Exact Renormalization Scheme for Quantum Anosov Maps

نویسنده

  • Itzhack Dana
چکیده

An exact renormalization scheme is introduced for quantum Anosov maps (QAMs) on a torus for general boundary conditions (BCs), whose number is always finite. Given a QAM Û with k BCs and Planck’s constant h̄ = 2π/p (p integer), its nth renormalization iterate Û (n) = Rn(Û) is associated with k BCs for all n and with a Planck’s constant h̄ = h̄/kn. It is shown that the quasienergy eigenvalue problem for Û (n) for all k BCs is equivalent to that for Û (n+1) at some fixed BCs, corresponding, for n > 0, to either strict periodicity for kp even or antiperiodicity for kp odd. The quantum cat maps are, in general, fixed points of either R or R2. The Hannay-Berry results turn out then to be significant also for general BCs. PACS numbers: 05.45.+b, 03.65.Ca, 03.65.Sq Typeset using REVTEX 1 Nonintegrable systems whose dynamics can be reduced to a 2D torus in phase space have attracted much attention in the quantum-chaos literature. When quantizing such a system on the torus, the admissible quantum states must satisfy proper boundary conditions (BCs), i.e., they have to be periodic on the torus up to constant Bloch phase factors specified by a Bloch wave vector w. If the Hamiltonian of the system is periodic in phase space, such as, for example, the kicked Harper model [1–4], its classical dynamics can be reduced to the toral phase space of one unit cell of periodicity, and all Bloch wave vectors w in some Brillouin zone (BZ) are allowed. When studying the quantum-chaos problem for such a system, it is natural and important to consider the sensitivity of the eigenstates to continuous variation of w in the BZ [1–4]. This sensitivity is usually strong for eigenstates spread over the chaotic region and weak for eigenstates localized on stability islands. In general, however, the Hamiltonian of a system whose dynamics can be reduced to a torus is not periodic in phase space. Simple and well known examples are the purely chaotic, Anosov “cat maps” [5–13], whose Hamiltonians are quadratic in the phase-space variables [11]. When quantizing these systems, it turns out that only a finite set of w’s in the BZ is allowed [10,12,13], see Eq. (2) below, but this set increases with increasing chaotic instability. For a large class of cat maps, the value w = 0, corresponding to strictly periodic quantum states on the torus, is allowed. This class of maps was first quantized, for w = 0, in the well known work of Hannay and Berry [6]. As a matter of fact, almost all the investigations of the quantum cat maps have been confined to this class with w = 0. Recently [13], the case of antiperiodic BCs (the quantum state assumes values of opposite signs on opposite sides of the torus) has been studied in some detail. The results of Hannay and Berry [6] and of Keating [11] revealed a very atypical feature of quantum cat maps, i.e., the high degeneracy in their spectra, which increases in the semiclassical limit. Typical spectral properties, fitting generic eigenvalue statistics, are already found by quantizing torus maps that are very slight perturbations of the cat maps [14–16]. According 2 to Anosov’s theorem [5], these maps have essentially the same classical dynamics, in particular they are purely chaotic, as the unperturbed cat maps, which are structurally stable. This ceases to be the case for larger perturbations that cause bifurcations generating elliptic islands [16]. However, the quantum BCs for a perturbed cat map are the same as those of the unperturbed cat map, independently of the size of the perturbation [17], see Eq. (2) below. Because of this reason and to simplify terms, general perturbed cat maps are referred to as Anosov maps in this paper. The importance of these maps is in that they may be viewed as generic torus maps on the basis of a general expression for a smooth torus map derived recently [17], see below. Understanding the properties of quantum Anosov maps (QAMs) for general toral BCs is essentially an open problem, since almost all the investigations have been confined to the strict-periodicity case, w = 0. In this paper, we introduce an exact renormalization scheme for QAMs for general BCs on the torus. We show that the spectrum and eigenstates of a general QAM for all BCs, and therefore all its quantum properties, can be fully reproduced from those of the renormalized QAM at some special, fixed BCs. Thus, the general BCs are practically “eliminated” by the renormalization. Specifically, consider a QAM given by the evolution operator Û quantizing a classical Anosov map. Quantization on a torus requires a Planck’s constant h̄ to satisfy 2π/h̄ = p, an integer. The finite number of BCs is denoted by k, which depends only on the classical unperturbed cat map. We define a renormalization transformation R generating by iteration a sequence of QAMs Û (n) = Rn(Û) on the same torus. The number of BCs for Û (n) is k for all n, and Û (n) is associated with a renormalized Planck’s constant h̄ = h̄/k. Thus, Û (n) has kp eigenstates at fixed BCs. The quantum cat maps are fixed points of either R or R2, so that general Û (n) or Û (2n) represent perturbations of a given quantum cat map in its classical limit n → ∞. We then show that the quasienergy eigenvalue problem for Û (n) for all k BCs is equivalent, by a unitary transformation accompanied by a scaling of variables, to that for Û (n+1) at some fixed BCs. The latter can be of four types for n = 0 (Û (0) = Û), but, for n > 0, they can be only of two types, i.e., strict periodicity for kp even 3 and antiperiodicity for kp odd. Thus, the total (all BCs) spectrum of Û , n = 0, ..., n− 1 (n > 1), coincides with a fraction k of the spectrum of Û (n) for one of these two types of BCs, and the corresponding eigenstates are related by the transformation above. In particular, the total spectrum of a quantum cat map for h̄ = 2π/p coincides with a fraction k of its fixed-BCs spectrum for h̄ = 2π/(kp), with arbitrary or even n > 0. It is interesting to note that the two types of BCs above are precisely those that have been studied in detail in the literature [6,7,10–16], so that several results, in particular those of Hannay and Berry [6] for the quantum cat maps, turn out now to be significant also for general BCs. We denote by (u, v) the phase-space variables, [û, v̂] = ih̄, and we assume that the classical dynamics can be reduced to a 2π×2π torus T , where it is described by an Anosov map M . In general, a smooth torus map M can be expressed uniquely as the composition of two maps, M = MA ◦M1 [17]. Here MA is a cat map, MA(z) = A · z mod 2π, where z is the column vector (u, v) and A is a 2× 2 integer matrix with det(A) = 1; by “Anosov” we just mean that |Tr(A)| > 2, a condition generically satisfied by A. The map M1 is defined by M1(z) = z + F(z) mod 2π, where F(z) is a 2π-periodic vector function of z. The QAM corresponding to M = MA ◦ M1 is the unitary operator Û = ÛAÛ1 [17], where ÛA is the “quantum cat map”, whose u representation is [6]

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تاریخ انتشار 1999