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Planck’s Quantum-Driven Integer Quantum Hall Effect in Chaos Yu Chen and Chushun Tian Institute for Advanced Study, Tsinghua University, Beijing 100084, People’s Republic of China (Received 20 June 2014; published 21 November 2014) We find in a canonical chaotic system, the kicked spin-1=2 rotor, a Planck’s quantum(he )-driven phenomenon bearing a close analogy to the integer quantum Hall effect but of chaos origin. Specifically, the rotor’s energy growth is unbounded (“metallic” phase) for a discrete set of critical values of he , but otherwise bounded (“insulating” phase). The latter phase is topological and characterized by a quantum number (“quantized Hall conductance”). The number jumps by unity whenever he passes through each critical value as it decreases. Our findings indicate that rich topological quantum phenomena can emerge from chaos. DOI: 10.1103/PhysRevLett.113.216802
PACS numbers: 73.43.-f, 05.45.Mt
Introduction.—The integer quantum Hall effect (IQHE) heralded a revolution in condensed matter physics [1,2]. The physics of IQHE includes two key ingredients. One is the integer filling of Landau bands due to the Pauli principle. This gives a quantized spectrum, Z (the integer set), of the Hall conductance. The other is the topological nature of IQHE [3]: the theory of Thouless and co-workers [4] unveils deep connections between transport quantization and the (topological) Chern number, opening a route to the quantum anomalous Hall effect [5] and various topological quantum phenomena [6]. Recently, the Chern number approach has been applied to explore topological phenomena in driven systems [7–9]. The Chern number description is, in essence, expressed in terms of an integral in the space composed of certain good quantum numbers such as Bloch momentum and (or) external parameters changing adiabatically and forming a cycle (see, e.g., Refs. [6,9]). In chaotic systems such quantum numbers and (or) adiabatic parameters are often absent and, consequently, the Chern number description does not apply. Moreover, chaotic and topological phenomena have totally opposite characteristics: the former are sensitive to disturbances, while the latter exhibit robustness. Can topological quantum phenomena arise in these systems? If yes, what are the roles of chaos? These fundamental questions are explored in this work. We find, surprisingly, that a novel topological mechanism emerges from chaos that gives rise to a phenomenon bearing a close analogy to IQHE in a genuine single-particle system in which both the Pauli principle and the magnetic field are absent. We focus on a canonical chaotic driven system, the kicked rotor [10–13]. Despite its simple construction—a particle moving on a ring under the influence of a sequential driving force (“kicking”), this system exhibits a wealth of phenomena. Its realization in atom optics [14] has triggered a renewal of studies of chaotic dynamics [15–21]. An important parameter governing the interplay 0031-9007=14=113(21)=216802(5)
between chaoticity and quantum interference is Planck’s quantum, he ≡ τℏ=I, with ℏ being Planck’s constant, I the particle’s moment of inertia, and τ the kicking period [11,22,23]. The common wisdom is that tuning he alters the kicked rotor’s symmetries, causing dramatic changes in the system’s behavior [11,12,19,24–33]. Here we report a novel he -driven phenomenon totally beyond this common wisdom: it is of a topological nature [34]. Specifically, we find that in quasiperiodically kicked spin-1=2 rotors there are an infinite number of topological quantum phases classified by Z, and decreasing he triggers sequential transitions between them. Strikingly, although the system—as simple as being single particle, one dimensional (1D), and free of a magnetic field—is totally different from genuine quantum Hall systems, the sequential transitions between topological phases resemble (in the sense of mathematical equivalence) IQHE. Our finding is a novel-type IQHE, with h−1 e mimicking the “filling fraction,” and is dubbed “the Planck’s quantum-driven IQHE.” It has no classical correspondence. Model and main results.—Consider a spin-1=2 (chargeless) particle moving on a ring and subject to a potential switched on at integer times. The potential V ≡ ~ i , with θ1 being the particle’s angular V i ðθ1 ; θ2 þ ωtÞσ position, θ2 an arbitrarily prescribed phase, σ i ; i ¼ 1; 2; 3 the Pauli matrices, and the Einstein summation convention ~ incommensurate with used, is modulated at a frequency ω, the kicking frequency 2π. Without loss of generality, we set pffiffiffi ~ ¼ 2π= 5 unless otherwise specified. V 1 has odd (even) ω parity in θ1 ðθ2 Þ and vice versa for V 2 ; V 3 has even parity in θ1;2 [35]. V couples the rotor’s spin and angular position degrees of freedom. This type of kicking was introduced by Scharf [36] and has been extensively studied [8,37,38]. The particle’s motion is described by ˆ ψ~ t ; ihe ∂ t ψ~ t ¼ HðtÞ
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X ˆ ~ δðt − sÞ; HðtÞ ¼ −ðhe ∂ θ1 Þ2 þ Vðθ1 ; θ2 þ ωtÞ
ð1Þ
s
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FIG. 1 (color online). Schematic representation of the main results. Red: for a discrete set of critical values of h−1 e , EðtÞ t→∞ increases linearly at large t (metal), EðtÞ=t ¼ σ ; for other values t→∞ of h−1 e , EðtÞ saturates at large t (insulator), EðtÞ=t ¼ 0. Blue: the insulator is characterized by quantized σ H that jumps by unity whenever h−1 increases, passing through each critical value. e (Practically, t is finite and the transitions are smeared.)
with ψ~ t a two-component spinor. We use the rotor’s energy, EðtÞ ≡ − 12 hhψ~ t j∂ 2θ1 jψ~ t iiθ2 , with h·iθ2 being the average over θ2 to probe the system’s phases. If the energy growth rate EðtÞ=t is zero (nonzero) at large t, the rotor would exhibit bounded (extended) motion in angular momentum space and would simulate an insulator (metal) in disordered electronic systems. In this work we predicted analytically and confirmed numerically that the system of Eq. (1) exhibits rich quantum phase structures as h−1 increases. The main e results are schematically shown in Fig. 1. The red curve in Fig. 1 shows that there is an infinite discrete set of critical values of h−1 e . At each critical value, the rotor behaves as a metal with a quantum growth rate of energy σ (namely, the value of the peaks in Fig. 1), which is universal, independent of the system’s details such as specific forms of V i and critical values; for other values of h−1 e insulating phases result. Furthermore, we found that these insulating phases are topological, characterized by a quantum number σ H ∈ Z. The blue curve shows that whenever h−1 e increases passing through each critical value, σ H jumps by unity. Therefore, the profiles of EðtÞ=t and σ H versus h−1 e bear a close analogy to those of IQHE: h−1 mimics the filling e fraction, EðtÞ=t the longitudinal conductance, and σ H the quantized Hall conductance. Topological mechanism.—By using the transformation ~ θ2 ˆ −ωt∂ ~ θ2 ˆ → eωt∂ [39], H ψ~ t ≡ ψ t , we reduce He ~ θ2 , ψ~ t → e−ωt∂ Eq. (1) to a two-dimensional (2D) periodically driven system, which gives ˆ tψ 0; ψt ¼ U
ˆ ≡ e−ði=he ÞVðθ1 ;θ2 Þ e−iðhe nˆ 21 þ2ω~ nˆ 2 Þ U
ð2Þ
for integer times, where nˆ 1;2 are canonically conjugates of θ1;2 and Vðθ1 ; θ2 Þ ¼ V i ðθ1 ; θ2 Þσ i . V 1 σ 1 þ V 2 σ 2 mimics a 2D “spin-orbit coupled” electron, with Θ ≡ ðθ1 ; θ2 Þ as its “momentum,” while V 3 σ 3 generates a “Zeeman coupling”
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FIG. 2 (color online). (a) Physics of the 2D system (2) is governed by the interference between the advanced (solid curve) and retarded (dashed curve) quantum amplitudes. (b) When this system is chaotic, the phase coherent propagation of these two amplitudes is encoded by a field over N which maps the 2D angular momentum space into the target space H2 × S2 .
breaking the (effective) time-reversal symmetry iσ 2 K; K is the combination of complex conjugation and the operation: ˆ θ1;2 → −θ1;2 , nˆ 1;2 → nˆ 1;2 . The second exponent of U ~ is incomoscillates in angular momenta, (recall that ω mensurate with 2π) mimicking “disorder” located at “position” N ≡ ðn1 ; n2 Þ. As we explained above, the canonical topological mechanism for conventional IQHE, namely, the Chern number description [4,6] does not apply to the present system. A question then arises: why is the predicted phenomenon topological? To understand this we note that properties of the 2D evolution (2) are governed by the interference between the advanced and retarded quantum amplitudes [Fig. 2(a)]. The phase coherent propagation of these two amplitudes is characterized generally by the position N and the momentum Θ. Chaoticity erases the memory of Θ, and information on the propagation is encoded in a matrix field over N, [this field, denoted as Zb;b ðNÞ × Zf;f ðNÞ, “emerges” from some mathematical facts and we refer readers to the microscopic theory below], and the matrix is ≃H2 × S2 , with H2 (S2 ) the 2hyperboloid (2-sphere). This field maps the 2D angular momentum space into the target space H2 × S2 [Fig. 2(b)]. Its homotopic classification is given by π 2 ðH 2 × S2 Þ ¼ π 2 ðS2 Þ ¼ Z, from which a topological θ term results. Such a topologically nontrivial field is the foundation of the Planck’s quantum-driven IQHE, existing also in disordered quantum Hall systems [40,41]. When the system (2) is regular, even partially, the above mapping no longer exists. Hence, the novel IQHE dist→∞ appears, EðtÞ=t ¼ 0 (cf. S3 of the Supplemental Material [42]). We see that the topological mechanism here is conceptually different from the Chern number description
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[8,9] which does not distinguish between chaotic and regular motions (see S2B of the Supplemental Material [42] for further discussions). Microscopic theory.—To derive the main results summarized above, we introduce K ω ðNsþ s− ;N 0 s0þ s0− Þ ≡ 0 ˆ ˆ † ÞjNs− iiω , hhNsþ j1=ð1−eiωþ UÞjN sþ 0 ihN 0 s0− j1=ð1−e−iω− U 0 which describes the interference between the advanced and retarded quantum amplitudes [cf. Fig. 2(a)]. Here, s , s0 are spin indices, ω ≡ω R 0 ðω=2Þ with ω understood as ω þ i0, and h·iω0 ≡ 02π dω0 =2πð·Þ. Then, EðtÞ ¼ 1 ˆ 21 K ω ψ 0 ⊗ ψ †0 Þ, where the trace “Tr” includes the 2 Trðn angular momentum and spin indices, and ψ 0 is uniform in Θ representation. Following Refs. [26,28], we express K ω as an exact integral over a supermatrix field, Z ≡ fZNsα;N 0 s0 α0 g with s the spin index and αð¼ b; fÞ denoting the supersymmetric space: the entries with α ¼ α0 (α ≠ α0 ) are commuting (anticommuting) variables. The integral reads K ω ¼ R ~ −S ½ð1−ZZÞ ~ −1 ZNs b;Ns b ½ð1−ZZÞ ~ −1 Z ~ N 0 s0 b;N 0 s0 b . DðZ;ZÞe − þ − þ Here, ~ þ Str lnð1 − eiω UZ ˆ U ˆ † ZÞ ~ S ¼ −Str lnð1 − ZZÞ
ð3Þ
with the supertrace “Str” including the angular momentum. Z is constrained by Z~ b;b ¼ Z†b;b, Z~ f;f ¼ −Z†f;f , and jZb;b Z†b;b j < 1. From Eq. (3), we find that the massless modes are proportional to the unit matrix in spin space. Therefore, hereafter, Z carries no spin indices. representation of Z, i.e., PThe iΔNΘWigner e Z , describes the phase coherNþðΔN=2Þ;N−ðΔN=2Þ ΔN ent propagation, with the position N and the momentum Θ. We see that information on the momentum relaxation is encoded in the components off-diagonal in angular momentum. Since chaoticity erases the memory of the momentum, all these components are eliminated, i.e., Z ¼ fZNα;Nα0 g ≡ ZðNÞ. Taking this into account we simω≪1
plify Eq. (3) to S ¼ Str lnðϵi σ i þ iQÞ − ðiω=2ÞStrðQΛÞ, where ϵi ¼ cotðjVj=2he ÞðV i =jVjÞ, and QðNÞ is a 4 × 4 supermatrix field constructed from ZðNÞ, Q≡
1 Z~
−1 1 Λ 1 Z~
Z
Z 1
;
Λ¼
1
0
0 −1
:
ð4Þ
Performing the hydrodynamic expansion for simplified S, we find 1 S ¼ Strð−σð∇QÞ2 þ σ H Q∇1 Q∇2 Q − 2iωQΛÞ: 4
ð5Þ
ZZ σ H ¼ 4εijk
σ¼2
dθ1 dθ2 ∂ θ1 ϵi ∂ θ1 ϵi ; 2π 2π ðϵ2 þ 1Þ2
dθ1 dθ2 ˆ ∂ θ1 ϵj ∂ θ2 ϵk ; O 2π 2π ðϵ2 þ 1Þ2
t→∞
unstable in the σ H direction. In this phase, EðtÞ=t ¼ σ , which is a main characteristic of metals. This metal has a small, universal energy growth rate σ ¼ Oð1Þ and is of a quantum nature. From Eq. (7), we find the asymptotic behavior of σ H at small he (see S2A of the Supplemental Material [42] for the derivations), σ H ∝ h−1 e ;
ð6Þ
ð7Þ
where εijk is the Rtotally antisymmetric tensor, and the ˆ ≡ ϵi þ dμ∂ μ ϵi. In deriving these results we operator O ˆ acts on a Hilbert space composed of wave note that U functions which vanish at infinity. To satisfy this condition, we allow V to depend smoothly on a certain parameter μ on a boundary strip of angular momentum space. This leads to ˆ Despite the μ dependence of ϵi and the μ integral in O. sharp differences between the kicked rotor and quantum Hall systems, Eqs. (5)–(7) strongly resemble the Pruisken’s field theory of conventional IQHE [2,40], with σ and σ H mimicking the unrenormalized longitudinal and Hall conductances, respectively. This analogy may be attributed to the similarities between the present system and a spin-orbit coupled electron subject to a magnetic field and a disordered potential. As shown in S1 of the Supplemental Material [42], Zb;b ðNÞ ≃ H2 and Zf;f ðNÞ ≃ S2 [43] and, therefore, Zb;b ðNÞ × Zf;f ðNÞ maps the 2D angular momentum space into the target space H 2 × S2 [Fig. 2(b)]; furthermore, the second term of Eq. (5), the topological θ term, characterizes the degree of such mappings, namely, the homotopic classification Z. To find EðtÞ at small t, we expand Eq. (5) in Z. Combined with the Z-integral expression of K ω , the leading expansion gives EðtÞ ¼ σt, which implies the absence of interference effects and is a manifestation of chaoticity (cf. Fig. 3). At large t quantum interference strongly renormalizes σ and σ H . We apply the renormalization group analysis to Eq. (5). A two-parameter renormalization group flow [2,44] [Fig. 4(a)] follows. It has two types of fixed points: (σ H ¼ n ∈ Z; 0) and (n þ 12, σ ). The former is stable, with a vanishing (static) conductance which is a main characteristic of insulators and implies that EðtÞ saturates at large t. These insulators are conceptually different from usual rotor insulators [11–14,20,26,28,39]: they are characterized by the quantum number σ H , which is endowed with a topological nature by the θ term. The latter type of fixed points are stable along the critical line of σ H ¼ n þ 12 but
The bare (unrenormalized) coefficients are ZZ
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ð8Þ
with the proportionality coefficient depending only on V i . Since σ H increases unboundedly with h−1 e , there is an
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infinite discrete set of critical values of h−1 e for which σ H is a half integer and the system is at a critical point. When h−1 e falls between two nearest critical values, the renormalization group flow drives the system to the same insulating phase, giving rise to a plateau in Fig. 1. Whenever h−1 e increases passing through each critical value, σ H jumps by unity (the plateau transition) and simultaneously the system exhibits a metal-insulator transition in the same universality class as the integer quantum Hall transition. Moreover, because of Eq. (8), large critical values of h−1 e are equally spaced along the h−1 e axis. Numerical confirmation.—For simulations we use V ¼ ð2 arctan 2d=dÞd · σ; d ¼ ( sin θ1 ; sin θ2 ; 0.8ðμ − cos θ1 − cos θ2 Þ). This was previously [8] used to find a μ-driven (he set to unity) effect analogous to that predicted in Ref. [45]. Here, because we are interested in the he dependent physics, we fix V and set μ ¼ 1. A manifestation of chaoticity is an early linear energy growth for small he [11,22]. To test this, we simulate the short-time 1D evolution (1) for 5 × 10−3 ≤ he ≤ 5 × 10−1 by standard fast Fourier transform techniques. The simulation results, averaged over 102 values of θ2 show that EðtÞ indeed grows linearly at small t [Fig. 3(a)], with the growth rate EðtÞ=t in excellent agreement with the analytic formula (6) [Fig. 3(b)]. Furthermore, we calculate the ˆ given in Eq. (2) by numerical quasienergy spectrum of U
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diagonalization. The Hilbert space is composed of wave functions on the angular momentum lattice of 64 × 64 sites satisfying periodic boundary conditions. We find that the quasi-energy spectrum is chaotic with the level spacing distribution, PðsÞ, obeying the Wigner surmise for the circular unitary ensemble [46] [Fig. 3(c)]. This confirms that the equivalent 2D system (2) is chaotic. To confirm the predicted novel-type IQHE, we simulate the long-time evolution (1) for different he . Because of difficulties of long-time simulations for small he , we set he ≥ 0.23. By calculating Eq. (7) explicitly, which gives the solid curve in Fig. 4(b), we find four insulating phases for he ≥ 0.23, corresponding, respectively, to the fixed points of σ H ¼ 0; 1; 2; 3 of the renormalization group flow and the three transitions at h−1 e ¼ 0.73, 2.19, and 3.60 [the intersections of solid curve and dashed lines in Fig. 4(b)], corresponding, respectively, to the critical points of σ H ¼ 12, 32, and 52. Figure 4(c) is the simulation result for EðtÞ=t at t ¼ 6 × 105 for 0.23 ≤ he ≤ 1.50. The results exhibit sharp peaks at h−1 e ¼ 0.77, 2.13, and 3.45, in excellent agreement with analytic predictions. The peaks are approximately uniformly spaced with a spacing ≈1.34, in agreement with the prediction of Eq. (8). The peak values (a)
(a)
(b)
(b)
(c)
(c)
FIG. 3 (color online). (a) Simulations of Eq. (1) show that EðtÞ grows linearly at small t. (b) The numerical (circles) and analytic (solid curve) results for the growth rate σ are in excellent ˆ has a chaotic quasienergy agreement. (c) The Floquet operator U spectrum, with the level spacing distribution (histograms) obeying the Wigner surmise (solid curve).
FIG. 4 (color online). Combined with the renormalization group flow (a), Eq. (7) (solid curve) predicts three transitions (the intersections of solid curve and dashed lines) for he ≥ 0.23 (b) Long-time simulations confirm this, with critical values in agreement with analytic predictions, and that the rotor is metallic at transitions with a universal quantum growth rate and otherwise is insulating with a vanishing growth rate (c).
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are approximately the same, EðtÞ=t ≈ 0.35, signaling a rotor metal with universal quantum growth rate. (The positions and magnitudes of these peaks are both stabilized at t ¼ 6 × 105 .) Between the peaks, EðtÞ=t is fully suppressed indicating a rotor insulator. Thus, simulations confirm the novel-type IQHE. Conclusion.—We have shown analytically and confirmed numerically a chaos-induced IQHE in a simple— 1D, single-particle, and free of magnetic field—system. This novel-type IQHE exists in a large class of driven chaotic systems. This shows that rich topological quantum phenomena can emerge from chaos. An interesting subject for future studies is the analog of the fractional quantum Hall effect in chaotic systems. We are grateful to I. Guarneri, G. Casati, and J. Wang for stimulating discussions and to A. Kamenev for useful comments on the manuscript. This work was supported by the NSFC (Grant No. 11174174) and by the Tsinghua Univ. ISRP.
[1] K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980). [2] The Quantum Hall Effect, edited by R. E. Prange and S. M. Girvin (Springer, New York, 1987). [3] R. B. Laughlin, Phys. Rev. B 23, 5632 (1981). [4] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982); Q. Niu, D. J. Thouless, and Y.-S. Wu, Phys. Rev. B 31, 3372 (1985). [5] F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988). [6] D. Xiao, M. C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010). [7] N. H. Lindner, G. Refael, and V. Galitski, Nat. Phys. 7, 490 (2011). [8] J. P. Dahlhaus, J. M. Edge, J. Tworzydlo, and C. W. J. Beenakker, Phys. Rev. B 84, 115133 (2011). [9] D. Y. H. Ho and J. Gong, Phys. Rev. Lett. 109, 010601 (2012). [10] B. V. Chirikov, Phys. Rep. 52, 263 (1979). [11] F. M. Izrailev, Phys. Rep. 196, 299 (1990). [12] G. Casati, B. V. Chirikov, J. Ford, and F. M. Izrailev, in Stochastic Behavior of Classical and Quantum Hamiltonian Systems, Lecture Notes in Physics Vol. 93, edited by G. Casati and J. Ford (Springer, New York, 1979). [13] S. Fishman, D. R. Grempel, and R. E. Prange, Phys. Rev. Lett. 49, 509 (1982). [14] F. L. Moore, J. C. Robinson, C. F. Bharucha, B. Sundaram, and M. G. Raizen, Phys. Rev. Lett. 75, 4598 (1995). [15] S. A. Gardiner, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 79, 4790 (1997). [16] H. Ammann, R. Gray, I. Shvarchuck, and N. Christensen, Phys. Rev. Lett. 80, 4111 (1998). [17] W. K. Hensinger et al., Nature (London) 412, 52 (2001). [18] D. A. Steck, W. H. Oskay, and M. G. Raizen, Science 293, 274 (2001).
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[19] C. Ryu, M. Andersen, A. Vaziri, M. d’Arcy, J. Grossman, K. Helmerson, and W. Phillips, Phys. Rev. Lett. 96, 160403 (2006). [20] J. Chabé, G. Lemarié, B. Grémaud, D. Delande, P. Szriftgiser, and J. C. Garreau, Phys. Rev. Lett. 101, 255702 (2008). [21] S. Chaudhury, A. Smith, B. E. Anderson, S. Ghose, and P. S. Jessen, Nature (London) 461, 768 (2009). [22] C. Tian, A. Kamenev, and A. Larkin, Phys. Rev. Lett. 93, 124101 (2004); Phys. Rev. B 72, 045108 (2005). [23] Hereafter we rescale the time by τ and the rotor’s angular momentum by I=τ. [24] F. M. Izrailev and D. L. Shepelyansky, Teor. Mat. Fiz. 43, 417 (1980) [Theor. Math. Phys. 43, 553 (1980)]. [25] V. V. Sokolov, O. V. Zhirov, D. Alonso, and G. Casati, Phys. Rev. Lett. 84, 3566 (2000). [26] C. Tian and A. Altland, New J. Phys. 12, 043043 (2010). [27] M. Sadgrove and S. Wimberger, Adv. At. Mol. Opt. Phys. 60, 315 (2011). [28] C. Tian, A. Altland, and M. Garst, Phys. Rev. Lett. 107, 074101 (2011). [29] J. Wang, C. Tian, and A. Altland, Phys. Rev. B 89, 195105 (2014). [30] M. B. d’Arcy, R. M. Godun, M. K. Oberthaler, D. Cassettari, and G. S. Summy, Phys. Rev. Lett. 87, 074102 (2001). [31] J. F. Kanem, S. Maneshi, M. Partlow, M. Spanner, and A. M. Steinberg, Phys. Rev. Lett. 98, 083004 (2007). [32] S. Fishman, I. Guarneri, and L. Rebuzzini, Phys. Rev. Lett. 89, 084101 (2002). [33] J. Wang, I. Guarneri, G. Casati, and J. B. Gong, Phys. Rev. Lett. 107, 234104 (2011). [34] Throughout this work we do not consider the nongeneric case of rational he =ð4πÞ. [35] To simplify technical discussions we further assume V 2 ðθ1 ; θ2 Þ ¼ V 1 ðθ2 ; θ1 Þ and V 3 ðθ1 ; θ2 Þ ¼ V 3 ðθ2 ; θ1 Þ. They render the effective field theory (5) isotropic. When these conditions are released the same phenomenon occurs. [36] R. Scharf, J. Phys. A 22, 4223 (1989). [37] M. Thaha and R. Blümel, Phys. Rev. Lett. 72, 72 (1994). [38] A. Ossipov, D. M. Basko, and V. E. Kravtsov, Eur. Phys. J. B 42, 457 (2004). [39] G. Casati, I. Guarneri, and D. L. Shepelyansky, Phys. Rev. Lett. 62, 345 (1989). [40] A. M. M. Pruisken, Nucl. Phys. B235, 277 (1984). [41] K. B. Efetov, Supersymmetry in Disorder and Chaos (Cambridge University Press, Cambridge, England, 1997). [42] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.113.216802 for technical details and extended discussions. [43] M. R. Zirnbauer, J. Phys. A 29, 7113 (1996). [44] A. M. M. Pruisken, Nucl. Phys. B240, 71 (1984). [45] X. L. Qi, Y. S. Wu, and S. C. Zhang, Phys. Rev. B 74, 085308 (2006). [46] F. Haake, Quantum Signatures of Chaos, 2nd ed. (SpringerVerlag, Berlin, 2001).
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Planck’s Quantum-Driven Integer Quantum Hall Effect in Chaos Yu Chen and Chushun Tian Institute for Advanced Study, Tsinghua University, Beijing 100084, People’s Republic of China (Received 20 June 2014; published 21 November 2014) We find in a canonical chaotic system, the kicked spin-1=2 rotor, a Planck’s quantum(he )-driven phenomenon bearing a close analogy to the integer quantum Hall effect but of chaos origin. Specifically, the rotor’s energy growth is unbounded (“metallic” phase) for a discrete set of critical values of he , but otherwise bounded (“insulating” phase). The latter phase is topological and characterized by a quantum number (“quantized Hall conductance”). The number jumps by unity whenever he passes through each critical value as it decreases. Our findings indicate that rich topological quantum phenomena can emerge from chaos. DOI: 10.1103/PhysRevLett.113.216802
PACS numbers: 73.43.-f, 05.45.Mt
Introduction.—The integer quantum Hall effect (IQHE) heralded a revolution in condensed matter physics [1,2]. The physics of IQHE includes two key ingredients. One is the integer filling of Landau bands due to the Pauli principle. This gives a quantized spectrum, Z (the integer set), of the Hall conductance. The other is the topological nature of IQHE [3]: the theory of Thouless and co-workers [4] unveils deep connections between transport quantization and the (topological) Chern number, opening a route to the quantum anomalous Hall effect [5] and various topological quantum phenomena [6]. Recently, the Chern number approach has been applied to explore topological phenomena in driven systems [7–9]. The Chern number description is, in essence, expressed in terms of an integral in the space composed of certain good quantum numbers such as Bloch momentum and (or) external parameters changing adiabatically and forming a cycle (see, e.g., Refs. [6,9]). In chaotic systems such quantum numbers and (or) adiabatic parameters are often absent and, consequently, the Chern number description does not apply. Moreover, chaotic and topological phenomena have totally opposite characteristics: the former are sensitive to disturbances, while the latter exhibit robustness. Can topological quantum phenomena arise in these systems? If yes, what are the roles of chaos? These fundamental questions are explored in this work. We find, surprisingly, that a novel topological mechanism emerges from chaos that gives rise to a phenomenon bearing a close analogy to IQHE in a genuine single-particle system in which both the Pauli principle and the magnetic field are absent. We focus on a canonical chaotic driven system, the kicked rotor [10–13]. Despite its simple construction—a particle moving on a ring under the influence of a sequential driving force (“kicking”), this system exhibits a wealth of phenomena. Its realization in atom optics [14] has triggered a renewal of studies of chaotic dynamics [15–21]. An important parameter governing the interplay 0031-9007=14=113(21)=216802(5)
between chaoticity and quantum interference is Planck’s quantum, he ≡ τℏ=I, with ℏ being Planck’s constant, I the particle’s moment of inertia, and τ the kicking period [11,22,23]. The common wisdom is that tuning he alters the kicked rotor’s symmetries, causing dramatic changes in the system’s behavior [11,12,19,24–33]. Here we report a novel he -driven phenomenon totally beyond this common wisdom: it is of a topological nature [34]. Specifically, we find that in quasiperiodically kicked spin-1=2 rotors there are an infinite number of topological quantum phases classified by Z, and decreasing he triggers sequential transitions between them. Strikingly, although the system—as simple as being single particle, one dimensional (1D), and free of a magnetic field—is totally different from genuine quantum Hall systems, the sequential transitions between topological phases resemble (in the sense of mathematical equivalence) IQHE. Our finding is a novel-type IQHE, with h−1 e mimicking the “filling fraction,” and is dubbed “the Planck’s quantum-driven IQHE.” It has no classical correspondence. Model and main results.—Consider a spin-1=2 (chargeless) particle moving on a ring and subject to a potential switched on at integer times. The potential V ≡ ~ i , with θ1 being the particle’s angular V i ðθ1 ; θ2 þ ωtÞσ position, θ2 an arbitrarily prescribed phase, σ i ; i ¼ 1; 2; 3 the Pauli matrices, and the Einstein summation convention ~ incommensurate with used, is modulated at a frequency ω, the kicking frequency 2π. Without loss of generality, we set pffiffiffi ~ ¼ 2π= 5 unless otherwise specified. V 1 has odd (even) ω parity in θ1 ðθ2 Þ and vice versa for V 2 ; V 3 has even parity in θ1;2 [35]. V couples the rotor’s spin and angular position degrees of freedom. This type of kicking was introduced by Scharf [36] and has been extensively studied [8,37,38]. The particle’s motion is described by ˆ ψ~ t ; ihe ∂ t ψ~ t ¼ HðtÞ
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X ˆ ~ δðt − sÞ; HðtÞ ¼ −ðhe ∂ θ1 Þ2 þ Vðθ1 ; θ2 þ ωtÞ
ð1Þ
s
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FIG. 1 (color online). Schematic representation of the main results. Red: for a discrete set of critical values of h−1 e , EðtÞ t→∞ increases linearly at large t (metal), EðtÞ=t ¼ σ ; for other values t→∞ of h−1 e , EðtÞ saturates at large t (insulator), EðtÞ=t ¼ 0. Blue: the insulator is characterized by quantized σ H that jumps by unity whenever h−1 increases, passing through each critical value. e (Practically, t is finite and the transitions are smeared.)
with ψ~ t a two-component spinor. We use the rotor’s energy, EðtÞ ≡ − 12 hhψ~ t j∂ 2θ1 jψ~ t iiθ2 , with h·iθ2 being the average over θ2 to probe the system’s phases. If the energy growth rate EðtÞ=t is zero (nonzero) at large t, the rotor would exhibit bounded (extended) motion in angular momentum space and would simulate an insulator (metal) in disordered electronic systems. In this work we predicted analytically and confirmed numerically that the system of Eq. (1) exhibits rich quantum phase structures as h−1 increases. The main e results are schematically shown in Fig. 1. The red curve in Fig. 1 shows that there is an infinite discrete set of critical values of h−1 e . At each critical value, the rotor behaves as a metal with a quantum growth rate of energy σ (namely, the value of the peaks in Fig. 1), which is universal, independent of the system’s details such as specific forms of V i and critical values; for other values of h−1 e insulating phases result. Furthermore, we found that these insulating phases are topological, characterized by a quantum number σ H ∈ Z. The blue curve shows that whenever h−1 e increases passing through each critical value, σ H jumps by unity. Therefore, the profiles of EðtÞ=t and σ H versus h−1 e bear a close analogy to those of IQHE: h−1 mimics the filling e fraction, EðtÞ=t the longitudinal conductance, and σ H the quantized Hall conductance. Topological mechanism.—By using the transformation ~ θ2 ˆ −ωt∂ ~ θ2 ˆ → eωt∂ [39], H ψ~ t ≡ ψ t , we reduce He ~ θ2 , ψ~ t → e−ωt∂ Eq. (1) to a two-dimensional (2D) periodically driven system, which gives ˆ tψ 0; ψt ¼ U
ˆ ≡ e−ði=he ÞVðθ1 ;θ2 Þ e−iðhe nˆ 21 þ2ω~ nˆ 2 Þ U
ð2Þ
for integer times, where nˆ 1;2 are canonically conjugates of θ1;2 and Vðθ1 ; θ2 Þ ¼ V i ðθ1 ; θ2 Þσ i . V 1 σ 1 þ V 2 σ 2 mimics a 2D “spin-orbit coupled” electron, with Θ ≡ ðθ1 ; θ2 Þ as its “momentum,” while V 3 σ 3 generates a “Zeeman coupling”
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FIG. 2 (color online). (a) Physics of the 2D system (2) is governed by the interference between the advanced (solid curve) and retarded (dashed curve) quantum amplitudes. (b) When this system is chaotic, the phase coherent propagation of these two amplitudes is encoded by a field over N which maps the 2D angular momentum space into the target space H2 × S2 .
breaking the (effective) time-reversal symmetry iσ 2 K; K is the combination of complex conjugation and the operation: ˆ θ1;2 → −θ1;2 , nˆ 1;2 → nˆ 1;2 . The second exponent of U ~ is incomoscillates in angular momenta, (recall that ω mensurate with 2π) mimicking “disorder” located at “position” N ≡ ðn1 ; n2 Þ. As we explained above, the canonical topological mechanism for conventional IQHE, namely, the Chern number description [4,6] does not apply to the present system. A question then arises: why is the predicted phenomenon topological? To understand this we note that properties of the 2D evolution (2) are governed by the interference between the advanced and retarded quantum amplitudes [Fig. 2(a)]. The phase coherent propagation of these two amplitudes is characterized generally by the position N and the momentum Θ. Chaoticity erases the memory of Θ, and information on the propagation is encoded in a matrix field over N, [this field, denoted as Zb;b ðNÞ × Zf;f ðNÞ, “emerges” from some mathematical facts and we refer readers to the microscopic theory below], and the matrix is ≃H2 × S2 , with H2 (S2 ) the 2hyperboloid (2-sphere). This field maps the 2D angular momentum space into the target space H2 × S2 [Fig. 2(b)]. Its homotopic classification is given by π 2 ðH 2 × S2 Þ ¼ π 2 ðS2 Þ ¼ Z, from which a topological θ term results. Such a topologically nontrivial field is the foundation of the Planck’s quantum-driven IQHE, existing also in disordered quantum Hall systems [40,41]. When the system (2) is regular, even partially, the above mapping no longer exists. Hence, the novel IQHE dist→∞ appears, EðtÞ=t ¼ 0 (cf. S3 of the Supplemental Material [42]). We see that the topological mechanism here is conceptually different from the Chern number description
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[8,9] which does not distinguish between chaotic and regular motions (see S2B of the Supplemental Material [42] for further discussions). Microscopic theory.—To derive the main results summarized above, we introduce K ω ðNsþ s− ;N 0 s0þ s0− Þ ≡ 0 ˆ ˆ † ÞjNs− iiω , hhNsþ j1=ð1−eiωþ UÞjN sþ 0 ihN 0 s0− j1=ð1−e−iω− U 0 which describes the interference between the advanced and retarded quantum amplitudes [cf. Fig. 2(a)]. Here, s , s0 are spin indices, ω ≡ω R 0 ðω=2Þ with ω understood as ω þ i0, and h·iω0 ≡ 02π dω0 =2πð·Þ. Then, EðtÞ ¼ 1 ˆ 21 K ω ψ 0 ⊗ ψ †0 Þ, where the trace “Tr” includes the 2 Trðn angular momentum and spin indices, and ψ 0 is uniform in Θ representation. Following Refs. [26,28], we express K ω as an exact integral over a supermatrix field, Z ≡ fZNsα;N 0 s0 α0 g with s the spin index and αð¼ b; fÞ denoting the supersymmetric space: the entries with α ¼ α0 (α ≠ α0 ) are commuting (anticommuting) variables. The integral reads K ω ¼ R ~ −S ½ð1−ZZÞ ~ −1 ZNs b;Ns b ½ð1−ZZÞ ~ −1 Z ~ N 0 s0 b;N 0 s0 b . DðZ;ZÞe − þ − þ Here, ~ þ Str lnð1 − eiω UZ ˆ U ˆ † ZÞ ~ S ¼ −Str lnð1 − ZZÞ
ð3Þ
with the supertrace “Str” including the angular momentum. Z is constrained by Z~ b;b ¼ Z†b;b, Z~ f;f ¼ −Z†f;f , and jZb;b Z†b;b j < 1. From Eq. (3), we find that the massless modes are proportional to the unit matrix in spin space. Therefore, hereafter, Z carries no spin indices. representation of Z, i.e., PThe iΔNΘWigner e Z , describes the phase coherNþðΔN=2Þ;N−ðΔN=2Þ ΔN ent propagation, with the position N and the momentum Θ. We see that information on the momentum relaxation is encoded in the components off-diagonal in angular momentum. Since chaoticity erases the memory of the momentum, all these components are eliminated, i.e., Z ¼ fZNα;Nα0 g ≡ ZðNÞ. Taking this into account we simω≪1
plify Eq. (3) to S ¼ Str lnðϵi σ i þ iQÞ − ðiω=2ÞStrðQΛÞ, where ϵi ¼ cotðjVj=2he ÞðV i =jVjÞ, and QðNÞ is a 4 × 4 supermatrix field constructed from ZðNÞ, Q≡
1 Z~
−1 1 Λ 1 Z~
Z
Z 1
;
Λ¼
1
0
0 −1
:
ð4Þ
Performing the hydrodynamic expansion for simplified S, we find 1 S ¼ Strð−σð∇QÞ2 þ σ H Q∇1 Q∇2 Q − 2iωQΛÞ: 4
ð5Þ
ZZ σ H ¼ 4εijk
σ¼2
dθ1 dθ2 ∂ θ1 ϵi ∂ θ1 ϵi ; 2π 2π ðϵ2 þ 1Þ2
dθ1 dθ2 ˆ ∂ θ1 ϵj ∂ θ2 ϵk ; O 2π 2π ðϵ2 þ 1Þ2
t→∞
unstable in the σ H direction. In this phase, EðtÞ=t ¼ σ , which is a main characteristic of metals. This metal has a small, universal energy growth rate σ ¼ Oð1Þ and is of a quantum nature. From Eq. (7), we find the asymptotic behavior of σ H at small he (see S2A of the Supplemental Material [42] for the derivations), σ H ∝ h−1 e ;
ð6Þ
ð7Þ
where εijk is the Rtotally antisymmetric tensor, and the ˆ ≡ ϵi þ dμ∂ μ ϵi. In deriving these results we operator O ˆ acts on a Hilbert space composed of wave note that U functions which vanish at infinity. To satisfy this condition, we allow V to depend smoothly on a certain parameter μ on a boundary strip of angular momentum space. This leads to ˆ Despite the μ dependence of ϵi and the μ integral in O. sharp differences between the kicked rotor and quantum Hall systems, Eqs. (5)–(7) strongly resemble the Pruisken’s field theory of conventional IQHE [2,40], with σ and σ H mimicking the unrenormalized longitudinal and Hall conductances, respectively. This analogy may be attributed to the similarities between the present system and a spin-orbit coupled electron subject to a magnetic field and a disordered potential. As shown in S1 of the Supplemental Material [42], Zb;b ðNÞ ≃ H2 and Zf;f ðNÞ ≃ S2 [43] and, therefore, Zb;b ðNÞ × Zf;f ðNÞ maps the 2D angular momentum space into the target space H 2 × S2 [Fig. 2(b)]; furthermore, the second term of Eq. (5), the topological θ term, characterizes the degree of such mappings, namely, the homotopic classification Z. To find EðtÞ at small t, we expand Eq. (5) in Z. Combined with the Z-integral expression of K ω , the leading expansion gives EðtÞ ¼ σt, which implies the absence of interference effects and is a manifestation of chaoticity (cf. Fig. 3). At large t quantum interference strongly renormalizes σ and σ H . We apply the renormalization group analysis to Eq. (5). A two-parameter renormalization group flow [2,44] [Fig. 4(a)] follows. It has two types of fixed points: (σ H ¼ n ∈ Z; 0) and (n þ 12, σ ). The former is stable, with a vanishing (static) conductance which is a main characteristic of insulators and implies that EðtÞ saturates at large t. These insulators are conceptually different from usual rotor insulators [11–14,20,26,28,39]: they are characterized by the quantum number σ H , which is endowed with a topological nature by the θ term. The latter type of fixed points are stable along the critical line of σ H ¼ n þ 12 but
The bare (unrenormalized) coefficients are ZZ
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ð8Þ
with the proportionality coefficient depending only on V i . Since σ H increases unboundedly with h−1 e , there is an
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infinite discrete set of critical values of h−1 e for which σ H is a half integer and the system is at a critical point. When h−1 e falls between two nearest critical values, the renormalization group flow drives the system to the same insulating phase, giving rise to a plateau in Fig. 1. Whenever h−1 e increases passing through each critical value, σ H jumps by unity (the plateau transition) and simultaneously the system exhibits a metal-insulator transition in the same universality class as the integer quantum Hall transition. Moreover, because of Eq. (8), large critical values of h−1 e are equally spaced along the h−1 e axis. Numerical confirmation.—For simulations we use V ¼ ð2 arctan 2d=dÞd · σ; d ¼ ( sin θ1 ; sin θ2 ; 0.8ðμ − cos θ1 − cos θ2 Þ). This was previously [8] used to find a μ-driven (he set to unity) effect analogous to that predicted in Ref. [45]. Here, because we are interested in the he dependent physics, we fix V and set μ ¼ 1. A manifestation of chaoticity is an early linear energy growth for small he [11,22]. To test this, we simulate the short-time 1D evolution (1) for 5 × 10−3 ≤ he ≤ 5 × 10−1 by standard fast Fourier transform techniques. The simulation results, averaged over 102 values of θ2 show that EðtÞ indeed grows linearly at small t [Fig. 3(a)], with the growth rate EðtÞ=t in excellent agreement with the analytic formula (6) [Fig. 3(b)]. Furthermore, we calculate the ˆ given in Eq. (2) by numerical quasienergy spectrum of U
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diagonalization. The Hilbert space is composed of wave functions on the angular momentum lattice of 64 × 64 sites satisfying periodic boundary conditions. We find that the quasi-energy spectrum is chaotic with the level spacing distribution, PðsÞ, obeying the Wigner surmise for the circular unitary ensemble [46] [Fig. 3(c)]. This confirms that the equivalent 2D system (2) is chaotic. To confirm the predicted novel-type IQHE, we simulate the long-time evolution (1) for different he . Because of difficulties of long-time simulations for small he , we set he ≥ 0.23. By calculating Eq. (7) explicitly, which gives the solid curve in Fig. 4(b), we find four insulating phases for he ≥ 0.23, corresponding, respectively, to the fixed points of σ H ¼ 0; 1; 2; 3 of the renormalization group flow and the three transitions at h−1 e ¼ 0.73, 2.19, and 3.60 [the intersections of solid curve and dashed lines in Fig. 4(b)], corresponding, respectively, to the critical points of σ H ¼ 12, 32, and 52. Figure 4(c) is the simulation result for EðtÞ=t at t ¼ 6 × 105 for 0.23 ≤ he ≤ 1.50. The results exhibit sharp peaks at h−1 e ¼ 0.77, 2.13, and 3.45, in excellent agreement with analytic predictions. The peaks are approximately uniformly spaced with a spacing ≈1.34, in agreement with the prediction of Eq. (8). The peak values (a)
(a)
(b)
(b)
(c)
(c)
FIG. 3 (color online). (a) Simulations of Eq. (1) show that EðtÞ grows linearly at small t. (b) The numerical (circles) and analytic (solid curve) results for the growth rate σ are in excellent ˆ has a chaotic quasienergy agreement. (c) The Floquet operator U spectrum, with the level spacing distribution (histograms) obeying the Wigner surmise (solid curve).
FIG. 4 (color online). Combined with the renormalization group flow (a), Eq. (7) (solid curve) predicts three transitions (the intersections of solid curve and dashed lines) for he ≥ 0.23 (b) Long-time simulations confirm this, with critical values in agreement with analytic predictions, and that the rotor is metallic at transitions with a universal quantum growth rate and otherwise is insulating with a vanishing growth rate (c).
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are approximately the same, EðtÞ=t ≈ 0.35, signaling a rotor metal with universal quantum growth rate. (The positions and magnitudes of these peaks are both stabilized at t ¼ 6 × 105 .) Between the peaks, EðtÞ=t is fully suppressed indicating a rotor insulator. Thus, simulations confirm the novel-type IQHE. Conclusion.—We have shown analytically and confirmed numerically a chaos-induced IQHE in a simple— 1D, single-particle, and free of magnetic field—system. This novel-type IQHE exists in a large class of driven chaotic systems. This shows that rich topological quantum phenomena can emerge from chaos. An interesting subject for future studies is the analog of the fractional quantum Hall effect in chaotic systems. We are grateful to I. Guarneri, G. Casati, and J. Wang for stimulating discussions and to A. Kamenev for useful comments on the manuscript. This work was supported by the NSFC (Grant No. 11174174) and by the Tsinghua Univ. ISRP.
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