On solving many-body Lindblad equation and quantum phase transition far from equilibrium
|
|
- Alberta Hawkins
- 6 years ago
- Views:
Transcription
1 On solving many-body Lindblad equation and quantum phase transition far from equilibrium Department of Physics, FMF, University of Ljubljana, SLOVENIA MECO 35, Pont-a-Mousson
2 Outline of the talk One dimensional quantum many body systems far from equilibrium: Paradigmatic models: open quantum spin chains with boundary openings On efficient computational methods: Analytical: Diagonalization of quantum Liouvillians in terms of Fock space of operators Numerical: Time-dependent DMRG in the Liouville space Idea: Non-equilibrium steady state may have similar computational complexity as the "ground state" of one-dimensional Hamiltonian models. On new exotic physics: Quantum phase transitions far-from equilibrium (example: anisotropic XY spin chain with transverse magnetic field) Finite size scaling offers surprising connection to wave resonators!
3 Outline of the talk One dimensional quantum many body systems far from equilibrium: Paradigmatic models: open quantum spin chains with boundary openings On efficient computational methods: Analytical: Diagonalization of quantum Liouvillians in terms of Fock space of operators Numerical: Time-dependent DMRG in the Liouville space Idea: Non-equilibrium steady state may have similar computational complexity as the "ground state" of one-dimensional Hamiltonian models. On new exotic physics: Quantum phase transitions far-from equilibrium (example: anisotropic XY spin chain with transverse magnetic field) Finite size scaling offers surprising connection to wave resonators!
4 Outline of the talk One dimensional quantum many body systems far from equilibrium: Paradigmatic models: open quantum spin chains with boundary openings On efficient computational methods: Analytical: Diagonalization of quantum Liouvillians in terms of Fock space of operators Numerical: Time-dependent DMRG in the Liouville space Idea: Non-equilibrium steady state may have similar computational complexity as the "ground state" of one-dimensional Hamiltonian models. On new exotic physics: Quantum phase transitions far-from equilibrium (example: anisotropic XY spin chain with transverse magnetic field) Finite size scaling offers surprising connection to wave resonators!
5 Outline of the talk One dimensional quantum many body systems far from equilibrium: Paradigmatic models: open quantum spin chains with boundary openings On efficient computational methods: Analytical: Diagonalization of quantum Liouvillians in terms of Fock space of operators Numerical: Time-dependent DMRG in the Liouville space Idea: Non-equilibrium steady state may have similar computational complexity as the "ground state" of one-dimensional Hamiltonian models. On new exotic physics: Quantum phase transitions far-from equilibrium (example: anisotropic XY spin chain with transverse magnetic field) Finite size scaling offers surprising connection to wave resonators!
6 Outline of the talk One dimensional quantum many body systems far from equilibrium: Paradigmatic models: open quantum spin chains with boundary openings On efficient computational methods: Analytical: Diagonalization of quantum Liouvillians in terms of Fock space of operators Numerical: Time-dependent DMRG in the Liouville space Idea: Non-equilibrium steady state may have similar computational complexity as the "ground state" of one-dimensional Hamiltonian models. On new exotic physics: Quantum phase transitions far-from equilibrium (example: anisotropic XY spin chain with transverse magnetic field) Finite size scaling offers surprising connection to wave resonators!
7 Outline of the talk One dimensional quantum many body systems far from equilibrium: Paradigmatic models: open quantum spin chains with boundary openings On efficient computational methods: Analytical: Diagonalization of quantum Liouvillians in terms of Fock space of operators Numerical: Time-dependent DMRG in the Liouville space Idea: Non-equilibrium steady state may have similar computational complexity as the "ground state" of one-dimensional Hamiltonian models. On new exotic physics: Quantum phase transitions far-from equilibrium (example: anisotropic XY spin chain with transverse magnetic field) Finite size scaling offers surprising connection to wave resonators!
8 Many-body Lindblad equation The central equation we address is the Lindblad equation for the many-body density operator ρ(t): dρ dt = ˆLρ := i[h, ρ] + X µ 2L µρl µ {L µl µ, ρ} where H is a many-body (Hamiltonian) with local couplings, and L µ are Lindblad bath operators which act locally on degrees of freedom near each bath.
9 Many-body Lindblad equation The central equation we address is the Lindblad equation for the many-body density operator ρ(t): dρ dt = ˆLρ := i[h, ρ] + X µ 2L µρl µ {L µl µ, ρ} where H is a many-body (Hamiltonian) with local couplings, and L µ are Lindblad bath operators which act locally on degrees of freedom near each bath. The most general representation of infinitesimal generator of completely positive, Markovian dynamics (Lindblad 1976/Gorini, Kosakowski, Sudarshan 1976)
10 Solvable example: open XY quantum spin chains Consider magnetic and heat transport of a Heisenberg XY spin 1/2 chain, with arbitrary either homogeneous or positionally dependent (e.g. disordered) nearest neighbour interaction Xn 1 H = `Jx m σmσ x m+1 x + Jmσ y mσ y y m+1 + m=1 nx h mσm z (1) which is coupled to two thermal/magnetic baths at the ends of the chain, generated by two pairs of canonical Lindblad operators L 1 = 1 q Γ L 1 2 σ 1 L 3 = 1 q Γ R 1 2 σ n L 2 = 1 q Γ L 2 2 σ+ 1 L 4 = 1 q Γ R 2 2 σ+ n (2) where σ m ± = σm x ± iσm y and Γ L,R 1,2 are positive coupling constants related to bath temperatures/magnetizations. e.g. if spins were non-interacting the bath temperatures T L,R would be given with Γ L,R 2 /Γ L,R 1 = exp( 2h 1,n /T L,R ). m=1
11 Solvable example: open XY quantum spin chains Consider magnetic and heat transport of a Heisenberg XY spin 1/2 chain, with arbitrary either homogeneous or positionally dependent (e.g. disordered) nearest neighbour interaction Xn 1 H = `Jx m σmσ x m+1 x + Jmσ y mσ y y m+1 + m=1 nx h mσm z (1) which is coupled to two thermal/magnetic baths at the ends of the chain, generated by two pairs of canonical Lindblad operators L 1 = 1 q Γ L 1 2 σ 1 L 3 = 1 q Γ R 1 2 σ n L 2 = 1 q Γ L 2 2 σ+ 1 L 4 = 1 q Γ R 2 2 σ+ n (2) where σ m ± = σm x ± iσm y and Γ L,R 1,2 are positive coupling constants related to bath temperatures/magnetizations. e.g. if spins were non-interacting the bath temperatures T L,R would be given with Γ L,R 2 /Γ L,R 1 = exp( 2h 1,n /T L,R ). Similar models were recently considered e.g. in Karevski and Platini PRL 2009, and Clark, Prior, Hartmann, Jaksch and Plenio, PRL2009 & m=1
12 Quantum phase transition far from equilibrium in XY chain TP & I. Pižorn, PRL 101, (2008) Jm x = (1 + γ)/2 Jm y = (1 γ)/2, h m = h C(j,k) = σj z σk z σj z σk z 1 Cres h 0.9 hc h 0.3 hc h ImΒ Γ ReΒ 0 Π Π Φ ReΒ h c = 1 γ 2
13 Analytical solution: "Third quantization" TP, New J. Phys. 10, (2008) Consider a general solution of the Lindblad equation: dρ dt = ˆLρ := i[h, ρ] + X µ 2L µρl µ {L µl µ, ρ} for a general quadratic system of n fermions, or n qubits (spins 1/2) H = 2nX j,k=1 w j H jk w k = w H w L µ = 2nX j=1 l µ,j w j = l µ w where w j, j = 1, 2,..., 2n, are abstract Hermitian Majorana operators {w j,w k } = 2δ j,k j, k = 1,2,..., 2n
14 Analytical solution: "Third quantization" TP, New J. Phys. 10, (2008) Consider a general solution of the Lindblad equation: dρ dt = ˆLρ := i[h, ρ] + X µ 2L µρl µ {L µl µ, ρ} for a general quadratic system of n fermions, or n qubits (spins 1/2) H = 2nX j,k=1 w j H jk w k = w H w L µ = 2nX j=1 l µ,j w j = l µ w where w j, j = 1, 2,..., 2n, are abstract Hermitian Majorana operators Two physical realizations: {w j,w k } = 2δ j,k j, k = 1,2,..., 2n canonical fermions c m, w 2m 1 = c m + c m,w 2m = i(c m c m),m = 1,..., n. spins 1/2 with canonical Pauli operators σ m, m = 1,..., n, Y w 2m 1 = σj x σ z m w2m = Y σy m m <m m <m σ z m
15 Analytical solution: "Third quantization" TP, New J. Phys. 10, (2008) Consider a general solution of the Lindblad equation: dρ dt = ˆLρ := i[h, ρ] + X µ 2L µρl µ {L µl µ, ρ} for a general quadratic system of n fermions, or n qubits (spins 1/2) H = 2nX j,k=1 w j H jk w k = w H w L µ = 2nX j=1 l µ,j w j = l µ w where w j, j = 1, 2,..., 2n, are abstract Hermitian Majorana operators Two physical realizations: {w j,w k } = 2δ j,k j, k = 1,2,..., 2n canonical fermions c m, w 2m 1 = c m + c m,w 2m = i(c m c m),m = 1,..., n. spins 1/2 with canonical Pauli operators σ m, m = 1,..., n, Y w 2m 1 = σj x σ z m w2m = Y σy m m <m m <m σ z m
16 Analytical solution: "Third quantization" TP, New J. Phys. 10, (2008) Consider a general solution of the Lindblad equation: dρ dt = ˆLρ := i[h, ρ] + X µ 2L µρl µ {L µl µ, ρ} for a general quadratic system of n fermions, or n qubits (spins 1/2) H = 2nX j,k=1 w j H jk w k = w H w L µ = 2nX j=1 l µ,j w j = l µ w where w j, j = 1, 2,..., 2n, are abstract Hermitian Majorana operators Two physical realizations: {w j,w k } = 2δ j,k j, k = 1,2,..., 2n canonical fermions c m, w 2m 1 = c m + c m,w 2m = i(c m c m),m = 1,..., n. spins 1/2 with canonical Pauli operators σ m, m = 1,..., n, Y w 2m 1 = σj x σ z m w2m = Y σy m m <m m <m σ z m
17 Fock space of operators Let us associate a Hilbert space structure x x to a linear 2 2n = 4 n dimensional space K of operators, with basis P α1,α 2,...,α 2n := w α 1 1 w α 2 2 w α 2n 2n α j {0, 1} orthogonal with respect to an inner product x y = tr x y
18 Fock space of operators Let us associate a Hilbert space structure x x to a linear 2 2n = 4 n dimensional space K of operators, with basis P α1,α 2,...,α 2n := w α 1 1 w α 2 2 w α 2n 2n α j {0, 1} orthogonal with respect to an inner product x y = tr x y K is just a usual Fock space with an unusual physical interpretation.
19 Fock space of operators Let us associate a Hilbert space structure x x to a linear 2 2n = 4 n dimensional space K of operators, with basis P α1,α 2,...,α 2n := w α 1 1 w α 2 2 w α 2n 2n α j {0, 1} orthogonal with respect to an inner product x y = tr x y K is just a usual Fock space with an unusual physical interpretation. Define a set of 2n adjoint annihilation linear maps ĉ j over K ĉ j P α = δ αj,1 w j P α and derive the actions of their Hermitian adjoints - the creation linear maps ĉ, P α ĉ j Pα = Pα ĉ j P α = δ α j,1 P α w j P α = δ αj,0 P α w j P α : ĉ j Pα = δα j,0 w jp α
20 Fock space of operators Let us associate a Hilbert space structure x x to a linear 2 2n = 4 n dimensional space K of operators, with basis P α1,α 2,...,α 2n := w α 1 1 w α 2 2 w α 2n 2n α j {0, 1} orthogonal with respect to an inner product x y = tr x y K is just a usual Fock space with an unusual physical interpretation. Define a set of 2n adjoint annihilation linear maps ĉ j over K ĉ j P α = δ αj,1 w j P α and derive the actions of their Hermitian adjoints - the creation linear maps ĉ, P α ĉ j Pα = Pα ĉ j P α = δ α j,1 P α w j P α = δ αj,0 P α w j P α : ĉ j Pα = δα j,0 w jp α Clearly, ĉ j,ĉ j satisfy canonical anti-commutation relations {ĉ j,ĉ k } = 0 {ĉ j,ĉ k } = δ j,k j, k = 1,2,..., 2n
21 Representing quantum Liouvillean in 3rd quantization picture The key is to reallize now that quatum Liouville oprator ˆL of a general quadratic system can be represented as a quadratic form in a-fermi maps ĉ j,ĉ j.
22 Representing quantum Liouvillean in 3rd quantization picture The key is to reallize now that quatum Liouville oprator ˆL of a general quadratic system can be represented as a quadratic form in a-fermi maps ĉ j,ĉ j. This we do in two steps, decomposing the Liouvillean as ˆL = ˆL 0 + ˆL bath where ˆL 0 is the unitary part ˆL 0ρ := i[h, ρ] and ˆL bath = P ˆL µ µ is a sum of dissipative bath operators ˆL µρ := 2L µρl µ {L µl µ, ρ}
23 Step 1: Unitary part Let us define a Lie derivative on K: ad A B := [A, B].
24 Step 1: Unitary part Let us define a Lie derivative on K: ad A B := [A, B]. Then: ad w j w k P α = w j w k P α P αw j w k = = 2(δ αj,1δ αk,0 + δ αj,0δ αk,1) w j w k P α = = 2(ĉ j ĉ k + ĉ j ĉk) P α = 2(ĉ j ĉk ĉ kĉj) P α
25 Step 1: Unitary part Let us define a Lie derivative on K: ad A B := [A, B]. Then: ad w j w k P α = w j w k P α P αw j w k = = 2(δ αj,1δ αk,0 + δ αj,0δ αk,1) w j w k P α = = 2(ĉ j ĉ k + ĉ j ĉk) P α = 2(ĉ j ĉk ĉ kĉj) P α Extending this relation by linearity to an arbitrary element of K, it follows that for an arbitrary quadratic Hamiltonian its Lie derivative has a very similar quadratic form in a-fermi maps ˆL 0 = iad H = 4i 2nX j,k=1 ĉ j H jkĉ k = 4iĉ H ĉ
26 Step 2: Dissipative part ˆL µρ = 2L µρl µ {L µl µ, ρ} = 2nX j,k=1 where we write ˆL j,k ρ := 2w j ρw k w k w j ρ ρw k w j. l µ,j l µ,k ˆL j,k ρ
27 Step 2: Dissipative part ˆL µρ = 2L µρl µ {L µl µ, ρ} = 2nX j,k=1 where we write ˆL j,k ρ := 2w j ρw k w k w j ρ ρw k w j. l µ,j l µ,k ˆL j,k ρ Straighforward calculation shows that, if we restrict ˆL j,k to the subspace of K with even number a fermions, i.e. subspace spanned by P α with even α := α α 2n, then ˆL j,k = 4ĉ j ĉ k 2ĉ j ĉk 2ĉ kĉj
28 Liouvillean in 3rd quantization: result Putting 1 and 2 together we find the original problem formulated as a quadratic problem in 2n a-fermions d dt ρ = ˆLρ namely ˆL = 2ĉ (2iH + M + M T ) ĉ + 2ĉ (M M T )ĉ where M = M is a complex Hermitian matrix parametrizing the Lindblad operators M jk = P µ l µ,jl µ,k.
29 Liouvillean in 3rd quantization: result Putting 1 and 2 together we find the original problem formulated as a quadratic problem in 2n a-fermions d dt ρ = ˆLρ namely ˆL = 2ĉ (2iH + M + M T ) ĉ + 2ĉ (M M T )ĉ where M = M is a complex Hermitian matrix parametrizing the Lindblad operators M jk = P µ l µ,jl µ,k. The question which remains is whether a linear canonical transformation to a new set of canonical a-fermi maps, we shall call them normal master modes ˆb j, so that ˆL becomes diagonal quadratic form in ˆb j?
30 Liouvillean in 3rd quantization: result Putting 1 and 2 together we find the original problem formulated as a quadratic problem in 2n a-fermions d dt ρ = ˆLρ namely ˆL = 2ĉ (2iH + M + M T ) ĉ + 2ĉ (M M T )ĉ where M = M is a complex Hermitian matrix parametrizing the Lindblad operators M jk = P µ l µ,jl µ,k. The question which remains is whether a linear canonical transformation to a new set of canonical a-fermi maps, we shall call them normal master modes ˆb j, so that ˆL becomes diagonal quadratic form in ˆb j? Below we provide a positive constructive answer to the question above, in 2 steps.
31 Reduction to normal master modes: Step 1 Following Lieb, Schultz and Mattis (1961), generalizing to non-hermitean case:
32 Reduction to normal master modes: Step 1 Following Lieb, Schultz and Mattis (1961), generalizing to non-hermitean case: Define 4n adjoint Hermitian Majorana maps â r = â r, r = 1,2,..., 4n: â 2j 1 = 1 2 (ĉ j + ĉ j ) â 2j = i 2 (ĉ j ĉ j ) satisfying the anti-commutation relations {â r,â s} = δ r,s in terms of which the Liouvillean can be rewritten as ˆL = â A â A 0ˆ½ (3) where A 0 = 2 P 2n j=1 M jj = 2trM and A antisymmetric complex 4n 4n matrix A 2j 1,2k 1 = 2iH jk M jk + M kj A 2j 1,2k = 2iM kj A 2j,2k 1 = 2iM jk A 2j,2k = 2iH jk + M jk M kj
33 Reduction to normal master modes: Step 2 Lemma: Diagonalizable complex anti-symmetrix matrix can be written as β β A = V T BV where B = β β 2 0 C A where the set of 2n c-numbers ordered as Reβ 1 Reβ 2... β 2n 0 shall be called rapidities, and eigenvector matrix V satisfies VV T = J = C A
34 Reduction to normal master modes: result Using the previous decomposition of A we rewrite the Liouvillean as ˆL = â V T BVâ A 0ˆ½ = (Vâ) B(Vâ) A 0ˆ½
35 Reduction to normal master modes: result Using the previous decomposition of A we rewrite the Liouvillean as ˆL = â V T BVâ A 0ˆ½ = (Vâ) B(Vâ) A 0ˆ½ Let us define the Normal-master-mode (NMM) maps ˆb := (ˆb 1, ˆb 1, ˆb 2, ˆb 2,..., ˆb 2n, ˆb 2n) := Vâ which, due to orthogonality of matrix V satisfy canonical relations {ˆb j, ˆb k } = 0 {ˆb j, ˆb k} = δ j,k {ˆb j, ˆb k} = 0
36 Reduction to normal master modes: result Using the previous decomposition of A we rewrite the Liouvillean as ˆL = â V T BVâ A 0ˆ½ = (Vâ) B(Vâ) A 0ˆ½ Let us define the Normal-master-mode (NMM) maps ˆb := (ˆb 1, ˆb 1, ˆb 2, ˆb 2,..., ˆb 2n, ˆb 2n) := Vâ which, due to orthogonality of matrix V satisfy canonical relations {ˆb j, ˆb k } = 0 {ˆb j, ˆb k} = δ j,k {ˆb j, ˆb k} = 0 ˆb j could be interpreted as annihilation map and ˆb j as a creation map of jth NMM, but we should note that ˆb j is in general not the Hermitian adjoint of ˆb j.
37 Reduction to normal master modes: result Using the previous decomposition of A we rewrite the Liouvillean as ˆL = â V T BVâ A 0ˆ½ = (Vâ) B(Vâ) A 0ˆ½ Let us define the Normal-master-mode (NMM) maps ˆb := (ˆb 1, ˆb 1, ˆb 2, ˆb 2,..., ˆb 2n, ˆb 2n) := Vâ which, due to orthogonality of matrix V satisfy canonical relations {ˆb j, ˆb k } = 0 {ˆb j, ˆb k} = δ j,k {ˆb j, ˆb k} = 0 ˆb j could be interpreted as annihilation map and ˆb j as a creation map of jth NMM, but we should note that ˆb j is in general not the Hermitian adjoint of ˆb j. In terms of NMM the Liouvillean achieves a very convenient normal form ˆL = 2 2nX j=1 β jˆb jˆbj
38 Non-equilibrium steady state: Liouvillean vacuum
39 Non-equilibrium steady state: Liouvillean vacuum From ˆL = 2ĉ (2iH+M+M T ) ĉ +2ĉ (M M T ) ĉ it follows immediately that 1 = P (0,0...,0) is left-annihilated by ˆL, 1 ˆL = 0.
40 Non-equilibrium steady state: Liouvillean vacuum From ˆL = 2ĉ (2iH+M+M T ) ĉ +2ĉ (M M T ) ĉ it follows immediately that 1 = P (0,0...,0) is left-annihilated by ˆL, 1 ˆL = 0. So we have just shown that 0 is always an eigenvalue of ˆL +, hence there should also exist the corresponding right eigenvector NESS, normalized as 1 NESS = tr ρ NESS = 1, which represents density matrix of non-equilibrium steady state ˆL NESS = 0
41 Non-equilibrium steady state: Liouvillean vacuum From ˆL = 2ĉ (2iH+M+M T ) ĉ +2ĉ (M M T ) ĉ it follows immediately that 1 = P (0,0...,0) is left-annihilated by ˆL, 1 ˆL = 0. So we have just shown that 0 is always an eigenvalue of ˆL +, hence there should also exist the corresponding right eigenvector NESS, normalized as 1 NESS = tr ρ NESS = 1, which represents density matrix of non-equilibrium steady state ˆL NESS = 0 Like any Fock vacuum NESS is uniquely determined thru the relations ˆb j NESS = 0, j = 1,2,... 2n
42 Non-equilibrium steady state: Liouvillean vacuum From ˆL = 2ĉ (2iH+M+M T ) ĉ +2ĉ (M M T ) ĉ it follows immediately that 1 = P (0,0...,0) is left-annihilated by ˆL, 1 ˆL = 0. So we have just shown that 0 is always an eigenvalue of ˆL +, hence there should also exist the corresponding right eigenvector NESS, normalized as 1 NESS = tr ρ NESS = 1, which represents density matrix of non-equilibrium steady state ˆL NESS = 0 Like any Fock vacuum NESS is uniquely determined thru the relations ˆb j NESS = 0, j = 1,2,... 2n And similarly we have for the left-vacuum state 1 ˆb j = 0, j = 1,2,... 2n
43 Complete excitation spectrum of quantum Liouvillean The complete excitation spectrum and the corresponding left/right eigenvectors of the Liouvillean are given in terms of a sequence of 2n binary integers (NMM occupation numbers) ν = (ν 1, ν 2,..., ν 2n), ν j {0, 1}, namely λ ν = 2 Θ L ν ˆL = λ ν Θ L ν ˆL Θ R ν = λ ν Θ R ν 2nX j=1 β j ν j Θ L ν = 1 ˆb ν 2n 2n ˆb ν 2 ˆb ν Θ R ν = ˆb ν 1 1 ˆb ν 2 2 ˆb ν 2n 2n NESS where by construction, left and right eigenvectors satisfy the bi-orthonormality relation Θ L ν ΘR ν = δ ν,ν.
44 NESS expectation values of physical observables Assume that all the rapidities are strictly away from the real line Reβ j > 0. Then: 1 NESS is unique. 2 Any initial density matrix relaxes to NESS with an exponential rate = 2min Reβ j. 3 Then the expectation value of any quadratic observable w j w k in a (unique) NESS can be explicitly computed as where w j w k NESS = δ j,k + 1 ĉ j ĉ k NESS = δ j,k + 2 = δ j,k 1 π 2nX m=1 Z G(ω) = (A iω½) 1 V 2m,2j 1 V 2m 1,2k 1 dω G 2j 1,2k 1 (ω)
45 NESS expectation values of physical observables Assume that all the rapidities are strictly away from the real line Reβ j > 0. Then: 1 NESS is unique. 2 Any initial density matrix relaxes to NESS with an exponential rate = 2min Reβ j. 3 Then the expectation value of any quadratic observable w j w k in a (unique) NESS can be explicitly computed as where w j w k NESS = δ j,k + 1 ĉ j ĉ k NESS = δ j,k + 2 = δ j,k 1 π 2nX m=1 Z G(ω) = (A iω½) 1 V 2m,2j 1 V 2m 1,2k 1 dω G 2j 1,2k 1 (ω)
46 NESS expectation values of physical observables Assume that all the rapidities are strictly away from the real line Reβ j > 0. Then: 1 NESS is unique. 2 Any initial density matrix relaxes to NESS with an exponential rate = 2min Reβ j. 3 Then the expectation value of any quadratic observable w j w k in a (unique) NESS can be explicitly computed as where w j w k NESS = δ j,k + 1 ĉ j ĉ k NESS = δ j,k + 2 = δ j,k 1 π 2nX m=1 Z G(ω) = (A iω½) 1 V 2m,2j 1 V 2m 1,2k 1 dω G 2j 1,2k 1 (ω)
47 NESS expectation values of physical observables Assume that all the rapidities are strictly away from the real line Reβ j > 0. Then: 1 NESS is unique. 2 Any initial density matrix relaxes to NESS with an exponential rate = 2min Reβ j. 3 Then the expectation value of any quadratic observable w j w k in a (unique) NESS can be explicitly computed as where w j w k NESS = δ j,k + 1 ĉ j ĉ k NESS = δ j,k + 2 = δ j,k 1 π 2nX m=1 Z G(ω) = (A iω½) 1 V 2m,2j 1 V 2m 1,2k 1 dω G 2j 1,2k 1 (ω)
48 Coming back to XY chain 0 1 B L h 1R R R T 1 h 2R R A = 0 R T 2 h 3R., A 0 = Γ L + + Γ R +, C Rn 1 A 0 0 R T n 1 B R h nr where B L := B Γ L +,Γ L, BR := B Γ R +,ΓR, ΓL,R ± R m := := ΓL,R 2 ± Γ L,R 1, and Jm y B Jm y Jm x A, R := B A, 0 Jm x i 0 2 Γ+ 1 i 2 Γ 1 2 Γ B Γ+,Γ := B i 2 Γ Γ i 2 i 2 Γ 1 2 Γ 0 C i 2 Γ+ A 1 2 Γ i 2 Γ i 2 Γ+ 0
49 Transverse Ising chain First we discuss a simple special case of homogeneous transverse Ising chain, say with numerical values J x m 1.5, J y m 0, h m 1.0 and take non-equilibrium bath couplings Γ L 1 = 1,Γ L 2 = 0.6, Γ R 1 = 1, Γ R 2 = 0.3
50 Transverse Ising chain First we discuss a simple special case of homogeneous transverse Ising chain, say with numerical values J x m 1.5, J y m 0, h m 1.0 and take non-equilibrium bath couplings Γ L 1 = 1,Γ L 2 = 0.6, Γ R 1 = 1, Γ R 2 = 0.3
51 Transverse Ising chain First we discuss a simple special case of homogeneous transverse Ising chain, say with numerical values ImΒ 2 J x m 1.5, J y m 0, h m 1.0 and take non-equilibrium bath couplings Γ L 1 = 1,Γ L 2 = 0.6, Γ R 1 = 1, Γ R 2 = ImΒ ReΒ Rapidity spectrum for different n: (note analytic results for n = with colored symbols!) ReΒ 2 ImΒ ReΒ 2
52 Spectral gap of Liouvillean The rate of relaxation to NESS is given by the spectral gap of ˆL.
53 Spectral gap of Liouvillean The rate of relaxation to NESS is given by the spectral gap of ˆL. We find explicit analytical result: = C(J x,h, Γ L,R 1,2 ) n 3
54 Spectral gap of Liouvillean The rate of relaxation to NESS is given by the spectral gap of ˆL. We find explicit analytical result: Comparing to numerics: 15 = C(J x,h, Γ L,R 1,2 ) n 3 10 n 3 5 Log n 3 C Log n n
55 Complete spectrum of Liouvillean for n = ImΛ ReΛ
56 Quantum phase transition far from equilibrium in XY chain TP & I. Pižorn, PRL 101, (2008) Jm x = (1 + γ)/2 Jm y = (1 γ)/2, h m = h C(j,k) = σj z σk z σj z σk z 1 Cres h 0.9 hc h 0.3 hc h ImΒ Γ ReΒ 0 Π Π Φ ReΒ h c = 1 γ 2
57 Spin-spin correlations in NESS (XY chain) Saturation vs. exponential decay & power law critical scaling at the critical point. ln C r a r n ln C r Α ln n h h c n ln r r b h h c lnξ ln h
58 Fluctuation of spin-spin correlation in NESS and "wave resonators" Near QPT: Scaling variable z = (h c h)n 2
59 Fluctuation of spin-spin correlation in NESS and "wave resonators" Near QPT: Scaling variable z = (h c h)n 2 Scaling ansatz: C 2j+α,2k+β = Ψ α,β (x = j/n, y = k/n,z)
60 Fluctuation of spin-spin correlation in NESS and "wave resonators" Near QPT: Scaling variable z = (h c h)n 2 Scaling ansatz: C 2j+α,2k+β = Ψ α,β (x = j/n, y = k/n,z) Certain combination Ψ(x,y) = ( / x + / y)(ψ 0,0 (x,y) + Ψ 1,1 (x,y)) obeys Helmoltz equation!!! «2 x y + 4z Ψ = octopole antenna sources 2
61 Operator space entanglement entropy of NESS (XY chain) Von Neumann entropy of a bipartition of NESS as an element of a Fock space Drastically different behaviour than for entanglement entropy of ground states of 1D critical/non-critical models! S n ln slope ln h c h 10 h 0.5 h h h n
62 Summary Method for exact solution of many-body open quantum systems has been outlined. Quantum phase transition far from equilibrium discovered - experimental tests desirable! Possible new realizations of wave/quantum chaos!?
63 Summary Method for exact solution of many-body open quantum systems has been outlined. Quantum phase transition far from equilibrium discovered - experimental tests desirable! Possible new realizations of wave/quantum chaos!?
64 Summary Method for exact solution of many-body open quantum systems has been outlined. Quantum phase transition far from equilibrium discovered - experimental tests desirable! Possible new realizations of wave/quantum chaos!?
65 Summary Method for exact solution of many-body open quantum systems has been outlined. Quantum phase transition far from equilibrium discovered - experimental tests desirable! Possible new realizations of wave/quantum chaos!?
(Quantum) chaos theory and statistical physics far from equilibrium:
(Quantum) chaos theory and statistical physics far from equilibrium: Introducing the group for Non-equilibrium quantum and statistical physics Department of physics, Faculty of mathematics and physics,
More informationIntroduction to the Mathematics of the XY -Spin Chain
Introduction to the Mathematics of the XY -Spin Chain Günter Stolz June 9, 2014 Abstract In the following we present an introduction to the mathematical theory of the XY spin chain. The importance of this
More informationarxiv: v2 [quant-ph] 12 Aug 2008
Complexity of thermal states in quantum spin chains arxiv:85.449v [quant-ph] Aug 8 Marko Žnidarič, Tomaž Prosen and Iztok Pižorn Department of physics, FMF, University of Ljubljana, Jadranska 9, SI- Ljubljana,
More informationQuantization of the Spins
Chapter 5 Quantization of the Spins As pointed out already in chapter 3, the external degrees of freedom, position and momentum, of an ensemble of identical atoms is described by the Scödinger field operator.
More informationMagnets, 1D quantum system, and quantum Phase transitions
134 Phys620.nb 10 Magnets, 1D quantum system, and quantum Phase transitions In 1D, fermions can be mapped into bosons, and vice versa. 10.1. magnetization and frustrated magnets (in any dimensions) Consider
More informationClassical and quantum simulation of dissipative quantum many-body systems
0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 Classical and quantum simulation of dissipative quantum many-body systems
More informationarxiv: v2 [cond-mat.stat-mech] 8 Aug 2011
Non-equilibrium phase transition in a periodically driven XY spin chain Tomaž Prosen 1,2 and Enej Ilievski 1 1 Department of Physics, FMF, University of Ljubljana, 1000 Ljubljana, Slovenia and 2 Institute
More informationRandom Fermionic Systems
Random Fermionic Systems Fabio Cunden Anna Maltsev Francesco Mezzadri University of Bristol December 9, 2016 Maltsev (University of Bristol) Random Fermionic Systems December 9, 2016 1 / 27 Background
More informationMultipartite entanglement in fermionic systems via a geometric
Multipartite entanglement in fermionic systems via a geometric measure Department of Physics University of Pune Pune - 411007 International Workshop on Quantum Information HRI Allahabad February 2012 In
More informationProblem Set No. 3: Canonical Quantization Due Date: Wednesday October 19, 2018, 5:00 pm. 1 Spin waves in a quantum Heisenberg antiferromagnet
Physics 58, Fall Semester 018 Professor Eduardo Fradkin Problem Set No. 3: Canonical Quantization Due Date: Wednesday October 19, 018, 5:00 pm 1 Spin waves in a quantum Heisenberg antiferromagnet In this
More information3 Symmetry Protected Topological Phase
Physics 3b Lecture 16 Caltech, 05/30/18 3 Symmetry Protected Topological Phase 3.1 Breakdown of noninteracting SPT phases with interaction Building on our previous discussion of the Majorana chain and
More informationSecond Quantization: Quantum Fields
Second Quantization: Quantum Fields Bosons and Fermions Let X j stand for the coordinate and spin subscript (if any) of the j-th particle, so that the vector of state Ψ of N particles has the form Ψ Ψ(X
More informationarxiv: v2 [quant-ph] 17 Jan 2014
Nonequilibrium Quantum Phase Transitions in the XY model: comparison of unitary time evolution and reduced density operator approaches arxiv:1309.4725v2 [quant-ph] 17 Jan 2014 Shigeru Ajisaka, Felipe Barra,
More informationOn the Random XY Spin Chain
1 CBMS: B ham, AL June 17, 2014 On the Random XY Spin Chain Robert Sims University of Arizona 2 The Isotropic XY-Spin Chain Fix a real-valued sequence {ν j } j 1 and for each integer n 1, set acting on
More informationAttempts at relativistic QM
Attempts at relativistic QM based on S-1 A proper description of particle physics should incorporate both quantum mechanics and special relativity. However historically combining quantum mechanics and
More informationThe Postulates of Quantum Mechanics
p. 1/23 The Postulates of Quantum Mechanics We have reviewed the mathematics (complex linear algebra) necessary to understand quantum mechanics. We will now see how the physics of quantum mechanics fits
More informationAlgebraic Theory of Entanglement
Algebraic Theory of (arxiv: 1205.2882) 1 (in collaboration with T.R. Govindarajan, A. Queiroz and A.F. Reyes-Lega) 1 Physics Department, Syracuse University, Syracuse, N.Y. and The Institute of Mathematical
More information10. Zwanzig-Mori Formalism
University of Rhode Island DigitalCommons@URI Nonequilibrium Statistical Physics Physics Course Materials 205 0. Zwanzig-Mori Formalism Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative
More informationThe AKLT Model. Lecture 5. Amanda Young. Mathematics, UC Davis. MAT290-25, CRN 30216, Winter 2011, 01/31/11
1 The AKLT Model Lecture 5 Amanda Young Mathematics, UC Davis MAT290-25, CRN 30216, Winter 2011, 01/31/11 This talk will follow pg. 26-29 of Lieb-Robinson Bounds in Quantum Many-Body Physics by B. Nachtergaele
More information2.4.8 Heisenberg Algebra, Fock Space and Harmonic Oscillator
.4. SPECTRAL DECOMPOSITION 63 Let P +, P, P 1,..., P p be the corresponding orthogonal complimentary system of projections, that is, P + + P + p P i = I. i=1 Then there exists a corresponding system of
More informationEntanglement Dynamics for the Quantum Disordered XY Chain
Entanglement Dynamics for the Quantum Disordered XY Chain Houssam Abdul-Rahman Joint with: B. Nachtergaele, R. Sims, G. Stolz AMS Southeastern Sectional Meeting University of Georgia March 6, 2016 Houssam
More informationEntanglement Entropy in Extended Quantum Systems
Entanglement Entropy in Extended Quantum Systems John Cardy University of Oxford STATPHYS 23 Genoa Outline A. Universal properties of entanglement entropy near quantum critical points B. Behaviour of entanglement
More information1 Quantum field theory and Green s function
1 Quantum field theory and Green s function Condensed matter physics studies systems with large numbers of identical particles (e.g. electrons, phonons, photons) at finite temperature. Quantum field theory
More informationFermionic coherent states in infinite dimensions
Fermionic coherent states in infinite dimensions Robert Oeckl Centro de Ciencias Matemáticas Universidad Nacional Autónoma de México Morelia, Mexico Coherent States and their Applications CIRM, Marseille,
More information2. Introduction to quantum mechanics
2. Introduction to quantum mechanics 2.1 Linear algebra Dirac notation Complex conjugate Vector/ket Dual vector/bra Inner product/bracket Tensor product Complex conj. matrix Transpose of matrix Hermitian
More informationTheoretical design of a readout system for the Flux Qubit-Resonator Rabi Model in the ultrastrong coupling regime
Theoretical design of a readout system for the Flux Qubit-Resonator Rabi Model in the ultrastrong coupling regime Ceren Burçak Dağ Supervisors: Dr. Pol Forn-Díaz and Assoc. Prof. Christopher Wilson Institute
More informationarxiv: v2 [quant-ph] 19 Sep 2011
Exact nonequilibrium steady state of a strongly driven open XXZ chain Tomaž Prosen Department of Physics, FMF, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia (Dated: September 20, 2011)
More information10. Zwanzig-Mori Formalism
University of Rhode Island DigitalCommons@URI Nonequilibrium Statistical Physics Physics Course Materials 0-9-205 0. Zwanzig-Mori Formalism Gerhard Müller University of Rhode Island, gmuller@uri.edu Follow
More information(1.1) In particular, ψ( q 1, m 1 ; ; q N, m N ) 2 is the probability to find the first particle
Chapter 1 Identical particles 1.1 Distinguishable particles The Hilbert space of N has to be a subspace H = N n=1h n. Observables Ân of the n-th particle are self-adjoint operators of the form 1 1 1 1
More informationRydberg Quantum Simulation Ground State Preparation by Master Equation
Rydberg Quantum Simulation Ground State Preparation by Master Equation Henri Menke 5. Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany (Talk: January 29, 2015,
More informationS.K. Saikin May 22, Lecture 13
S.K. Saikin May, 007 13 Decoherence I Lecture 13 A physical qubit is never isolated from its environment completely. As a trivial example, as in the case of a solid state qubit implementation, the physical
More informationarxiv: v2 [quant-ph] 4 Dec 2008
Matrix product simulations of non-equilibrium steady states of quantum spin chains arxiv:0811.4188v2 [quant-ph] 4 Dec 2008 Submitted to: J. Stat. Mech. 1. Introduction Tomaž Prosen and Marko Žnidarič Physics
More information1 More on the Bloch Sphere (10 points)
Ph15c Spring 017 Prof. Sean Carroll seancarroll@gmail.com Homework - 1 Solutions Assigned TA: Ashmeet Singh ashmeet@caltech.edu 1 More on the Bloch Sphere 10 points a. The state Ψ is parametrized on the
More informationQuantum Entanglement and the Bell Matrix
Quantum Entanglement and the Bell Matrix Marco Pedicini (Roma Tre University) in collaboration with Anna Chiara Lai and Silvia Rognone (La Sapienza University of Rome) SIMAI2018 - MS27: Discrete Mathematics,
More informationMean-Field Theory: HF and BCS
Mean-Field Theory: HF and BCS Erik Koch Institute for Advanced Simulation, Jülich N (x) = p N! ' (x ) ' 2 (x ) ' N (x ) ' (x 2 ) ' 2 (x 2 ) ' N (x 2 )........ ' (x N ) ' 2 (x N ) ' N (x N ) Slater determinant
More informationPh 219/CS 219. Exercises Due: Friday 20 October 2006
1 Ph 219/CS 219 Exercises Due: Friday 20 October 2006 1.1 How far apart are two quantum states? Consider two quantum states described by density operators ρ and ρ in an N-dimensional Hilbert space, and
More informationSystems of Identical Particles
qmc161.tex Systems of Identical Particles Robert B. Griffiths Version of 21 March 2011 Contents 1 States 1 1.1 Introduction.............................................. 1 1.2 Orbitals................................................
More informationSpin-Boson Model. A simple Open Quantum System. M. Miller F. Tschirsich. Quantum Mechanics on Macroscopic Scales Theory of Condensed Matter July 2012
Spin-Boson Model A simple Open Quantum System M. Miller F. Tschirsich Quantum Mechanics on Macroscopic Scales Theory of Condensed Matter July 2012 Outline 1 Bloch-Equations 2 Classical Dissipations 3 Spin-Boson
More informationFrustration-free Ground States of Quantum Spin Systems 1
1 Davis, January 19, 2011 Frustration-free Ground States of Quantum Spin Systems 1 Bruno Nachtergaele (UC Davis) based on joint work with Sven Bachmann, Spyridon Michalakis, Robert Sims, and Reinhard Werner
More informationQuantum Many Body Systems and Tensor Networks
Quantum Many Body Systems and Tensor Networks Aditya Jain International Institute of Information Technology, Hyderabad aditya.jain@research.iiit.ac.in July 30, 2015 Aditya Jain (IIIT-H) Quantum Hamiltonian
More informationDensity Matrices. Chapter Introduction
Chapter 15 Density Matrices 15.1 Introduction Density matrices are employed in quantum mechanics to give a partial description of a quantum system, one from which certain details have been omitted. For
More informationFinite temperature form factors in the free Majorana theory
Finite temperature form factors in the free Majorana theory Benjamin Doyon Rudolf Peierls Centre for Theoretical Physics, Oxford University, UK supported by EPSRC postdoctoral fellowship hep-th/0506105
More informationShared Purity of Multipartite Quantum States
Shared Purity of Multipartite Quantum States Anindya Biswas Harish-Chandra Research Institute December 3, 2013 Anindya Biswas (HRI) Shared Purity December 3, 2013 1 / 38 Outline of the talk 1 Motivation
More informationTopological Phases in One Dimension
Topological Phases in One Dimension Lukasz Fidkowski and Alexei Kitaev arxiv:1008.4138 Topological phases in 2 dimensions: - Integer quantum Hall effect - quantized σ xy - robust chiral edge modes - Fractional
More information2 The Density Operator
In this chapter we introduce the density operator, which provides an alternative way to describe the state of a quantum mechanical system. So far we have only dealt with situations where the state of a
More informationEfficient time evolution of one-dimensional quantum systems
Efficient time evolution of one-dimensional quantum systems Frank Pollmann Max-Planck-Institut für komplexer Systeme, Dresden, Germany Sep. 5, 2012 Hsinchu Problems we will address... Finding ground states
More informationQuantum field theory and Green s function
1 Quantum field theory and Green s function Condensed matter physics studies systems with large numbers of identical particles (e.g. electrons, phonons, photons) at finite temperature. Quantum field theory
More informationPhysics 4022 Notes on Density Matrices
Physics 40 Notes on Density Matrices Definition: For a system in a definite normalized state ψ > the density matrix ρ is ρ = ψ >< ψ 1) From Eq 1 it is obvious that in the basis defined by ψ > and other
More informationVariational analysis of dissipative Ising models
GRK Workshop Hildesheim 2016 Institute for Theoretical Physics Leibniz University Hannover 08.02.2016 Outline 1 Dissipative quantum systems 2 Variational formulation and dynamics 3 Non-Markovianity 4 Steady
More informationAnderson Localization Looking Forward
Anderson Localization Looking Forward Boris Altshuler Physics Department, Columbia University Collaborations: Also Igor Aleiner Denis Basko, Gora Shlyapnikov, Vincent Michal, Vladimir Kravtsov, Lecture2
More informationarxiv:quant-ph/ v5 10 Feb 2003
Quantum entanglement of identical particles Yu Shi Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom and Theory of
More informationarxiv: v1 [cond-mat.stat-mech] 7 Nov 2018
An Extension of ETH to Non-Equilibrium Steady States arxiv:1813114v1 [cond-mat.stat-mech] 7 Nov 018 Sanjay Moudgalya, 1 Trithep Devakul, 1, D. P. Arovas, 3 and S. L. Sondhi 1 1 Department of Physics, Princeton
More informationQuantum error correction for continuously detected errors
Quantum error correction for continuously detected errors Charlene Ahn, Toyon Research Corporation Howard Wiseman, Griffith University Gerard Milburn, University of Queensland Quantum Information and Quantum
More informationFermions in Quantum Complexity Theory
Fermions in Quantum Complexity Theory Edwin Ng MIT Department of Physics December 14, 2012 Occupation Number Formalism Consider n identical particles in m modes. If there are x j particles in the jth mode,
More informationJournal Club: Brief Introduction to Tensor Network
Journal Club: Brief Introduction to Tensor Network Wei-Han Hsiao a a The University of Chicago E-mail: weihanhsiao@uchicago.edu Abstract: This note summarizes the talk given on March 8th 2016 which was
More informationQuantum Mechanics Solutions. λ i λ j v j v j v i v i.
Quantum Mechanics Solutions 1. (a) If H has an orthonormal basis consisting of the eigenvectors { v i } of A with eigenvalues λ i C, then A can be written in terms of its spectral decomposition as A =
More informationLecture notes on topological insulators
Lecture notes on topological insulators Ming-Che Chang Department of Physics, National Taiwan Normal University, Taipei, Taiwan Dated: May 8, 07 I. D p-wave SUPERCONDUCTOR Here we study p-wave SC in D
More informationMP 472 Quantum Information and Computation
MP 472 Quantum Information and Computation http://www.thphys.may.ie/staff/jvala/mp472.htm Outline Open quantum systems The density operator ensemble of quantum states general properties the reduced density
More informationSECOND QUANTIZATION. notes by Luca G. Molinari. (oct revised oct 2016)
SECOND QUANTIZATION notes by Luca G. Molinari (oct 2001- revised oct 2016) The appropriate formalism for the quantum description of identical particles is second quantisation. There are various equivalent
More informationA Simple Model of Quantum Trajectories. Todd A. Brun University of Southern California
A Simple Model of Quantum Trajectories Todd A. Brun University of Southern California Outline 1. Review projective and generalized measurements. 2. A simple model of indirect measurement. 3. Weak measurements--jump-like
More information4.3 Lecture 18: Quantum Mechanics
CHAPTER 4. QUANTUM SYSTEMS 73 4.3 Lecture 18: Quantum Mechanics 4.3.1 Basics Now that we have mathematical tools of linear algebra we are ready to develop a framework of quantum mechanics. The framework
More informationBRST and Dirac Cohomology
BRST and Dirac Cohomology Peter Woit Columbia University Dartmouth Math Dept., October 23, 2008 Peter Woit (Columbia University) BRST and Dirac Cohomology October 2008 1 / 23 Outline 1 Introduction 2 Representation
More informationUniversal Quantum Simulator, Local Convertibility and Edge States in Many-Body Systems Fabio Franchini
New Frontiers in Theoretical Physics XXXIV Convegno Nazionale di Fisica Teorica - Cortona Universal Quantum Simulator, Local Convertibility and Edge States in Many-Body Systems Fabio Franchini Collaborators:
More informationLecture 6 The Super-Higgs Mechanism
Lecture 6 The Super-Higgs Mechanism Introduction: moduli space. Outline Explicit computation of moduli space for SUSY QCD with F < N and F N. The Higgs mechanism. The super-higgs mechanism. Reading: Terning
More informationTensor operators: constructions and applications for long-range interaction systems
In this paper we study systematic ways to construct such tensor network descriptions of arbitrary operators using linear tensor networks, so-called matrix product operators (MPOs), and prove the optimality
More information4 Matrix product states
Physics 3b Lecture 5 Caltech, 05//7 4 Matrix product states Matrix product state (MPS) is a highly useful tool in the study of interacting quantum systems in one dimension, both analytically and numerically.
More informationTime Evolving Block Decimation Algorithm
Time Evolving Block Decimation Algorithm Application to bosons on a lattice Jakub Zakrzewski Marian Smoluchowski Institute of Physics and Mark Kac Complex Systems Research Center, Jagiellonian University,
More informationQuantum Mechanics I Physics 5701
Quantum Mechanics I Physics 5701 Z. E. Meziani 02/24//2017 Physics 5701 Lecture Commutation of Observables and First Consequences of the Postulates Outline 1 Commutation Relations 2 Uncertainty Relations
More information(Dynamical) quantum typicality: What is it and what are its physical and computational implications?
(Dynamical) : What is it and what are its physical and computational implications? Jochen Gemmer University of Osnabrück, Kassel, May 13th, 214 Outline Thermal relaxation in closed quantum systems? Typicality
More informationMP463 QUANTUM MECHANICS
MP463 QUANTUM MECHANICS Introduction Quantum theory of angular momentum Quantum theory of a particle in a central potential - Hydrogen atom - Three-dimensional isotropic harmonic oscillator (a model of
More information1 Fundamental physical postulates. C/CS/Phys C191 Quantum Mechanics in a Nutshell I 10/04/07 Fall 2007 Lecture 12
C/CS/Phys C191 Quantum Mechanics in a Nutshell I 10/04/07 Fall 2007 Lecture 12 In this and the next lecture we summarize the essential physical and mathematical aspects of quantum mechanics relevant to
More informationEntanglement in Many-Body Fermion Systems
Entanglement in Many-Body Fermion Systems Michelle Storms 1, 2 1 Department of Physics, University of California Davis, CA 95616, USA 2 Department of Physics and Astronomy, Ohio Wesleyan University, Delaware,
More informationThe Quantum Heisenberg Ferromagnet
The Quantum Heisenberg Ferromagnet Soon after Schrödinger discovered the wave equation of quantum mechanics, Heisenberg and Dirac developed the first successful quantum theory of ferromagnetism W. Heisenberg,
More informationQuantum Field Theory
Quantum Field Theory PHYS-P 621 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory 1 Attempts at relativistic QM based on S-1 A proper description of particle physics
More informationMixing Quantum and Classical Mechanics: A Partially Miscible Solution
Mixing Quantum and Classical Mechanics: A Partially Miscible Solution R. Kapral S. Nielsen A. Sergi D. Mac Kernan G. Ciccotti quantum dynamics in a classical condensed phase environment how to simulate
More informationwhere P a is a projector to the eigenspace of A corresponding to a. 4. Time evolution of states is governed by the Schrödinger equation
1 Content of the course Quantum Field Theory by M. Srednicki, Part 1. Combining QM and relativity We are going to keep all axioms of QM: 1. states are vectors (or rather rays) in Hilbert space.. observables
More informationarxiv: v2 [math-ph] 2 Jun 2014
Spin Operators Pauli Group Commutators Anti-Commutators Kronecker product and Applications Willi-Hans Steeb and Yorick Hardy arxiv:45.5749v [math-ph] Jun 4 International School for Scientific Computing
More informationQuantum Mechanics C (130C) Winter 2014 Assignment 7
University of California at San Diego Department of Physics Prof. John McGreevy Quantum Mechanics C (130C) Winter 014 Assignment 7 Posted March 3, 014 Due 11am Thursday March 13, 014 This is the last problem
More informationPHY 396 K. Problem set #5. Due October 9, 2008.
PHY 396 K. Problem set #5. Due October 9, 2008.. First, an exercise in bosonic commutation relations [â α, â β = 0, [â α, â β = 0, [â α, â β = δ αβ. ( (a Calculate the commutators [â αâ β, â γ, [â αâ β,
More informationSupplementary Information for Experimental signature of programmable quantum annealing
Supplementary Information for Experimental signature of programmable quantum annealing Supplementary Figures 8 7 6 Ti/exp(step T =.5 Ti/step Temperature 5 4 3 2 1 0 0 100 200 300 400 500 Annealing step
More informationLecture 5. Hartree-Fock Theory. WS2010/11: Introduction to Nuclear and Particle Physics
Lecture 5 Hartree-Fock Theory WS2010/11: Introduction to Nuclear and Particle Physics Particle-number representation: General formalism The simplest starting point for a many-body state is a system of
More informationNumerical Simulation of Spin Dynamics
Numerical Simulation of Spin Dynamics Marie Kubinova MATH 789R: Advanced Numerical Linear Algebra Methods with Applications November 18, 2014 Introduction Discretization in time Computing the subpropagators
More informationThe Dirac Equation. Topic 3 Spinors, Fermion Fields, Dirac Fields Lecture 13
The Dirac Equation Dirac s discovery of a relativistic wave equation for the electron was published in 1928 soon after the concept of intrisic spin angular momentum was proposed by Goudsmit and Uhlenbeck
More informationEntanglement spectrum and Matrix Product States
Entanglement spectrum and Matrix Product States Frank Verstraete J. Haegeman, D. Draxler, B. Pirvu, V. Stojevic, V. Zauner, I. Pizorn I. Cirac (MPQ), T. Osborne (Hannover), N. Schuch (Aachen) Outline Valence
More informationMatrix Product States
Matrix Product States Ian McCulloch University of Queensland Centre for Engineered Quantum Systems 28 August 2017 Hilbert space (Hilbert) space is big. Really big. You just won t believe how vastly, hugely,
More informationSECOND QUANTIZATION. Lecture notes with course Quantum Theory
SECOND QUANTIZATION Lecture notes with course Quantum Theory Dr. P.J.H. Denteneer Fall 2008 2 SECOND QUANTIZATION 1. Introduction and history 3 2. The N-boson system 4 3. The many-boson system 5 4. Identical
More informationQuantum Entanglement in Exactly Solvable Models
Quantum Entanglement in Exactly Solvable Models Hosho Katsura Department of Applied Physics, University of Tokyo Collaborators: Takaaki Hirano (U. Tokyo Sony), Yasuyuki Hatsuda (U. Tokyo) Prof. Yasuhiro
More informationQuantum Information & Quantum Computing
Math 478, Phys 478, CS4803, February 9, 006 1 Georgia Tech Math, Physics & Computing Math 478, Phys 478, CS4803 Quantum Information & Quantum Computing Problems Set 1 Due February 9, 006 Part I : 1. Read
More informationLie algebraic aspects of quantum control in interacting spin-1/2 (qubit) chains
.. Lie algebraic aspects of quantum control in interacting spin-1/2 (qubit) chains Vladimir M. Stojanović Condensed Matter Theory Group HARVARD UNIVERSITY September 16, 2014 V. M. Stojanović (Harvard)
More informationParticles I, Tutorial notes Sessions I-III: Roots & Weights
Particles I, Tutorial notes Sessions I-III: Roots & Weights Kfir Blum June, 008 Comments/corrections regarding these notes will be appreciated. My Email address is: kf ir.blum@weizmann.ac.il Contents 1
More information1 Mathematical preliminaries
1 Mathematical preliminaries The mathematical language of quantum mechanics is that of vector spaces and linear algebra. In this preliminary section, we will collect the various definitions and mathematical
More informationExercises Symmetries in Particle Physics
Exercises Symmetries in Particle Physics 1. A particle is moving in an external field. Which components of the momentum p and the angular momentum L are conserved? a) Field of an infinite homogeneous plane.
More informationReflections in Hilbert Space III: Eigen-decomposition of Szegedy s operator
Reflections in Hilbert Space III: Eigen-decomposition of Szegedy s operator James Daniel Whitfield March 30, 01 By three methods we may learn wisdom: First, by reflection, which is the noblest; second,
More informationPropagation bounds for quantum dynamics and applications 1
1 Quantum Systems, Chennai, August 14-18, 2010 Propagation bounds for quantum dynamics and applications 1 Bruno Nachtergaele (UC Davis) based on joint work with Eman Hamza, Spyridon Michalakis, Yoshiko
More informationSolution Set 3. Hand out : i d dt. Ψ(t) = Ĥ Ψ(t) + and
Physikalische Chemie IV Magnetische Resonanz HS Solution Set 3 Hand out : 5.. Repetition. The Schrödinger equation describes the time evolution of a closed quantum system: i d dt Ψt Ĥ Ψt Here the state
More informationDecoherence and Thermalization of Quantum Spin Systems
Copyright 2011 American Scientific Publishers All rights reserved Printed in the United States of America Journal of Computational and Theoretical Nanoscience Vol. 8, 1 23, 2011 Decoherence and Thermalization
More informationFermionic tensor networks
Fermionic tensor networks Philippe Corboz, Institute for Theoretical Physics, ETH Zurich Bosons vs Fermions P. Corboz and G. Vidal, Phys. Rev. B 80, 165129 (2009) : fermionic 2D MERA P. Corboz, R. Orus,
More informationC/CS/Phys C191 Quantum Mechanics in a Nutshell 10/06/07 Fall 2009 Lecture 12
C/CS/Phys C191 Quantum Mechanics in a Nutshell 10/06/07 Fall 2009 Lecture 12 In this lecture we summarize the essential physical and mathematical aspects of quantum mechanics relevant to this course. Topics
More informationA classification of gapped Hamiltonians in d = 1
A classification of gapped Hamiltonians in d = 1 Sven Bachmann Mathematisches Institut Ludwig-Maximilians-Universität München Joint work with Yoshiko Ogata NSF-CBMS school on quantum spin systems Sven
More informationHopping transport in disordered solids
Hopping transport in disordered solids Dominique Spehner Institut Fourier, Grenoble, France Workshop on Quantum Transport, Lexington, March 17, 2006 p. 1 Outline of the talk Hopping transport Models for
More information