DOI 10.1007/s10773-015-2541-2
Quantum Fisher Information of Localization Transitions in One-Dimensional Systems
X. M. Liu·Z. Z. Du·W. W. Cheng·J.-M. Liu
Received: 20 September 2014 / Accepted: 16 January 2015 / Published online: 4 February 2015
© Springer Science+Business Media New York 2015
Abstract The concept of quantum Fisher information (QFI) is used to characterize the localization transitions in three representative one-dimensional models. It is found that the localization transition in each model can be distinctively illustrated by the evolution of QFI.
For the Aubry-Andr´e model, the QFI exhibits an inflexion at the boundary between the extended states and localized ones. In thet1 −t2 model, the QFI has a transition point separating the extended states from the localized states, while the mobility edge of the QFI is energy dependent. Furthermore, nine energy bands in the Soukoulis-Economou (S-E) model can be clearly revealed by the QFI with global mobility edges and local mobility edges. The present work demonstrates the implication of the QFI as a general fingerprint to characterize the localization transitions.
Keywords Quantum fisher information·Localization transiton·Mobility edges· Localized states
1 Introduction
The Anderson localization is a nontrivial quantum phenomenon induced by interference in disordered potentials. For one-dimensional disorder systems, all the states are localized for any finite disorder and there is not a true quantum phase transition connected with the localization. However, for certain one-dimensional quasi-periodic and aperiodic systems [1,2], there exists a localization transition from extended states to localized ones. Generally, the transition arises from incommensurate potentials or long-range correlated potentials.
X. M. Liu ()·Z. Z. Du·W. W. Cheng·J.-M. Liu
Laboratory of Solid State Microstructure and Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
e-mail: [email protected] X. M. Liu
Institute of Mathematical and Physical Sciences, Jiangsu University of Science and Technology, Zhenjiang 212003, China
This interesting property attracts much attention and extensive theoretical and experimental studies of disorder effects in quantum systems [3–18]. Recently, due to the advances in manipulation of ultra-cold atoms, some experiments have realized the quasi-periodic model in optical lattices and observed the localization transitions [19,20]. In fact, there are many theoretical studies with respect to localization transitions in one-dimensional systems, for example, by transfer-matrix method and other numerical approaches or from the perspective of quantum information [21].
On the other hand, Fisher information (FI) is an important concept in quantum estima- tion and quantum information theory [22,23]. It gives the information we can extract from a system and thus brings rich fingerprints for evolutions and transitions. Moreover, the FI characterizes the sensitivity of a state with respect to perturbation of the parameters. Ulti- mately, the precision of parameter estimation is limited by the quantum Cramer-Rao bound [22–24]. This limit is imposed by the quantum Fisher information (QFI) [25,26]. More- over, a big QFI represents a high estimation precision. Therefore, to improve the precision, it is necessary to use quantum resources such as coherence, entanglement and so on, so that the quantum states of a system can be characterized and manipulated with high pre- cision. In fact, much work about the QFI has been concerned with the entanglement and spin squeezing [27–31]. In addition, the QFI has been applied to various quantum systems in decoherence channels to estimate the loss parameter [32,33] and noisy quantum sys- tems to obtain an optimal quantum measurement [34]. Recently, the problem of parameter estimation in systems with quantum phase transition attracts much attention [35]. In partic- ular, for those quantum systems with the localization transitions, the implication of the QFI as a probe of the localization transitions is an issue of interest. In this work, from the QFI perspective, we focus on the localization transitions in three well known one-dimensional tight-binding quantum models, i.e. the Aubry-Andr´e model [36], thet1−t2model [4] and the Soukoulis-Economou (S-E) model [37–44]. It will be demonstrated that the QFI in characterizing these localization transitions is highly appreciated.
The rest of this work is organized as follows. In Section2, we review the basic concepts related to the QFI. The main results of calculation in dealing with the localization transitions and QFI in the three models will be presented in Section3. Finally, we conclude with a summary and discussion in Section4.
2 Quantum Fisher Information
In this section, we recall the basic concepts relevant with the QFI. For a density matrix ρ(θ )which evolves with a parameterθ, the precision of theθ estimation is bounded by the quantum Cramer-Rao inequality [22–24],
θˆ≥θQCB≡ 1
√νF, F =tr[ρ(θ )L2θ] (1) whereνis the number of independent measurements andF is the so-called QFI. The so- called symmetric logarithmic derivativeLθ is given by
∂
∂θρ(θ )= 1
2[ρ(θ )Lθ+Lθρ(θ )]. (2) Here, we propose the system evolution by an SU(2) rotation
ρθ =UθρinUθ†, Uθ =exp(iθ J−→n) (3)
in which
J−→n =−→J−→n =
α=x,y,z
1
2nασα, (4)
is the angular momentum operator in the normalized −→n direction, andσα are the Pauli matrices. For the pure states, the QFI is proportional to the variance of J−→n and can be expressed asF (ρin, J−→n)=4(J−→n)2. Generally, the QFI is determined by the input states and the rotated direction. We can also improve the estimation precision by finding a proper input states for a givenJ−→n or by choosing a proper−→n for a givenρin. It is well known that the entanglement of the input states is a critical resource to obtain high estimation precision, which allows us to take this advantage. Therefore, for the pure states, we can search for the maximally entangled state to improve the QFI and thus to obtain the high precision.
However, once the input stateρinis given, we can adjust the rotated direction to optimize the QFI. The mixed state is the case, to be shown below.
For a mixed stateρθ, we firstly derive the expression ofLθ, then obtain QFI:
F (ρθ, J−→n)=
i=j
2(pi−pj)2
pi+pj |i|J−→n|j|2= −→n C−→nT (5) in whichpi and|iare the eigenvalue and eigenvector ofρin. The matrix elements of the symmetric matrix C is expressed as
Ckl =
i=j
(pi−pj)2
pi+pj [i|Jk|j j|Jl|i + i|Jl|j j|Jk|i] (6) Therefore, we can use a unitary matrixOto diagonalize the matrixCand find the direction
−
→n that corresponds to the maximal QFI,
Cd =OTCO=diag(E1, E2, E3), (7) Here, we propose E1 ≥ E2 ≥ E3. Obviously, we can conclude the maximal Fisher information isE1. A realistic scheme for calculating the QFI is also given.
3 The QFI and Localization Transitions
In our calculations, for simplicity, we propose a single particle moving in the one- dimensional lattice with L sites as dimension. Generally, the tight binding Hamiltonian can be written:
H= L
i
εic†ici−t L
i
c†ici+H.c.
(8) where εi is the on-site potential, t is the hopping term representing tunneling between nearest-neighboring sites,c†i(ci)is a creation (annihilation) operator of theithsite. By con- vention, we taket = 1. In the occupation number representation, the basis of eigenstates can be described by|n1, n2, ..., ni, ..., nL =
c†1n1 c2†n2
...
c†ini
...
c†LnL
|0, where ni = 0,1, and|0is the vacuum state. Moreover, for the single particle system, we have L
i=1ni =1. Therefore, it can be further simplified as|i = |0, ...,1i, ...,0 =ci†|0. The eigenstate|φβcorresponding to eigenenergyEβfor the Hamiltonian can be described by
|ϕβ = L
i
ϕβi |i = L
i
ϕiβc†i |0, (9)
whereϕiβ is the strength of theβth wave function at ith side. Thereafter, by diagonal- izing the Hamiltonian numerically with the open boundary condition, we can obtain all eigenenergiesEβand eigenstates|ϕβ, and thus the QFI can be calculated.
For simplicity, we calculate the QFI of the single site. For each site of the single particle system, there exist two local states|0and|1, which respectively represent the states with and without a particle at theithsite. If we consider the single sitei, in the basis|0and|1, the reduced density matrix is
ρi =
ρ11i 0 0 ρ22i
(10) whereρi11=1−φiφi∗andρ22i =φiφ∗i. Similar treatment can be done for the other sites.
Again, we only investigate the single site, and the analytic expression of the QFI is:
F =
ρ11i −ρ22i 2
(11) However, it should be mentioned that for non-uniform systems, the value of QFI depends on the site positioni. We then obtain the site-averaged QFI as a replacement:
F = 1 L
L i=1
F (ρi) (12)
Meanwhile, it is easy to find that the site-averaged QFI depends on the system sizeLfor a given energy levelEβ. When the eigenstate is localized completely,F¯derives the maximum.
However, when the eigenstate is extended completely,F¯reach the minimum. In general, we normalize the site-averaged QFI by the corresponding minimum and maximum for different system sizes L.
3.1 The Aubry-Andr´e Model
In the quasi-periodic Aubry-Andr´e model [36], the on-site potentialεifor the tight-binding Hamiltonian of (8) is:
εi =β0+λcos(2π iς ) (13)
whereβ0is the single-site energy of the non-perturbed periodic lattice. As we all know, if ς is an irrational Diophantine number [45,46] which is a necessary and sufficient condi- tion for a phase transition in this model, there will appear a localization phase transition at a modulation strengthλ = 2. Here, we takeς =(√
5−1)/2 which is a conventional choice for the Aubry-Andr´e model. Moreover, the Aubry-Andr´e Hamiltonian is self-dual at this point and coincides with the Harper Hamiltonian that exhibits the famous Hofstadter butterfly fractal spectrum.
For the ground state φgβ
, we present the quantum Fisher information as the function ofλ for different system sizes L in Fig.1. As expected, we observe that the QFI is relatively small in the extended state regionλ < 2, while it increases rapidly in the localized state region λ > 2. This difference becomes even more significant at big L. A clean and well defined transition point can thus be identified atL→ ∞. It means that the pseudo localization transition point will approach the real critical point when size L tends to infinity. In Fig.2, we present the QFI spectrum for all energy level states|φβfor different λvalues. The behaviors of the QFI for all other eigenstates are very similar to the counterpart in the ground state, i.e., the QFI are relatively small in the extended states and relatively big in the localized states. There exists an obvious dividing line at the pointλ=2, implying that the QFI as a fingerprint of this localization transition is significant.
(a)
(b)
Fig. 1 (a) QFI for the ground state with respect to parameterλin the Aubry-Andr´e model under different system size L. Here,ς=(√
5−1)/2. (b) First derivative of QFI with respect to parameterλas the system size L increases
3.2 Thet1−t2Model
Thet1-t2model [4] is the extension of the Aubry-Andr´e model taking into account of the next-neareat-neighboring(NNN) interactions and has the form
t2(un+2+un−2)+t1(un+1+un−1) +Vcos(2π αn+δ)un
=Eun (14)
Fig. 2 QFI with respect to eigenenergiesEβand different potential strengthλin the Aubry-Andr´e model.
Here,L=500. The dash lineλ=2 is the boundary between extended states and localized states
whereui is the amplitude of the wave function of the sitei,V is the strength of the on- site potential. Ast2/t1 is small, the energy mobility edge is approximately given by the linear relationship: cosh(p)= E+tV wherep = ln(t1/t2),t = t1ep), which is obtained in the exponential hopping model [37]. For bigt2/t1(≥ 0.3), however, the boundary differs considerably from the linear relationship and becomes nonlinear. We only deal with the cases with smallt2/t1ratios.
In Fig.3, the calculated QFI data as a function of parameterV /t1 for the ground state φβg
att2/t1 = 0.1 are plotted. The observed behaviors are similar to the situation in the A-A model. Moreover, the QFI exhibits an inflexion point nearV /t1 =1.5. Similarly, the QFI spectrum in the space of energy eigenvalues and potential (V,E) is presented in Fig.4.
Different from the Aubry-Andr´e model, here the mobility edge shows an energy-dependent behavior. The dash line represents the boundary defined by the relationship cosh(p)=EV+t. On the other hand, we find all the curves of the QFI exhibit very similar behavior for a fixed potential, i.e., they are relatively small for the energy above the boundary and are relatively big for energies below the boundary.
3.3 The Soukoulis-Economou (S-E) Model
In the Soukoulis-Economou (S-E) model [37], the on-site potential for the tight-binding Hamiltonian shown in (4) has the following form
εi =V0[cos(Qi)+V1cos(2Qi)], (15) whereQis incommensurate toπ. In the present calculation, we takeV0 = 1.9, V1 = 13 andQ = 0.7 as given in Refs [37–44], so that a comprehensive comparison with those results in literature. This model was numerically solved by Soukoulis and Economou, and the main feature is a 9-band energy spectrum structure [37]. Subsequently, Sun and Wang found that every band contains many sub-bands [38]. As shown in Fig.5, we can call these
(a)
(b)
Fig. 3 (a) QFI for the ground state with respect to parameterV /t1under different system size in thet1-t2
model. Here,t2/t1= 0.1. (b) First derivative of QFI with respect to parameterV /t1as the system size L increases
bands from the lower energy to the high energy the 1st−9thbands. All the works support the existence of the mobility edges. However, the locations of mobility edges are authors- dependent, which have been summarized in Ref. [39].
In detail, by studying the transmission coefficients and spatial behaviors of the eigen- states, a global mobility edge separating the 6th and 7th bands was claimed [37]. All eigenstates in the lower six bands are extended, while all eigenstates in the upper three bands are localized. Using the data on conductivity obtained by the Landauer formula, Liu and Chao claimed another global mobility edge between the 2nd and 3rd bands [40]. At
Fig. 4 QFI with respect to all eigenenergies in thet1-t2model under different potentialV whent2/t1= 0.1. The shading of QFI curves indicates the magnitude of the values of QFI for the corresponding wave functions. Here,L = 500. Thedash linerepresents the boundary between localized states and extended states, cosh(p)=(E+t)/V(p=ln(t1/t2))
the same time, there is a pair of local mobility edges in the 3rd −6thbands, respectively.
Zheng and Zhu used the criterion of self-energy convergent length and found a pair of local mobility edges in the 1st and 2ndbands [41,42]. On the other hand, Liu and Zhou revealed
Fig. 5 QFI of the individual state with respect to the eigenenergies in the Soukoulis-Economou (S-E) model
Fig. 6 QFI of the individual state for the different band. Here Fig. 6(i) is for the ith band, where i=1,2, ...,9, respectively. The system sizes L=1000 (black), 2000 (red) and 6000 (blue), respectively
no local mobility edge [43], while Sun reported that some local mobility edges exist but no global ones are identified [44]. In spite of these scattering claims and results, the 9-band structure seems to be the common feature.
In Fig.5we plot the calculated QFI of the individual state against eigenenergiesEβ in the S-E model. Indeed, the energy spectrum consists of 9-bands. From the lower energy states to the high energy ones, we call them the 1st−9thbands respectively. In general, we observe that the QFI are relatively small (near zero) in the 1st−6thbands and relatively big in the 7th−9thbands. Moreover the 6thband is essentially different from the 7thband.
One understands that the behaviors of QFI in response toEβ shown in Figs.6(1)–(6) are similar. The QFI has a plateau at these band center regions. The values of these QFI are relatively small when the system size is big. The characteristic is similar as that for extended states in the slowly varying potential model [21]. Therefore, the corresponding eigenstates in the 1st- 6thbands are extended or marginally extended.
As shown in Figs. 6(7)–(9), all the QFI in the 7th−9th bands are relatively big. The variations of QFI with respect toEβ in these bands are similar and show roughly size- independent. The eigenstates in the 7th−9th bands are the localized states, which is in agreement with all the existing results about the three bands [37–44]. In a rough sense, the global mobility edges between the 6thand 7thbands show the mixed characteristics of extended states and localized states, and thus represent the fingerprints separating the two types of states in this specific model.
4 Conclusion
In this work, we investigate the localization transition in three typical one-dimension sys- tems, i.e. the Aubry-Andr´e model, thet1−t2 model and the Soukoulis-Economou (S-E)
model. For the Aubry-Andr´e model, the QFI exhibits an sharp transition at the boundary between the extended states and localized ones as the system size increases. In thet1−t2
model, the QFI has a critical point separating the extended states from the localized states, however the mobility edge of the QFI is energy dependent in comparison with that of the Aubry-Andr´e model. In the Soukoulis-Economou (S-E) model, nine energy bands can be clearly revealed by the QFI with global mobility edges and local mobility edges. Moreover, we compare the results from QFI with that from other methods. Generally, in these mod- els, we find that the QFI can well manifest the degree of localization of the quantum states and characterize the localization transition of these systems. The QFI is relatively big at the localized states and relatively small at the extended ones. Moreover, we observe that when the system size increases, the transition of the QFI becomes sharp. All the results show that quantum Fisher information is a good approach to detect localization transition in one- dimension systems. We predict that the quantum Fisher information can be applied to the other quantum phase transition in one-dimension systems.
Acknowledgments This work was supported by the National 973 Projects of China (Grants No.
2011CB922101), the Natural Science Foundation of China (Grants Nos. 11234005, 11374147), and the Priority Academic Program Development of Jiangsu Higher Education Institutions, China.
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