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Journal of Physics A Mathematical and General

Narrowly avoided crossings

Michael Wilkinson

Published Feb 21, 1987• 42 citations
Publisher

Abstract

In order to create a degeneracy in a quantum mechanical system without symmetries one must vary two parameters in the Hamiltonian. When only one parameter, lambda say, is varied, there is only a finite closest approach Delta E of two eigenvalues, and never a crossing. Often the gaps Delta E in these avoided crossings are much smaller than the mean spacing between the eigenvalues, and it has been conjectured that in this case the gap results from tunnelling through classically forbidden regions of phase space and decreases exponentially as h(cross) to 0: Delta E=Ae -S / h(cross) . The author reports the results of numerical calculations on a system with two parameters, epsilon , lambda , which is completely integrable when epsilon =0. It if found that the gaps Delta E obtained by varying lambda decrease exponentially as h(cross) to 0, consistent with the tunnelling conjecture. When epsilon =0, Delta E=0 because the system is completely integrable. As epsilon to 0, the gaps do not vanish because the prefactor A vanishes; instead it is found that S diverges logarithmically. Also, keeping h(cross) fixed, the gaps are of size Delta E=O( epsilon nu ), where nu is usually very close to an integer. Theoretical arguments are presented which explain this result.

DOI
10.1088/0305-4470/20/3/026

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What is a quantum phase transition? Consider a Hamiltonian, H ( g ), whose degrees of freedom reside on the sites of a lattice, and which varies as a function of a dimensionless coupling, g . Let us follow the evolution of the ground state energy of H ( g ) as a function of g . For the case of a finite lattice, this ground state energy will generically be a smooth, analytic function of g . The main possibility of an exception comes from the case when g couples only to a conserved quantity (i.e. H ( g ) = H 0 + gH 1 , where H 0 and H 1 commute). This means that H 0 and H 1 can be simultaneously diagonalized and so the eigenfunctions are independent of g even though the eigenvalues vary with g ; then there can be a level-crossing where an excited level becomes the ground state at g = gc (say), creating a point of nonanalyticity of the ground state energy as a function of g (see Fig. 1.1). The possibilities for an infinite lattice are richer. An avoided level-crossing between the ground and an excited state in a finite lattice could become progressively sharper as the lattice size increases, leading to a nonanalyticity at g = gc in the infinite lattice limit. We shall identify any point of nonanalyticity in the ground state energy of the infinite lattice system as a quantum phase transition: The nonanalyticity could be either the limiting case of an avoided level-crossing or an actual level-crossing.

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What Is a Quantum Phase Transition? Consider a Hamiltonian, H(g) , whose degrees of freedom reside on the sites of a lattice, and which varies as a function of a dimensionless coupling g . Let us follow the evolution of the ground state energy of H(g) as a function of g . For the case of a finite lattice, this ground state energy will generically be a smooth, analytic function of g . The main possibility of an exception comes from the case when g couples only to a conserved quantity (i.e., H(g) = H 0 + gH 1 , where H 0 and H 1 commute). This means that H 0 and H 1 can be simultaneously diagonalized and so the eigenfunctions are independent of g even though the eigenvalues vary with g ; then there can be a level-crossing where an excited level becomes the ground state at g = g c (say), creating a point of nonanalyticity of the ground state energy as a function of g (see Fig. 1.1). The possibilities for an infinite lattice are richer. An avoided level-crossing between the ground and an excited state in a finite lattice could become progressively sharper as the lattice size increases, leading to a nonanalyticity at g = g c in the infinite lattice limit. We shall identify any point of nonanalyticity in the ground state energy of the infinite lattice system as a quantum phase transition: The nonanalyticity could be either the limiting case of an avoided level-crossing or an actual level-crossing.

Cambridge University Press eBooks•Jan 13, 2000•0 citations
In order to create a degeneracy in a quantum mechanical system without symmetries one must vary two parameters in the Hamiltonian. When only one parameter, lambda say, is varied, there is only a finite closest approach Delta E of two eigenvalues, and never a crossing. Often the gaps Delta E in these avoided crossings are much smaller than the mean spacing between the eigenvalues, and it has been conjectured that in this case the gap results from tunnelling through classically forbidden regions of phase space and decreases exponentially as h(cross) to 0: Delta E=Ae -S / h(cross) . The author reports the results of numerical calculations on a system with two parameters, epsilon , lambda , which is completely integrable when epsilon =0. It if found that the gaps Delta E obtained by varying lambda decrease exponentially as h(cross) to 0, consistent with the tunnelling conjecture. When epsilon =0, Delta E=0 because the system is completely integrable. As epsilon to 0, the gaps do not vanish because the prefactor A vanishes; instead it is found that S diverges logarithmically. Also, keeping h(cross) fixed, the gaps are of size Delta E=O( epsilon nu ), where nu is usually very close to an integer. Theoretical arguments are presented which explain this result.
For a polynomial potential with resonant fundamental frequencies (1:2 and 1:3 resonances), quantum avoided crossings can occur when quantum eigenvalues are plotted versus a parameter in the Hamiltonian. In the present paper, primitive (EBK) semiclassical behavior at the quantum avoided crossing is reinvestigated, using the exact analytical calculation of the action integrals, which was devised recently [Chem. Phys. 185, 263 (1994)] for an approximate resonance Hamiltonian that can be deduced from the exact polynomial Hamiltonian by low order perturbation theory. The previously reported behavior, that is semiclassical levels passing through the intersection instead of avoiding each other, is shown to happen if there exist two superimposed branches in the plot of the second action integral ℐ2 as a function of the energy. These results are interpreted in terms of semiclassical diabatic basis and of quantum dynamical tunneling. In contrast, if the semiclassical system enters the (anti)crossing region with semiclassical quantum numbers ℐ2 which do not lie on superimposed branches of the plot, it is shown that at least one, and possibly two, level(s) must cross the separatrix, that is pass from the inside to the outside of the resonance region (or conversely) in order to adapt to the quantum avoided crossing. This causes (i) corresponding semiclassical quantum number ℐ2 to change (ii) the close correspondence between quantum and semiclassical mechanics to break down.
Two recent preprints [B. Altshuler, H. Krovi, and J. Roland, "Quantum adiabatic optimization fails for random instances of NP-complete problems", arXiv:0908.2782 and "Anderson localization casts clouds over adiabatic quantum optimization", arXiv:0912.0746] argue that random 4th order perturbative corrections to the energies of local minima of random instances of NP-complete problem lead to avoided crossings that cause the failure of quantum adiabatic algorithm (due to exponentially small gap) close to the end, for very small transverse field that scales as an inverse power of instance size N. The theoretical portion of this work does not to take into account the exponential degeneracy of the ground and excited states at zero field. A corrected analysis shows that unlike those in the middle of the spectrum, avoided crossings at the edge would require high [O(1)] transverse fields, at which point the perturbation theory may become divergent due to quantum phase transition. This effect manifests itself only in large instances [exp(0.02 N) >> 1], which might be the reason it had not been observed in the authors' numerical work. While we dispute the proposed mechanism of failure of quantum adiabatic algorithm, we cannot draw any conclusions on its ultimate complexity.
The evolution mod Psi (t)) of a system with slowly-varying Hamiltonian H( Omega t) depends not only on the slowness parameter Omega but also on Planck's constant h(cross). For systems with two (or more) classically separated phase-space regions which are quantally connected by tunnelling, the curves of instantaneous energy levels display hyperbolic near-degeneracies rather than crossings. In such cases the limits Omega to 0 (h(cross) fixed and small) and h(cross) to 0 ( Omega fixed and small) lead to opposite behaviour of mod Psi (t)). As an illustration, the uniform semiclassical adiabatic behaviour ( Omega and h(cross) small, Omega /h(cross) arbitrary) is calculated exactly for a double-well potential for which one well gets shallower as the other gets deeper.
Using the Wehrl entropy, we study the delocalization in phase space of energy eigenstates in the vicinity of avoided crossings in the Lipkin-Meshkov-Glick model. These avoided crossings, appearing at intermediate energies in a certain parameter region of the model, originate classically from pairs of trajectories lying in different phase-space regions which, contrary to the low-energy regime, are not connected by the discrete parity symmetry of the model. As coupling parameters are varied, a sudden increase of the Wehrl entropy is observed for eigenstates participating in avoided crossings that are close to the critical energy of the excited-state quantum phase transition. This allows us to detect when an avoided crossing is accompanied by a superposition of the pair of classical trajectories in the Husimi function of eigenstates. This superposition yields an enhancement of dynamical tunneling, which is observed by considering initial Bloch states that evolve partially into the partner region of the paired classical trajectories, thus breaking the quantum-classical correspondence in the evolution of observables.
What is a quantum phase transition? Consider a Hamiltonian, H ( g ), whose degrees of freedom reside on the sites of a lattice, and which varies as a function of a dimensionless coupling, g . Let us follow the evolution of the ground state energy of H ( g ) as a function of g . For the case of a finite lattice, this ground state energy will generically be a smooth, analytic function of g . The main possibility of an exception comes from the case when g couples only to a conserved quantity (i.e. H ( g ) = H 0 + gH 1 , where H 0 and H 1 commute). This means that H 0 and H 1 can be simultaneously diagonalized and so the eigenfunctions are independent of g even though the eigenvalues vary with g ; then there can be a level-crossing where an excited level becomes the ground state at g = gc (say), creating a point of nonanalyticity of the ground state energy as a function of g (see Fig. 1.1). The possibilities for an infinite lattice are richer. An avoided level-crossing between the ground and an excited state in a finite lattice could become progressively sharper as the lattice size increases, leading to a nonanalyticity at g = gc in the infinite lattice limit. We shall identify any point of nonanalyticity in the ground state energy of the infinite lattice system as a quantum phase transition: The nonanalyticity could be either the limiting case of an avoided level-crossing or an actual level-crossing.
What Is a Quantum Phase Transition? Consider a Hamiltonian, H(g) , whose degrees of freedom reside on the sites of a lattice, and which varies as a function of a dimensionless coupling g . Let us follow the evolution of the ground state energy of H(g) as a function of g . For the case of a finite lattice, this ground state energy will generically be a smooth, analytic function of g . The main possibility of an exception comes from the case when g couples only to a conserved quantity (i.e., H(g) = H 0 + gH 1 , where H 0 and H 1 commute). This means that H 0 and H 1 can be simultaneously diagonalized and so the eigenfunctions are independent of g even though the eigenvalues vary with g ; then there can be a level-crossing where an excited level becomes the ground state at g = g c (say), creating a point of nonanalyticity of the ground state energy as a function of g (see Fig. 1.1). The possibilities for an infinite lattice are richer. An avoided level-crossing between the ground and an excited state in a finite lattice could become progressively sharper as the lattice size increases, leading to a nonanalyticity at g = g c in the infinite lattice limit. We shall identify any point of nonanalyticity in the ground state energy of the infinite lattice system as a quantum phase transition: The nonanalyticity could be either the limiting case of an avoided level-crossing or an actual level-crossing.