Model von Neumann entropy

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Model von Neumann entropy Remarks Entropy Density of Quasifree Fermionic States supported by Left/Right Movers T1 T2 Walter H. Aschbacher (Ecole Polytechnique) Open Quantum Systems, Grenoble, December 2010 Entropy Density of Quasifree Fermionic States supported by Left/Right Movers 1/20

model von

spin spin

density sn

j1 ≈

specific model

quasifree fermionic

neumann entropy

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École polytechnique

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Publié par	chaeh
Nombre de lectures	44
Langue	Español

Extrait

Model
vonNeumannentropy
Remarks
Entropy Density of Quasifree Fermionic States supported by
Left/Right Movers
T2T1
Walter H. Aschbacher (Ecole Polytechnique)
Open Quantum Systems, Grenoble, December 2010
EntropyDensityofQuasifreeFermionicStatessupportedbyLeft/RightMovers 1/20
Model
vonNeumannentropy
Remarks
Contents
1. Model
1.1 Quasifree setting
1.2 NESS
1.3 Left/Right movers
2. von Neumann entropy
2.1 Toeplitz Majorana correlation
2.2 Reduced density matrix
2.3 Asymptotics
3. Remarks
EntropyDensityofQuasifreeFermionicStatessupportedbyLeft/RightMovers 2/20Model
vonNeumannentropy
Remarks
What are we physically interested in?
Question:
1 A sample is suitably coupled to thermal reservoirs s.t., for large times, the
system approaches a nonequilibrium steady state (NESS).
2 We consider a class of quasifree fermionic NESS over the discrete line
which are supported by so-called Left/Right movers.
13 We ask: What is the von Neumann entropy densitys = tr(% log% )n n nn
of the reduced density matrix% of such NESS restricted to a ﬁnite stringn
of large lengthn?
6666666666666r r r r r r r r r r r r r
| {z }
n
4 The prominent XY chain will serve as illustration (in the fermionic picture).
Remark Several other correlators can be treated similarly (e.g. spin-spin, EFP).
EntropyDensityofQuasifreeFermionicStatessupportedbyLeft/RightMovers 3/20Model
vonNeumannentropy
Remarks
What are we physically interested in?
Speciﬁc model: XY chain [Lieb et al. 61, Araki 84]
The Heisenberg Hamiltonian density reads
X
(x) (x+1) (x)H = J + ;x n n n 3
n=1;2;3
and the XY chain is the special case withJ = 0.3
Experiments SrCuO , Sr CuO [Sologubenko et al. 01] withJ = 02 2 3 3
PrCl [D’Iorio et al. 83, Culvahouse et al. 69] withJ =J ,J 0, i.e.,1 2 33
2J =J 10 , and = 03 1
EntropyDensityofQuasifreeFermionicStatessupportedbyLeft/RightMovers 4/206Model
vonNeumannentropy
Remarks
Formalism of quantum statistical mechanics
Rigorous foundation in the early 1930s:
1 An observableA is a selfadjoint operator on the Hilbert space of the system.
2 The dynamics of the system is determined by a distinguished selfadjoint operator
itH itHH, called the Hamiltonian, throughA7!A = e Ae .t
3 A pure state is a vector in the Hilbert space, and the expectation value of the
measurement ofA in the state is( ;A ).
Algebraic reformulation and generalization (von Neumann, Jordan, Wigner, ...):
Observables
C algebraA
Dynamics
t (Strongly) continuous group of -automorphisms onA
States
Normalized positive linear functionals! onA, denoted byE(A).
t itH itHExample A =L(h), (A) = e Ae , and!(A) = tr(%A) with density matrix%
EntropyDensityofQuasifreeFermionicStatessupportedbyLeft/RightMovers 5/20Model Quasifreesetting
vonNeumannentropy NESS
Remarks Left/Rightmovers
1.1 Quasifree setting
Observables
The total observable algebra is:
CAR algebra
2The CAR algebraA over the one-particle Hilbert spaceh =‘ (Z) is
theC algebra generated by anda(f) withf2h satisfying:
a(f) is antilinear inf
fa(f);a(g)g = 0
fa(f);a (g)g = (f;g)
The ﬁnite subalgebra of observables on the string is:
Stringa
2Let h = ‘ (Z ) be the one-particle subspace over the ﬁnite stringn n
Z =f1; 2;:::;ng. The string algebraA is theC subalgebra ofAn n
generated bya(f) withf2h .n
By the Jordan-Wigner transformation, we have the isomorphism
n n22A ’C :n
EntropyDensityofQuasifreeFermionicStatessupportedbyLeft/RightMovers 6/2011Model Quasifreesetting
vonNeumannentropy NESS
Remarks Left/Rightmovers
Quasifree setting
States
2ForF = [f ;f ]2h , we deﬁneJF = [cf ;cf ] with conjugationc, and1 2 2 1

B(F ) =a (f ) +a(cf ):1 2
The two-point function is characterized as follows:
Density
2The density of a state!2E(A) is the operatorR2L(h ) satisfying
20R 1 andJRJ = 1 R, and, for allF;G2h ,
!(B (F )B(G)) = (F;RG):
The class of states we are concerned with is:
Quasifree state
2A state !2E(A) with density R2L(h ) is called quasifree if it
vanishes on odd polynomials in the generators and if
2n!(B(F ):::B(F )) = pf [(JF ;RF )] :1 2n i j i;j=1
EntropyDensityofQuasifreeFermionicStatessupportedbyLeft/RightMovers 7/20Model Quasifreesetting
vonNeumannentropy NESS
Remarks Left/Rightmovers
1.2 Nonequilibrium steady states (NESS)
States
For the nonequilibrium situation, we use:
NESS [Ruelle 01]
tA NESS w.r.t theC -dynamical system (A; ) with initial state! 20
tE(A) is a large time weak- limit point of! (suitably averaged).0
The nonequilibrium setting for the XY chain is:
Theorem: XY NESS [Dirren et al. 98, Araki-Ho 00, A-Pillet 03]
tLeth = Re(u) [ Re(u)] generate the coupled dynamics . Then,
Q 10the decoupled quasifree initial state with density R = (1 + e )0
tand Q = 0 h h converges under to the unique0 L L R R
Qh 1quasifree NESS with densityR = (1 + e ) , where
ith ith
Q =