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PT symmetric Aubry–Andre model
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Physics Department, Anadolu University, Eskisehir, Turkey
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Article history: Received 22 January 2014 Received in revised form 25 April 2014 Accepted 9 May 2014 Available online xxxx Communicated by P.R. Holland
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PT symmetric Aubry–Andre model describes an array of N coupled optical waveguides with posi tion-dependent gain and loss. We show that the reality of the spectrum depends sensitively on the degree of quasi-periodicity for small number of lattice sites. We obtain the Hofstadter butterfly spec trum and discuss the existence of the phase transition from extended to localized states. We show that rapidly changing periodical gain/loss materials almost conserve the total intensity. © 2014 Published by Elsevier B.V.
Keywords: Non-Hermitian Hamiltonian PT symmetric optical system
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1. Introduction In 1998, PT symmetric non-Hermitian Hamiltonians with real spectra were introduced [1,2]. Recent experimental realization of PT symmetric systems with balanced gain and loss has attracted a lot of attention [3–8]. The PT symmetric optical systems lead to interesting results such as unconventional beam refraction and power oscillation [9–11], nonreciprocal Bloch oscillations [12], uni directional invisibility [13], an additional type of Fano resonance [14], and chaos [15]. In the PT symmetric optical systems, the net gain or loss of particles vanishes due to the balanced gain and loss mechanism. These systems are described by non-Hermitian Hamil tonian with real energy eigenvalues provided that non-Hermitian degree is below than a critical number, γ P T . If it is beyond the critical number, spontaneous PT symmetry breaking occurs. It implies the eigenfunctions of the Hamiltonian are no longer si multaneous eigenfunctions of PT operator and consequently the energy spectrum becomes either partially or completely complex. The critical number of non-Hermitian degree is shown to be differ ent for planar and circular array configurations [16] and it can be increased if impurities and tunneling energy are made position dependent in an extended lattice [17]. However, γ P T decreases with increasing the lattice sites [18–21], hence the PT symmetric phase is fragile. An important consequence of PT symmetric opti cal systems is the power oscillations. It was shown that the beam power in a one dimensional tight binding chain doesn’t depend on the microscopic details such as disorder and periodicity [22]. The probability-preserving time evolution in terms of the Dirac inner product for PT symmetric tight-binding ring was considered [23]. It is interesting to note that the PT operator coincides with time
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evolution operator at some certain time allows perfect state trans fer in the PT symmetric optical lattice with position-dependent tunneling energy [24]. The equivalent Hermitian Hamiltonian for a tight-binding chain can also be constructed to understand the non-Hermitian system [25]. In PT symmetric optical system, disorder plays important roles. To model disorder, complex impurities can be chosen ran domly with zero mean. In this case, the system would be no longer PT symmetric as a whole (global symmetry), but possesses a lo cal Pd T symmetry [22]. Therefore the system admits real energy spectrum. In this paper, we investigate quasi-periodical array of PT symmetric tight binding chain [26–29]. It is well known that disorder and quasi-periodical potentials in quantum mechanical systems induce localization. Here, we show that localization occurs if some certain conditions are satisfied. Our system is described by the PT symmetric extension of the Aubry–Andre model [30]. The energy spectrum associated with Hermitian Aubry–Andre model at certain strength of parameters has fractal structure, which is known as the Hofstadter butterfly spectrum [31,32]. We also in vestigate the Hofstadter butterfly spectrum in the presence of non Hermitian impurities.
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E-mail address: [email protected]. http://dx.doi.org/10.1016/j.physleta.2014.05.005 0375-9601/© 2014 Published by Elsevier B.V.
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Consider an array of N coupled optical waveguides with posi tion-dependent gain and loss and constant tunneling amplitude J through which light is transferred from site to site. We adopt open boundary conditions. The beam propagation in the tight-binding structure can be described by a set of equations for the electric field amplitudes cn ,
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Fig. 1. The PT symmetric Hofstadter butterfly spectrum for the real part of the energy eigenvalues at N = 50 and at β = 1/2.
γ0 = 2 (a), γ0 = 0.1 (b). The graph has a line of reflection
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where n = 1, 2, ..., N is the waveguide number and the posi tion-dependent non-Hermitian degree γn describes the strength of N gain/loss material that is assumed to be balanced, i.e., n=1 γn = 0. The field amplitude transforms as cn → c N −n+1 under parity trans formation and the complex number transforms as i → −i under anti-linear time reversal transformation. Thus the global PT sym metry is lost unless a precise relation between γn holds. Here, we consider a quasi-periodical system with global PT symmetry and study localization. Consider the following gain/loss parameter
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i γn = V cos(2π β n + φ N ) + i γ0 sin(2π β n + φ N ),
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where V and γ0 are constants, β determines the degree of the quasi-periodicity and φ N is the constant phase difference which depends on the total number of sites. We require that gain and loss are balanced, so we demand φ N = −π β( N + 1) + φ0 , where the constant φ0 is an integer multiple of π . Without loss of gen erality, we take φ0 = 0. We emphasize that the system is PT symmetric globally. Eq. (1) with (2) can be called the PT sym metric Aubry–Andre model [30], which can now be engineered experimentally [3–5]. The most interesting result of the Hermi tian Aubry–Andre model is that the states at the center of the lattice is localized (Anderson localization) for irrational values of β when V > 2. Apparently, the non-Hermitian character of the Aubry–Andre equation (1) could change the physics of this sys tem dramatically. Note also that Aubry–Andre model coincides with the Harper model when V = 2 and γ0 = 0 and the en ergy spectrum as a function of β is known as Hofstadter butterfly spectrum, which is an example of fractal structure that appears in physics [31,32]. Here, we study the Hofstadter butterfly spec trum and localization effect for the PT symmetric Aubry–Andre model. It is s fficient to analyze the region 0 < β < 1 since the system repeats itself in equal intervals of β . Furthermore, the energy spec trum is symmetric with respect to β = 0.5 axis and the spectrum does not depend on the sign of γ0 . As a special case, if β is either 0 or 1, then the gain/loss terms vanish. If β = 1/2 and N is even, the system has gain and loss with amplitudes ∓i γ0 at alternating lattice sites. The gain/loss materials change periodically if β is a ra tional number and quasi-periodically if β is an irrational number. In the latter case, the gain/loss impurities are quasi-periodically ordered. Note that β can be given with a finite number of dig its in a real experiment. To increase the incommensurability of β = p /q (p, q are two coprime positive integers), one can choose s fficiently large p and q.
Fig. 2. The imaginary part of the energy eigenvalues at N = 50, V = 0 and as a function of β .
γ0 = 2
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We look for stationary solutions of Eq. (1). Suppose first that V = 0. In the absence of gain and loss, the system has the well known energy spectrum of width 4 J : E = −2 J cos nπ / N. In the presence of gain and loss, the real part of the energy eigenvalues, R{ E }, are still contained in [−2 J , 2 J ] for any N. More precisely, the energy width is a decreasing function |γ0 |. The distribution of R{ E } crucially depends on the strength of quasi-periodical or der through the value of β . It consists of a finite number of bands when β is rational. In this case, R{ E } is the union of bands and the length of the gap between any two bands depends on q (β = p /q). On the other hand, the fractal structure appears and the spectrum is a Cantor set when β is irrational (for the mathematicians, this property is known as the 10 Martinis conjecture [34] in the Hermi tian limit). Such a fractal structure can be seen in Fig. 1, where we plot the PT symmetric Hofstadter butterfly spectrum at γ0 = 2, V = 0 (a) and at V = 2, γ0 = 0.1 (b). An important difference be tween V = 0 and V = 0 cases is that the symmetry with respect to zero energy axis is lost for the latter case. However, the real part of the energy eigenvalues is symmetric with respect to β = 0.5 axis for any V . Note also that the width of R{ E } increases with V and it takes its maximum value when γ0 = 0. As can be seen below, the PT symmetry breaking point is very small for large N and thus the corresponding energy spectrum is not real. However, there exists some special values of β for Fig. 1b with entirely real spectrum. For example, the spectrum is real when β = 1/5. To this end, we stress that the nice fractal picture can be seen not only for the real part of the spectrum but also for the imaginary part of the spectrum. Fig. 2 plots it at V = 0 and γ0 = 2. To gain more insight on the role of quasi-periodicity, let us study how the real and imaginary parts of the energy change with
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√ Fig. 3. The real part of energy eigenvalues as a function of γ0 at N = 30, V = 0 and J = 1. We take β = 0.6 and β = ( 5 − 1)/2 ≈ 0.618.
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Fig. 4.√The imaginary part of energy eigenvalues as a function of γ0 at J = 1, V = 0 and N = 10, 100. The degree of the quasi-periodicity changes β = ( 5 − 1)/2 is nearly 1.6 times greater than γ P T for β = 6/10 when N = 25. For large number of lattice sites, γ P T → 0.
γ P T sign ficantly. γ P T for
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γ0 for weakly and strongly disordered system. Fig. 3 plots R{ E } as a function γ0 for a weak β = 1/3 and strong disorders β = 11/30
when N = 30. As can be seen from the figures, the degree of quasi periodicity in the lattice has a dramatic effect for large values of γ0 . The real part of energy shrink to zero (they become degen erate and the energy width becomes zero) for very large values of γ0 if the system is periodical while this is not the case if it is quasi-periodical. However, the corresponding imaginary part of en ergy eigenvalues are different from zero for such a large number of γ0 . If the impurity strength, γ0 , exceeds a critical point, γ P T , PT symmetry is spontaneously broken and thus the energy eigenvec tors are not simultaneous eigenvectors of the Hamilton and PT operators. In this case, the energy eigenvalues become partially or entirely complex. An important consequence for our system is that strong disorder increases the critical point γ P T considerably. Fig. 4 plots the imaginary part of the energy eigenvalues for var ious values of N at β = 0.6 and the inverse of the golden ratio √ β = ( 5 − 1)/2 ≈ 0.618, which is the common choice in the study of the Aubry–Andre model. We numerically find that due to the quasi-periodicity, γ P T increases by a factor of nearly 1.6 when N = 25. However, the number of lattice sites N has dominant ef fect on γ P T and thus increasing the degree of the quasi-periodicity has slightly changed γ P T for large N. The critical point decreases
with increasing N and approaches zero when N is large. The PT symmetric phase is said to be fragile since γ P T is zero as N →∞ [18]. It is well known that the Hermitian Aubry–Andre model, γ0 = 0, displays a phase transition from extended to exponen tially localized states (it is also known as metal–insulator transition [33]). Of particular importance is the self-dual point V / J = 2 where the localization transition occurs [30]. Let us now study whether localization takes place for the PT symmetric Aubry Andre model. Suppose first V = 0. We take the inverse of the √ 5−1 , 2
golden ratio β = J = 1 and N = 49 with the initial condi tion |cn ( z = 0)|2 = δn,25 . We find numerically that initially localized wave packet delocalizes in time when γ0 = 2. We repeat numerical solution for large values of γ0 , but exponentially localized states do not emerge for the PT symmetric Aubry–Andre model. This result is interesting that quasi-periodicity does not induce localization for the PT symmetric Aubry–Andre model contrary to Hermitian one. This is because of the fragile nature of the PT symmetric phase. Before metal–insulator phase transition takes place, the system en ters broken PT symmetric phase and the corresponding intensity N grows exponentially, where the intensity is given by I = n=1 |cn |2 and satisfies
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Fig. 5. The total intensity as a function of z for the parameters ω = 0 (dashed) and ω = 3 (solid) at fixed V = 4. It grows exponentially for the static case while it is almost constant for large values of ω . We plot σ ( z) for V = 4 (solid) √ and V = 0 (dashed) at fixed ω = 3. Localization takes place when σ ( z) oscillates (non-periodically) and ballistic
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The intensity grows exponentially in the broken PT symmetric case while it oscillates when the energy spectrum is entirely real. Suppose now V = 0. We find numerically the time evolution of the single site excitation. It is well known that the metal–insulator transition occurs at V = 2 and γ0 = 0 and the wave packet is lo calized around the single site when V > 2. If V < 2, the proba bility |c 25 |2 goes to zero rapidly with z. The presence of gain/loss change the dynamics sign ficantly. Although the probability |c 25 |2 doesn’t rapidly go to zero when γ0 = 0, this cannot be consid ered localization in the rigorous sense. This is because the in troduction of gain/loss to the system does not conserve the total intensity and the generated particles enter the system not only n = 25-th lattice site but also the other lattice sites. Thus the oc cupation on the waveguide away from n = 25 is not negligible. For large z, the particles are generated even at the edges of the sys tem. A question arises. Does the phase transition from extended to exponentially localized state occurs if we somehow find a way to make the total intensity bounded for large values of γ0 ? To answer this question, consider z-dependent periodic impurity strength [15, 35–39]
i γn = V cos(2π β n + φ N ) + i γ0 cos(2πω z) sin(2π β n + φ N ), (4) where ω is a constant. Note that the corresponding Hamiltonian is still PT invariant. The gain and loss are also locally balanced after one period. The intensity oscillates in time when ω = 0 if γ0 < γ P T . The os cillation is not in general periodic. Introducing periodically chang ing impurity, ω = 0, makes the oscillation periodical with z. In creasing ω decreases the period of the intensity. We assert that the intensity is in principle conserved in the limit ω → ∞ since impurities do not have enough time to transfer intensity to the system. So, we expect that rapidly changing impurities practically conserves the intensity. To check this argument, we solve Eq. (1) numerically. We find that the intensity is almost constant when ω = 3 as can be seen from Fig. 5. The quasi-periodicity has nothing to do with the intensity conservation and the intensity is almost conserved for any values of β . To predict localization, let usdefine the variance of the prob ability distribution as
σ ( z) =
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2 n n|cn | / P is the average site z-dependent occupation [39]. We plot the variance in Fig. 5. The linearly increasing σ ( z) with re spect to z implies that the wave packet delocalizes (spreads ballis tically with z). On the contrary, oscillating σ ( z) shows us that the wave packet is localized. We find that the onset of local ization appears for PT symmetric Aubry–Andre model provided
that V / J > 2 and the system is quasi-periodical. We emphasize that localization doesn’t take place if the system is ordered, i.e. β is a rational number. In the localization regime, the occupa tion at n = 25 – the well oscillates periodically with z and is almost constant for large values of V . We note that the underlying mechanism of localization studied here is essentially the same of Anderson localization. To summarize, we have studied PT symmetric tight binding optical lattice with quasi-periodically ordered impurities by con sidering the complex extension Aubry–Andre model. We have plot ted complex Hofstatder butterfly spectrum and shown that the re ality of the spectrum depends sensitively on the impurity strength and quasi-periodicity. We have shown that the critical point γ P T increases with the increasing degree of quasi-periodicity. We have demonstrated that the transition from extended to localized states does not occur for the system described by PT symmetric Aubry Andre model. The metal–insulator transition occurs if the impuri ties changes periodically with z at each site. We have also shown that rapidly changing periodical impurities conserves the total in tensity.
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Physics Department, Anadolu University, Eskisehir, Turkey
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Article history: Received 22 January 2014 Received in revised form 25 April 2014 Accepted 9 May 2014 Available online xxxx Communicated by P.R. Holland
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PT symmetric Aubry–Andre model describes an array of N coupled optical waveguides with posi tion-dependent gain and loss. We show that the reality of the spectrum depends sensitively on the degree of quasi-periodicity for small number of lattice sites. We obtain the Hofstadter butterfly spec trum and discuss the existence of the phase transition from extended to localized states. We show that rapidly changing periodical gain/loss materials almost conserve the total intensity. © 2014 Published by Elsevier B.V.
Keywords: Non-Hermitian Hamiltonian PT symmetric optical system
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1. Introduction In 1998, PT symmetric non-Hermitian Hamiltonians with real spectra were introduced [1,2]. Recent experimental realization of PT symmetric systems with balanced gain and loss has attracted a lot of attention [3–8]. The PT symmetric optical systems lead to interesting results such as unconventional beam refraction and power oscillation [9–11], nonreciprocal Bloch oscillations [12], uni directional invisibility [13], an additional type of Fano resonance [14], and chaos [15]. In the PT symmetric optical systems, the net gain or loss of particles vanishes due to the balanced gain and loss mechanism. These systems are described by non-Hermitian Hamil tonian with real energy eigenvalues provided that non-Hermitian degree is below than a critical number, γ P T . If it is beyond the critical number, spontaneous PT symmetry breaking occurs. It implies the eigenfunctions of the Hamiltonian are no longer si multaneous eigenfunctions of PT operator and consequently the energy spectrum becomes either partially or completely complex. The critical number of non-Hermitian degree is shown to be differ ent for planar and circular array configurations [16] and it can be increased if impurities and tunneling energy are made position dependent in an extended lattice [17]. However, γ P T decreases with increasing the lattice sites [18–21], hence the PT symmetric phase is fragile. An important consequence of PT symmetric opti cal systems is the power oscillations. It was shown that the beam power in a one dimensional tight binding chain doesn’t depend on the microscopic details such as disorder and periodicity [22]. The probability-preserving time evolution in terms of the Dirac inner product for PT symmetric tight-binding ring was considered [23]. It is interesting to note that the PT operator coincides with time
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evolution operator at some certain time allows perfect state trans fer in the PT symmetric optical lattice with position-dependent tunneling energy [24]. The equivalent Hermitian Hamiltonian for a tight-binding chain can also be constructed to understand the non-Hermitian system [25]. In PT symmetric optical system, disorder plays important roles. To model disorder, complex impurities can be chosen ran domly with zero mean. In this case, the system would be no longer PT symmetric as a whole (global symmetry), but possesses a lo cal Pd T symmetry [22]. Therefore the system admits real energy spectrum. In this paper, we investigate quasi-periodical array of PT symmetric tight binding chain [26–29]. It is well known that disorder and quasi-periodical potentials in quantum mechanical systems induce localization. Here, we show that localization occurs if some certain conditions are satisfied. Our system is described by the PT symmetric extension of the Aubry–Andre model [30]. The energy spectrum associated with Hermitian Aubry–Andre model at certain strength of parameters has fractal structure, which is known as the Hofstadter butterfly spectrum [31,32]. We also in vestigate the Hofstadter butterfly spectrum in the presence of non Hermitian impurities.
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E-mail address: [email protected]. http://dx.doi.org/10.1016/j.physleta.2014.05.005 0375-9601/© 2014 Published by Elsevier B.V.
i
dz
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Consider an array of N coupled optical waveguides with posi tion-dependent gain and loss and constant tunneling amplitude J through which light is transferred from site to site. We adopt open boundary conditions. The beam propagation in the tight-binding structure can be described by a set of equations for the electric field amplitudes cn ,
dcn
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2. Model
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Fig. 1. The PT symmetric Hofstadter butterfly spectrum for the real part of the energy eigenvalues at N = 50 and at β = 1/2.
γ0 = 2 (a), γ0 = 0.1 (b). The graph has a line of reflection
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where n = 1, 2, ..., N is the waveguide number and the posi tion-dependent non-Hermitian degree γn describes the strength of N gain/loss material that is assumed to be balanced, i.e., n=1 γn = 0. The field amplitude transforms as cn → c N −n+1 under parity trans formation and the complex number transforms as i → −i under anti-linear time reversal transformation. Thus the global PT sym metry is lost unless a precise relation between γn holds. Here, we consider a quasi-periodical system with global PT symmetry and study localization. Consider the following gain/loss parameter
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i γn = V cos(2π β n + φ N ) + i γ0 sin(2π β n + φ N ),
(2)
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where V and γ0 are constants, β determines the degree of the quasi-periodicity and φ N is the constant phase difference which depends on the total number of sites. We require that gain and loss are balanced, so we demand φ N = −π β( N + 1) + φ0 , where the constant φ0 is an integer multiple of π . Without loss of gen erality, we take φ0 = 0. We emphasize that the system is PT symmetric globally. Eq. (1) with (2) can be called the PT sym metric Aubry–Andre model [30], which can now be engineered experimentally [3–5]. The most interesting result of the Hermi tian Aubry–Andre model is that the states at the center of the lattice is localized (Anderson localization) for irrational values of β when V > 2. Apparently, the non-Hermitian character of the Aubry–Andre equation (1) could change the physics of this sys tem dramatically. Note also that Aubry–Andre model coincides with the Harper model when V = 2 and γ0 = 0 and the en ergy spectrum as a function of β is known as Hofstadter butterfly spectrum, which is an example of fractal structure that appears in physics [31,32]. Here, we study the Hofstadter butterfly spec trum and localization effect for the PT symmetric Aubry–Andre model. It is s fficient to analyze the region 0 < β < 1 since the system repeats itself in equal intervals of β . Furthermore, the energy spec trum is symmetric with respect to β = 0.5 axis and the spectrum does not depend on the sign of γ0 . As a special case, if β is either 0 or 1, then the gain/loss terms vanish. If β = 1/2 and N is even, the system has gain and loss with amplitudes ∓i γ0 at alternating lattice sites. The gain/loss materials change periodically if β is a ra tional number and quasi-periodically if β is an irrational number. In the latter case, the gain/loss impurities are quasi-periodically ordered. Note that β can be given with a finite number of dig its in a real experiment. To increase the incommensurability of β = p /q (p, q are two coprime positive integers), one can choose s fficiently large p and q.
Fig. 2. The imaginary part of the energy eigenvalues at N = 50, V = 0 and as a function of β .
γ0 = 2
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We look for stationary solutions of Eq. (1). Suppose first that V = 0. In the absence of gain and loss, the system has the well known energy spectrum of width 4 J : E = −2 J cos nπ / N. In the presence of gain and loss, the real part of the energy eigenvalues, R{ E }, are still contained in [−2 J , 2 J ] for any N. More precisely, the energy width is a decreasing function |γ0 |. The distribution of R{ E } crucially depends on the strength of quasi-periodical or der through the value of β . It consists of a finite number of bands when β is rational. In this case, R{ E } is the union of bands and the length of the gap between any two bands depends on q (β = p /q). On the other hand, the fractal structure appears and the spectrum is a Cantor set when β is irrational (for the mathematicians, this property is known as the 10 Martinis conjecture [34] in the Hermi tian limit). Such a fractal structure can be seen in Fig. 1, where we plot the PT symmetric Hofstadter butterfly spectrum at γ0 = 2, V = 0 (a) and at V = 2, γ0 = 0.1 (b). An important difference be tween V = 0 and V = 0 cases is that the symmetry with respect to zero energy axis is lost for the latter case. However, the real part of the energy eigenvalues is symmetric with respect to β = 0.5 axis for any V . Note also that the width of R{ E } increases with V and it takes its maximum value when γ0 = 0. As can be seen below, the PT symmetry breaking point is very small for large N and thus the corresponding energy spectrum is not real. However, there exists some special values of β for Fig. 1b with entirely real spectrum. For example, the spectrum is real when β = 1/5. To this end, we stress that the nice fractal picture can be seen not only for the real part of the spectrum but also for the imaginary part of the spectrum. Fig. 2 plots it at V = 0 and γ0 = 2. To gain more insight on the role of quasi-periodicity, let us study how the real and imaginary parts of the energy change with
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√ Fig. 3. The real part of energy eigenvalues as a function of γ0 at N = 30, V = 0 and J = 1. We take β = 0.6 and β = ( 5 − 1)/2 ≈ 0.618.
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Fig. 4.√The imaginary part of energy eigenvalues as a function of γ0 at J = 1, V = 0 and N = 10, 100. The degree of the quasi-periodicity changes β = ( 5 − 1)/2 is nearly 1.6 times greater than γ P T for β = 6/10 when N = 25. For large number of lattice sites, γ P T → 0.
γ P T sign ficantly. γ P T for
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γ0 for weakly and strongly disordered system. Fig. 3 plots R{ E } as a function γ0 for a weak β = 1/3 and strong disorders β = 11/30
when N = 30. As can be seen from the figures, the degree of quasi periodicity in the lattice has a dramatic effect for large values of γ0 . The real part of energy shrink to zero (they become degen erate and the energy width becomes zero) for very large values of γ0 if the system is periodical while this is not the case if it is quasi-periodical. However, the corresponding imaginary part of en ergy eigenvalues are different from zero for such a large number of γ0 . If the impurity strength, γ0 , exceeds a critical point, γ P T , PT symmetry is spontaneously broken and thus the energy eigenvec tors are not simultaneous eigenvectors of the Hamilton and PT operators. In this case, the energy eigenvalues become partially or entirely complex. An important consequence for our system is that strong disorder increases the critical point γ P T considerably. Fig. 4 plots the imaginary part of the energy eigenvalues for var ious values of N at β = 0.6 and the inverse of the golden ratio √ β = ( 5 − 1)/2 ≈ 0.618, which is the common choice in the study of the Aubry–Andre model. We numerically find that due to the quasi-periodicity, γ P T increases by a factor of nearly 1.6 when N = 25. However, the number of lattice sites N has dominant ef fect on γ P T and thus increasing the degree of the quasi-periodicity has slightly changed γ P T for large N. The critical point decreases
with increasing N and approaches zero when N is large. The PT symmetric phase is said to be fragile since γ P T is zero as N →∞ [18]. It is well known that the Hermitian Aubry–Andre model, γ0 = 0, displays a phase transition from extended to exponen tially localized states (it is also known as metal–insulator transition [33]). Of particular importance is the self-dual point V / J = 2 where the localization transition occurs [30]. Let us now study whether localization takes place for the PT symmetric Aubry Andre model. Suppose first V = 0. We take the inverse of the √ 5−1 , 2
golden ratio β = J = 1 and N = 49 with the initial condi tion |cn ( z = 0)|2 = δn,25 . We find numerically that initially localized wave packet delocalizes in time when γ0 = 2. We repeat numerical solution for large values of γ0 , but exponentially localized states do not emerge for the PT symmetric Aubry–Andre model. This result is interesting that quasi-periodicity does not induce localization for the PT symmetric Aubry–Andre model contrary to Hermitian one. This is because of the fragile nature of the PT symmetric phase. Before metal–insulator phase transition takes place, the system en ters broken PT symmetric phase and the corresponding intensity N grows exponentially, where the intensity is given by I = n=1 |cn |2 and satisfies
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Fig. 5. The total intensity as a function of z for the parameters ω = 0 (dashed) and ω = 3 (solid) at fixed V = 4. It grows exponentially for the static case while it is almost constant for large values of ω . We plot σ ( z) for V = 4 (solid) √ and V = 0 (dashed) at fixed ω = 3. Localization takes place when σ ( z) oscillates (non-periodically) and ballistic
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σ (z) increases linearly. We take β =
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γ0 = 2 for both plots. We assume that only n = 25-th well is occupied initially.
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dI dz
=2
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(3)
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The intensity grows exponentially in the broken PT symmetric case while it oscillates when the energy spectrum is entirely real. Suppose now V = 0. We find numerically the time evolution of the single site excitation. It is well known that the metal–insulator transition occurs at V = 2 and γ0 = 0 and the wave packet is lo calized around the single site when V > 2. If V < 2, the proba bility |c 25 |2 goes to zero rapidly with z. The presence of gain/loss change the dynamics sign ficantly. Although the probability |c 25 |2 doesn’t rapidly go to zero when γ0 = 0, this cannot be consid ered localization in the rigorous sense. This is because the in troduction of gain/loss to the system does not conserve the total intensity and the generated particles enter the system not only n = 25-th lattice site but also the other lattice sites. Thus the oc cupation on the waveguide away from n = 25 is not negligible. For large z, the particles are generated even at the edges of the sys tem. A question arises. Does the phase transition from extended to exponentially localized state occurs if we somehow find a way to make the total intensity bounded for large values of γ0 ? To answer this question, consider z-dependent periodic impurity strength [15, 35–39]
i γn = V cos(2π β n + φ N ) + i γ0 cos(2πω z) sin(2π β n + φ N ), (4) where ω is a constant. Note that the corresponding Hamiltonian is still PT invariant. The gain and loss are also locally balanced after one period. The intensity oscillates in time when ω = 0 if γ0 < γ P T . The os cillation is not in general periodic. Introducing periodically chang ing impurity, ω = 0, makes the oscillation periodical with z. In creasing ω decreases the period of the intensity. We assert that the intensity is in principle conserved in the limit ω → ∞ since impurities do not have enough time to transfer intensity to the system. So, we expect that rapidly changing impurities practically conserves the intensity. To check this argument, we solve Eq. (1) numerically. We find that the intensity is almost constant when ω = 3 as can be seen from Fig. 5. The quasi-periodicity has nothing to do with the intensity conservation and the intensity is almost conserved for any values of β . To predict localization, let usdefine the variance of the prob ability distribution as
σ ( z) =
n (n
− n¯ )2 |cn |2 / P , where n¯ =
2 n n|cn | / P is the average site z-dependent occupation [39]. We plot the variance in Fig. 5. The linearly increasing σ ( z) with re spect to z implies that the wave packet delocalizes (spreads ballis tically with z). On the contrary, oscillating σ ( z) shows us that the wave packet is localized. We find that the onset of local ization appears for PT symmetric Aubry–Andre model provided
that V / J > 2 and the system is quasi-periodical. We emphasize that localization doesn’t take place if the system is ordered, i.e. β is a rational number. In the localization regime, the occupa tion at n = 25 – the well oscillates periodically with z and is almost constant for large values of V . We note that the underlying mechanism of localization studied here is essentially the same of Anderson localization. To summarize, we have studied PT symmetric tight binding optical lattice with quasi-periodically ordered impurities by con sidering the complex extension Aubry–Andre model. We have plot ted complex Hofstatder butterfly spectrum and shown that the re ality of the spectrum depends sensitively on the impurity strength and quasi-periodicity. We have shown that the critical point γ P T increases with the increasing degree of quasi-periodicity. We have demonstrated that the transition from extended to localized states does not occur for the system described by PT symmetric Aubry Andre model. The metal–insulator transition occurs if the impuri ties changes periodically with z at each site. We have also shown that rapidly changing periodical impurities conserves the total in tensity.
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