An exactly solvable model for the integrability–chaos transition in rough quantum billiards

  • A Corrigendum to this article was published on 05 February 2013

Abstract

A central question of dynamics, largely open in the quantum case, is to what extent it erases a system's memory of its initial properties. Here we present a simple statistically solvable quantum model describing this memory loss across an integrability–chaos transition under a perturbation obeying no selection rules. From the perspective of quantum localization–delocalization on the lattice of quantum numbers, we are dealing with a situation where every lattice site is coupled to every other site with the same strength, on average. The model also rigorously justifies a similar set of relationships, recently proposed in the context of two short-range-interacting ultracold atoms in a harmonic waveguide. Application of our model to an ensemble of uncorrelated impurities on a rectangular lattice gives good agreement with ab initio numerics.

Introduction

There are two basic prototypes of nondissipative dynamics, corresponding to whether time propagation retains a very strong or a very weak memory of the initial conditions: integrable dynamics, where time evolution conserves as many independent quantities as there are degrees of freedom; and completely chaotic dynamics, where all the traces of the initial state of the system—except for its energy and perhaps very few other quantities—are erased exponentially fast. As an integrable system is perturbed away from integrability, its many conserved quantities stop being conserved, and a fundamental problem is to understand the manner and the mechanism by which that happens.

In the case of classical systems, this problem was highlighted by the Fermi–Pasta–Ulam (FPU) 'paradox' and was central for the development of several subfields of classical mechanics, including the theories of solitons and of dynamical chaos1. The most celebrated result is the Kolmogorov–Arnold–Moser (KAM) theorem, which explains the gradual disappearance of quasiperiodic orbits as the perturbation increases. In quantum mechanics, there are several relevant experimental2,3,4 and numerical5 studies, all in the context of studies of relaxation, as well as attempts to formulate a quantum KAM theorem6. Nevertheless, the problem remains largely open.

For isolated quantum-chaotic systems, the key property governing the memory of the initial state under time propagation is eigenstate thermalization, which is defined as follows: consider the values 〈α|Â|α〉, which are the quantum expectation values of an observable  with respect to the eigenstates |α〉, with the corresponding eigenenergies Eα, of the system Hamiltonian Ĥ. The system has the eigenstate thermalization property with respect to the observable Â, if, in any limit in which the density of states increases without bound, and in any microcanonical energy window, all the values 〈α|Â|α〉 in the window become equal to each other (here 〈α|Â|α〉 is said to be in a microcanonical energy window, if the eigenenergy Eα, corresponding to the state |α〉, lies within some specified narrow energy interval). This fact was first discovered in the context of studying the transition from quantum to classical behaviour, in the initiating paper by Shnirelman7, and in the body of work in mathematics and mathematical physics that immediately followed (see ref. 8 and the references therein). Eigenstate thermalization is now believed to be the main explanation for why large quantum systems behave thermodynamically9,10,11. In the present context, however, of greatest interest is the residual variation among the 〈α|Â|α〉 values within an energy window—a consequence of the fact that the density of states is not actually infinite. This variation has been linked to a physically relevant autocorrelation function12 and studied in many examples8,13,14,15; however, the most relevant point for the present purposes is that the residual variation provides an upper limit on how much difference there can be between the following two quantities: the first is the infinite time average of the quantum expectation value of an observable  with respect to the time-evolving state |ψ(t)〉,

where is the initial state, and the second quantity is the microcanonical average of 〈α|Â|α〉. The latter is simply the average of the 〈α|Â|α〉 values over all states |α〉 in a narrow energy window centred at the mean energy of the initial state. (The occupation numbers are non-negligible generally only for eigenstates |α〉 from a narrow energy window centred at the mean energy Ē (see the Supplementary Discussion of ref. 11)). Now, if there is some non-zero residual variation among the values of 〈α|Â|α〉 within a microcanonical window, then one can pick an initial state whose occupation numbers tend to be, say, larger for states |α〉 whose values 〈α|Â|α〉 are larger. Then the infinite time average Ā will be larger than the microcanonical average of the 〈α|Â|α〉 values. In contrast, as pointed out in refs 9, 10, 11, if all the values 〈α|Â|α〉 are the same, then Ā and the microcanonical average of the 〈α|Â|α〉 values are the same.

In the other extreme, in the case of integrable systems, time evolution preserves the values of all quantum numbers2, although it has been shown that, at least in the case of large systems, the infinite time average is able to erase all the information about the relative phases of the eigenstates composing the initial state16,17,18. Eigenstate thermalization fails for integrable systems.

A real micro- or mesoscopic physical system is usually neither completely integrable nor completely chaotic. On the other hand, it may often be described as an integrable system perturbed by an integrability-breaking potential. It is therefore of central importance to understand how the dynamics of the system changes as the strength of the perturbing potential is increased from zero to the point where all resemblance to the original integrable system is lost.

In the present work, we show that the gradual disappearance of the integrals of motion under perturbation can, for a conceptually important class of quantum systems, be completely characterized by a simple and universal relation. Our approach is to develop an exactly solvable statistical model that captures the essence of the physics, and then test its predictions—in the full range of parameters from the integrable regime through the well-developed chaos—against the exact time dynamics of a deformed random Gaussian model19, a Šeba-type billiard20, and a two-dimensional Anderson model21,22,23. All three are examples of 'rough billiards,' by which we mean that, in their interiors, they have scatterers that are completely decorrelated, and, as a result, the perturbation has large matrix elements between any two momentum states. Correspondingly, in our statistical model, we will be considering perturbing potentials that obey no selection rules. From the point of view of 'localization–delocalization in the space of quantum numbers, our model leads to a lattice with extremely long-range hoppings. On the other end of the spectrum, one finds the kicked rotor24 and the Bunimovich stadium25, with a Coulomb scatterer in a box as an intermediate case26. In the thermodynamic limit, our model always leads to delocalized eigenstates: the phenomena described in this article are thus mesoscopic finite-size effects. We show that in our model, the memory of the initial conditions is describable in terms of a simple analytic expression that is valid through the full range of perturbation strengths, from the integrable to the completely quantum-chaotic extremes, and that features a great deal of universality. This type of universality in the memory of the initial conditions in quantum systems was first noticed in the context of two short-range-interacting ultracold atoms in a harmonic waveguide27. In the present paper, we give another physically relevant example of a perturbation with no selection rules: a lattice billiard with uncorrelated impurities. Here the absence of long-range correlation allows the perturbation to couple any momentum state with any other, with, on average, the same strength.

Results

Formulation of the problem and intuitive analysis

We consider the Hamiltonian of an integrable quantum system with an integrability-breaking perturbation . Here the integrable part of the Hamiltonian is a function H0 of all the members of the complete set of the integrals of motion, where the latter are labelled ; d is the number of the degrees of freedom. The eigenstates and eigenvalues of , respectively; the corresponding quantities for Ĥ are |α〉 and Eα, as before. We assume that both the and the Eα spectra are free of degeneracies. The non-degenerate spectrum is a generic property of an integrable Hamiltonian unless it has mutually non-commuting integrals of motion or commensurate frequencies28,29. However, because of the absence of level repulsion, the energy levels are often near-degenerate: they are distributed as if by a Poisson process, resulting in the exponential distribution of level spacings30.

Now, we imagine that the system is prepared in some initial nonequilibrium state , and then allowed to evolve according to the Hamiltonian Ĥ. Following a transient period, the expectation value of a generic observable  will, overwhelming majority of the time, fluctuate around the infinite-time average given in equation (1). Figure 1 gives an example of such a process, in the regime intermediate between fully integrable and fully chaotic, showing a partial retention of the memory of the initial state.

Figure 1: An example of a partially retained memory of the initial state.
figure1

Shown are the x and y components of the hopping energy (respectively: Ex in solid red, with initially positive value, and Ey in dotted blue, with initially negative value), as a function of time, of a two-dimensional 33×33-site noninteracting Anderson model in the presence of an Aharonov–Bohm (A–B) flux (see the first subsection of the Methods section). A fully chaotic system would obey the equipartition theorem, predicting that the infinite time averages of the two components should be equal, = 0. However, for intermediate strengths of the integrability-breaking perturbation, a residual deviation from equipartition—strongly correlated with , the initial deviation—remains. The two straight horizontal lines on the right-hand side of the plot are the exact values of , computed from equation (1). In terms of the Anderson model parameters introduced in the text following the paragraph containing equation (3), this system has ɛ=1.0, corresponding to the Anderson's disorder parameter of W/J=0.87. The initial state was represented by an eigenstate of the impurity-free lattice, |ψinit〉=|nx=−9, ny=+3〉, whose energy was E=−1.46 J, where J is the hopping constant (see the first subsection of the Methods section). At zero A–B flux, the ground state is |nx = 0, ny= 0〉. The A–B fluxes φx and φy through a complete loop along the x and the y directions, respectively, were chosen to be far from approximately commensurate; we had , where φ0 is the elementary quantum of magnetic flux, ϕ = 1.618… is the golden ratio, and e=2.718... is the base of the natural logarithm. The apparent correlation between the two energies is due to the conservation of the total energy.

As we are concentrating on the disappearance of the integrals of motion, we take our observable of interest to be diagonal in the eigenbasis of , the integrable part of the Hamiltonian: . Let us choose the initial state to be an eigenstate, . In the discussion that follows, it will help to imagine that we chose a state with a 'highly improbable' (meaning: very different from the microcanonical average) value of the observable of interest Â. According to equation (1), in the course of time evolution, the initial state gets transformed into a superposition, with essentially random phase relationships, of the eigenstates |α〉 of the perturbed Hamiltonian, Ĥ, such that each |α〉 in the superposition has a substantial overlap with , that is, a large corresponding occupation number.

As the eigenstates (let us call these the 'unperturbed eigenstates') form a basis, we may expand the eigenstates |α〉 of the full, perturbed Hamiltonian Ĥ (the 'perturbed eigenstates') as linear combinations of the unperturbed eigenstates. The stronger the perturbing potential , the more different the perturbed eigenstates from the unperturbed ones, and the greater the number of unperturbed eigenstates that must be linearly combined (with appreciable weights) to build any perturbed eigenstate. A standard measure of the latter number is the so-called number of principal components, to be defined below. Among the perturbed eigenstates |α〉 whose overlap with is non-negligible, a typical one will be such that when it is expanded in terms of the unperturbed eigenstates, the weight of the state is of the order of , which is the so-called inverse participation ratio (IPR)15. The number of principal components mentioned above is then given by ; in this case, it is the number of principal unperturbed components.

As we said, we will assume that the matrix elements of the integrability-breaking perturbation obey no selection rules. We will also assume that the equi-energy surface is 'sufficiently irrational,' which is a concept that needs a bit of explaining: as is well known, energy eigenstates in an integrable system are completely characterized by their quantum numbers (which we can take to be integers) corresponding to all the conserved quantities. In a typical integrable system, for sufficiently high excited states, if one of the quantum numbers is changed by 1, while all the others are held fixed, the change in the energy of the eigenstate will, in general, be large compared with the energy level spacing, that is, there will be many eigenstates whose energies lie in between the energy of the original state and the energy of the state that differs from the original by 1 in a single quantum number. Eigenstates that are neighbours in energy generally have very different values of all quantum numbers. In fact, if one orders the eigenstates by increasing energy and looks at how their quantum numbers change from one eigenstate to the next in the sequence, these changes may appear quite random; if they do, we say that the equi-energy surface is 'sufficiently irrational.' More precisely, let be the sequence of the values of the kth quantum number as one is going from one eigenstate of to the next in the order of increasing energy, labelled by ν. The equi-energy surface is said to be sufficiently irrational, if, for every k, the sequence passes any simple statistical test for randomness. This property seems to be quite generic for integrable systems with incommensurable frequencies; see Supplementary Figure S2 for an example.

These two assumptions (the absence of selection rules in the perturbing potential and the sufficient irrationality of the equi-energy surface of ) imply that the presence of the state as one of the components of the state |α〉 does not 'bias the selection' of the other states that enter the expansion of |α〉. This means that the expansion of |α〉 appears as if the states (other than ) that enter into it were indiscr iminately chosen from a microcanonical shell around . This would make—were it not for the 'systematic' presence of the initial state in the expansion of |α〉—the value of Â, on average, equal to the microcanonical average. The state becomes 'the only' component of the state |α〉 that 'remembers' the initial value of Â. Because on average unperturbed states enter the expansion of a perturbed state—and out of these unperturbed states that enter the expansion, usually, exactly one has the initial value for the observable —we obtain the estimate that the infinite time average of  is the weighted average of the initial value and the microcanonical value of Â, where the weight ratio is in favor of the microcanonical value: is the quantum expectation value of  with respect to the initial state, and AMC is the microcanonical average, that is, the average over all the values such that the unperturbed eigenenergies of lie in a narrow window centred at the mean energy of the system.

A statistically solvable model

Guided by this reasoning, we have constructed a statistical model that captures the essential physics and can be solved exactly. The model has two principal ingredients. First, the unperturbed Hamiltonian Ĥ0 is replaced by an ensemble of Hamiltonians, each member of which has the same eigenenergies and eigenstates as Ĥ0, but which eigenstate corresponds to which eigenvalue is chosen at random. It is convenient to think of these random assignments as permuting the eigenvalues, while the eigenstates remain fixed; so each permutation σ, which sends the N-tuple (1, 2,..., N) to (σ(1), σ(2),...,σ(N)), says that the jth largest eigenvalue corresponds to the eigenstate that in the original Hamiltonian corresponded to the σ−1(j)th largest eigenvalue. Second, the perturbation is replaced by an ensemble of perturbations; the distribution of the perturbations (viewed as matrix elements between various eigenstates) is assumed to be invariant under the σ of the eigenstates. Here we are permuting only the unperturbed eigenstates whose unperturbed eigenvalues lie within a microcanonical energy window WMC (EE centred at the mean energy E of the initial state. Also, the perturbations are truncated so that they couple only the eigenstates within that energy window. The following result then follows (see the Supplementary Methods for the details of the derivation):

Here N gives the number of the states in the microcanonical window WMC (EE, and 〈…〉σ,V̂ stands for an average over the uniformly distributed permutations σ and the permutation-invariant-distributed perturbations V^. It will be convenient to introduce, for any quantity that depends on the values of the quantum numbers , the microcanonical average as

where N is the number of unperturbed eigenstates whose eigenvalues are in the microcanonical energy window. Then the typical number of the (interacting) principal components is given by is the microcanonical average of the IPR of the noninteracting eigenstates over the interacting ones is the value of  in the initial state (it is written as an expectation value even though, given our choices, the initial state is also an eigenstate of Â), and is the microcanonical average for the observable.

What we suggest now is to treat the 'real-world' Hamiltonian Ĥ as a single instance, of the ensemble of Hamiltonians considered above. In this case, the ensemble average on the left-hand side of equation (2) will serve as the 'best predictor' for the 'real-world' value of the infinite-time average of the observable, Ā. Notice that, conversely, one of the constituents on the right-hand side of equation (2) will, in its turn, also have to be estimated by a 'best predictor.' Namely, while Ā and AMC are the same for all (σ, V^)-instances and thus can be extracted from a single instance (which can be the 'real-world' instance), the quantity (on which Npc depends) varies from instance to instance, and so its averaging over instances has a nontrivial effect. Because, however, we only have access to a single realization of Ĥ, we must use the 'real-world' value of as the 'best estimate' for the ensemble average:

where is the 'real-world' microcanonical average of .

An undesirable feature of the expression in equation (2) is that it depends on the number of states in the microcanonical window, N. However, in the limit where N greatly exceeds the number of principal components, N Npc, N disappears from the expression. We finally obtain the desired relationship between the infinite-time average and the initial value of our observable of interest:

This is almost our original estimate on the basis of intuitive arguments, except that instead of the typical value of the IPR of the perturbed over the unperturbed states, , we have the microcanonical average of the IPR of the unperturbed states over the perturbed states. Note the high degree of universality: the same parameter is used regardless of what observable  one is interested in (which means—as  is diagonal in the unperturbed basis—regardless of which integral of motion of one is interested in), or with which initial state one starts from among the unperturbed eigenstates.

Testing the analytical prediction

The relationships in equations (2) and (3) are our central results. We now test them against exact time dynamics of particular physical systems. Our integrable system will always be a 33×33-site two-dimensional lattice. We also assume that both x and y cycles of the lattice are threaded by an Aharonov–Bohm (A–B) solenoid each31,32,33. If the lattice is imagined to cover the surface of a torus, the flux is produced by two solenoids: one toroidal, contained within the torus of the lattice, and one straight, passing through the hole of the torus. The corresponding magnetic field fluxes are assumed to be highly irrational but weak (that is, of the order of one) multiples of the elementary magnetic flux quantum; they are also presumed to be mutually irrational—see the first subsection of the Methods section, below. Their purpose is to destroy the dihedral symmetries of the square lattice while preserving the conservation of momentum in both x and y directions. This lifts the degeneracies in the unperturbed spectrum and randomizes the sequence of appearance of the quantum numbers (two components of the momentum vector) along the energy axis. We investigated three different types of the integrability-breaking perturbation. The first perturbation (Fig. 2a) completes the lattice to a single instance of a deformed (real) Gaussian random matrix model19 that couples the eigenstates of the unperturbed lattice. The second perturbation (Fig. 2b) is a single impurity of a fixed strength. This system is a lattice version of Šeba billiards34, which are known to lie in between integrable and completely quantum-chaotic systems34. The third example (Fig. 2c) is a single instance of the conventional Anderson disorder21, with a rectangular distribution of the strength in each of the impurities. (We should mention that ergodicity in the context of an Anderson lattice was also studied in ref. 35) Whereas the first and the second examples correspond to permutation-invariant perturbations, the third one does not. In particular, in comparison to the first two types of perturbation, in the Anderson case, there are much stronger correlations between certain kinds of matrix elements. Namely, for each matrix element, consider the momentum difference between the states that the matrix element connects. If one looks at the set of matrix elements that have the same momentum difference, one finds that from one realization of the perturbation to the next, their values all change by the same factor. The observable of interest is the 'equipartition measure'—the difference between the x and y hopping energies, Ex and Ey—whose value in a state of a thermal equilibrium is always zero, thanks to the xy symmetry. The infinite-time average of the 'equipartition measure' ExEy is shown as a function of its initial value. As the governing parameter ɛ, we have chosen the ratio between the root mean square of the off-diagonal matrix elements (the same for all the pairs ) and the typical energy spacing at the energy of interest: is the density of states at the energy E.

Figure 2: Testing the predictive power of equation (3).
figure2

Our integrable system is a single particle on a two-dimensional lattice with an Aharonov–Bohm flux. Lattice parameters are the same as for Figure 1. Each row of plots corresponds to a different type of non-integrable perturbation, as follows. (a,d) A real Gaussian random matrix perturbation acting between the eigenstates of the unperturbed lattice. (b,e) A single impurity of fixed strength. (c,f) An Anderson-type disorder. In plots (a,d, and c,f), the data points marked with (red) squares are for the case ɛ=0.25; with (blue) circles, ɛ=0.5; and with (green) triangles, ɛ=1. In plots b,e, the (red) squares are also for ɛ=0.25, while the (green) triangles are for ɛ=2×105. Plots a,b,c show the infinite time average, obtained from exact time dynamics, of the 'equipartition measure' ExEy versus the value, this measure had in the initial state. The straight solid lines are the predictions of equation (3). Included as insets are the representations of the lattice and its perturbation. In plots d,e and f, we show the level-spacing histograms for the perturbed Hamiltonians, which show how close the systems presented in plots a,b and c are to integrability or to well-developed chaos. The horizontal axis, Δe, is the level spacing in the unfolded spectrum. The closer the actual distribution to the curve representing the Poissonian level statistics, the more integrable the system; the closer the actual distribution to the curve representing the GOE or Šeba or the GUE, the more chaotic the system. See the main text for details.

For each type of perturbation, we time-evolved 201 different initial states, taken to be all the eigenstates of the unperturbed lattice whose eigenvalues came from a representative microcanonical energy window; the middle eigenstate (the 101st one in the order of increasing energy within the window) had the energy of −1.5 J, where J is the hopping constant (see the first subsection of the Methods section). A single instance of the corresponding Hamiltonian (which was random for the cases a and c) was used in all three cases. The different initial states have different values for , lying between some minimum and maximum values; we needed to group the initial states into sets with similar -values. Thus, we partitioned the interval from the minimum to the maximum -value into subintervals centred at −3J, −2.5J,...3J, each of width . All the initial states whose -values fell into the same subinterval constituted a 'group of initial states with similar -values.' The points shown in plots a, b, c correspond to the groups: the x value of a point is the centre of the subinterval defining the group, whereas the y value is the group average of . The theoretical curves, produced by equation (3), also involved an averaging: the value of which is fed into equation (3) was, for each group, the group average of rather than the centre of the subinterval; this explains why the theoretical curves are not exactly straight lines. As Figure 2a–c shows, the prediction of equation (3) agrees very well with the numerical results.

Figure 2d–f is there to show where, on the continuum between integrability and well-developed chaos, our various systems lie. The discrete points are the plots of the actual level spacing statistics of our systems, properly unfolded36; for the cases d and f, we also average over 16 realizations of the perturbation V^. For comparison, we also plot the curves corresponding to the completely integrable and (the appropriate) completely quantum-chaotic systems. The curves labelled 'Poisson' arise for Poisson level statistics, which is characteristic for integrable systems. The most recognizable feature is the non-zero value at zero spacing, representing the absence of level repulsion in integrable systems. The Gaussian orthogonal ensemble (GOE) curve is valid for systems with well-developed quantum chaos in the presence of time-reversal invariance; the Šeba distribution34 holds for systems with singular perturbations; and the Gaussian unitary ensemble (GUE) is used in the cases of quantum chaos without time-reversal invariance36. We see that when ɛ=0.25, the level-spacing statistics of our systems is intermediate between those of the integrable-like and the appropriate completely quantum-chaotic-like distributions. As ɛ increases, the level spacing distributions for the real Gaussian, singular, and Anderson perturbations converge, respectively, to the GOE, Šeba and GUE predictions. (The reason for the GUE statistics in the Anderson case is the Aharonov–Bohm flux, which breaks the time-reversal invariance32. In the case of the first model, the statistics remains of a GOE type, because the perturbation matrix elements were artificially fixed to real values.) The expression in equation (3) is thus confirmed in the full range from the integrable regime all the way to the well-developed quantum chaos.

To relate our predictions to the system parameters, we further connect the IPR —otherwise empirically irrelevant—to the governing parameter ɛ. This allows us to trace the integrability-to-chaos transition, that is, express the memory of the initial state through the strength of the non-integrable perturbation. Figure 3 shows, for a particular group of the initial states, the values of the 'equipartition measure' as a function of the parameter ɛ. The numerical values are drawn from the set used to produce Figure 2. To obtain the theoretical prediction, we tabulate numerically the IPR , averaged over the unperturbed states |α0〉, for a deformed (complex) Gaussian random matrix model. For large values of ɛ, we also use known theoretical results37,38 for the so-called strength function, , combined with the assumption 39 of the Gaussian character of the fluctuation of 〈α0〉:

Figure 3: The memory of the initial state as a function of the strength of the integrability-breaking perturbation.
figure3

For the case of Anderson disorder, we compute, for three similar initial states, , the infinite time average of the 'equipartition measure' ExEy as a function of the disorder strength ɛ. The red crosses correspond to the averages over the this group of states. The red solid line corresponds to the formula (3) combined with numerically tabulated values of the IPR for the case of the deformed (complex) Gaussian random matrix model. Finally, the black dotted line is given by the asymptotic analytic prediction in equation (4). The lattice parameters are the same as for Figure 1.

where q=3 for a non-integrable perturbation that belongs to the Gaussian orthogonal class, and q=2 in the Gaussian unitary case. (The Aharonov-Bohm flux present in the example of the Figure 3 implies the latter.) The result in equation (4) for the Gaussian orthogonal case can be also confirmed directly38, without the assumption of Gaussianity of 〈α0〉. The strength function for an equidistant spectrum perturbed by a real matrix whose matrix elements have the same magnitude and random signs—the case closely related to a Poisson spectrum perturbed by a real Gaussian matrix—first appears in Wigner's classical paper40. The strength function for the latter system per se is predicted in ref. 41. (See the second subsection of the Methods section for more details.)

Discussion

We have thus demonstrated that, in the case, when the integrability-breaking perturbation obeys no selection rules and when the spectrum of the underlying integrable system is 'sufficiently irrational,' it is possible to characterize the memory of the initial values of the unperturbed integrals of motion, as the strength of the perturbation is increased, by a simple and universal expression. The expression was verified for three different types of perturbations away from integrability, including a deformed Gaussian random matrix Model, an isolated impurity, and the two-dimensional Anderson model; in each case, the expression works for a full range of perturbation strengths from completely integrable to completely chaotic.

Our predictions should be testable experimentally with techniques that are already available, or nearly so; probable contexts include those of the investigations of Anderson localization with cold gases in optical lattices22,23, or those of quantum mirage configurations of a quantum corral42, modified to a Šeba-type billiard. In general, memory effects should be visible as soon as the discreteness of energy levels is discernible.

Future work must try to come to terms with cases where the partial preservation of the integrals of motion is enhanced via some selection rules obeyed by the perturbing potential. The two most empirically relevant examples of selection rules come from the limitations imposed by the few-site nature of the hopping terms in typical lattice Hamiltonians, and by the few-body nature of interactions in many-body systems. Encouragingly, some work relevant to these cases has already been done; ref. 43 studied the role of the former type of selection rules, whereas the effects of the latter type on the structure of the eigenstates has been investigated in a number of works; for a review, see ref. 19 and the references therein, particularly ref. 44. More generally, one should start systematically increasing the complexity of the topology of the network of transitions.

Methods

Numerical models used to verify equation (3)

As an example of an unperturbed Hamiltonian H×0, we use a Nx×Ny (33×33 in the numerical examples considered) two-dimensional lattice with periodic boundary conditions and odd Nx, Ny:

where |jx, jy〉 are the eigenstates of position. Both x and y hopping constants have the same amplitude J. We also assume that both x and y cycles of the lattice are threaded by an Aharonov–Bohm (A–B) solenoid each. This leads to the hopping constants acquiring complex phase factors, with phases , respectively. Here is the magnetic flux through the x cycle is the elementary quantum of magnetic flux, c is the speed of light, and q is the electric charge of the lattice particle. The flux is assumed to be very weak, Δ nx Δ ny) 1. (After a suitable gauge transformation, the complex hoppings can be replaced by real ones, supplemented by twisted boundary conditions31,32,33.) The eigenstates of the unperturbed Hamiltonian are plane waves |(nx, ny)〉,

The linear momentum quantum numbers nx and ny (|nx|≤(Nx−1)/2 and |ny|≤(Ny−1)/2) constitute a set of the integrals of motion. Eigenenergies of the unperturbed Hamiltonian are

The purpose of introducing the A–B flux is to generate an equi-energy surface E n x , n y = E = E that is sufficiently irrational with respect to the lattice of the integer quantum numbers nx and ny. To this end, the 'defects' Δnx and Δny must be both irrational and mutually irrational (see the first subsection of the first section of the Supplementary Methods and Supplementary Fig. S2). In our numerical calculations, we used Δnx=ϕ/4 and Δny=e/10, where ϕ=1.618... is the golden ratio and e=2.718... is the base of the natural logarithm.

As the first example of an integrability-breaking perturbation V̂^^^, we considered a member of a Gaussian orthogonal ensemble acting between the eigenstates of the unperturbed lattice (Fig. 2a):

where the independent real Gaussian-distributed random variables of unit variance and zero mean, with the remaining matrix elements controlling the hermiticity of .

The second example (Fig. 2b) is a singular perturbation,

which produces a matrix with all-equal matrix elements:

In this case, the eigenstates of the perturbed Hamiltonian can be found exactly. They read

where the corresponding eigenenergies Eα are the solutions of the algebraic equation

and the normalization factor is defined by

A solution of this form was at first obtained by Šeba20, for a flat continuous billiard. A similar problem involving a two-dimensional lattice with periodic boundary conditions in one direction and a trapping potential in another was recently analysed45.

Finally, we consider an Anderson-type disorder (Fig. 2c):

where the are NxNy real independent variables, distributed uniformly between −1 and +1; W is the Anderson disorder parameter. The interaction strength parameter V0 used in the previous cases corresponds to the r.m.s. of the (generally complex) off-diagonal matrix elements

.

Relevant results on the IPR in random matrix models

To obtain the solid line in Figure 3, we computed the IPR , averaged over all unperturbed states |α0〉, for a N×N deformed Gaussian random matrix ĥ = ĥ0 + v̂, with N=2000. The 'integrable' part of the matrix, Ĥ0, was represented by a diagonal matrix whose N diagonal entries were given by N-independent real random numbers uniformly distributed in the interval . Here ρ is the density of states. The 'non-integrable part', v̂, was a random matrix drawn from the Gaussian unitary ensemble of random Hermitian matrices: (real) diagonal matrix elements and the real and imaginary parts of the off-diagonal matrix elements above the diagonal (those below the diagonal then being fixed by hermiticity) were given by independent, Gaussian-distributed random numbers with standard deviations , respectively. For completeness, the Gaussian orthogonal case would give . In both cases, V0 fixes the mean square of the off-diagonal matrix elements:

Equation (4) uses the following asymptotic expression for the IPR η:

where q=3 for a non-integrable perturbation that belongs to the Gaussian orthogonal class, and q=2 in the Gaussian unitary case. In refs 37,38,40, 41, it was shown that for ɛ 1, the strength function converges to

in both Gaussian Orthogonal and Gaussian unitary cases. Here Γ is determined by the Fermi golden rule: . Also, it is known39 that in the same limit, the individual coefficients 〈α0〉 behave as independent Gaussian random variables, with zero mean and with a standard deviation governed by the expression (7). Recall that the fourth moment of a Gaussian distribution of a variable ξ is related to the second one as46 . Also, it is important that when ɛ 1, the energy width Γ in equation (7) contains many levels, and thus the sum in the definition of the IPR, , can be replaced by an integral. The asymptotic formula (6) for the IPR immediately follows.

Ab initio computations

All numerical time evolution was computed from full exact diagonalization of the Hamiltonians using the Mathematica computer package.

Additional information

How to cite this article: Olshanii, M. et al. An exactly solvable model for the integrability–chaos transition in rough quantum billiards. Nat. Commun. 3:641 doi: 10.1038/ncomms1653 (2012).

Change history

  • 05 February 2013

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Acknowledgements

We. are grateful to F. Werner and D. Cohen for enlightening discussions on the subject. Supported by the Office of Naval Research grant N00014-09-1-0502 (M.O. and V.D.) and the National Science Foundation grants PHY-1019197 (M.O. and V.D.) and PHY-0902906 (K.J.).

Author information

Relevant numerics were run by M.O., K.J., M.R. and H.K. The building of the analytic model and of the models to be studied numerically, and the analysis and the interpretation of numerical and analytical data, was a close collaboration of all the authors.

Correspondence to Maxim Olshanii.

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Supplementary Figures S1-S2 and Supplementary Methods (PDF 71 kb)

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Olshanii, M., Jacobs, K., Rigol, M. et al. An exactly solvable model for the integrability–chaos transition in rough quantum billiards. Nat Commun 3, 641 (2012). https://doi.org/10.1038/ncomms1653

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