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Recently, there has been renewed interest in the physics of quantum non-integrable systems - systems that lack or have very few conservation laws - with implications discovered across diverse fields including quantum information, statistical mechanics, condensed matter and quantum gravity. For instance, highly excited states of non-integrable systems can store quantum information in a local noise-resistant manner, mimic equilibrium statistical ensembles, capture finite temperature phase transitions and harbor correlations that resemble those outside a blackhole embedded in hyperbolic space. Moreover, they have been predicted to exhibit various features in their temporal dynamics, including thermalization at long times and a quantum version of the "butterfly effect" that certifies classical chaos at short times. However, traditional algorithms for studying physics in this regime have exponential complexity, making access to it extremely challenging. We strive to develop techniques for studying non-integrable systems, unearthing their hidden equilibrium and dynamical properties and exploring their broader ramifications on other fields of physics.
Schematic showing that random structures (black strings) on states ψ and operators O are correlated (colored strings)
Unifying chaos in states and operators
The emergence of irreversibility and long-time properties independent of the fine details of the initial conditions from perfectly reversible quantum evolution has mystified physicists for decades. The eigenstate thermalization hypothesis (ETH), developed in the 1990s, describes how thermal equilibrium at long times arises from a chaotic, random nature of the matrix elements of physical observables in energy eigenstates of generic quantum systems. A more recent idea, the ergodic bipartition (EB), points out that similar randomness in the eigenstates themselves leads to entanglement dynamics largely independent of the initial conditions. We unify the ETH and the EB and develop an elegant diagrammatic prescription for calculating correlations, à la Feynman diagrams, that rests on the sound mathematical framework of free probability theory. Learn even more
Probing chaotic eigenstates in polynomial-time
Eigenstates of non-integrable quantum systems - sometimes called chaotic eigenstates for their connections to quantum chaos - demand computational resources that grow exponentially in the system size for exact solutions. We develop an algorithm, the orthogonal operator polynomial expansion (OOPEX), that approximates expectation values in these states in polynomial time by exploiting the fact that such expectation values vary smoothly with the eigenstate energy. As a result, we can easily access system sizes and correlation lengths that are well beyond the reach of exact diagonalization. Learn even more
Figure shows agreement between exact (ragged lines) and approximate (smooth lines with symbols) expectation values for several observables for the non-integrable Ising chain.
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