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arXiv:1703.00201 · Journal of Mathematical Physics

Signatures of quantum phase transitions from the boundary of the numerical range

Ilya M. Spitkovsky, Stephan Weis

Published Dec 1, 2018• 6 citations
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Abstract

We analyze the smoothness of the ground state energy of a one-parameter Hamiltonian by studying the differential geometry of the numerical range and continuity of the maximum-entropy inference. The domain of the inference map is the numerical range, a convex compact set in the plane. We show that its boundary, viewed as a manifold, has the same order of differentiability as the ground state energy. We prove that discontinuities of the inference map correspond to C1-smooth crossings of the ground state energy with a higher energy level. Discontinuities may appear only at C1-smooth points of the boundary of the numerical range. Discontinuities exist at all C2-smooth non-analytic boundary points and are essentially stronger than at analytic points or at points which are merely C1-smooth (non-exposed points).

DOI
10.1063/1.5017904

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We analyze the smoothness of the ground state energy of a one-parameter Hamiltonian by studying the differential geometry of the numerical range and continuity of the maximum-entropy inference. The domain of the inference map is the numerical range, a convex compact set in the plane. We show that its boundary, viewed as a manifold, has the same order of differentiability as the ground state energy. We prove that discontinuities of the inference map correspond to C1-smooth crossings of the ground state energy with a higher energy level. Discontinuities may appear only at C1-smooth points of the boundary of the numerical range. Discontinuities exist at all C2-smooth non-analytic boundary points and are essentially stronger than at analytic points or at points which are merely C1-smooth (non-exposed points).
We study the continuity of an abstract generalization of the maximum-entropy inference—a maximizer. It is defined as a right-inverse of a linear map restricted to a convex body which uniquely maximizes on each fiber of the linear map a continuous function on the convex body. Using convex geometry we prove, amongst others, the existence of discontinuities of the maximizer at limits of extremal points not being extremal points themselves and apply the result to quantum correlations. Further, we use numerical range methods in the case of quantum inference which refers to two observables. One result is a complete characterization of points of discontinuity for 3 × 3 matrices.
In this paper, we discuss the connection between two genuinely quantum phenomena—the discontinuity of quantum maximum entropy inference and quantum phase transitions at zero temperature. It is shown that the discontinuity of the maximum entropy inference of local observable measurements signals the non-local type of transitions, where local density matrices of the ground state change smoothly at the transition point.Wethen propose to use the quantum conditional mutual information of the ground state as an indicator to detect the discontinuity and the non-local type of quantum phase transitions in the thermodynamic limit.
In this paper, we discuss the connection between two genuinely quantum phenomena—the discontinuity of quantum maximum entropy inference and quantum phase transitions at zero temperature. It is shown that the discontinuity of the maximum entropy inference of local observable measurements signals the non-local type of transitions, where local density matrices of the ground state change smoothly at the transition point. We then propose to use the quantum conditional mutual information of the ground state as an indicator to detect the discontinuity and the non-local type of quantum phase transitions in the thermodynamic limit.
We revisit the maximum-entropy inference of the state of a finite-level quantum system under linear constraints. The constraints are specified by the expected values of a set of fixed observables. We point out the existence of discontinuities in this inference method. This is a pure quantum phenomenon since the maximum-entropy inference is continuous for mutually commuting observables. The question arises why some sets of observables are distinguished by a discontinuity in an inference method which is still discussed as a universal inference method. In this paper we make an example of a discontinuity and we explain a characterization of the discontinuities in terms of the openness of the (restricted) linear map that assigns expected values to states.
Quantum phase transitions manifest as an abrupt change in the ground state of a many-body system; yet it is an open question whether this sudden change necessarily precludes a continuous dynamical connection between the two phases. We introduce a classification of quantum phase transitions based on this geometric aspect of the ground-state manifold, that differs from known classifications. By leveraging the framework of counterdiabatic driving, we explicitly construct schedules that dynamically connect one phase to another. This strategy allows us to uncover a large class of quantum phase transitions, where the states on both sides are separated only by a finite geometric distance in the thermodynamic limit. We term such transitions traversable, since exact counterdiabatic driving links the two phases via a finite dynamical protocol in the thermodynamic limit. We show that multiple known transitions fall into this class -- e.g., symmetry-breaking transitions obeying hyperscaling and discontinuous transitions with an enhanced continuous symmetry. We further show the existence of quantum phase transitions that cannot be crossed dynamically even with the help of nonlocal counterdiabatic driving, as they would require divergent amplitudes and frequencies. Geometrically, these nontraversable transitions correspond to an infinite distance separating the two phases of matter; we show that the class comprises continuous transitions exhibiting mean-field universality, and discontinuous transitions arising from the competition between metastable minima. Our geometric classification goes beyond the known taxonomy, is independent of local order parameters and renormalization group fixed points, and has direct implications for the complexity of state preparation and adiabatic quantum computation.
It is shown that the gap that limits the speed of a quantum annealing process can take three salient morphologies: (i) the supersteep gap where both the ground and the first excited eigenenergy level curves have topologically related pairs of nearby inflection points giving both the maximum and the minimum a steep aspect, (ii) the steep gap where only the first excited eigenenergy level curve has a pair of inflection points giving its minimum a steep aspect while the maximum of the ground level does not exhibit inflection points, and (iii) the mild gap that has no related inflection points. Classification of the various singularities betrayed by the inflection points relies on the critical value curves of the quadratic numerical range mapping of the matrix H0+iH1, where H0 is the transverse field Hamiltonian and H1 the problem Hamiltonian. It is shown that the ground level is mapped to the generically smooth boundary of the numerical range, while the first excited level is mapped to an interior non-smooth critical value curve exhibiting swallow tails. The major result is that the position of the swallow tails relative to the boundary allows the supersteep versus steep discrimination, while the absence of swallow tail-boundary interaction characterizes the mild gap. As a corollary of the singularity analysis, the highly structured initial and final Hamiltonians of the Grover search create unstable singularities that break into stable swallow tails under perturbation, with the consequence of invalidating the gap scaling estimates computed around the unstable singularity. Classification of all stable singularities from a global viewpoint requires the Legendrian approach where the energy level curves become Legendrian knots in the contact space. Last but not least, it will be shown that a supersteep swallow tail previews tunneling.