Kicked Harper model
| Roberto Artuso (2011), Scholarpedia, 6(10):10462. | doi:10.4249/scholarpedia.10462 | revision #137310 [link to/cite this article] |
The kicked Harper model is a widely studied quantum mechanical system. It may be viewed either as the quantum version of the classical two-dimensional area preserving map\[\tag{1} p_{n+1}\,=\,p_n +K\sin (x_n)\qquad x_{n+1}\,=\,x_n -L\sin(p_{n+1}) \quad (mod\,(2 \pi)), \]
or as a pulsed version of Harper's Hamiltonian \(\tag{2} \hat{H}\,=\,L\cos(\hat{p})+ K \cos(\hat{x})\delta_1(t) \)
where the periodic delta function is defined as\[\tag{3} \delta_1(t)=\sum_{n=-\infty}^{\infty}\, \delta(t-n)\,\,\,\,\hbox{and}\,\,\,\,\hat{p}=-i \hbar \frac{\partial}{\partial x}. \]
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Nature of the model
The way of introducing the model by Eq. (1) focuses the interest on general "quantum chaos" issues, as the classical map possesses rich dynamical features: in particular, when considering the symmetric case (K=L), a transition from an integrable regime to fully developed chaos can be observed as the parameter is increased; intermediate values yield a mixed phase space, which is the typical situation with Hamiltonian systems. On the other side the Hamiltonian Eq. (2) generalizes the well known, and vastly investigated, Harper model (which is simply obtained from Eq. (2) if one omits the periodic delta function), which provides an approximation to electron motion in a two-dimensional crystal in the presence of a perpendicular magnetic field, under suitable approximations (Peierls' substitution from a single band expression). In the framework of quantum chaos, the model attracted an early interest as a counterexample to dynamical localization, namely the suppression of classical chaotic diffusion due to quantum interference, as in the paradigmatic case of Chirikov standard map. Take for instance the symmetric case \(K=L\ :\) when the parameter value is sufficiently high (see Figure 1, where \(K=L=1.1\))
the phase space presents the coexistence of (non-isolating) invariant circles, and a stochastic layer that allows unbounded transport, if the map is considered on the plane or on a cylinder (\(x\) takes then the role of an angle). By going to the quantum dynamics the possibility of delocalization has been suggested by Leboeuf et al. (1990), and indeed quantum diffusion was observed, together with examples of quantum ballistic motion and dynamical localization by Lima and Shepelyansky (1991) (for non-symmetric cases).
Spectral and dynamical features
Peculiarities of quantum dynamics were early recognized to be related to fractal properties of the spectrum (Geisel et al. (1991) and Ketzmerick et al. (1992)): since the Hamiltonian Eq. (2) is time-dependent (and periodic), the relevant object ruling quantum dynamical evolution is the Floquet unitary propagator over one period \(\tag{4} \hat{U}_{L,K}\,=\,\exp \left({-i\frac{L}{\hbar} \cos (\hbar \hat{n})}\right) \exp \left({-i\frac{K}{\hbar}\cos(x)} \right) \)
where \(\hat{n}=-i \partial/\partial x\) and periodic boundary conditions have been imposed.
The term spectrum in this context designates the quasi energy spectrum, which is determined by the eigenvalue equation \( \hat{U}_{L,K} \psi_{\omega}\,=\,e^{-i \omega} \psi_{\omega}. \) When the effective Planck's constant is rational \( \hbar=2 \pi p/q\) there is a \(q\) band Bloch spectrum: the general case of an irrational \(\hbar\) will be considered henceforth. As transport is here studied along (angular) momentum it is convenient to use the Fourier representation. \(\tag{5} \psi_{\omega}(x)=\sum_{m=-\infty}^{\infty}\, e^{i m x}\, \phi_m \qquad \phi_m=(2 \pi)^{-1}\, \int_{0}^{2 \pi}\,dx\,\psi_{\omega}(x)e^{imx}. \)
In particular, if we denote by \( \psi_0\) the initial state of the system, the displacement along the momentum basis at later times \(m\) is described by the probability distribution \(\tag{6} p_m(k)=\left| < \phi_k | \hat{U}_{L,K}^m \psi_0 > \right|^2; \)
in particular the spreading of the state is quantified by the variance \( \sigma_2(m)\,=\,\sum_k |k|^2 p_m(k). \) If the system exhibits dynamical localization then \(\sigma_2(m)\) reproduces the classical linear growth up to a break time \(t_B\ ,\) and after that it fluctuates around a constant value, while quantum diffusion corresponds to the asymptotic behavior \( \sigma_2(m)\sim m^{\gamma} \) where \(\gamma=1\) means normal diffusion, \(\gamma=2\) ballistic motion (which is an upper bound for \(\gamma\)) and non-zero, non-integer values of \(\gamma\) correspond to quantum anomalous diffusion.
Another important dynamical indicator is the integrated correlation function \(\tag{7} C_{int}(m)\,=\,\frac{1}{m} \sum_{k=0}^{m-1}\, \left| <\psi_0 | \hat{U}_{L,K}^k \psi_0> \right|^2. \)
Since \(\hat{U}_{L,K}\) is unitary (and so its spectrum is included in the unit circle) we may express Eq. (7) in terms of the spectral measure, as \( <\psi_0 \, |\, \hat{U}_{L,K}^k \, \psi_0 > \,=\, \int_{0}^{2\pi}\,d\mu_{\psi_0}(\lambda)\,e^{- i \lambda k}. \) The behavior of integrated correlations is thus intimately linked to the nature of the spectral measure: if \(C_{int}(m)\) has a non-zero asymptotic limit this indicates a pure point component in the spectrum, while, if the spectrum is purely continuous, we have a power law decay (Ketzmerick et al. (1992)) \(\tag{8} C_{int}(m) \sim m^{-D_2(\mu_{\psi_0})} \)
where \(D_2(\mu_{\psi_0})\) is a fractal index, namely the correlation dimension of the spectral measure. Though the variance growth is also connected to the nature of the spectrum, no precise statement as Eq. (8) is in general possible: rigorous results typically are in the form of lower bounds, and involve spectral indices as the upper Hausdorff dimension of the measure (Guarneri (1989), Last (1996)): such bounds are extremely important, as they dictate unbounded spreading when specific fractal exponents are non-zero.
Phase diagram of the model
In the case of strongly irrational modulation the phase diagram of the model presents many interesting features (Artuso et al. (1992a), Artuso et al. (1994)). The conventional (unkicked) Harper model presents the following structure: localization and pure point spectrum for \(L<K\ ;\) ballistic spreading and extended states for \(L>K\ ;\) diffusive spreading and singular continuous spectrum for \(L=K\ .\) These features are intimately connected to Aubry-Andre' duality (Aubry and Andre' (1980)): namely if we have an eigenfunction for a parameter pair \((\tilde{K},\tilde{L})\ ,\) its Fourier transform is an eigenfunction for the reversed pair \((\tilde{L},\tilde{K})\ :\) so under exchange of parameter values we pass from localized eigenfunctions to extended states. A form of duality is present also in the kicked Harper model (Guarneri and Borgonovi (1993)), and this is reflected in the structure of the phase diagram, which, however, displays a more complicated structure than the Harper's one. Along the critical line \(K=L\) the observed behavior is quantum diffusion, generally of anomalous kind (Artuso et al. (1992b)).
The overall phase diagram leads to the definition of dual regions pairs (\(I\) and \(I^*\)) and (\(II\) and \(II^*\)), see (Figure 2)
The first pair of regions reproduces the conventional Harper picture: dynamical localization and pure point spectrum in \(I\ ,\) ballistic spreading and extended states in \(I^*\ .\) In the complementary regions (\(II\) and \(II^*\)) the evidence is for a mixed character of the spectrum: a pure point component is always present, as integrated correlations do not vanish asymptotically: by duality also continuous spectrum is not empty, and, accordingly, unbounded spreading is observed, of ballistic type.
Efficient numerical methods for analysis of fractal properties of spectrum of evolution operator have been developed in (Ketzmerick, Kruse and Geisel (1999),Prosen, Satija and Shah (2001)).
Related models
The kicked Harper model at \(K=L\) is a special case of a kicked harmonic oscillator, known as Zaslavsky web map, when the ratio between oscillator and kick periods is four. The properties of dynamics of the quantum kicked harmonic oscillator are described in (Kells, Twamley and Heffernan (2004)).
References
- Leboeuf, P; Kurchan, J; Feingold, M and Arovas, D P (1990). Phase-space localization: topological aspects of quantum chaos Phys.Rev.Lett. 65: 3076-3079. doi:10.1103/physrevlett.65.3076.
- Lima(1991). Fast delocalization in a model of quantum kicked rotator Phys.Rev.Lett. 67: 1377-1380. doi:10.1103/physrevlett.67.1377.
- Geisel, T; Ketzmerick, R and Petschel, G (1991). Metamorphosis of a Cantor spectrum due to classical chaos Phys.Rev.Lett. 67: 3635-3638. doi:10.1103/physrevlett.67.3635.
- Ketzmerick, R; Petschel, G and Geisel, T (1992). Slow decay of temporal correlations in quantum systems with Cantor spectra Phys.Rev.Lett. 69: 695-698. doi:10.1103/physrevlett.69.695.
- Guarneri, I (1989). Spectral properties of quantum diffusion on discrete lattices Europhys.Lett. 10: 95. doi:10.1209/0295-5075/10/2/001.
- Last, Y (1996). Quantum dynamics and decompositions of singular continuous spectra J.Funct.Anal. 142: 406. doi:10.1006/jfan.1996.0155.
- Artuso, R; Borgonovi, F; Guarneri, I; Rebuzzini, L and Casati, G (1992a). Phase diagram in the kicked Harper model Phys.Rev.Lett. 69: 3302-3305. doi:10.1103/physrevlett.69.3302.
- Artuso, R; Casati, G; Borgonovi, F; Rebuzzini, L and Guarneri, I (1994). Fractal and dynamical properties of the kicked Harper model Int.J.Mod.Phys. B 8(3): 207-235. doi:10.1142/s0217979294000099.
- Aubry(1980). Analyticity breaking and Anderson localization in incommensurate lattices Ann.Israel Phys.Soc. 3: 133-164.
- Guarneri(1993). Generic properties of a class of translation invariant quantum maps J.Phys. A 26: 119-132. doi:10.1088/0305-4470/26/1/014.
- Artuso, R; Casati, G and Shepelyansky, D (1992b). Fractal spectrum and anomalous diffusion in the kicked Harper model Phys.Rev.Lett. 68: 3826. doi:10.1103/physrevlett.68.3826.
- Ketzmerick, R; Kruse, K and Geisel, T (1999). Efficient diagonalization of kicked quantum systems Physica D 31: 247. doi:10.1016/s0167-2789(98)00230-9.
- Prosen, T; Satija, I.I and Shah, N (2001). Dimer decimation and intricately nested localized-ballistic phases of a kicked Harper model Phys. Rev. Lett. 87: 066601. doi:10.1103/physrevlett.87.066601.
- Kells, G; Twamley, J and Heffernan, D (2004). Dynamical properties of the delta kicked harmonic oscillator Phys. Rev. E 70: 015203. doi:10.1103/physreve.70.015203.
Recommended Reading
L.E.Reichl, "The Transition to chaos in conservative classical systems and quantum manifestations", Springer, Berlin (2004).
External links
[1] A visualization program for two-dimensional maps, written by James Meiss:
(Figure 1) was generated with this program.
See also
Chirikov standard map, Spectral properties of quantum diffusion, Anderson localization and quantum chaos maps, Zaslavsky web map
