on the domain Λ, where Λ ⊂ Z 2 is the centred square of side-length L with the vertical edges identified.

1. Example and setup As a microscopic model for a quantum Hall system consider a system of interacting fermions on the domain Λ, where Λ ⊂ Z 2 is the centred square of side-length L with the vertical edges identified.

1. Example and setup As a microscopic model for a quantum Hall system consider a system of interacting fermions on the domain Λ, where Λ ⊂ Z 2 is the centred square of side-length L with the vertical edges identified. A typical Hamiltonian could be of the form Λ H Λ � a ∗ � a ∗ = x T ( x − y ) a y + x φ ( x ) a x 0 ( x , y ) ∈ Λ 2 x ∈ Λ � a ∗ x a x W ( d Λ ( x , y )) a ∗ + y a y − µ N Λ , { x , y }⊂ Λ where a ∗ x , i and a x , i are standard fermionic creation and annihilation operators at the sites x ∈ Λ.

1. Example and setup As a microscopic model for a quantum Hall system consider a system of interacting fermions on the domain Λ, where Λ ⊂ Z 2 is the centred square of side-length L with the vertical edges identified. A typical Hamiltonian could be of the form Λ H Λ � a ∗ � a ∗ = x T ( x − y ) a y + x φ ( x ) a x 0 ( x , y ) ∈ Λ 2 x ∈ Λ � a ∗ x a x W ( d Λ ( x , y )) a ∗ + y a y − µ N Λ , { x , y }⊂ Λ where a ∗ x , i and a x , i are standard fermionic creation and annihilation operators at the sites x ∈ Λ. In the following by a “local Hamiltonian” we mean a family A = { A Λ } Λ of self-adjoint operators A Λ indexed by the system size Λ and possibly other parameters that is a “sum of local terms”. Typically � A Λ � ∼ | Λ | = L d .

1. Example and setup Assume that H 0 = { H Λ 0 } has a ground state that is gapped uniformly in the system size | Λ | , i.e. � � E Λ 0 , σ ( H Λ 0 ) \ { E Λ inf Λ dist 0 } = g > 0 .

1. Example and setup Assume that H 0 = { H Λ 0 } has a ground state that is gapped uniformly in the system size | Λ | , i.e. � � E Λ 0 , σ ( H Λ 0 ) \ { E Λ inf Λ dist 0 } = g > 0 . Now add the potential of an electric field of magnitude ε pointing in the 2-direction, V ε, Λ := � ε x 2 a ∗ x a x . x ∈ Λ

1. Example and setup Assume that H 0 = { H Λ 0 } has a ground state that is gapped uniformly in the system size | Λ | , i.e. � � E Λ 0 , σ ( H Λ 0 ) \ { E Λ inf Λ dist 0 } = g > 0 . Now add the potential of an electric field of magnitude ε pointing in the 2-direction, V ε, Λ := � ε x 2 a ∗ x a x . x ∈ Λ Note that the potential difference of ε L at the two edges is, for sufficiently large system size L , larger than the spectral gap g . Thus, the perturbed Hamiltonian H ε, Λ := H Λ 0 + V ε, Λ no longer has a meaningful gap above the ground state.

1. Example and setup Assume that initially the perturbation V ε, Λ is switched-off and the system is in its ground state P Λ 0 .

1. Example and setup Assume that initially the perturbation V ε, Λ is switched-off and the system is in its ground state P Λ 0 . Then slowly turn on the electric field.

1. Example and setup Assume that initially the perturbation V ε, Λ is switched-off and the system is in its ground state P Λ 0 . Then slowly turn on the electric field. Once the field has reached its final value, one expects that the system is in a (almost) stationary state that, in particular, could carry a stationary, non-vanishing Hall current along the closed direction of the cylinder.

1. Example and setup Assume that initially the perturbation V ε, Λ is switched-off and the system is in its ground state P Λ 0 . Then slowly turn on the electric field. Once the field has reached its final value, one expects that the system is in a (almost) stationary state that, in particular, could carry a stationary, non-vanishing Hall current along the closed direction of the cylinder. This state is not the ground state of H ε, Λ , nor is it any other equilibrium state of H ε, Λ , since, for example, the local Fermi-levels at the opposite edges are expected to be different.

1. Example and setup

1. Example and setup Heuristic picture suggesting the existence of a non-equilibrium almost-stationary state (NEASS):

2. Results Let H 0 and H 1 be families of self-adjoint local Hamiltonians, let H 0 have a gapped ground state, let V v be a slowly varying potential, and put H := H 0 + V v + ε H 1 .

2. Results Let H 0 and H 1 be families of self-adjoint local Hamiltonians, let H 0 have a gapped ground state, let V v be a slowly varying potential, and put H := H 0 + V v + ε H 1 . Theorem (Non-equilibrium almost-stationary states) There is a sequence of self-adjoint local Hamiltonians S n , such that for any n ∈ N the projector := e i ε S ε, Λ 0 e − i ε S ε, Λ Π ε, Λ P Λ n n n satisfies n , H ε, Λ ] = ε n +1 [Π ε, Λ [Π ε, Λ n , R ε, Λ n ] for some local R n . . . .

2. Results Theorem (Non-equilibrium almost-stationary states) There is a sequence of self-adjoint local Hamiltonians S n , such that for any n ∈ N the projector := e i ε S ε, Λ 0 e − i ε S ε, Λ Π ε, Λ P Λ n n n satisfies n , H ε, Λ ] = ε n +1 [Π ε, Λ [Π ε, Λ n , R ε, Λ n ] for some local R n . Let ρ ε, Λ ( t ) be the solution of the Schr¨ odinger equation i d d t ρ ε, Λ ( t ) = [ H ε, Λ , ρ ε, Λ ( t )] ρ ε, Λ (0) = Π ε, Λ with . n Then there is a constant C independent of Λ such that for any local Hamiltonian B it holds that � � ρ ε, Λ ( t ) B Λ � � n B Λ �� � ≤ C ε n +1 | t | (1 + | t | d ) �| B �| . 1 Π ε, Λ sup − tr � tr � � | Λ | Λ

2. Results Let f : R → [0 , 1] be a smooth “switching” function, i.e. f ( t ) = 0 for t ≤ 0 and f ( t ) = 1 for t ≥ T > 0, and define H ( t ) := H 0 + f ( t )( V v + ε H 1 ) .

2. Results Let f : R → [0 , 1] be a smooth “switching” function, i.e. f ( t ) = 0 for t ≤ 0 and f ( t ) = 1 for t ≥ T > 0, and define H ( t ) := H 0 + f ( t )( V v + ε H 1 ) . Theorem (Adiabatic switching) The solution of the adiabatic time-dependent Schr¨ odinger equation i ε d d t ρ ε, Λ ( t ) = [ H ε, Λ ( t ) , ρ ε, Λ ( t )] ρ ε, Λ (0) = P Λ with 0 satisfies for all t ≥ T that for any n ∈ N there exists a constant C such that for any local Hamiltonian B � n B Λ �� � ρ ε, Λ ( t ) B Λ � � � ≤ C ε n − d | t | (1 + | t | d ) �| B �| . 1 Π ε, Λ sup − tr � tr � � | Λ | Λ

3. Example continued In the quantum Hall example from the beginning take the current operator J Λ 1 = ∂ α 1 H Λ 0 ( α ) | α =0 as the observable.

3. Example continued In the quantum Hall example from the beginning take the current operator J Λ 1 = ∂ α 1 H Λ 0 ( α ) | α =0 as the observable. Then the Hall current density satisfies 1 � � j Λ tr (Π ε, Λ n J Λ 1 ) − tr ( P Λ 0 J Λ + O ( ε n − 2 ) = 1 ) Hall , 1 | Λ |

3. Example continued In the quantum Hall example from the beginning take the current operator J Λ 1 = ∂ α 1 H Λ 0 ( α ) | α =0 as the observable. Then the Hall current density satisfies 1 � � j Λ tr (Π ε, Λ n J Λ 1 ) − tr ( P Λ 0 J Λ + O ( ε n − 2 ) = 1 ) Hall , 1 | Λ | ε | Λ | tr ( P ε, Λ 1 J Λ 1 ) + O ( ε 2 ) , = uniformly in the system size.

3. Example continued In the quantum Hall example from the beginning take the current operator J Λ 1 = ∂ α 1 H Λ 0 ( α ) | α =0 as the observable. Then the Hall current density satisfies 1 � � j Λ tr (Π ε, Λ n J Λ 1 ) − tr ( P Λ 0 J Λ + O ( ε n − 2 ) = 1 ) Hall , 1 | Λ | ε | Λ | tr ( P ε, Λ 1 J Λ 1 ) + O ( ε 2 ) , = uniformly in the system size. Inserting the explicit expression for P ε, Λ 1 , we obtain for the Hall conductivity Kubo’s “current-current-correlation” formula j Λ i � � � ��� Hall , 1 σ Λ P Λ ∂ α 1 P Λ X 2 , P Λ Hall := = 0 ( α ) | α =0 , + O ( ε ) . | Λ | tr 0 0 ε

4. Remarks ◮ If the perturbation and/or the observable are localized, the result holds with the corresponding normalisation of the trace.

4. Remarks ◮ If the perturbation and/or the observable are localized, the result holds with the corresponding normalisation of the trace. ◮ We actually prove a general space-time adiabatic theorem , similar to what we called space-adiabatic perturbation theory long ago ( Panati, Spohn, T. (2003) ).

4. Remarks ◮ If the perturbation and/or the observable are localized, the result holds with the corresponding normalisation of the trace. ◮ We actually prove a general space-time adiabatic theorem , similar to what we called space-adiabatic perturbation theory long ago ( Panati, Spohn, T. (2003) ). ◮ The new proof in the context of interacting systems and error bounds uniform in the system size is inspired by the recent adiabatic theorem of Bachmann, de Roeck, Fraas (2017) .