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Monday, 11 November 2019
Time Speaker Title Resources
09:30 to 11:00 Peter Reimann Analytical approaches on thermalization (Lecture 1)
11:30 to 12:10 Hal Tasaki A Lieb-Schultz-Mattis type theorem without continuous symmetry

By a Lieb-Schultz-Mattis type theorem, we mean a no-go theorem that rules out the existence of a unique gapped ground state in a certain class of quantum many-body systems.  The original Lieb-Schultz-Mattis theorem, which was proved in 1961 for the antiferromagnetic Heisenberg chain, and its various extensions rely essentially on the U(1) symmetry of the models.  Recently it was realized that Lieb-Schultz-Mattis type statements can be shown for one-dimensional quantum many-body systems that have discrete on-site symmetry whose action forms a nontrivial projective representation of the symmetry group.  We here present a fully rigorous version of such a Lieb-Schultz-Mattis type theorem for one-dimensional quantum spin systems.  Rather surprisingly (at least to the present speaker), the operator algebraic formulation of spin chains seems to be mandatory for the proof.  In particular the notion of the von Neumann algebra (especially the Cuntz algebra) plays an essential role in the proof.
The new theorem can be stated as a no-go theorem for the existence of a translation invariant area-law states (which is not necessarily a ground state), and hence can be interpreted as a kind of "no-scar theorem".
The talk is based on a joint work with Yoshiko Ogata in arXiv:1808.08740 (Comm. Math. Phys. 2019).

12:10 to 12:50 Subroto Mukerjee Transport, multifractality, and scaling at the localization transition in quasiperiodic systems

There has been a revival of interest in localization phenomena in quasiperiodic systems with a view to examining how they differ fundamentally from such phenomena in random systems. Motivated by this, we study transport in the quasiperiodic, one-dimentional (1d) Aubry-Andre model and its generalizations to 2d and 3d. We study the conductance of open systems, connected to leads, as well as the Thouless conductance, which measures the response of a closed system to boundary perturbations. We find that these conductances show signatures of a metal-insulator transition from an insulator, with localized states, to a metal, with extended states having (a) ballistic transport (1d), (b) superdiffusive transport (2d), or (c) diffusive transport (3d); precisely at the transition, the system displays sub-diffusive critical states.  We calculate the beta function and show that, in 1d and 2d, single-parameter scaling is unable to describe the transition. Further-more, the conductances show strong non-monotonic variations with L and an intricate structure of resonant peaks and subpeaks. In 3d, our results are well approximated by single parameter scaling.

14:30 to 15:10 Rajiv Singh Local entanglement and confinement transitions in random quantum spin-ice models

We discuss our Numerical Linked Cluster Expansion (NLC) and Exact Diagonalization (ED) based recent studies of local entanglement and confinement transitions in random quantum spin-ice models. Nature of the different phases is determined through correlation functions as well as through the behavior of entanglement entropy associated with local regions.

15:10 to 15:50 Subhro Bhattacharjee Many-body chaos in classical spin systems

In this talk, I shall discuss our recent numerical results to characterise spatio-temporal chaos interacting classical spin systems with local interactions in presence and absence of competing interactions. In particular, I shall discuss the temperature dependence of the measures of spatio-temporal chaos-- the Lyapunov exponent and the butterfly velocity, and indicate their possible connection with transport coefficients.

Tuesday, 12 November 2019
Time Speaker Title Resources
09:30 to 11:00 Peter Reimann Analytical approaches on thermalization (Lecture 2)
11:30 to 12:10 Krishnendu Sengupta A path integral approach to entanglement and non-equilibrium dynamics

We present a path integral formalism for expressing matrix elements of the density matrix of a quantum many-body system between any two coherent states in terms of standard Matsubara action with periodic(anti-periodic) boundary conditions on bosonic(fermionic) fields. We show that this enables us to express several entanglement measures for bosonic/fermionic many-body systems described by a Gaussian action in terms of its Matsubara Green function. We apply this formalism to compute various entanglement measures for the Bose-Hubbard model in the strong-coupling regime, both in the presence and absence of Abelian and non-Abelian synthetic gauge fields, within a strong coupling mean-field theory. In addition, our method also provides an alternative formalism for addressing time evolution of quantum-many body systems\blue{, with Gaussian actions,} driven out of equilibrium without the use of Keldysh technique. We demonstrate this by deriving analytical expressions of the return probability and the counting statistics of several operators for a class of integrable models represented by free Dirac fermions subjected to a periodic drive in terms of the elements of their Floquet Hamiltonians. We provide a detailed comparison of our method with the earlier, related, techniques used for similar computations, discuss the significance of our results, and chart out other systems where our formalism can be used.

12:10 to 12:50 Anatoli Polkovnikov High temperature adiabatic flows in ergodic systems

I will discuss the structure of local generators of adiabatic transformations in ergodic Hamiltonians. These generators play a key role in many different setups like finding quantum speed limit bounds, quantum information geometry, shortcuts to adiabaticity, Schrieffer Wolff transformations, Floquet Hamiltonians, Wilson-Wegner flows and many more. While mathematically these generators are ill defined in quantum chaotic systems, they can allow for very accurate local approximations in certain directions. Using an example of a specific nonintegrable Ising chain I will show that these generators are highly anisotropic allowing one to define adiabatic flows connecting families of Hamiltonians. These flows are very reminiscent of Renormalization Group flows. I will also show that near singular (massively degenerate) points one can define special (dark) states which are very stable to adiabatic deformations. Such special states are analogous to recently discovered quantum scars.

14:30 to 15:10 Marko Medenjak Diffusion from Convection

In my talk I will introduce a non-trivial contribution to the diffusion constant in generic many-body systems arising from the quadratic fluctuations of their ballistically propagating, i.e. convective, modes. The result is obtained by expanding the current operator in the vicinity of equilibrium states in terms of powers of local and quasi-local conserved quantities. Tthe only finite contribution to diffusion constant arises from the second-order terms in this expansion. One of the consequences of the result is that whenever there are at least two coupled modes with degenerate velocities, the system behaves super-diffusively in accordance with the non-linear fluctuating hydrodynamics theory. Furthermore the expression saturates the exact diffusion constants in quantum and classical interacting integrable systems.

15:10 to 15:50 Mohammad Maghrebi Non-equilibrium criticality in quench dynamics of infinite-range spin models

Long-range interacting spin systems are ubiquitous in physics and exhibit a variety of ground state disorder-to-order phase transitions. In this talk, I consider a prototype of infinite-range interacting models known as the Lipkin-Meshkov-Glick (LMG) model describing the collective interaction of N spins, and investigate the dynamical properties of fluctuations and correlations after a sudden quench of the Hamiltonian. Specifically, I focus on critical quenches, where the initial state and/or the post-quench Hamiltonian are critical. Depending on the type of quench, I identify three distinct behaviors where both the short-time dynamics and the stationary state at long times are effectively thermal, quantum, and genuinely non-equilibrium, characterized by distinct universality classes and  static and dynamical critical exponents. I will argue that these behaviors can be identified by an infrared effective temperature that is finite, zero, and infinite (the latter scaling with the system size as N^1/3), respectively. Our analytical findings are confirmed by exact numerical simulation.

16:20 to 17:00 Mahendra Verma Boltzmann equation and Hydrodynamic equations: Exploration of equilibrium and nonequilibrium regimes

In this talk I will discuss the equilibrium and nonequilibrium aspects of hydrodynamic equations, and relate it to kinetic theory. Under equilibrium, the velocity field is a uncorrelated random variable with $k^2$ spectrum. In numerical simulations, Fourier modes with equal amplitudes but random phases produces the $k^2$ spectrum. However, if the initial velocity field contains only large scales excitations, then simulations produce Kolmogorov's $k^{-5/3}$ spectrum in the presences of viscosity, and a combination of $k^{-5/3}$ and $k^2$ spectra in the absence of viscosity. Thus, initial condition plays an important role in determining the behaviour of fluid systems.

I will also describe how energy flux can be used to determine arrow of time. In turbulence (more generally, driven-dissipative nonequilibrium systems) forced at large scale, the energy typically flows from large scales to dissipative scales. This generic and multiscale process breaks time reversal symmetry and principle of detailed balance, thus can yield an arrow of time. Also, the conversion of large-scale coherence to small-scales decoherence could be treated as a dissipation mechanism for generic physical systems.

Wednesday, 13 November 2019
Time Speaker Title Resources
09:30 to 11:00 Marcos Rigol Generalized Thermalization in Integrable Lattice Systems
11:30 to 12:10 Peter Reimann Relaxation theory for perturbed many-body quantum systems

We develop an analytic prediction for the temporal relaxation of isolated many-body quantum systems  subject to weak-to-moderate perturbations. Provided that the unperturbed behavior is known, we employ a typicality approach, modeling the essential characteristics of the perturbation operator, to describe the time evolution of expectation values in the perturbed system. Our predictions are validated by comparison with various numerical and experimental results from the literature.

12:10 to 12:50 Arnab Das Dynamical Scarring of an Interacting Floquet System: Resonance versus Emergent Conservation Laws

A generic, interacting, isolated quantum system has only one conserved quantity, its energy, that restricts its dynamics. This hypothesis forms the plinth of statistical mechanics, leading to the Gibbs' distribution. This readily implies the system should heat up indefinitely towards a featureless, locally infinite temperature-like state when driven externally by modulating a parameter of the system with time, as there will be no conservation law left to prevent that. Here we show, such unbounded thermalizations can be prevented under strong periodic drive. In this regime, for certain choice of the drive parameters (the scars" ), a local conserved quantity emerges, and prevents the system from heating up by fracturing the Hilbert space into dynamically disjoint sectors. Since our undriven system is disorder-free, interacting, and non-integrable, it provides an example where application of a founding assumption of statistical mechanics fails for a generic interacting system. We devised a perturbation theory, which predicts isolated and sharp resonances in the parameter space, implying existence of parameter regimes without resonant heating. A Magnus expansion in a time-dependent frame reveals emergence of an integrable structure at the scar points up to second order.  Excellent agreements of both these series with the exact  numerical results at leading orders indicate those are asymptotic in nature. Along with the finite-size analysis, they strongly indicate persistence of the phenomenon in the thermodynamic limit.

14:30 to 15:10 Gunter Schutz Charge-current correlation equalities far from thermal equilibrium

We prove that a recently derived correlation equality between conserved charges and their associated conserved currents for quantum systems far from equilibrium [O.A. Castro-Alvaredo, B. Doyon, and T. Yoshimura, Phys. Rev. X 6, 041065 (2016)], is valid under more general conditions than assumed so far. Similar correlation identities turn out to hold also in the absence of translation invariance, for lattice models, and in any space dimension, and to imply a symmetry of the non-equilibrium linear response functions. In generalized Gibbs ensembles these charge-current correlation equalities give rise to a current symmetry somewhat reminiscent of the Onsager relations, both for quantum and classical Markovian dynamics [R. Grisi and G.M.S., J. Stat. Phys. 145, 1499-1512 (2011)].

15:10 to 15:50 Amit Dutta "Dynamical preparation of a topological state in SSH models under periodic driving"

Exploiting the possibility of temporal variation of the winding number, we have prepared a SSH chain in its stroboscopic topological state, starting from the trivial one, by application of a periodic perturbation. The periodic driving, we employ here, is adiabatically switched on to break the particle-hole symmetry and generate a chiral mass term in the effective Floquet Hamiltonian; consequently the Floquet Hamiltonian also gets deformed without crossing the gapless quantum critical point. The particle hole symmetry is subsequently restored in the Floquet Hamiltonian by adiabatically switching off a part of the periodic potential. Thereafter, the Floquet Hamiltonian develops a symmetry protected non-trivial topological winding number. Furthermore, we also observe stroboscopic topologically protected localised edge states in a long open chain and show that a bulk boundary correspondence survives a unitary non-equilibrium situation in 1D BDI Hamiltonians.  We also generalise the present protocol to extended SSH chain with higher neighbour hopping.

Thursday, 14 November 2019
Time Speaker Title Resources
09:30 to 11:00 Marcos Rigol Entanglement entropy of highly excited eigenstates of many-body lattice Hamiltonians
11:30 to 12:10 Takashi Mori Steady states of open systems and eigenstate thermalization hypothesis

I discuss steady states of macroscopic quantum systems under dissipation not obeying the detailed balance condition. I argue that the Gibbs state at an effective temperature gives a good description of the steady state provided that the system Hamiltonian obeys the eigenstate thermalization hypothesis (ETH) and the perturbation theory in the weak system-environment coupling is valid in the thermodynamic limit. I explain our result on the criterion to guarantee the validity of the perturbation theory, which implies that the perturbation theory works in the thermodynamic limit when the Liouvillian is gapped for bulk-dissipated systems. While, the perturbation theory breaks down in boundary-dissipated chaotic systems due to the presence of diffusive transports. We numerically confirm these theoretical predictions. This work suggests a connection between steady states of macroscopic open quantum systems and the ETH.

12:10 to 12:50 Marcos Rigol Prethermalization and Thermalization in Isolated Quantum Systems

"Prethermalization has been extensively studied in systems close to integrability. We discuss a more general, yet conceptually simpler, setup for this phenomenon. We consider a--possibly nonintegrable--reference dynamics, weakly perturbed so that the perturbation breaks at least one conservation law. We argue then that the evolution of the system proceeds via intermediate (generalized) equilibrium states of the reference dynamics. The motion on the manifold of equilibrium states is governed by an autonomous equation, flowing towards global equilibrium in a time of order 1/g^2, where g is the perturbation strength. We also describe the leading correction to the time- dependent reference equilibrium state, which is, in general, of order g [1]. The theory is well confirmed in numerical calculations of model Hamiltonians in the context of quantum quenches [1] and driven systems [2], for which we use numerical linked cluster expansions and full exact diagonalization. For the driven systems, we discuss the relationship between heating rates and, within the eigenstate thermalization hypothesis, the smooth function that characterizes the off-diagonal matrix elements of the drive operator in the eigenbasis of the static Hamiltonian. We show that such a function, in nonintegrable and (remarkably) integrable Hamiltonians, can be probed experimentally by studying heating rates as functions of the frequency of the drive.

References:
[1] K. Mallayya, M. Rigol, and W. De Roeck, Prethermalization and
Thermalization in Isolated Quantum Systems, Phys. Rev. X 9, 021027 (2019).
[2] K. Mallayya and M. Rigol, Heating Rates in Periodically Driven Strongly
Interacting Quantum Many-Body Systems, arXiv:1907.04261 (2019)."

14:30 to 15:10 Samriddhi Sankar Ray Thermalisation, Many-Body Chaos, and Weak Solutions: The Galerkin-truncated Inviscid Burgers Equation

The Galerkin-truncated inviscid equations of hydrodynamics play a key role in developing theories for turbulent flows. In recent years, these truncated equations, which unlike the parent partial differential equations (PDE) thermalise, have served as a useful tool to understand the interplay of statistical physics and turbulence. By using the example of the one-dimensional Burgers equation, we will first show how thermalisation sets-in in such systems and once thermalised how they serve as a useful microscopic model to understand the classical analogues of out-of-time-ordered correlators (OTOCs) and many-body chaos. Finally, we will discuss a new technique---tyger purging---which can be used to suppress thermalisation, without resorting to viscosity, and obtain the so-called weak solutions of the parent PDE. These results have an important bearing on the celebrated blow-up problem of the three dimensional Euler equation.

15:10 to 15:50 Enej Ilievski The equilibrium landscape of the Heisenberg spin chain

Focusing on the isotropic Heisenberg spin-1/2 chain, the paradigm for integrable integrable quantum models, we introduce the notion of the equilibrium landscape and explicit construction of the entire manifold of local equilibrium ensembles. We outline two complementary approaches, the functional integral Thermodynamic Bethe Ansatz approach and the lattice regularization transfer matrix approach, and establish equivalence between the two. As we go along, we clarify several subtle features such as the null modes and explain the breakdown of the canonical Y-system functional identities.

Friday, 15 November 2019
Time Speaker Title Resources
09:40 to 10:20 Takahiro Sagawa Characterizing complexity of many-body quantum dynamics by higher-order eigenstate thermalization

Complexity of dynamics is at the core of quantum many-body chaos and exhibits a hierarchical feature: higher-order complexity implies more chaotic dynamics. Conventional ergodicity in thermalization processes can be regarded as manifestation of the lowest order complexity, which is represented by the eigenstate thermalization hypothesis (ETH) stating that individual energy eigenstates are thermal.
In this talk, we propose a higher-order generalization of the ETH, named the k-ETH (k = 1, 2, 3,… ), to quantify higher-order complexity of quantum many body dynamics at the level of individual energy eigenstates, where the lowest order ETH (1-ETH) is the conventional ETH. We show that the 2-ETH implies a universal behavior of the 2-Renyi entanglement entropy of individual energy eigenstates. In particular, we show that the Page correction of the entanglement entropy originates from the 2-ETH, while as is well known, the volume law can be accounted for by the 1-ETH.  We numerically verified that the 2-ETH approximately holds for non-integrable spin models, but does not hold in the integrable case.  In addition, we discuss a quantum-information theoretic aspect of the k-ETH in terms of unitary k-designs. Reference: Kazuya Kaneko, Eiki Iyoda, and Takahiro Sagawa, to appear in arXiv soon.

10:20 to 11:00 Keiji Saito Weak and strong ETH from the clustering property

Clustering of an equilibrium bipartite correlation is widely observed in non-critical many-body quantum systems. Herein, we consider the thermalization phenomenon in generic finite systems exhibiting clustering. We discuss the weak and strong version of eigenstate thermalization, depending on the energy regime with different density of states.

11:30 to 12:10 Christian Mendl Scrambling and thermalization in a diffusive quantum many-body system

Out-of-time ordered (OTO) correlation functions describe scrambling of information in correlated quantum matter. They are of particular interest in incoherent quantum systems lacking well defined quasi-particles. Thus far, it is largely elusive how OTO correlators spread in incoherent systems with diffusive transport governed by a few globally conserved quantities. Here, we study the dynamical response of such a system using high-performance matrix-product-operator techniques. Specifically, we consider the non-integrable, one-dimensional Bose-Hubbard model in the incoherent high-temperature regime. Our system exhibits diffusive dynamics in time-ordered correlators of globally conserved quantities, whereas OTO correlators display a ballistic, light-cone spreading of quantum information. The slowest process in the global thermalization of the system is thus diffusive, yet information spreading is not inhibited by such slow dynamics. We furthermore develop an experimentally feasible protocol to overcome some challenges faced by existing proposals and to probe time-ordered and OTO correlation functions. Our study opens new avenues for both the theoretical and experimental exploration of thermalization and information scrambling dynamics. (arXiv: 1612.02434, DOI: 10.1088/1367-2630/aa719b)

12:10 to 12:50 Dibyendu Roy Pancharatnam-Zak phase

The geometric phase gained by an electron in a one-dimensional periodic lattice due to weak electric perturbation is found and referred to as the Pancharatnam-Zak phase. The underlying mathematical structure responsible for this phase is unveiled. As opposed to the well known Zak phase, the Pancharatnam-Zak phase is a gauge invariant observable phase, and correctly characterizes the energy bands of the lattice. We demonstrate the gauge invariance of the Pancharatnam-Zak phase in two celebrated models displaying topological phases. A filled band generalization of this geometric phase is constructed and is observed to be sensitive to the Fermi-Dirac statistics of the band electrons. The measurement of the single-particle Pancharatnam-Zak phase in individual topological phases, as well as the statistical contribution in its many-particle generalization, should be accessible in various controlled quantum experiments.

14:30 to 15:10 Marko Znidaric Localized and ballistic eigenstates in chaotic spin ladders and the Fermi-Hubbard model

I will introduce a class of spin ladder models that is in general chaotic, can be disordered, and includes the integrable Hubbard chain as a special case. Nevertheless, one can analytically show the existence of exponentially many atypical eigenstates that are populated by noninteracting excitations. Depending on parameters they can e.g. exhibit Anderson localization, or, surprisingly, ballistic transport at any disorder strength. These properties differ strikingly from those of typical eigenstates nearby in energy which give rise to diffusion. Results have implications for possible localization in the presence of non-Abelian symmetries, as well as for recent interest in states avoiding thermalization.

15:10 to 15:50 Anatoly Dymarsky Euclidean operator growth and quantum chaos

Operator growth in Euclidean time differs substantially from its Minkowski counterpart. While the latter is known to conform to Lieb-Robinson bound, much less is known about the Euclidean case. We will introduce a new bound which establishes Euclidean version of the Lieb-Robinson result and discuss how Euclidean growth is related to quantum chaos and connects the out of time ordered correlators (OTOC) and eigenstate thermalization hypothesis (ETH). We will also show that the rate of Euclidean growth can be connected to the grow of Lancsoz coefficients, which were recently proposed to encode chaotic behavior. Using this relation we propose a new bound on Lyapunov exponents, which is valid for both low and high temperatures

Monday, 18 November 2019
Time Speaker Title Resources
09:30 to 11:00 Herbert Spohn Classical Toda chain as paradigm for the hydrodynamics of integrable systems (Lecture 1)

The experts in the field agree that the hydrodynamics for integrable systems depends only on the two-particle phase shift and whether particles are classical, bosons, or fermions. I will use the classical Toda as an example to illustrate the general structure.

11:30 to 12:10 Wen Wei Ho Long-lived interacting phases of matter protected by multiple time-translation symmetries in quasiperiodically-driven systems

The discrete time-translation symmetry of a periodically-driven (Floquet) system allows for the existence of novel, nonequilibrium interacting phases of matter. A well-known example is the discrete time crystal, a phase characterized by the spontaneous breaking of this time-translation symmetry. In this talk, I will show that the presence of *multiple* time-translation symmetries, realized by quasiperiodically driving a system with two or more incommensurate frequencies, leads to a panoply of novel non-equilibrium phases of matter, both spontaneous symmetry-breaking ("discrete time quasi-crystals") and topological. I will demonstrate that these phases are stable in a long-lived, 'preheating' regime, by outlining rigorous mathematical results establishing slow heating at high driving frequencies. As a byproduct, I will introduce the notion of many-body localization (MBL) in quasiperiodically-driven systems.

12:10 to 12:50 Sho Sugiura Many-body scar state intrinsic to Floquet system

I will talk about the violation of the Floquet version of eigenstate thermalization hypothesis. We consider experimentally realistic Hamiltonians based on PXP type interactions without disorder. We exactly prove the existence of many-body scar states in the Floquet eigenstates, by showing the explicit expressions of the wave functions. Using the underlying physical mechanism, various driven Hamiltonians with Floquet-scar states can be systematically engineered.

14:30 to 15:10 Arnab Sen Collapse and revival of quantum many-body scars via Floquet engineering

The presence of quantum scars, athermal eigenstates of a many-body Hamiltonian with finite energy density, leads to absence of ergodicity and long-time coherent dynamics in closed quantum systems starting from simple initial states. Such non-ergodic coherent dynamics, where the system does not explore its entire phase space, has been experimentally observed in a chain of ultracold Rydberg atoms. We show, via study of a periodically driven Rydberg chain, that the drive frequency acts as a tuning parameter for several reentrant transitions between ergodic and non-ergodic regimes. The former regime shows rapid thermalization of correlation functions and absence of scars in the spectrum of the system's Floquet Hamiltonian. The latter regime, in contrast, has scars in its Floquet spectrum which control the long-time coherent dynamics of correlation functions. Our results open a new possibility of drive frequency-induced tuning between ergodic and non-ergodic dynamics in experimentally realizable disorder-free quantum many-body systems.
Reference: Mukherjee, Nandy, Sen, Sen, Sengupta, arxiv:1907.08212

15:30 to 16:30 Christian Maes Statistical mechanical perspectives on cosmological puzzles

We review some well-known paradoxes in cosmology and give a statistical mechanics reading.  Puzzles include the horizon and the flatness problem, the information paradox and the dark energy problem and the origin of the so called space roar.  Each time, we emphasize the role of statistical arguments to complement the dynamical understanding.

Tuesday, 19 November 2019
Time Speaker Title Resources
09:30 to 11:00 Herbert Spohn Classical Toda chain as paradigm for the hydrodynamics of integrable systems (Lecture 2)

The experts in the field agree that the hydrodynamics for integrable systems depends only on the two-particle phase shift and whether particles are classical, bosons, or fermions. I will use the classical Toda as an example to illustrate the general structure.

11:30 to 12:10 Stefano Olla Thermal diffusion and energy transport for system with more conserved quantities

Energy transport can be influenced by the presence of other conserved quantities. We consider here diffusive systems where energy and the other conserved quantities evolve macroscopically on the same diffusive space–time scale. In these situations, the Fourier law depends also on the gradient of the other conserved quantities

12:10 to 12:50 François Huveneers Slow thermalization in a chain of classical anharmonic oscillators

I will consider a chain of oscillators in the infinite volume limit. There is no proper thermalization if the chain is purely harmonic. Instead, as soon as some anharmonic interaction is added, the chain is generically expected to equilibrate on the so-called kinetic time scales (in the regime of weak anharmonic interactions). In some cases, however, recent numerical results confirming the predictions from the Boltzmann-Peierls equation, indicate that the proper equilibrium state is only reached on much larger time scales. This is due to the presence of a pseudo-conserved quantity that can be identified as the ""number of phonons"" in the chain. I will analyze the origin of this phenomenon and discuss predictions for the time scales on which genuine thermalization is reached.
Joint work with J. Lukkarinen

14:30 to 15:10 Rajdeep Sensarma Memories of initial states and density imbalance in dynamics of interacting disordered systems

Strongly disordered systems do not thermalize when they are initialized to arbitrary initial states; rather they retain the memory of initial states at long times For disordered interacting systems, this is a characteristic of many body localized phase. Using a newly developed field theoretic method which can keep track of arbitrary initial conditions in non-equilibrium dynamics, we study the experimentally relevant problem of a disordered interacting system starting from an initial density pattern of 1 and 0 particles on different lattice sites. For non-interacting systems, we derive a universal relation between long time memory of the initial state (as encoded in density imbalance), and the localization length. For interacting systems, we show that the memory of the initial state remains encoded in the noise correlations of the bath of excitations created during the evolution. This leads to a finite density imbalance in interacting strongly disordered systems at long times.

15:10 to 15:50 Marin Bukov (Pre)thermalization in periodically driven systems: a quantum or classical phenomenon?

I will give a brief introduction to prethermalization in quantum Floquet systems and compare the phenomenon to their classical counterparts, reviewing the basic concepts and presenting supporting numerical evidence. Thermalization in closed nonintegrable Floquet systems happens in four stages: (i), the system undergoes constrained thermalization with respect to inverse-frequency Floquet expansion, before (ii) it enters a prethermal plateau whose temperature is given by the energy density of the initial state. In the prethermal plateau, energy absorption is exponentially suppressed due to the existence of quasi-conserved local integrals of motion. The system eventually enters a second transient of unconstrained thermalization with respect to the (possibly nonlocal) exact Floquet Hamiltonian (iii), before it eventually heats up to an infinite-temperature state (iv). Curiously, the same behavior is observed in classical many-body spin systems of hundreds of spins, which allows to reach regimes inaccessible in quantum simulations, and raises the question as to how much of this exciting physics is actually driven by quantum effects.

Wednesday, 20 November 2019
Time Speaker Title Resources
09:30 to 11:00 Tomaz Prosen Solvable models of diffusion and many-body chaos (Lecture 1)
11:30 to 12:10 Herbert Spohn On the collision rate assumption for the classical Toda chain

For the hydrodynamics of integrable systems, a major breakthrough resulted from the availability of a formula for the GGE averaged currents. I will discuss the validity of the formula, paying particular attention to the Toda chain

12:10 to 12:50 Jerome Dubail Generalized Hydrodynamics on an Atom Chip

I will give a short introduction to the hydrodynamic description of one-dimensional integrable quantum systems, established in 2016 [1,2]. I will contrast this new framework with the standard hydrodynamic description of fluids which relies on the assumption of local equilibration. Then I will describe the recent cold atom experiment (done by I. Bouchoule and M. Schemmer in Palaiseau) which demonstrates that the new theory provides the correct description of the dynamics of the one-dimensional Bose gas.

[1] O. A. Castro-Alvaredo et al., Emergent Hydrodynamics in Integrable Quantum Systems Out of Equilibrium, Phys. Rev. X 6, 041065 (2016)
[2] B. Bertini et al., Transport in Out-of-Equilibrium XXZ Chains: Exact Profiles of Charges and Currents, Phys. Rev. Lett. 117, 207201 (2016)
[3] M. Schemmer et al., Generalized Hydrodynamics on an Atom Chip, Phys. Rev. Lett. 122, 090601 (2019).

14:30 to 15:10 Boris V. Fine Chaotic properties of spin lattices at high temperatures

I review our old and new investigations of matters relevant to the behavior of out-of-time-order correlators (OTOCs) in spin lattices including (i) systematic investigations of  largest Lyapunov exponents and Lyapunov spectra for classical spin lattices [1,2]; (ii) behavior of Lyapunov spectra near second-order phase transitions [3]; (iii) relation between Loschmidt echoes, largest Lyapunov exponents and OTOCs in lattice models[4,5,6]; and (iv) extracting system's ergodization time from OTOCs[7].

[1] A. S. de Wijn, B. Hess and B. V. Fine, Phys. Rev. Lett. 109, 034101 (2012).
[2]  A. S. de Wijn, B. Hess, and B. V. Fine, J. Phys. A: Math. Theor. 46, 254012 (2013).
[3] A. S. de Wijn, B. Hess, B. V. Fine, Phys. Rev. E 92, 062929 (2015).
[4] B. V. Fine, T. A. Elsayed, C. M. Kropf, and A. S. de Wijn, Phys. Rev. E 89, 012923 (2014).
[5] T. A. Elsayed and B. V. Fine, Phys. Scr. 2015, 014011, (2015) (eprint arXiv:1409.4763).
[6] A. E. Tarkhov, S. Wimberger and B. V. Fine, Phys. Rev. A 96, 023624 (2017).
[7] A. E. Tarkhov, B. V. Fine,  New J. Phys. 20, 123021 (2018).

15:10 to 15:50 Soumya Bera Dynamical properties of many-body localized phase

We will present some recent results about the dynamical properties of many-body localization transition in both ergodic and localized phase. Here we investigate the charge relaxation in quantum-wires of spin-less disordered fermions (t−V-model). Our observable is the time-dependent density propagator at different disorder strengths and at finite interaction. In the ergodic phase, we observe slow charge relaxation, however with a strong indication that it could only be a transient phenomenon. In the localized phase, we observe even slower, possibly logarithmic, decay of charge, which suggests that the many-body localized phase is of different type compare to conventional Anderson localized phase.

16:20 to 17:00 Cédric Bernardin Gamma-convergence approach for large deviations problems in interacting diffusion systems

We consider extended slow-fast systems of N interacting diffusions. The typical behavior of the empirical density is described by a nonlinear McKean-Vlasov equation depending on a scaling parameter separating the time scale of the slow variable from the time scale of the fast variable. Its atypical behavior is encapsulated in a large N Large Deviation Principle (LDP) with a rate functional. We study the Γ-convergence of as the scaling parameter goes to 0 and show it converges to the rate functional appearing in the Macroscopic Fluctuations Theory (MFT) for diffusive systems.

Thursday, 21 November 2019
Time Speaker Title Resources
09:30 to 11:00 Tomaz Prosen Solvable models of diffusion and many-body chaos (Lecture 2)
11:30 to 13:00 Benjamin Doyon Generalised hydrodynamics (Lecture 1)

Hydrodynamics is a powerful theory for describing the emergent dynamics, away from equilibrium, at large scales in space and time. In recent years, the theory adapting hydrodynamics to integrability, called generalised hydrodynamics” (GHD), has been developed. It turns out to be extremely general, applicable to quantum and classical chains, field theories and gases. These lectures will cover the fundamental concepts of this theory. Starting from an overview of hydrodynamics, I will describe the most important aspects of integrability, from which I will derive the basic GHD equations. I will emphasise the universality of GHD and the physical interpretation of its various ingredients, and if time allows, explain some of its applications and more advanced concepts.  Only very basic knowledge of hydrodynamics and integrability is assumed, and I will keep everything as non-technical as possible.

14:30 to 15:10 Tomaz Prosen TBA
15:10 to 15:50 Manas Kulkarni Kardar Parisi Zhang (KPZ) scaling in non-integrable and integrable classical models

We will discuss spatio-temporal correlations in classical non-integrable and integrable models. We start with earlier works on discrete Nonlinear Schrodinger Equation [1,2] and then present results on equilibrium spatio-temporal correlations in classical non-integrable and integrable spin chains. For the non-integrable case, we consider the classical XXZ model (Lattice Landau Lifshitz model) and show regimes where we find KPZ scaling [3]. We explain it using the framework of nonlinear fluctuating hydrodynamics (NFH). To our surprise, we find that a classical integrable spin chain (Integrable Lattice Landau Lifshitz model) also has regimes in which it displays KPZ behaviour [4]. Our findings are along the lines of what was recently found in quantum integrable spin chains thereby providing strong evidence for a classical-quantum correspondence.

[1] Manas Kulkarni, Austen Lamacraft, Phys. Rev. A 88, 021603, Rapid Communications [2013]
[2] Manas Kulkarni, David Huse, Herbert Spohn, Phys. Rev. A 92, 043612 [2015]
[3] Avijit Das, Kedar Damle, Abhishek Dhar, David A. Huse, Manas Kulkarni, Christian B.
Mendl, Herbert Spohn, J Stat Phys,  https://doi.org/10.1007/s10955-019-02397-y [2019]
[4] Avijit Das, Manas Kulkarni, Herbert Spohn, Abhishek Dhar, Phys. Rev. E 100, 042116 [2019]

16:20 to 17:00 Anupam Kundu Equilibrium density of confined particles with repulsive interaction

Systems of interacting particles confined in external potentials are ubiquitous in nature. Particularly, pairwise repulsive interactions with power-law divergences have taken a special place in physics and mathematics. I will discuss a system of N classical particles interacting via pairwise repulsive interaction potential. They are also confined by an external potential which tries to pull them to its minimum. On the other hand the interaction and the entropy try to spread them apart. As a result they settle down over some region with an inhomogeneous density. By deriving a large-N macroscopic description in terms of free energy functional (action) and then minimising it, it is possible to find various interesting distributions of the particles. I will show how the range of interaction as well as the form of the external potential affect this density.

Friday, 22 November 2019
Time Speaker Title Resources
09:30 to 11:00 Benjamin Doyon Generalised hydrodynamics (Lecture 2)

Hydrodynamics is a powerful theory for describing the emergent dynamics, away from equilibrium, at large scales in space and time. In recent years, the theory adapting hydrodynamics to integrability, called generalised hydrodynamics” (GHD), has been developed. It turns out to be extremely general, applicable to quantum and classical chains, field theories and gases. These lectures will cover the fundamental concepts of this theory. Starting from an overview of hydrodynamics, I will describe the most important aspects of integrability, from which I will derive the basic GHD equations. I will emphasise the universality of GHD and the physical interpretation of its various ingredients, and if time allows, explain some of its applications and more advanced concepts.  Only very basic knowledge of hydrodynamics and integrability is assumed, and I will keep everything as non-technical as possible.

11:30 to 12:10 Maurizio Fagotti Entanglement evolution, generalized hydrodynamics, and invariant subspaces

I examine the space where time evolution takes place in a quantum many-body system. After having showed that it is generally tiny in comparison with the entire Hilbert space, I identify some invariant subspaces that are particularly relevant in the presence of inhomogeneities. I characterise such subspaces in a generalised quantum XY model and show that generalised hydrodynamics (GHD) is nothing but the Schrödinger equation in the invariant subspace of the so-called ""locally quasi-stationary states"". The validity of this picture in interacting integrable systems is an open question, and, for example, I show that the recent findings on the time evolution of the entanglement entropy of subsystems provide both promising and discouraging hints.

REFERENCES:
- M. Fagotti, SciPost Phys. 6, 059 (2019);
- M. Fagotti, arXiv:1910.01046;
- B. Bertini, M. Fagotti, L. Piroli, and P. Calabrese, J. Phys. A: Math. Theor. 51, 39LT01 (2018);
- V. Alba, B. Bertini, and M. Fagotti, SciPost Phys. 7, 005 (2019).

12:10 to 12:50 Vir Bulchandani Anomalous transport in one-dimensional quantum systems

We discuss novel mechanisms for anomalous transport in clean, one-dimensional quantum systems. First, we present analytical and numerical evidence that integrability-breaking interactions in one-dimensional metals give rise to superdiffusive transport of energy at low temperatures, that is characterized by a single, non-universal, dynamical exponent. This unexpected violation of Fourier's law should be visible in time-resolved experiments on quantum wires. We next present a generic mechanism for the emergence of Kardar-Parisi-Zhang physics in isotropic spin chains, focusing on the role played by nonlinear, torsional modes of the local magnetization.

14:30 to 15:10 Ravin Bhatt Many-Body Localization in the Quantum Hall Regime

While Anderson’s famous 1958 paper [1] used a model of non-interacting electrons in a disordered potential to discover the phenomenon of localization, the experiments that motivated his study involved quantum spin diffusion in disordered media, a topic that was addressed several decades later [2]. Even more intriguing was the suggestion of many-body localization in interacting fermionic models [3], a topic which, because of its wide implication for fundamental concepts such as thermalization in isolated quantum systems, has given rise to extensive research in the intervening decade and a half.

The study of many-body localization (MBL) is complicated both on the analytical front as well as numerical. Nevertheless, definite progress is being made. It seems to be clear that MBL exists in one-dimensional spin models [4], as first suggested by numerical simulations [5]. Obtaining results in higher dimensions with the same amount of rigor appears distant at present, though studies [6] have shown that a proper treatment of rare fluctuations is crucial for answering this issue.

On the numerical front, models in higher dimensions as well as fully interacting fermionic models suffer the problem of the explosion of the Hilbert space with size. Consequently, we recently explored the possibility of MBL in a system of two-dimensional electrons (interacting via a Coulomb interaction) in the presence of disorder, when placed in a large perpendicular magnetic field in the extreme quantum (lowest Landau level) limit [7-9]. While this situation is significantly more challenging than one-dimensional spin models, we are able to consider both lattice and continuum models; the latter bypass commensurability requirements imposed by lattices in higher dimensions. Using eigenvalue statistics as well as time evolution methods to study several different cases, we find that MBL is very strongly affected by topology, even more so than earlier analytic arguments suggest [10].

This research was supported by US Department of Energy, Office of Basic Energy Sciences.

References:

[1] P. W. Anderson, Physical Review 109, 1492 (1958).
[2] R. N. Bhatt and P. A. Lee, Physical Review Letters 48, 344 (1982); M. Milovanovic, S. Sachdev & R. N. Bhatt,
Physical Review Letters 63, 82 (1989); R. N. Bhatt and D. S. Fisher, Physical Review Letters 68, 3072 (1992).
[3] D. M. Basko, I. L. Aleiner and B. L. Altshuler, Annals of Physics 321, 1126 (2006).
[4] John Z. Imbrie, Journal of Statistical Physics 163, 998 (2016).
[5] Arijeet Pal and David A. Huse, Physical Review B 82, 174411 (2010).
[6] Wojciech De Roeck and Francois Huveneers, Physical Review B 95, 155129 (2017).
[7] Scott D. Geraedts and R. N. Bhatt, Physical Review B 95, 054303 (2017).
[8] Akshay Krishna, Matteo Ippoliti and R. N. Bhatt, Physical Review B 99, 041111(R) (2019).
[9] Akshay Krishna, Matteo Ippoliti and R. N. Bhatt, Physical Review B 100, 054202 (2019).
[10] Rahul Nandkishore and Andrew C. Potter, Physical Review B 90, 195115 (2014).

15:10 to 15:50 Jacopo De Nardis Anomalous spin diffusion in one-dimensional antiferromagnets

The problem of characterizing low-temperature spin dynamics in antiferromagnetic spin chains has so far remained elusive. Here we reinvestigate it by focusing on isotropic antiferromagnetic chains whose low-energy effective field theory is governed by the quantum non-linear sigma model. Employing an exact non-perturbative theoretical approach, we analyze the low-temperature behaviour in the vicinity of non-magnetized states and obtain exact expressions for the spin diffusion constant and the NMR relaxation rate, which we compare with previous theoretical results in the literature. Surprisingly, in $SU(2)$-invariant spin chains in the vicinity of half-filling we find a crossover from the semi-classical regime to a strongly interacting quantum regime characterized by zero spin Drude weight and diverging spin conductivity, indicating super-diffusive spin dynamics. The dynamical exponent of spin fluctuations is argued to belong to the Kardar-Parisi-Zhang universality class. Furthermore, by employing numerical tDMRG simulations, we find robust evidence that the anomalous spin transport persists also at high temperatures, irrespectively of the spectral gap and integrability of the model.

Monday, 25 November 2019
Time Speaker Title Resources
09:30 to 11:00 Maksym Serbyn MBL and other mechanisms of ergodicity breaking (Lecture 1)
11:30 to 13:00 Maksym Serbyn MBL and other mechanisms of ergodicity breaking (Lecture 2)
14:30 to 15:10 Benjamin Doyon Diffusion and superdiffusion from hydrodynamic projection

Hydrodynamic projections, the projection onto conserved charges representing ballistic propagation of fluid waves, give exact transport results in hydrodynamic systems, such as the exact Drude weights. I show that this principle can be extended beyond the Euler scale, in particular to the diffusive scale. By hydrodynamic reduction, Hilbert spaces of observables are constructed that describe the finer scales of hydrodynamics. In particular, the “diffusive space” is the space of quadratically extensive charges that were first introduced by Prosen. In this space, the Green-Kubo formula for the Onsager matrix is easily formulated. The projection onto any given charge gives a lower bound for diffusion. Particular examples of such charges are products of linearly extensive charges. I show that they are covariant derivatives of vector fields on the manifold of maximal entropy states. Thus they are calculable from the thermodynamics, and give an explicit lower bound, which is saturated in integrable systems. They represent the contribution to diffusion from scattering of ballistic waves. Under certain conditions, co-propagating ballistic waves render diffusion infinite. I will finally explain how in these cases an analysis of fractionally extensive charges bound from below the superdiffusive exponent by that of the Kardar-Parisi-Zhang class predicted by nonlinear fluctuating hydrodynamics.

Tuesday, 26 November 2019
Time Speaker Title Resources
09:30 to 11:00 Romain Vasseur MBL and measurement-induced transitions (Lecture 1)
11:30 to 13:00 Romain Vasseur MBL and measurement-induced transitions (Lecture 2)
14:30 to 15:10 Sarang Gopalakrishnan Anomalous transport in integrable spin chains

"The XXZ spin chain is a canonical model of one-dimensional quantum magnetism. We consider spin transport at nonzero temperature in this model. Spin transport is ballistic in the easy-plane regime, diffusive in the easy-axis regime, and superdiffusive at the isotropic Heisenberg point. Energy transport, by contrast, is ballistic in all cases. The framework of generalized hydrodynamics offers an elementary explanation of these phenomena [1] in terms of the quasiparticle content of the XXZ model. It also predicts new transport anomalies, e.g., in local relaxation in the easy-axis regime in the presence of a magnetic field [2], and in the a.c. conductivity in the easy-plane regime [3]. These anomalies come from the asymptotic behavior of large bound states of many quasiparticles, as we will discuss.

[1] SG, R. Vasseur, PRL 122, 127202 (2019)
[2] SG, R. Vasseur, B. Ware, PNAS 116, 16250 (2019)
[3] U. Agrawal, SG, R. Vasseur, and B. Ware, arXiv:1909.05263"

15:10 to 15:50 Alexander Mirlin Anderson localization on random regular graphs: Toy-model of many body- localization

I will discuss Anderson (de-)localization on random regular graphs (RRG), which have locally the structure of a tree but do not have boundary (and thus possess large-scale loops).
Our analytical treatment of the RRG model uses a field-theoretical supersymmetry approach and the saddle-point analysis justified by large “volume” (number of sites) N. The resulting saddle-point equation can be efficiently solved numerically by population dynamics, and also analyzed analytically [1]. The obtained results are in perfect agreement with those of exact diagonalization. In the delocalized phase on RRG, eigenfunctions are ergodic in the sense that their inverse participation ratio scales as 1/N and the spectral statistics is Wigner Dyson in the large-N limit. This limit is reached via a finite-size crossover from small (N << N_c) to large (N >> N_c) system, where N_c is the correlation volume diverging exponentially at the transition, ln N_c ~ _c, where _c is the correlation length. A distinct feature of this crossover is a non-monotonicity of the spectral and wave-function statistics, which is related to properties of the critical phase in the RRG model [2]. We have also performed a detailed study of eigenfunction and energy level correlations, fully confirming the ergodicity of the delocalized phase in the large-N limit [1]. We further demonstrate numerically that the correlation length on the delocalized side, _c, diverges with the critical index _d = ½, in agreement with analytical result [3].
Importantly, properties of the RRG model differ crucially from those of a model on a finite Cayley tree, where wave function moments exhibit multifractal scaling with N in the limit of large N. The multifractality spectrum depends on disorder strength and on the position of the lattice, as we show both analytically and numerically [4, 5].
Finally, I will discuss connections between the (de-)localization on RRG and the many-body localization [1, 6].
[1] K. S. Tikhonov and A. D. Mirlin, Phys. Rev. B 99, 024202 (2019)
[2] K. S. Tikhonov, A. D. Mirlin, and M.A. Skvortsov, Phys. Rev. B 94, 220203 (2016)
[3] K. S. Tikhonov and A. D. Mirlin, Phys. Rev. B 99, 214202 (2019)
[4] K. S. Tikhonov and A. D. Mirlin, Phys. Rev B 94, 184203 (2016)
[5] M. Sonner, K. S. Tikhonov and A. D. Mirlin, Phys. Rev. B 96, 214204 (2017)
[6] K. S. Tikhonov and A. D. Mirlin, Phys. Rev. B 97, 214205 (2018) and in preparation

Wednesday, 27 November 2019
Time Speaker Title Resources
09:30 to 11:00 Wojciech De Roeck Mechanisms of slow thermalization (Lecture 1)
11:30 to 12:10 Maksym Serbyn Quantum many-body scars, mixed phase spaces and non-universal thermalization

The statistical mechanics description of many-particle systems rests on the assumption of ergodicity, the ability of a system to explore all allowed configurations in the phase space. For quantum many-body systems, statistical mechanics predicts the equilibration of highly excited non-equilibrium state towards a featureless thermal state. Hence, it is highly desirable to explore possible ways to avoid ergodicity in quantum systems. Many-body localization presents one generic mechanism for a strong violation of ergodicity relying on the presence of quenched disorder. In my talk I will discuss a different mechanism of the weak ergodicity breaking relevant for the experimentally realized Rydberg-atom quantum simulator [1]. This mechanism arises from the presence of special eigenstates in the many-body spectrum that are reminiscent of quantum scars in chaotic non-interacting systems [2], which were recently understood as emerging from the embedded SU(2) symmetric subspace [3]. In this talk I will concentrate on the variational description of unusual quantum many-body revivals that originate from the special eigenstates. I will relate this dynamics to presence of stable periodic trajectories within time-dependent variational principle (TDVP) description of dynamics. I will use TDVP to find new “scars” and explore their response to perturbations of the Hamiltonian. Finally, I will argue that the mixed phase space generally leads to non-universal dependence of thermalization on the initial state and discuss a new opportunities for the creation of novel states with long-lived coherence in systems that are now experimentally realizable [1].

[1] H. Bernien, et al., Nature 551, 579–584 (2017), arXiv:1707.04344
[2] C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, Z. Papić, Nature Physics (May 2018), arXiv:1711.03528 and Phys. Rev. B 98, 155134 (2018) arXiv:1806.10933
[3] S. Choi, et al., Phys. Rev. Lett. 122, 220603 (2019) arXiv:1812.05561
[4] A. A. Michailidis, C. J. Turner, Z. Papić, D. A. Abanin, M. Serbyn, arXiv:1905.08564

12:10 to 12:50 Arti Garg Many-body localization in the presence of long range interactions

In this talk I will discuss effects of long range interactions and long range hopping on a disordered fermionic system. Based on perturbative arguments there is a common belief that MBL can exist only in systems with short range interactions. We analyse effect of power-law interactions and power law hopping, on the system which has all single particle states localised as well as for the system that has single particle mobility edges in the absence of interactions. By mapping this model to an effective Anderson model on a complex graph in the Fock space, we calculated the probablity distribution for number of resonances upto third order. Though the most-probable value of the number of resonances diverge for the system with long-range hopping (t » t_0/r^{\alpha} with \alpha < 2), there is no enhancement of the number of resonances as the range of power-law interactions increases. Since delocalization is driven by proliferation of resonances in the Fock space, our analysis of resonance count indicates that the long-range hopping delocalizes the many-body localised system but in contrast to this there is no signature of delocalization in the presence of long-range interactions. We further provide support in favour of this anlysis based on quench dynamics,level spacing statistics, return probability and transport.

References:
1) S. Nag and A. Garg, Physical Review B 96 , 060203 (R) (2017)
2) S. Nag and A. Garg, Phys. Rev. B 99, 224203 (2019)
3) S. Mukherjee, S. Nag and A. Garg, Phys. Rev. B 97, 144202 (2018)
4) " Transport in many-body localized systems in the presence of long range interactions and long range hopping" (under preparation)

14:30 to 15:10 Augustine Kshetrimayum Time evolution of many-body localized systems in two spatial dimensions

Many-body localization is a striking mechanism that prevents interacting quantum systems from thermalizing. The absence of thermalization behaviour manifests itself, for example, in a remanence of local particle number configurations, a quantity that is robust over a parameter range -- unlike the situation in other instances of integrability. Local particle numbers are directly accessible in state-of-the-art quantum simulators, in systems of cold atoms even in two spatial dimensions. Yet, the classical simulation to benchmark such quantum simulators is highly challenging. In this work, we present a comprehensive tensor network simulation of a many-body localized systems in spatial dimensions using a variant of an iPEPS algorithm. The required translational invariance can be restored by implementing the disorder into an auxiliary spin system, providing an exact disorder average under dynamics. We observe signatures of many-body localization for the infinite system. Interestingly, in this setting of finitely many disorder values, localization emerges in the interacting regime, for which we provide an intuitive argument, while Anderson localization is absent. We benchmark our results against a simulation involving non-interacting fermions and find results compatible with an absence of localization.

15:10 to 15:50 Sumilan Banerjee Low-energy level spacing, residual entropy and thermalization across a non-Fermi liquid to Fermi liquid transition

In recent years, a zero-dimensional solvable model of interacting fermions, namely Sachdve-Ye-Kitaev (SYK) model, has emerged as a new paradigm to describe non-Fermi liquid metals as well as thermalization and many-body quantum chaos in interacting systems. I will discuss a quantum phase transition (QPT) between a Sachdev-Ye-Kitaev (SYK) non-Fermi liquid (NFL) and a Fermi liquid in a solvable large-N model where the zero-temperature residual entropy of the NFL vanishes continuously at the QPT in the large-N limit. We show via exact diagonalization that, even at finite-N, the QPT manifests itself in the system-size scaling of low-energy level spacings above the ground state. The evidence of this QPT is also directly visible in the single-particle Greens function and, indirectly, in the thermalization dynamics after a quench. Interestingly, we find that the QPT has little effect on the entanglement entropy of the ground state.

Thursday, 28 November 2019
Time Speaker Title Resources
09:30 to 11:00 Wojciech De Roeck Mechanisms of slow thermalization (Lecture 2)
11:30 to 12:10 Sergej Flach Dynamical Glass

Classical many body interacting systems are typically chaotic (nonzero Lyapunov exponents) and their micro-canonical dynamics ensures that time averages and phase space averages are identical (ergodic hypothesis). In proximity to an integrable limit the long- or short-range properties of the network of nonintegrable action space perturbations define the finite time relaxation properties of the system towards Gibbs equilibrium. Long range networks are characterized by a two-step ergodization whose details are controlled by the network size and the largest Lyapunov exponent. Short range networks show a novel diverging time scale related to a diffusion process which slows down upon approaching the integrable limit.

• Carlo Danieli, David K. Campbell and Sergej Flach, Phys. Rev. E 95, 060202(R) (2017)
• Mithun Thudiyangal, Yagmur Kati, Carlo Danieli and Sergej Flach, Phys. Rev. Lett. 120, 184101 (2018)
• Alexander Yu. Cherny, Thomas Engl and Sergej Flach, Phys. Rev. A 99 023603 (2019)
• Mithun Thudiyangal, Carlo Danieli, Yagmur Kati and Sergej Flach, Phys. Rev. Lett. 122 054102 (2019)
• Carlo Danieli, Thudiyangal Mithun, Yagmur Kati, David K. Campbell and Sergej Flach, Phys. Rev. E 100032217 (2019)

12:10 to 12:50 Wojciech De Roeck TBA
Friday, 29 November 2019
Time Speaker Title Resources
09:40 to 10:20 Bruno Bertini Non-equilibrium dynamics in dual-unitary quantum circuits

I will consider the non-equilibrium dynamics of a recently introduced class of "statistically solvable” many-body quantum systems: the dual-unitary circuits. These systems furnish a minimal modelling of generic locally interacting many-body quantum systems: they are generically non-integrable and include a quantum chaotic subclass. I will first discuss the dynamics of initial-state dependent quantities, such as the entanglement entropies and local correlators, showing that it is possible to find (and classify) a family of initial states in MPS form that allow for an exact solution of the dynamics (also in the presence of quantum chaos). Then I will discuss the dynamics of an initial-state independent quantity, the so called “operator entanglement”, that measures the growth of entanglement that the Heisenberg evolution induces in operator space. I will show that one can identify different subclasses of dual-unitary circuits. In particular I will describe a “maximally chaotic” subclass, where the entanglement of local operators grows linearly, and a “dynamically constrained” one, where the entanglement of local operators is bounded.

10:20 to 11:00 Lea F. Santos Time Scales and Manifestations of Chaos in Many-Body Quantum Dynamics

A major open question in studies of nonequilibrium quantum dynamics is the identification of the time scales involved in the relaxation process of isolated many-body systems. While there is consensus on what equilibration and thermalization mean in these systems, there is no agreement on how long they take to reach equilibrium. To answer this question, we look for dynamical manifestations of spectral correlations in different observables and use them to discuss a generalization of the Thouless time to interacting systems and to show that the relaxation time grows with system size. Our studies also include an analysis of the self-averaging properties of systems out of equilibrium. We show numerically and analytically that self-averaging properties depend not only on the presence or absence of chaos, but also on the quantity and the time scale considered.

11:30 to 12:10 Romain Vasseur Anomalous low-frequency conductivity in quantum spin chains

In this talk, I will explain how anomalous transport can emerge in one-dimensional quantum systems at finite temperature, due to hierarchies of quasiparticle excitations. I will describe how to attack this problem using a combination of analytical (generalized hydrodynamics) and numerical (matrix product operators) techniques.

12:10 to 12:50 Abhishek Dhar Signatures of MBL in a classical disordered nonlinear chain of oscillators

We explore signatures of many-body-localization (MBL) in classical systems through a study of thermal conductivity in a disordered nonlinear chain of oscillators. Through extensive numerical simulations of this system connected to heat baths at different temperatures, we computed the nonequilibrium steady state heat current and the temperature prole. We show evidence that, for any  T> 0, the thermal conductivity goes to zero faster than any power of T, in the T->0 limit. in fact the temperature dependence is well described by the Arrhenius  form \kappa ~ exp(-A D/T), where D is the disorder strength. A heuristic explanation based on the picture of chaotic islands, formed by two-oscillator resonances. is presented.


text:
## Moment of Inertia of a Circle

This tool calculates the moment of inertia I (second moment of area) of a circle. Enter the radius 'R' or the diameter 'D' below. The calculated result will have the same units as your input. Please use consistent units for any input.

 R = or... D = Results:

## Background

### Moment of inertia of circle

The moment of inertia of circle with respect to any axis passing through its centre, is given by the following expression:

$I = \frac{\pi R^4}{4}$

where R is the radius of the circle.

Expressed in terms of the circle diameter D, the above equation is equivalent to:

$I = \frac{\pi D^4}{64}$

### Units

The above equations for the moment of inertia of circle, reveal that the latter is analogous to the fourth power of circle radius or diameter. Since those are lengths, one can expect that the units of moment of inertia should be of the type:  $$[Length]^4$$. In fact, this is true for the moment of inertia of any shape, not just the circle. By definition, the moment of inertia is the second moment of area, in other words the integral sum of cross-sectional area times the square distance from the axis of rotation, hence its dimensions are  $$[Length]^4$$.

Typical units for the moment of inertia, in the imperial system of measurements are:

• in4 ,
• ft4

Typical units for the moment of inertia, in metric  are:

• mm4 ,
• cm4 ,
• m4

### Definition

The second moment of area of any planar, closed shape is given by the following integral:

$I=\iint_A y^2 dA$

where A is the area of the shape and y the distance of any point inside area A from a given axis of rotation.

From this definition it becomes clear that the moment of inertia is not a property of the shape alone but is always related to an axis of rotation. Often though, one may use the term "moment of inertia of circle", missing to specify an axis. Depending on the context, an axis passing through the center may be implied, however, for more complex shapes it is not guaranteed that the implied axis would be obvious.

From the definition also,  it is also apparent that the moment of inertia should always have a positive value, since there is only a squared term inside the integral.

### Finding the equation for the moment of inertia of a circle

Using the above definition, which applies for any closed shape, we will try to reach to the final equation for the moment of inertia of circle, around an axis x passing through its center. First we must define the coordinate system. Since we have a circular area, the Cartesian x,y system is not the best option. Instead we choose a polar system, with its pole O coinciding with circle center, and its polar axis L coinciding with the axis of rotation x , as depicted in the figure below. The independent variables are r and φ. Specifically, for any point of the plane, r is the distance from pole and φ is the angle from the polar axis L, measured in counter-clockwise direction.

With this coordinate system, the differential area dA now becomes: $$dA=dr\: ds = dr \:(rd\varphi)=r\:dr \:d\varphi$$, where ds is the differential arc length for differential angle dφ.

Furthermore, the area, enclosed by the circle, should have these boundaries:

• $$r\in [0,R]$$
• $$\varphi \in [0,2\pi]$$

Moreover, the coordinate y of any point, can be expressed in terms of the polar coordinates r and φ. To do so, we consider for the arbitrary point P (see figure) the blue colored right triangle and using simple trigonometry we find: $$y=r \sin\varphi$$

Therefore, the definite integral for the moment of inertia of the circle should be written as:

$I_x=\int^R_0\int^{2\pi}_0 r^2 \sin^2\varphi \:r\:dr d\varphi$

First, we integrate over variable φ.

$I_x=\int^R_0 \left(\int^{2\pi}_0 r^3 \sin^2\varphi\:d\varphi\right)dr\Rightarrow$

$I_x=\int^R_0 \left(r^3 \int^{2\pi}_0 \sin^2\varphi \:d\varphi\right)dr$

We take advantage of the trigonometric identity: $$\sin^2\varphi = \frac{1-\cos2\varphi}{2}$$ and we calculate the integral inside the parentheses like this:

$\int^{2\pi}_0 \sin^2\varphi \:d\varphi=\int^{2\pi}_0 \frac{1-\cos2\varphi}{2} d\varphi=$

$\left[\frac{1}{2} \varphi\right]_0^{2\pi} - \left[\frac{1}{4}\sin2 \varphi\right]_0^{2\pi} =$

$\left(\frac{2\pi}{2} -0\right) -\left(\frac{1}{4}\sin4 \pi -\frac{1}{4}\sin0\right)=$

$\pi -\left(0 -0\right)=\pi$

Substituting to the expression of Ix , we now have to integrate over variable r:

$I_x=\int^R_0 \left(r^3 \pi\right)dr\Rightarrow$

$I_x=\pi\int^R_0 \left(r^4 \over 4\right)'dr=\pi\left[ r^4 \over 4\right]^R_0\Rightarrow$

$I_x=\pi\left(\frac{R^4}{4} -0\right)\Rightarrow$

$I_x=\frac{\pi R^4}{4}$

### Parallel Axes Theorem

The moment of inertia of any shape, around an arbitrary, non centroidal axis, can be found if its moment of inertia around a centroidal axis, parallel to the first one, is known. The so-calledParallel Axes Theorem is given by the following equation:

$I' = I + A d^2$

where I' is the moment of inertia in respect to an arbitrary axis, I the moment of inertia in respect to a centroidal axis, parallel to the first one, d the distance between the two parallel axes and A the area of the shape.

For example, if we need the moment of inertia of a circle around an axis, that is tangent to the circumference, we make these considerations:

• d=R
• $$A=\pi R^2$$
• $$I = \frac{\pi R^4}{4}$$

Applying theParallel Axes Theorem we find:

$I'=\frac{\pi R^4}{4}+\pi R^2 R^2=\frac{5\pi R^4}{4}$

It turns out that the new moment of inertia is drastically increased compared to the centroidal one. This is a more general characteristic. Since the distance from centroid is squared, it affects the moment of inertia much more than the area A.

### Mass moment of inertia

In Physics the term moment of inertia has a different meaning. It is related with the mass distribution of an object (or multiple objects) about an axis. This is different from the definition usually given in Engineering disciplines (also in this page) as a property of the area of a shape, commonly a cross-section, about the axis. The termsecond moment of area seems more accurate in this regard.

### Applications

The moment of inertia (second moment or area) is used in beam theory to describe the rigidity of a beam against flexure (see beam bending theory). The bending moment M applied to a cross-section is related with its moment of inertia with the following equation:

$M = E\times I \times \kappa$

where E is the Young's modulus, a property of the material, and κ the curvature of the beam due to the applied load. Beam curvature κ describes the extent of flexure in the beam and can be expressed in terms of beam deflection w(x) along longitudinal beam axis x, as: $$\kappa = \frac{d^2 w(x)}{dx^2}$$. Therefore, it can be seen from the former equation, that when a certain bending moment M is applied to a beam cross-section, the developed curvature is reversely proportional to the moment of inertia I. Integrating curvatures over beam length, the deflection, at some point along x-axis, should also be reversely proportional to I.

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Under the auspices of the Computational Complexity Foundation (CCF)

REPORTS > KEYWORD > ALGEBRAIC GEOMETRY CODES:
Reports tagged with algebraic geometry codes:
TR12-148 | 7th November 2012
Eli Ben-Sasson, Ariel Gabizon, Yohay Kaplan, Swastik Kopparty, Shubhangi Saraf

#### A new family of locally correctable codes based on degree-lifted algebraic geometry codes

Revisions: 1

We describe new constructions of error correcting codes, obtained by "degree-lifting" a short algebraic geometry (AG) base-code of block-length $q$ to a lifted-code of block-length $q^m$, for arbitrary integer $m$. The construction generalizes the way degree-$d$, univariate polynomials evaluated over the $q$-element field (also known as Reed-Solomon codes) are "lifted" ... more >>>

TR13-085 | 13th June 2013
Eli Ben-Sasson, Yohay Kaplan, Swastik Kopparty, Or Meir, Henning Stichtenoth

#### Constant rate PCPs for circuit-SAT with sublinear query complexity

The PCP theorem (Arora et. al., J. ACM 45(1,3)) says that every NP-proof can be encoded to another proof, namely, a probabilistically checkable proof (PCP), which can be tested by a verifier that queries only a small part of the PCP. A natural question is how large is the blow-up ... more >>>

TR13-155 | 10th November 2013
Gil Cohen, Amnon Ta-Shma

#### Pseudorandom Generators for Low Degree Polynomials from Algebraic Geometry Codes

Revisions: 2

Constructing pseudorandom generators for low degree polynomials has received a considerable attention in the past decade. Viola [CC 2009], following an exciting line of research, constructed a pseudorandom generator for degree d polynomials in n variables, over any prime field. The seed length used is $O(d \log{n} + d 2^d)$, ... more >>>

TR20-165 | 6th November 2020
Sarah Bordage, Jade Nardi

#### Interactive Oracle Proofs of Proximity to Algebraic Geometry Codes

Revisions: 3

In this work, we initiate the study of proximity testing to Algebraic Geometry (AG) codes. An AG code $C = C(\mathcal C, \mathcal P, D)$ is a vector space associated to evaluations on $\mathcal P$ of functions in the Riemann-Roch space $L_\mathcal C(D)$. The problem of testing proximity to an ... more >>>

ISSN 1433-8092 | Imprint


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# Harvard Astronomy 201b

In Uncategorized on April 26, 2013 at 9:18 pm

How were the observations done?

If you are a theorist, you probably don’t like words like “bandpass”, and hate words like “calibration.”  If you are an observer, you probably know and love these words.  But even theorists have to understand how observations are done: they’re what keep science science.  Here I want to briefly explain some aspects of the paper’s discussion of how the data were taken that were on first read opaque.

First of all, given that the $NH_3$ rest frequency is roughly 24 GHz, we can find the energy of the transition: $E=h\nu = 10^{-4}\;\rm{eV}$.  This corresponds to an excitation temperature of 1.2 K, which we can also translate into a velocity via $v\sim (E/m)^{1/2} \sim 24\; {\rm m/s}$.  A typical molecular cloud temperature is on the order of 20 K, high enough that there should be many molecules in the upper energy state (think of it as a toy-model 2 level system) and hence many upper to lower transitions occurring (leading to emission at 24 GHz).

A 1 MHz window means the observations were done between 24 GHz – 500 MHz and 24 GHz + 500 MHz, and dividing this interval by 256 gives the 3.9 kHz per channel quoted.   This can be translated into a velocity by thinking of it using the small-z redshift formula $z\approx v/c$. Here, we have a fractional frequency shift $\Delta \nu / \nu \simeq v/c$, and setting $\Delta \nu / \nu$ =  3.9  kHZ/24 GHz leads to the velocity quoted (.049 km/s).

The baselines given (35 meters up to 1 km) determine the spatial resolution: just good old Carroll and Ostlie diffraction limit, $\theta = 1.22 \lambda/D$, where $\theta$ is the angular resolution, $\lambda$ the wavelength of light, and $D$ the diameter of the aperture (here, the baseline).  24 GHz has $\lambda = 1.2$ cm, so with a 1 km baseline the angular resolution is about .05’’ (for comparison, this is 1.3 times the angular resolution of the Hubble Space Telescope at 500 nm).

Nepers is a unit of optical depth; .0689 is optically thin (and measures the optical depth due to the Earth’s atmosphere, since the authors note this corresponds to fair weather; there is negligible optical depth from sources in space at this frequency).  It is interesting to note that the optical depth is given at 22 GHz, but the observations are made at 24 GHz; this is because 22 GHz is a standard fiducial frequency to allow comparison with the weather conditions under which other radio observations might have been made.

Now, calibration—theorists’ bane.  The paper’s discussion is compact and therefore somewhat confusing to those uninitiated into the secrets of radio astronomy, but understanding it is worthwhile for the insight it can yield into what actually goes on at these remote “large arrays” in the desert (alien storage?)

Possible very large array user? Image credit: http://www.123rf.com

In a radio interferometric observation, what you actually measure is the amplitude and phase of the signal in each channel (here, there are 256): i.e., at each different frequency corresponding to the channels in the window about 24 GHz.  (Note: interferometric means that phases are being compared between signal received at spatially separated points on Earth: this allows the difference in path length from the source to two antennae to be computed, and hence the direction to the source). The amplitude and phase at these different frequencies is desirable because it gives (eventually) intensity as a function of frequency, or equivalently a spectral energy distribution (SED).  This in turn allows inferences about the properties of the source (e.g. dust grain size from SEDs of debris disks, cf. problem set 2, problem 4 ($\beta$ describes the shape of the SED)).There will be some intrinsic noise in both amplitude and phase, which you want to eliminate.  For a source that has a flat spectrum, you know any bumps you see at a particular frequency are due to noise in the channel at that frequency.  This is the purpose of having an amplitude calibrator: typically quasars are used because they have flat spectra.

Now, the amplitude is measured by the amount of electrical current coming from a given channel: the more photons at that frequency hit the antenna, the more current.  But astronomers care about flux, not current.  The absolute flux calibrator is a source with a known flux. Measuring the voltage this source produces allows the conversion factor between voltage and flux to be deduced.  Using a source with known flux also allows one to calibrate the bandpass: the flux should be zero outside the window, but to know this is because the window is working, you have to be sure the flux is not just coincidentally zero at frequencies outside the window anyway.

Finally, the astute reader may notice that noise is reported as mJy/beam.  The noise should be linear in the area of the beam, so a larger beam would have more noise.  Thus two sets of observations done with different beam sizes would have different noise levels due to this; reporting noise/beam allows direct comparison of how good observations done with different beam sizes are.


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Multi-objective recursive least squares

1. May 2, 2010

Superfish

Is this possible?

I've computed a multiobjective least squares solution and want to make it able to be updated recursively but I get stuck at applying the woodbury matrix identity since it's no longer a rank 1 udpate. Are there any derivations of this anywhere or is this not possible? Thanks

2. May 5, 2010

jasonRF

Of course, if you have a rank 2 update, you can always use Woodbury's twice in a row, etc.

A generalization of Woodbury's identity is
$$(A+BCD)^{-1}=A^{-1} - A^{-1} B(D A^{-1} B + C^{-1} )^{-1} D A^{-1}$$
Where A is nxn, B is nxm, C is mxm, and D is mxn, so this can work for a rank m update. It isn't too suprising that you have to invert a rank m matrix to do it.

jason

EDIT: just realized that Woodbury's was never stated. In the above, if B is nx1 vector, D is the transpose of B and C is just the number 1, then you have Woodbury's identity. The generalization above is usually called the matrix inversion lemma

Last edited: May 5, 2010


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# Measure_test configuration

hi I use notepad++ as text editor for measure writiting. Nornally, i use OS Apply measure or Measure Tab, however, there are some limited with that approach : need refresh to ensure measure data, so many steps to get error codes in stderr file. I got instructions from Measure Writing Guide and link.However, there are still some issues i did not solve, I would like to know about :

1. Use Notepad++ as Ruby IDE and directly test and trace error like RubyMine in that link - How i use following ruby argument in notepad++ with nppExec like Rubymine ? -e $stdout.sync=true;$stderr.sync=true;load(\$0=ARGV.shift) -I 'C:\Program Files (x86)\OpenStudio 1.5.0\Ruby'

2. Measure test file structure :

a. seed OSM/Model : i need to apply many measures at same time to get result and trace error for tested measure - i often use OS measure tab to apply mauilple measure - how to do like this with notepad++ or command line ? should i need to apply measure to seed OSM then save it before run measure_test.rb?

b. Argument inputs for parallel applying measure: How to assign an hash with argument name and argument value to measure applied in seed OSM like tested measure in measure.rb

c. Which syntax should use to get trace on command line screen (notepad++) when running measure. i often use Puts and Runner.RegisterInfo. Which one i need to use or need to use another methods ?

d. if i need to include an .epw and other input file csv etc... or idf.file in seed model so which part (def) of measure_test.rb where i need make syntax for insert file ? Where to get OS API about file import/export ?

class NewMeasureTest < MiniTest::Unit::TestCase
def setup
#add setup for use last run file or rerun again  - reporting measure
end
def test_agruments
#test arguments here (right number, error with bad input, etc...)
end
def test run measure
# get result and assert it
end
end #class

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# Gearbox block #

#### Transmission Type #

Manual

Each gear is either engaged or disengaged. Smooth gear shifting requires engaging the clutch or configuring a Torque Converter.

The Auto shift option in the Gearbox configuration provides automatic gear shifting.

Automatic

Gears are engaged via clutches inside the gearbox, so shifting gears provide a smooth transition between them. Automatic transmissions are typically used in combination with a Torque Converter as clutch.

The [D] mode provides auto-shifting, while the [M] mode requires gears to be engaged manually.

#### Gear Ratios #

Forward Gear Ratios
Forward gears (positive ratios). 0 is the first gear, 1 the second gear, and so on. Ratios should be configured in descending order for the automatic shifting features to work.
Reverse Gear Ratios
Reverse gears (negative ratios). 0 is the first reverse gear, 1 the second reverse gear, and so on. If Auto-shift in Reverse is enabled then ratios should be specified in ascending order (i.e. -3, -2, -1).

#### Manual Transmission #

Each gear is either engaged or disengaged. Smooth gear shifting requires engaging the clutch or configuring a Torque Converter.

Gear Change Time
Time in seconds that takes shifting to current gear to another. Gearbox stays in Neutral during this time.
Auto Shift

Enables automatic gear shifting

Neutral Rpm
Engages Neutral when rpm falls below this value
First Gear Rpm
First Gear is engaged from Neutral above this rpm
Gear Down Rpm
Shifts gear down below this rpm
Gear Up Rpm
Shifts gear up above this rpm
Gear Up Interval
Minimum time in seconds between two consecutive gear up shifts.
Gear Down Interval
Minimum time in seconds between two consecutive gear down shifts.
2nd Gear Min Speed
Minimum speed of the vehicle in m/s to allow automatic gear shifting above the 2nd gear. Useful to prevent gear shifting when the drive wheels are heavily slipping in 1st gear.
Allow Park Mode

Park mode [P] is typical of automatic transmissions. This option enables park mode in the manual transmission.

Strict Park Mode
If enabled the park mode [P] can be engaged or disengaged only when the brake is strongly pressed.

#### Automatic Transmission #

Gears are engaged via clutches inside the gearbox, so shifting gears provide a smooth transition between them. Automatic transmissions are typically used in combination with a Torque Converter as clutch.

Gear Transition Time
Duration in seconds of the smooth transition from a gear to another.
Shift Interval
Minimum time in seconds between two consecutive gear changes.
Gear Down Rpm
A shorter gear is selected below this rpm.
Gear Up Rpm
A longer gear is selected above this rpm.
Initial Gear Forward
The forward gear to engage when selecting the [D] mode. Typically is 1, but some vehicles with many gears may engage the 2nd, 3rd or even 4rd. In these cases the shorter gears can be selected in [L] mode and with the shift-up / shift-down inputs.
Initial Gear Reverse
The reverse gear to engage when selecting the [R] mode. Typically is -1, but some vehicles with several reverse gears may engage any other initially. Other reverse gears may be selected with the shift-up / shift-down inputs.
Next Gear Min Speed
Minimum speed of the vehicle in m/s to allow automatic engagement of gears based on current speed. Note that above this min speed the system may select a gear shorter than the initial ones.
Auto-Shift In Reverse
Applies auto-shifting in reverse mode [R]. Reverse gear ratios should be in ascending order for this feature to work properly (i.e. -3, -2, -1).
Strict Park Mode
If enabled the park mode [P] can be engaged or disengaged only when the brake is strongly pressed.

### Data Bus parameters #

Data Bus Value Description Notes
InputData.ManualGear Manual gear lever position -1 (reverse), 0 (neutral), 1, 2, 3, ...
InputData.AutomaticGear Automatic transmission mode 1 0, 1, 2, 3, 4, 5 = M, P, R, N, D, L
InputData.GearShift Incremental gear shifting value 2 Add +1 for gear up or -1 for gear down
VehicleData.GearboxGear Currently engaged gear Negative = reverse, 0 = Neutral or Park, Positive = forward.
VehicleData.GearboxMode Actual transmission mode 0, 1, 2, 3, 4, 5 = M, P, R, N, D, L
VehicleData.GearboxShifting Is the gearbox in the middle of a gear shift operation? 0 = no, 1 = yes
SettingsData.AutoShiftOverride Auto-shift override setting 3 0 = no override, 1 = force auto shift, 2 = force manual shift

See Data Bus Reference for more information.

1 Automatic transmission modes:
• M (0): Manual: do not automatically engage gears. Use manual gear shifting.
• P (1): Park
• R (2): Reverse. Gear shifting is supported for more than one reverse gears.
• N (3): Neutral
• D (4): Drive: automatically engage forward gears. Gear shifting is supported for forward gears.
• L (5): Low: first gear only.

2 GearShift value is reset to 0 when the vehicle has acknowledged and processed the input. Successive gear shift commands can be grouped by adding/subtracting +-1 to this bus value.

3 AutoShiftOverride: See notes below.

#### The AutoShiftOverride value #

The automatic shifting feature in both Manual and Automatic transmissions may be overridden from scripting via Data Bus by modifying the AutoShiftOverride value in the Settings channel.

The effect of the AutoShiftOverride value depends on the vehicle's transmission type:

• Automatic transmission: the override value switches among modes Manual [M] and Drive [D], but does nothing if any other mode is selected (i.e. Neutral [N] or Reverse [R]). It's also subject to the conditions of each mode. For example, [D] can't be selected if the vehicle is moving backwards faster than a threshold speed.
• Manual transmission: the override value enables or disables the auto-shift feature.

AutoShiftOverride is intended to temporarily override auto-shifting in specific situations. For example, it's used for preventing gear shifting during replays, as the replay data already includes the currently engaged gear.

AutoShiftOverride is not a reliable way of implementing an auto-shift / manual-shift selector in the vehicle because the value might be overwritten by other components. Read below.

### Auto-shift / manual-shift selection #

Determining the transmission type

If you want to configure the shift mode externally to the vehicle (i.e. via UI or hotkey) then you should probably use a Manual transmission and modify the auto-shift flag. The Automatic transmission is intended to allow users to select among Drive [D] (automatic) or Manual [M] themselves using the vehicle input.

The behavior of the automated gear shifting is different between Automatic and Manual transmissions:

• Automatic Transmission: auto-shift is enabled in Drive [D] mode. Gears must be shifted manually in the Manual [M] mode.
• Manual Transmission: auto-shift is enabled with the Auto Shift flag in the Gearbox settings.

Example: Add-on component that modifies the Gearbox settings in runtime in any transmission type.

using UnityEngine;
using VehiclePhysics;

public class GearShiftModeSelector : VehicleBehaviour
{
public enum GearShiftMode { Manual, Automatic }
public GearShiftMode mode = GearShiftMode.Manual;

Gearbox.Settings m_gearboxSettings;

public override void OnEnableVehicle ()
{
// Retrieve a reference to the gearbox settings
var gearbox = vehicle.GetInternalObject(typeof(Gearbox)) as Gearbox;
if (gearbox == null)
{
// This vehicle doesn't have a gearbox. Self-disable this add-on.
enabled = false;
}

m_gearboxSettings = gearbox.settings;
}

public override void FixedUpdateVehicle ()
{
if (m_gearboxSettings.type == Gearbox.Type.Automatic)
{
// Automatic Transmission. Select the proper mode.
if (mode == GearShiftMode.Automatic)
{
// Force select mode D (4) unless current mode is R (2)
if (vehicle.data.Get(Channel.Input, InputData.AutomaticGear) != 2)
vehicle.data.Set(Channel.Input, InputData.AutomaticGear, 4);
}
else
{
// Force select mode M (0)
vehicle.data.Set(Channel.Input, InputData.AutomaticGear, 0);
}
}
else
{
// Manual gearbox. Just match the auto-shift mode.
m_gearboxSettings.autoShift = mode == GearShiftMode.Automatic;
}
}
}