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arXiv:1710.01289 · Physical Review Research

Quantum caustics and the hierarchy of light cones in quenched spin chains

W. Kirkby, J. Mumford, D. H. J. O'Dell

Published Nov 27, 2019• 20 citations
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Abstract

The authors demonstrate that the light cones which spread information in spin chains are examples of quantum caustics, i.e. quantum versions of a class of phenomena that includes ship's wakes and rainbows. Caustics form of a hierarchy, described by catastrophe theory, which gives a natural explanation of the emergence of universal scaling wave functions dressing light cones: the simplest is the Airy function while more general situations lead to Pearcey and Hyperbolic Umbilic functions. The presence of space-time vortices inside light cones is also predicted.

DOI
10.1103/physrevresearch.1.033135

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The authors demonstrate that the light cones which spread information in spin chains are examples of quantum caustics, i.e. quantum versions of a class of phenomena that includes ship's wakes and rainbows. Caustics form of a hierarchy, described by catastrophe theory, which gives a natural explanation of the emergence of universal scaling wave functions dressing light cones: the simplest is the Airy function while more general situations lead to Pearcey and Hyperbolic Umbilic functions. The presence of space-time vortices inside light cones is also predicted.
We show that the light cone-like structures that form in spin chains after a quench are quantum caustics. Their natural description is in terms of catastrophe theory and this implies: 1) a hierarchy of light cone structures corresponding to the different catastrophes; 2) dressing by characteristic wave functions that obey scaling laws determined by the Arnol'd and Berry indices; 3) a network of vortex-antivortex pairs in space-time inside the cone. We illustrate the theory by giving explicit calculations for the transverse field Ising model and the XY model, finding fold catastrophes dressed by Airy functions and cusp catastrophes dressed by Pearcey functions; multisite correlation functions are described by higher catastrophes such as the hyperbolic umbilic. Furthermore, we find that the vortex pairs created inside the cone are sensitive to phase transitions in these spin models with their rate of production being determined by the dynamical critical exponent. More broadly, this work illustrates how catastrophe theory can be applied to singularities in quantum fields.
We describe a new class of nonequilibrium quantum many-body phenomena in the form of networks of caustics that dominate the many-body wave function in the semiclassical regime following a sudden quench. It includes the light cone-like propagation of correlations as a particular case. Caustics are singularities formed by the birth and death of waves and form a hierarchy of universal patterns whose natural mathematical description is via catastrophe theory. Examples in classical waves range from rainbows and gravitational lensing in optics to tidal bores and rogue waves in hydrodynamics. Quantum many-body caustics are discretized by second quantization (``quantum catastrophes'') and live in Fock space, which can potentially have many dimensions. We illustrate these ideas using the Bose Hubbard dimer and trimer models, which are simple enough that the caustic structure can be elucidated from first principles and yet run the full range from integrable to nonintegrable dynamics. The dimer gives rise to discretized versions of fold and cusp catastrophes whereas the trimer allows for higher catastrophes including the codimension-3 hyperbolic and elliptic umbilics, which are organized by, and projections of, an 8-dimensional corank-2 catastrophe known as X9{X}_{9}X9​. These results describe a hitherto unrecognized form of universality in quantum dynamics organized by singularities that manifest as strong fluctuations in mode population probabilities.
As part of the quest to uncover universal features of quantum dynamics, we study catastrophes that form in simple many-body wave functions after a quench, focusing on two-mode systems that include the two-site Bose Hubbard model, and under some circumstances optomechanical systems and the Dicke model. When the wave function is plotted in Fock space plus time certain characteristic structures generically appear that we identify as cusp caustics. In the vicinity of such a catastrophe the wave function takes on a universal form described by the Pearcey function and obeys scaling relations which depend on the total number of particles NNN. In the thermodynamic limit (N→∞N \rightarrow \inftyN→∞) the cusp becomes singular, but at finite NNN it is decorated by an interference pattern. This pattern contains an intricate network of vortex-antivortex pairs, initiating a theory of topological structures in Fock space. In the case where the quench takes the form of a δ\deltaδ-kick we show how to analytically map the wave function onto the Pearcey function and hence obtain the scaling exponents for the size and position of the cusp, as well as those for the amplitude and characteristic length scales of its interference pattern.
Recent theoretical studies have predicted the existence of caustics in many-body quantum dynamics, where they manifest as extended regions of enhanced probability density that obey temporal and spatial scaling relations. Focusing on the transverse-field Ising model, we investigate the dynamics initiated by a local quench in a spin chain, resulting in outward-propagating excitations that create a distinct caustic pattern. We calculate the scaling of the first two maxima of the interference fringes dressing the caustic, finding a universal exponent of 2/3, associated with an Airy function catastrophe. We demonstrate that this property is universal in the entire paramagnetic phase of the model, and starts varying at the quantum phase transition (QPT). This robust scaling persists even under perturbations that break the integrability of the model. We additionally explore the effect of boundary conditions and find that open boundaries introduce significant edge effects, leading to complex interference patterns. Despite these edge-induced dynamics, the overall power-law scaling exponent remains robust. These findings highlight the potential of quantum caustics as a powerful diagnostic tool for QPTs, demonstrating resilience against integrability-breaking perturbations and boundary condition variations.
Caustics occur widely in dynamics and take on shapes classified by catastrophe theory. At finite wavelengths they produce interference patterns containing networks of vortices (phase singularities). Here we investigate caustics in quantized fields, focusing on the collective dynamics of quantum spins. We show that, following a quench, caustics are generated in the Fock space amplitudes specifying the many-body configuration and which are accessible in experiments with cold atoms, ions, or photons. The granularity of quantum fields removes all singularities, including phase singularities, converting point vortices into nonlocal vortices that annihilate in pairs as the quantization scale is increased. Furthermore, the continuous scaling laws of wave catastrophes are replaced by discrete versions. Such "quantum catastrophes" are expected to be universal dynamical features of quantized fields.
We investigate the quantum interference and energetic phenomena associated with classical caustics in the quantum bouncing ball problem, we refer to as quantum caustics. By considering an initial Gaussian wavepacket, we show that caustics associated with the underlying classical trajectory families are exhibited. We connect the associated phase singularity chains in the vicinity of the caustic with the semiclassical Pearcey function built on the cusp catastrophe lines. We also quantify the amount of superenergy exhibited in these solutions - regions of space where the local energy exceeds the largest constituent energy eigenvalue. We give a complimentary description of the caustic and superenergy behavior using the Madelung/Bohm trajectories, which gives additional insight about the energy of the trajectories and how they traverse the phase singularity chains.