Fidelity susceptibility in Gaussian Random Ensembles Marek Ku´ s* Piotr Sierant** Artur Maksymov** Jakb Zakrzewski** *Center for Theoretical Physics PAS, Warszawa, Poland **Marian Smoluchowski Instytute of Physics, Jagiellonin University, Cracow, Poland
Setting ◮ Parametric family of Hamiltonians H = H 0 + λ H 1 ◮ Pure state fidelity F = |� ψ ( 0 ) | ψ ( λ ) �| ◮ We are interested in fidelity of eigenstates: H ( λ ) | ψ n ( λ ) � = E n ( λ ) | ψ n ( λ ) � ◮ For small λ F = 1 − 1 2 χλ 2 + O ( λ 3 ) (first order terms vanish due to the normalization of eigenfunctions) ◮ Definition: χ - fidelity susceptibility (of H 0 )
Properties ◮ Alternatively χ = ∂ 2 F � � � ∂λ 2 � λ = 0 ◮ Hence χ = ( � ∂ λ ψ ( λ ) | ∂ λ ψ ( λ ) � − � ψ ( λ ) | ∂ λ ψ ( λ ) � − � ∂ λ ψ ( λ ) | ψ ( λ ) � ) | λ = 0 what exhibits a nice geometric picture - the real part of the natural Riemannian structure on a manifold of quantum states (Provost and Vallee, 1980) ◮ Quantum Fisher Information G G = 4 χ ◮ The n -th eigenastate fidelity susceptibility for H 0 can be alternatively expressed as | H 1 , nm | 2 � χ n = ( E n − E m ) 2 m � = n H 1 , nm = � ψ n ( 0 ) | H 1 | ψ n ( 0 ) � , E n = E n ( 0 )
Applications ◮ Quantum phase transitions ◮ Bose-Hubbard model (You, Li, Gu, Phys. Rev. E 76 , 022101, 2007) ◮ XY model (Zanardi, Paunkovi´ c, Phys. Rev. E 74 , 031123, 2006) ◮ Dicke model ( ibid. ) ◮ Quantum many-body localization (Hu et al. , Phys. Rev. E 94 , 052119, 2016; Maksymov, Sierant, Zakrzewski, in preparation .) ◮ Whenever you know an ingenuous application of the Quantum Fisher Information
Random Matrices ◮ Usually quite complicated Hamiltonians. A minimalistic assumption ◮ H a , a = 1 , 2 are random matrices from the classical Random ( N × N ) Matrix Gaussian ensembles with densities � − β � 4 J 2 Tr H 2 P ( H a ) ∼ exp a ◮ variance � H 2 nn � = 2 � H 2 mn � = 2 J 2 ◮ GOE ( β = 1 ), GUE ( β = 2 ), GSE ( β = 4 ) ◮ ultimately N → ∞ , J = O ( 1 / N )
Detour. Quantum Chaos. Level Curvature Distribution ◮ Similar quantity was thoroughly investigated in the context of disordered system and quantum chaos ◮ Level Curvature Distribution | H 1 , nm | 2 K n := ∂ E n ( λ ) � = − E n + ∂λ E m − E n m � = n ◮ Characterization of spectral fluctuations in quantum chaotic systems ◮ Conductance in disordered systems � g � ∼ �| K |� , ( λ - magnetic flux through the probe) ◮ The distribution of the curvature in quantum chaotic or disordered systems in Random Matrix Theory, as conjectured by Zakrzewski and Delande W ( K ) ∼ ( 1 + K 2 ) 1 + β/ 2 ◮ Later proved by von Oppen and Fyodorov & Sommers
Back to susceptibility Our task: the distribution of susceptibility at the energy E � N � 1 � P ( χ, E ) = δ ( χ − χ n ) δ ( E − E n ) N ρ ( E ) n = 1 where the averaging � � is over � − β �� � Tr H 2 0 + Tr H 2 P ( H 0 , H 1 ) ∼ exp 1 4 J 2 and ρ ( E ) - the density of states
Calculations. Few tricks ◮ Fourier representation of δ ( χ − χ n ) ◮ averaging over H 1 reduces to a Gaussian integral ◮ averaging over H 0 reduces to the one over the distribution of eigenvalues only, i.e. with the distribution | E k − E l | β e − β k E 2 � � P ( E 1 , E 2 , . . . , E N ) ∼ 4 J 2 k k < l ◮ integrating with δ ( E − E n ) and using the orthogonal/unitary invariance of the RMT distributions allows reducing the dimension N by 1 ◮ at the center of the spectrum E = 0 (can be relaxed) we arrive at β � � ∞ � det ¯ H 2 d ω e − i ωχ P ( χ ) ∼ � 1 � −∞ H 2 − 2 i ω J 2 ¯ 2 det β N − 1 where the averaging is now over ¯ H from ( N − 1 ) × ( N − 1 ) ensemble
Calculations. Some further tricks ◮ Gaussian integral representation � − β � H 2 − 2 i ω J 2 � � H 2 − 2 i ω J 2 � � 2 � ¯ − z † ¯ det ∼ d z exp z β β where z , a N − 1 -component real/complex vector, due to orthogonal/unitary invariance may be chosen as z = r [ 1 , 0 , 0 .. ] T ◮ We arrive at � ∞ � � � H 2 β e − r 2 X � drr s δ χ − 2 J 2 r 2 /β det ¯ P ( χ ) ∼ N − 1 0 ◮ Block matrix representation � H 11 � H 1 j ¯ H = H 1 k V and integration over H 1 m leaves the averaging over the ( N − 2 ) × ( N − 2 ) GOE/GUE matrix V
Asymptotic results GOE The asymptotic ( N → ∞ ) results for the scaled fidelity susceptibility x = χ/ N P O ( x ) = 1 1 � 1 + 1 � � − 1 � exp x 2 6 x 2 x P ( x ) P ( x ) b) N=200 a) 10 0 N=200 2 . 0 N=924 N=924 10 − 1 N=3432 1 . 5 N=3432 10 − 2 N=12870 N=12870 1 . 0 10 − 3 10 − 4 0 . 5 10 − 5 0 . 0 10 − 1 10 0 10 1 10 2 0 . 0 0 . 3 0 . 6 0 . 9 x x
Asymptotic results GUE � 3 1 1 4 + 1 x + 1 � � − 1 � P U ( x ) = 3 √ π exp x 5 / 2 x 2 x P ( x ) P ( x ) a) b) 10 0 N=924 N=924 2 N=3432 N=3432 10 − 2 N=12870 N=12870 1 10 − 4 10 − 6 0 10 − 1 10 0 10 1 10 2 0 . 0 0 . 5 1 . 0 1 . 5 x x
Arbitrary N . GOE � N − 2 � 1 � � N ( χ ) = C O � 2 � χ � � 1 1 + 2 χ + 1 1 1 2 2 N I O , 2 P O √ χ N − 2 1 + χ 1 + 2 χ 2 1 + χ � N + 2 N N + 3 / 2 , N even , I O , 2 = N N + 1 / 2 , odd . N P ( χ ) 10 1 N=2 N=3 10 − 1 N=4 N=5 10 − 3 N=10 N=20 10 − 3 10 − 2 10 − 1 10 0 10 1 10 2 χ
Arbitrary N . GUE � 1 � N − 2 � � χ 1 2 P U C U N ( χ ) = × N 1 + χ 1 + 2 χ � � � 2 � 2 � 4 3 � 1 + 3 1 � 1 N − 2 + 1 � 1 I U , 2 I U , 4 × N − 2 1 + 2 χ 1 + 2 χ 1 + χ 1 + χ 4 2 4 � � N 2 + 2 N , 1 3 N , even , even , N N I U , 2 I U , 4 = = N 2 + 4 N + 3 , N 1 N 3 ( N + 1 ) , N odd , N odd . 10 5 10 4 10 3 10 2 GUE, N=3x3 10 1 GUE, N=4x4 GUE, N=5x5 10 0 GUE, N=6x6 GUE, N=6x6 10 − 1 10 − 2 10 − 1 10 0 10 1 10 2
Recommend
More recommend
