Monday, 16 July 2018
We study the inverse problem of determining vector and scalar potentials appearing in the relativistic Schrödinger equation in space dimension 3 or higher from information about the solution on a suitable subset of the boundary. We prove unique determination of these potentials modulo a natural gauge invariance. This is joint work with Manmohan Vashisth.
Tuesday, 17 July 2018
We shall talk about the Witten conjecture which roughly says that the generating function of certain intersection numbers on the moduli space of curves satisfies the KdV equation. We shall also discuss a proof of the celebrated conjecture using the ESLV formula which relates them to Hurwitz numbers. Surprisingly the generating function for Hurwitz numbers satisfies the KP hierarchy.
Friday, 20 July 2018
After introducing controllability (reaching a desired state in finite time) and stabilizability (reaching a steady state as time tends to infinity), for ODE systems, I will discuss these issues for some PDE systems, arising in fluid models.
Laplacian growth is a problem of two-dimensional pattern formation, that has drawn a lot of attention in theoretical and experimental physics due to the spontaneous formation of fractal patterns in a non-equilibri um system. Although the problem has been shown to be integrable, albeit it in a somewhat unexpected and unconventional way, an explanation of the fractal properties of the growth has eluded theoretical study. Attempting to be methodical, I will first review the problem and its relation to integrable systems, then I will undoubtedly add to the confusion by suggesting tentative and speculative directions of research.
Tuesday, 24 July 2018
This will be an elementary introduction to Fatou-Bieberbach domains and Short C^k's.
Wednesday, 25 July 2018
Fractional derivatives are as old integer derivatives and yet they have no unique definition. In this talk, I'll introduce a new perspective on fractional derivatives highlighting their connection to boundary-value problems for partial differential equations. This perspective readily affords a general framework to analyse fractional differential equations. The motivating physical problem is that of a small heavy particle in a viscous fluid. The upshot of our analysis is a much simpler proof for existence of solutions to such equations as well as a new numerical method to compute solutions in an efficient and accurate manner.
I will describe three models of physical interest in which Integrability and Supersymmetry are entwined. We will start with two elementary examples and end with a model which is still a topic of current research.
We present results on a combinatorial problem which was solved recently using techniques from integrable systems. Specifically, we introduce a new class of square-ice configurations (also known as six-vertex model configurations) on triangular subsets of the square lattice, which we call Alternating Sign Triangles (ASTs). The proof of the enumeration of ASTs uses the integrability of the six-vertex (or square-ice) model. We will explain the origin of this problem and the ideas involved in the proof. Time permitting, we will also give product formulas for other classes of square-ice configurations using similar ideas. This is joint work with Ilse Fischer and Roger Behrend.
Thursday, 26 July 2018
The six Painleve equations are second-order ordinary differential equations whose solutions serve as nonlinear special functions for many problems in mathematical physics. We will give an overview of various applications of Painleve functions to integrable probability and nonlinear wave equations, and then discuss their integrable structure and asymptotic analysis.
The sine-Gordon equation is a semi-linear wave equation used to model many physical phenomenon like seismic events that includes earthquakes, slow slip and after-slip processes, dislocation in solids etc. Solution of homogeneous sine- Gordon equation exhibit soliton like structure that propagates without change in its shape and structure. The question whether solution of sine- Gordon equation still exhibit soliton like behaviour under an external forcing has been challenging as it is extremely difficult to obtain an exact solution even under simple forcing like constant. In this study solution to an inhomogeneous sine-Gordon equation with Heaviside forcing function is analysed. Various one- dimensional test cases like kink and breather with no flux and non reflecting boundary conditions are studied.
Friday, 27 July 2018
Let S be a closed oriented surface of genus at least two. The Teichmüller space T(S) is the universal cover of the moduli space of hyperbolic structures on S. For any choice of a complex structure on S, a theorem of M. Wolf, and independently N. Hitchin, identifies T(S) with the vector space of holomorphic quadratic differentials on the resulting Riemann surface X, that we denote by Q(X). An earlier result of Hubbard and Masur identifies Q(X) with the space of certain topological objects called measured foliations on S. I shall discuss these results, the relation between them, and their generalizations to the case of meromorphic quadratic differentials. All these spaces have some natural symplectic structures, and I shall mention some open questions concerning them.
We shall first talk about Kostant-Souriau's method of geometric quantization of coadjoint orbits. Adler had showed that the Toda system can be given a coadjoint orbit description. We shall talk about quantization of the Toda system by viewing it as a single orbit of a multiplicative group of lower triangular matrices of determinant one with positive diagonal entries. We get a unitary representation of the group with square integrable polarized sections of the quantization as the module . We find the Rawnsley coherent states after completion of the above space of sections. Finally we give an expression for the quantum Hamiltonian for the system.This is joint work with Dr. Saibal Ganguli.
Monday, 30 July 2018
Bukhvostov and Lipatov showed that weakly interacting instantons and anti-instantons in the O(3) NLSM in two dimensions are described by an exactly soluble model containing two coupled Dirac fermions. The model can be reformulated as a bosonic QFT with two interacting bosons, which upon an analytic continuation into a strong coupling regime, become equivalent to the originating O(3) NLSM. This observation provides a remarkable opportunity for the exact instanton counting. As an illustration, I will discuss the calculation of the vacuum energy of the O(3) NLSM with twisted boundary conditions.
These talks review various approaches to non-equilibrium processes in integrable models. Examples include the increase in understanding since 2011 of Drude weight and semiclassical kinetic theory in integrable models, and obtaining exact far-from-equilibrium results for some quantities in the XXZ model through expansion potentials. In many cases the predictions of theoretical approaches based on integrability can be confirmed due to progress in DMRG and other matrix product state algorithms, and the essentials of such algorithms will be reviewed.
Tuesday, 31 July 2018
These talks review various approaches to non-equilibrium processes in integrable models. Examples include the increase in understanding since 2011 of Drude weight and semiclassical kinetic theory in integrable models, and obtaining exact far-from-equilibrium results for some quantities in the XXZ model through expansion potentials. In many cases the predictions of theoretical approaches based on integrability can be confirmed due to progress in DMRG and other matrix product state algorithms, and the essentials of such algorithms will be reviewed.
The standard method for implementing algorithms for quantum computation is through quantum circuits. Such circuits typically contain quantum gates involving more than a single or two qubits. Multi-qubit gates can be decomposed into 1- and 2-qubit gates, but this is not necessarily the most efficient strategy. We present a framework for quantum control directly at the level of multiple qubits. A key ingredient is what we call an eigengate: a simple quantum circuit that maps computational basis states to eigenstates of a many-body hamiltonian. We show how to make it all work for a Krawtchouk qubit chain, and for operators associated to a spin chain with inverse square exchange, first introduced by Polychronakos.
Based on arXiv:1707.05144, Phys Rev A97.04232, with Koen Groenland
Wednesday, 01 August 2018
We address the Yang-Lee formalism for Integrable Quantum Field Theories, discussing various features of this approach to statistical physics and focusing the attention on the zeros of the grand canonical partition in the fugacity variable of the soliton sector of the Sine-Gordon model and of the simplest integrable quantum field theory, the so called Yang-Lee model.
These talks review various approaches to non-equilibrium processes in integrable models. Examples include the increase in understanding since 2011 of Drude weight and semiclassical kinetic theory in integrable models, and obtaining exact far-from-equilibrium results for some quantities in the XXZ model through expansion potentials. In many cases the predictions of theoretical approaches based on integrability can be confirmed due to progress in DMRG and other matrix product state algorithms, and the essentials of such algorithms will be reviewed.
Thursday, 02 August 2018
A review of the Calogero integrable model and its various generalizations will be given, with emphasis on physical properties and recent results. Topics will include integrability, the connection of these systems with fractional statistics, their matrix model and operator formulations, their various reductions and extensions, and their hydrodynamic description, properties and solitons.
A particle in a one-dimensional channel with excluded volume interaction displays anomalous diffusion with fluctuations scaling at t1/4, in the long-time limit. This phenomenon, seen in various experimental situations, is called single-file diffusion. In this talk, we shall present the exact formula for the distribution of a tracer and its large deviations in the one-dimensional symmetric simple exclusion process, a pristine model for single-file diffusion, thus answering a problem that has eluded solution for decades. We use the mathematical arsenal of integrable probabilities developed recently to solve the one-dimensional Kardar-Parisi-Zhang equation. Our results can be extended to situations where the system is far from equilibrium, leading to a Gallavotti-Cohen Fluctuation Relation and providing us with a highly nontrivial check of the Macroscopic Fluctuation Theory.
Joint work with Takashi Imamura (Chiba) and Tomohiro Sasamoto (Tokyo).
Lattice models for itinerant (spin-less) fermions can be tuned to display supersymmetry (susy): a fermionic symmetry that squares to the hamiltonian time evolution. Critical and massive phases of the susy Mk models on 1D lattices are described by minimal models of superconformal field theory or by (integrable) massive QFT. Many susy lattice models, including Nicolai and susy SYK models in 1D and the susy M1 model on 2D lattices) feature extensive degeneracies of supersymmetric ground states. We discuss how such ground states can be counted and explore the implications of their existence. We also present topological pumping protocols of 2-particle bound states that are protected by supersymmetry. Includes recent results obtained with Sergey Shadrin, Ruben La, and Bart van Voorden
We study non-equilibrium dynamics of integrable and non-integrable closed quantum systems whose unitary evolution is interrupted with stochastic resets, characterized by a reset rate $r$, that project the system to its initial state. We show that the steady state density matrix of a non-integrable system, averaged over the reset distribution, retains its off-diagonal elements for any finite $r$. Consequently a generic observable $\hat O$, whose expectation value receives contribution from these off-diagonal elements, never thermalizes under such dynamics for any finite $r$. We demonstrate this phenomenon by exact numerical studies of experimentally realizable models of ultracold bosonic atoms in a tilted optical lattice. For integrable Dirac-like fermionic models driven periodically between such resets, the reset-averaged steady state is found to be described by a family of generalized Gibbs ensembles (GGE s) characterized by $r$. We also study the spread of particle density of a non-interacting one-dimensional fermionic chain, starting from an initial state where all fermions occupy the left half of the sample, while the right half is empty. When driven by resetting dynamics, the density profile approaches at long times to a nonequilibrium stationary profile that we compute exactly. We suggest concrete experiments that can possibly test our theory.
Friday, 03 August 2018
A review of the Calogero integrable model and its various generalizations will be given, with emphasis on physical properties and recent results. Topics will include integrability, the connection of these systems with fractional statistics, their matrix model and operator formulations, their various reductions and extensions, and their hydrodynamic description, properties and solitons.
I will outline a way to make the parameters (e.g., the interaction strength) of certain quantum integrable models time-dependent without breaking their integrability. Interesting many-body models that emerge from this approach include a superconductor with the interaction strength inversely proportional to time, a Floquet BCS superconductor, and the problem of molecular production in an atomic Fermi gas swept through a Feshbach resonance as well as various models of multi-level Landau-Zener tunneling.
Amazingly, the non-stationary Schrodinger equation for all these models has a similar structure and is integrable with a similar technique as the famous Knizhnikov-Zamolodchikov equations of the Conformal Field Theory. I will use this to solve for their dynamics and also discuss some interesting physics that emerges at large times.
Sine-square deformation (SSD) is one example of smooth boundary conditions that have significantly smaller finite-size effects than open boundary conditions. In a one-dimensional system with SSD, the interaction strength varies smoothly from the center to the edges according to the sine-square function. Thus, the Hamiltonian of the system lacks translational symmetry. Nevertheless, previous studies have revealed that the SSD leaves the ground state of the uniform chain with periodic boundary conditions (PBC) almost unchanged for critical systems. In particular, I showed that the correspondence is exact for critical XY and quantum Ising chains. The same correspondence between SSD and PBC holds for Dirac fermions in 1+1 dimension and a family of more general conformal field theories. If time permits, I will also review more recent results and discuss the excited states of the SSD systems.
[1] H. Katsura, J. Phys. A: Math. Theor. 44, 252001 (2011); 45, 115003 (2012).
[2] I. Maruyama, H. Katsura, T. Hikihara, Phys. Rev. B 84, 165132 (2011)."
Monday, 06 August 2018
A review of the Calogero integrable model and its various generalizations will be given, with emphasis on physical properties and recent results. Topics will include integrability, the connection of these systems with fractional statistics, their matrix model and operator formulations, their various reductions and extensions, and their hydrodynamic description, properties and solitons.
Recently the upper bound on the Lyapunov exponent in thermal quantum systems was conjectured by Maldacena, Shenker and Stanford. I investigate this bound in a semi-classical dynamical system and argue the implication of this bound. Particularly I will show that this bound is related to a spontaneous energy emission in quantum mechanics. As an example, acoustic Hawking radiation in free fermi fluid will be explained through this energy emission.
Tuesday, 07 August 2018
"We consider anti-ferromagnetic spin chains with a weak frustration -just one bond in a large chain-, such as systems with an odd number of spins with periodic boundary conditions. We show that, in certain cases, a new quantum phase of matter arises in these systems. Such phase is extended, gapless, but not relativistic. The low-energy excitations have a quadratic (Galilean) spectrum. Locally, the correlation functions on the ground state do not show significant deviations compared to the not-frustrated case, but correlators involving a number of sites (or distances) scaling like the system size display new behaviors. In particular, the Von Neumann entanglement entropy is found to follow new rules, for which neither area law applies, nor one has a divergence of the entropy with the system size. Such very long range correlations are novel and of potential technological interest. We display such new phase in a few prototypical chains using numerical simulations and we study analytically the paradigmatic example of the Ising chain. Through these examples we argue that this phase emerges generally in (weakly) frustrated systems with discrete symmetries.
[1] S.M. Giampaolo, F. Ramos, and F. Franchini, In Preparation"
The talk will discuss the form of standard two-point equilibrium spatio-temporal correlations of observables in classical integrable systems such as the Toda chain. Next we will discuss the form of the classical analogue of the Out-of-Time-Ordered-Commutator [OTOC] in such systems.
The correspondence between the one-dimensional Calogero model and the two-dimensional lowest Landau level anyon model is known for some time, but the mapping between the models was so far rather formal. I will present an explicit mapping between the N-body harmonic Calogero eigenstates to the lowest Landau level eigenstates of N anyons in a harmonic trap. The mapping is achieved in terms of a convolution kernel that uses as input the scattering eigenstates of the free Calogero model on the infinite line, which are obtained in an operator formulation.
Wednesday, 08 August 2018
A review of the Calogero integrable model and its various generalizations will be given, with emphasis on physical properties and recent results. Topics will include integrability, the connection of these systems with fractional statistics, their matrix model and operator formulations, their various reductions and extensions, and their hydrodynamic description, properties and solitons.
In everyday fluids, the viscosity is the measure of resistance to the fluid flow and has a dissipative character. Avron, Seiler, and Zograf showed that viscosity of a quantum Hall (QH) fluid at zero temperature is non-dissipative. This non-dissipative viscosity (also known as ‘odd’ or ‘Hall’ viscosity) is the antisymmetric component of the total viscosity tensor and can be non-zero for parity violating fluids. I will discuss free surface dynamics of a two-dimensional incompressible fluid with the odd viscosity (not quite quantum Hall hydro). For the case of incompressible fluids, the odd viscosity manifests itself through the free surface (no stress) boundary conditions. We first find the free surface wave solutions of hydrodynamics in the linear approximation and study the dispersion of such waves. As expected, the surface waves are chiral. In the limit of vanishing shear viscosity and gravity, we derive effective nonlinear Hamiltonian equations for the surface dynamics, generalizing the linear solutions to the weakly nonlinear case. In a small surface angle approximation, the equation of motion results in a new class of non-linear chiral dynamics which we dub as {\it chiral Burgers} equation. I will briefly discuss how this program can be extended to the free surface of quantum Hall hydrodynamics.
Thursday, 09 August 2018
I will describe new approach to integrable models from the “four-dimensional Chern-Simons theory”, in collaboration with Kevin Costello and Edward Witten. This can be thought of as an analog of the Witten’s explanation of the knot invariants via the three-dimensional Chern-Simons theory. It would be interesting to put our work in broader context of statistical physics and condensed matter physics.
Friday, 10 August 2018
I will talk about the potential instability of many-body localized systems in the presence of an ‘ergodic bubble’, a finite but large delocalized region within an insulator. I will discuss some recent arguments suggesting that for two and higher dimensions a localized system is unstable even in the presence of a single ergodic bubble. To examine these arguments, we construct several models of a finite ergodic bubble coupled to an Anderson insulator of non-interacting fermions. We first describe the ergodic region using a GOE random matrix and perform an exact diagonalization study of small systems. I will show that the results are in excellent agreement with a refined theory of the thermalization, lending strong support to the avalanche scenario. I will then talk about the limit of large system sizes by modeling the ergodic region via a Hubbard model with all-to-all random hopping. The combined system, consisting of the bubble and the insulator, can be reduced to an effective Anderson impurity problem. We find that the spectral function of a local operator in the ergodic region changes dramatically when coupled to a large number of Anderson fermions, suggesting a possible way in which the instability argument can fail.