Periodic Reporting for period 1 - VERMOUTH (noVEl appRoaches to the quantuM bOUnds To cHaos)
Reporting period: 2022-10-01 to 2024-09-30
In recent years, experimental and technological progress in quantum simulation and computation has stimulated enormous theoretical interest in the coherent dynamics of complex quantum systems. This has motivated a renaissance for the study of quantum chaos, which has become a broad multidisciplinary field, with concepts and approaches stemming from condensed matter, high energy physics and quantum information theory. These different interdisciplinary approaches have emphasised the existence of a set of bounds imposed by quantum effects on the physical properties of many-body systems, such as viscosity, conductivity or Lyapunov exponent. They are characterized by a time scale which only depends on temperature and the Planck constant and for this reason they have been dubbed ``"Planckian bounds". Their remarkable feature is that they are all saturated by models of black holes.
However, there are several outstanding open questions regarding such quantum constraints. I) Are these bounds a coincidence or they are potentially indicative of a deep principle in quantum mechanics? II) What is the relation between the bound to chaos and the Planckian one? III) What are the physical properties of the systems saturating these bounds?
The purpose of this proposal is to tackle several issues at the intersection of these fields from the perspective of standard quantum statistical mechanics. In particular, the project intends to explain the bounds as a direct consequence of the quantum Fluctuation-Dissipation Theorem (FDT), one of the cornerstones of equilibrium physics.
However, there are several outstanding open questions regarding such quantum constraints. I) Are these bounds a coincidence or they are potentially indicative of a deep principle in quantum mechanics? II) What is the relation between the bound to chaos and the Planckian one? III) What are the physical properties of the systems saturating these bounds?
The purpose of this proposal is to tackle several issues at the intersection of these fields from the perspective of standard quantum statistical mechanics. In particular, the project intends to explain the bounds as a direct consequence of the quantum Fluctuation-Dissipation Theorem (FDT), one of the cornerstones of equilibrium physics.
This project has achieved three main scientific achievements.
1. Quantum bounds to the generalized Lyapunov exponents, with Jorge Kurchan
So far, most of the attention of studies on quantum chaos has focused on the maximal Lyapunov exponent. Classically, however, that are several quantifiers to study chaotic dynamics. For instance, the generalised Lyapunov exponent (GLE) accounts for chaotic fluctuations of the dynamical system. Its non-trivial behaviour is associated with multifractality. Together with my Host (Jorge Kurchan), we introduced the generalized quantum Lyapunov exponents Lq, defined from the growth rate of the powers of the square commutator. They may be related to an appropriately defined thermodynamic limit of the spectrum of the commutator, which plays the role of a large deviation function, obtained from the exponents Lq via a Legendre transform. We have shown that such exponents obey a generalized bound to chaos due to the fluctuation-dissipation theorem, as already discussed in the literature. The bounds for larger q are actually stronger, placing a limit on the large deviations of chaotic properties. We exemplified our findings at infinite temperature via a numerical study of the kicked top, a paradigmatic model of quantum chaos.
This activity was the content of Scientific Objective 1 (SO1) and it has been published in the Special Issue of Entropy: "Quantum chaos - dedicated to Professor Giulio Casati on the Occasion of His 80th Birthday" and it can be found at https://arxiv.org/abs/2212.10123(opens in new window)
2. The Eigenstate Thermalization Hypothesis and Free Probability Theory, with Laura Foini and Jorge Kurchan
Discussing the "Planckian bounds" requires an understanding of the equilibrium dynamics of two and more multi-time correlation functions.
The best way we have to understand quantum statistical mechanics is the Eigenstate Thermalization Hypothesis (ETH), which is an ansatz on the structure of observables when written in the eigenbasis of the Hamiltonian. At equilibrium, there are correlations between observables at different times, and ETH has been recently shown to encode all the relevant correlations of matrix elements. However, the structure of these correlations has been elusive. In this achievement, we uncover the close relationship between this perspective on ETH and Free Probability theory, as applied to a thermal ensemble or an energy shell. Free Probability is a branch of math which generalizes probability to non-commuting variables. This mathematical framework allows one to reduce in a straightforward way higher-order correlation functions to a decomposition given by minimal blocks, identified as free cumulants, for which we give an explicit formula. The present results uncover a direct connection between the Eigenstate Thermalization Hypothesis and the structure of Free Probability, widening considerably the latter’s scope and highlighting its relevance to quantum thermalization.
This activity was inspired by Scientific Objective 3 (SO3) and it has been published in Phys. Rev. Lett. 129, 170603 – 2022 and it can be found at https://arxiv.org/pdf/2204.11679.pdf(opens in new window)
3. General Eigenstate Thermalization via Free Cumulants in Quantum Lattice Systems, with Felix Fritzsch and Tomaz Prosen
The Eigenstate-Thermalization-Hypothesis (ETH) is the most successful way to understand quantum thermalization and dynamics at equilibrium. Only recently attention has been paid to so-called general ETH, which accounts for higher-order correlations among matrix elements. In the previous activity (2.) we rationalized theoretically the general ETH using the language of Free Probability. In this work, we perform the first numerical investigation of the general ETH in physical many-body systems with local interactions by testing the decomposition of higher-order correlators into free cumulants. We perform exact diagonalization on two classes of local non-integrable (chaotic) quantum many-body systems: spin chain Hamiltonians and Floquet brickwork unitary circuits. We show that the dynamics of four-time correlation functions are encoded in fourth-order free cumulants, as predicted by ETH. Their non-trivial frequency dependence encodes the physical properties of local many-body systems and distinguishes them from structureless, rotationally invariant ensembles of random matrices.
This activity is an application of Scientific Objective 3 (SO3) and it can be found at https://arxiv.org/pdf/2303.00713.pdf(opens in new window)
1. Quantum bounds to the generalized Lyapunov exponents, with Jorge Kurchan
So far, most of the attention of studies on quantum chaos has focused on the maximal Lyapunov exponent. Classically, however, that are several quantifiers to study chaotic dynamics. For instance, the generalised Lyapunov exponent (GLE) accounts for chaotic fluctuations of the dynamical system. Its non-trivial behaviour is associated with multifractality. Together with my Host (Jorge Kurchan), we introduced the generalized quantum Lyapunov exponents Lq, defined from the growth rate of the powers of the square commutator. They may be related to an appropriately defined thermodynamic limit of the spectrum of the commutator, which plays the role of a large deviation function, obtained from the exponents Lq via a Legendre transform. We have shown that such exponents obey a generalized bound to chaos due to the fluctuation-dissipation theorem, as already discussed in the literature. The bounds for larger q are actually stronger, placing a limit on the large deviations of chaotic properties. We exemplified our findings at infinite temperature via a numerical study of the kicked top, a paradigmatic model of quantum chaos.
This activity was the content of Scientific Objective 1 (SO1) and it has been published in the Special Issue of Entropy: "Quantum chaos - dedicated to Professor Giulio Casati on the Occasion of His 80th Birthday" and it can be found at https://arxiv.org/abs/2212.10123(opens in new window)
2. The Eigenstate Thermalization Hypothesis and Free Probability Theory, with Laura Foini and Jorge Kurchan
Discussing the "Planckian bounds" requires an understanding of the equilibrium dynamics of two and more multi-time correlation functions.
The best way we have to understand quantum statistical mechanics is the Eigenstate Thermalization Hypothesis (ETH), which is an ansatz on the structure of observables when written in the eigenbasis of the Hamiltonian. At equilibrium, there are correlations between observables at different times, and ETH has been recently shown to encode all the relevant correlations of matrix elements. However, the structure of these correlations has been elusive. In this achievement, we uncover the close relationship between this perspective on ETH and Free Probability theory, as applied to a thermal ensemble or an energy shell. Free Probability is a branch of math which generalizes probability to non-commuting variables. This mathematical framework allows one to reduce in a straightforward way higher-order correlation functions to a decomposition given by minimal blocks, identified as free cumulants, for which we give an explicit formula. The present results uncover a direct connection between the Eigenstate Thermalization Hypothesis and the structure of Free Probability, widening considerably the latter’s scope and highlighting its relevance to quantum thermalization.
This activity was inspired by Scientific Objective 3 (SO3) and it has been published in Phys. Rev. Lett. 129, 170603 – 2022 and it can be found at https://arxiv.org/pdf/2204.11679.pdf(opens in new window)
3. General Eigenstate Thermalization via Free Cumulants in Quantum Lattice Systems, with Felix Fritzsch and Tomaz Prosen
The Eigenstate-Thermalization-Hypothesis (ETH) is the most successful way to understand quantum thermalization and dynamics at equilibrium. Only recently attention has been paid to so-called general ETH, which accounts for higher-order correlations among matrix elements. In the previous activity (2.) we rationalized theoretically the general ETH using the language of Free Probability. In this work, we perform the first numerical investigation of the general ETH in physical many-body systems with local interactions by testing the decomposition of higher-order correlators into free cumulants. We perform exact diagonalization on two classes of local non-integrable (chaotic) quantum many-body systems: spin chain Hamiltonians and Floquet brickwork unitary circuits. We show that the dynamics of four-time correlation functions are encoded in fourth-order free cumulants, as predicted by ETH. Their non-trivial frequency dependence encodes the physical properties of local many-body systems and distinguishes them from structureless, rotationally invariant ensembles of random matrices.
This activity is an application of Scientific Objective 3 (SO3) and it can be found at https://arxiv.org/pdf/2303.00713.pdf(opens in new window)
The scientific results of the action go beyond the state of the art, identifying new bounded time scales (1.) and providing a new formalism - free probability - to tackle many-body correlations (2.-3.)
In particular, the second and third achievements open a new path in many-body physics, bringing a relatively new branch of math into the study of quantum correlations. From our work, it is clear that Free Probability allows one to deeply simplify the study of multipoint correlation functions. However, it is still not clear if it is "only" a useful mathematical tool, or if this would allow us to make ket novel discoveries. To fully explore the potentialities of this remarkable relation, further research on the topic of Free Probability in many-body physics shall be uptaken.
In particular, the second and third achievements open a new path in many-body physics, bringing a relatively new branch of math into the study of quantum correlations. From our work, it is clear that Free Probability allows one to deeply simplify the study of multipoint correlation functions. However, it is still not clear if it is "only" a useful mathematical tool, or if this would allow us to make ket novel discoveries. To fully explore the potentialities of this remarkable relation, further research on the topic of Free Probability in many-body physics shall be uptaken.