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arXiv:chao-dyn/9809008 · Physical Review Letters

Overcoming the Wall in the Semiclassical Baker's Map

Lev Kaplan, Eric J. Heller

Published Feb 26, 1996• 28 citations
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Abstract

A major barrier in semiclassical calculations for chaotic systems is the exponential increase in the number of terms at long times. Using an analogy with spin-chain partition functions, we overcome this ``exponential wall'' for the baker's map, reducing to order ${\mathrm{NT}}^{3/2}$ the number of operations needed to evolve an $N\ensuremath{-}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}$ system for $T$ time steps. This method enables us to obtain semiclassical results up to the Heisenberg time and beyond, providing new insight as to the accuracy of the semiclassical approximation. The semiclassical result is often correct; its breakdown is nonuniform.

DOI
10.1103/physrevlett.76.1453

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quant-pharXiv:2610.04170•Oct 6, 2026
A major barrier in semiclassical calculations for chaotic systems is the exponential increase in the number of terms at long times. Using an analogy with spin-chain partition functions, we overcome this ``exponential wall'' for the baker's map, reducing to order NT3/2{\mathrm{NT}}^{3/2}NT3/2 the number of operations needed to evolve an N\ensuremath−stateN\ensuremath{-}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}N\ensuremath−state system for TTT time steps. This method enables us to obtain semiclassical results up to the Heisenberg time and beyond, providing new insight as to the accuracy of the semiclassical approximation. The semiclassical result is often correct; its breakdown is nonuniform.
Because of a formal equivalence with the partition function of an Ising chain, the semiclassical traces of the quantum baker map can be calculated using the transfer-matrix method. We analyze the transfer matrices associated with the baker map and the symmetry-reflected baker map (the latter happens to be unitary but the former is not). In both cases simple quantum-circuit representations are obtained, which exhibit the typical structure of qubit quantum bakers. In the case of the baker map it is shown that nonunitarity is restricted to a one-qubit operator (close to a Hadamard gate for some parameter values). In a suitable continuum limit we recover the already known infinite-dimensional transfer operator. We devise truncation schemes allowing the calculation of long-time traces in regimes where the direct summation of Gutzwiller's formula is impossible. Some aspects of the long-time divergence of the semiclassical traces are also discussed.
Because of a formal equivalence with the partition function of an Ising chain, the semiclassical traces of the quantum baker map can be calculated using the transfer-matrix method. We analyze the transfer matrices associated with the baker map and the symmetry-reflected baker map (the latter happens to be unitary but the former is not). In both cases simple quantum-circuit representations are obtained, which exhibit the typical structure of qubit quantum bakers. In the case of the baker map it is shown that nonunitarity is restricted to a one-qubit operator (close to a Hadamard gate for some parameter values). In a suitable continuum limit we recover the already known infinite-dimensional transfer-operator. We devise truncation schemes allowing the calculation of long-time traces in regimes where the direct summation of Gutzwiller's formula is impossible. Some aspects of the long-time divergence of the semiclassical traces are also discussed.
The semiclassical limit of the coherent state propagator $⟨{\mathbf{z}}^{\ensuremath{''}}\ensuremath{\mid}{e}^{\ensuremath{-}i\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}T∕\ensuremath{\hbar}}\ensuremath{\mid}{\mathbf{z}}^{\ensuremath{'}}⟩$ involves complex classical trajectories of the Hamiltonian $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}(\mathbf{u},\mathbf{v})=⟨\mathbf{v}\ensuremath{\mid}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\ensuremath{\mid}\mathbf{u}⟩$ satisfying $\mathbf{u}(0)={\mathbf{z}}^{\ensuremath{'}}$ and $\mathbf{v}(T)={\mathbf{z}}^{\ensuremath{''}}*$. In this work we study mostly the case ${\mathbf{z}}^{\ensuremath{'}}={\mathbf{z}}^{\ensuremath{''}}$. The propagator is then the return probability amplitude of a wave packet. We show that a plot of the exact return probability brings out the quantal images of the classical periodic orbits. Then we compare the exact return probability with its semiclassical approximation for a soft chaotic system with two degrees of freedom. We find two situations where classical trajectories satisfying the correct boundary conditions must be excluded from the semiclassical formula. The first occurs when the contribution of the trajectory to the propagator becomes exponentially large as $\ensuremath{\hbar}$ goes to zero. The second occurs when the contributing trajectories undergo bifurcations. Close to the bifurcation the semiclassical formula diverges. More interestingly, in the example studied, after the bifurcation, where more than one trajectory satisfying the boundary conditions exist, only one of them in fact contributes to the semiclassical formula, a phenomenon closely related to Stokes lines. When the contributions of these trajectories are filtered out, the semiclassical results show excellent agreement with the exact calculations.
We consider the Wigner--Smith time-delay matrix of a chaotic cavity with broken time-reversal symmetry, a finite number MMM of equivalent open channels, and a uniform tunnel barrier of reflection probability RRR. Exact random-matrix results for nonideal coupling are known at the level of the joint matrix distribution, whereas explicit finite-channel formulas for Schur moments are much less developed. We show that every Schur moment of the nonideal Wigner--Smith matrix factorizes into its ideal-coupling inverse-Laguerre value and a barrier factor given by a circular unitary ensemble (CUE) average of a Schur polynomial. A dual-Cauchy generating function reduces the latter to a Toeplitz determinant with a rational symbol, placing the barrier problem within the classical theory of exact CUE characteristic-polynomial ratios. For one-row partitions, specializing the exact CUE ratio formula gives a closed generating function for all barrier coefficients and therefore all finite delay-time moments in the allowed range n≤Mn\le Mn≤M. The exact ratio formula decomposes into a sector visible to the semiclassical expansion and an additional contribution proportional to RM+1R^{M+1}RM+1, which is invisible to every algebraic order in 1/M1/M1/M. For the second moment, this contribution exactly reproduces the exponentially small term in the known random-matrix expression for ⟨τW2⟩\langleτ_W^2\rangle⟨τW2​⟩ that is absent from the semiclassical expansion. We also derive an exact finite sum for all one-column partitions and an exact complement symmetry for Schur averages inside the M×MM\times MM×M square. The new contribution therefore lies not in the underlying CUE ratio identities, which are classical, but in combining them with the exact nonideal Wigner--Smith mapping and extracting the resulting finite-channel time-delay formulas and corrections to semiclassical Schur-moment conjectures.