Tangent-space methods for matrix product states, part 2 Laurens Vanderstraeten University of Ghent Overview of part 2 Recap Variational optimization of ground states Time-dependent variational principle Elementary excitations Beyond Laurens Vanderstraeten Tangent-space methods for MPS, part 2 2/47 Recap Fixed points of transfer matrix Local observables Structure factor Laurens Vanderstraeten Tangent-space methods for MPS, part 2 3/47 Recap Tangent vector Reduced parametrization Overlap with MPS and norm Laurens Vanderstraeten Tangent-space methods for MPS, part 2 4/47 Overview of part 2 Recap Variational optimization of ground states Time-dependent variational principle Elementary excitations Beyond Laurens Vanderstraeten Tangent-space methods for MPS, part 2 5/47 Finding ground states Variational optimization of MPS tensor amounts to find a path through the manifold towards the optimum Laurens Vanderstraeten Tangent-space methods for MPS, part 2 6/47 Evaluating the gradient We need to sum all different relative positions! Laurens Vanderstraeten Tangent-space methods for MPS, part 2 7/47 Evaluating the gradient Disconnected contributions Laurens Vanderstraeten Tangent-space methods for MPS, part 2 8/47 Evaluating the gradient Disconnected contributions Laurens Vanderstraeten Tangent-space methods for MPS, part 2 9/47 Evaluating the gradient Connected contributions Laurens Vanderstraeten Tangent-space methods for MPS, part 2 10/47 Evaluating the gradient The full expression for the gradient is and allows us to find a path through the manifold towards the optimal MPS Laurens Vanderstraeten Tangent-space methods for MPS, part 2 11/47 Energy minimization Steepest descent Conjugate gradient with Convergence criterion!! Quasi-newton methods, Hessian methods, ... faster than - imaginary-time evolution (~ steepest descent) iTEBD - power methods iDMRG (not really...) Laurens Vanderstraeten Tangent-space methods for MPS, part 2 12/47 The gradient in the tangent space Can we interpret the gradient as a tangent vector?? Remember that we need to use the correct parametrization for the tangent space!! which we can use to write down the tangent-space gradient Laurens Vanderstraeten Tangent-space methods for MPS, part 2 13/47 Parameter-space gradient vs. tangent-space gradient Laurens Vanderstraeten Tangent-space methods for MPS, part 2 14/47 Overview of part 2 Recap Variational optimization of ground states Time-dependent variational principle Elementary excitations Beyond Laurens Vanderstraeten Tangent-space methods for MPS, part 2 15/47 Time evolution in the MPS manifold make MPS tensor time dependent flow equation for the tensor Laurens Vanderstraeten Tangent-space methods for MPS, part 2 16/47 Time evolution in the MPS manifold Start from Schrödinger equation If contained in the manifold, the left-hand side is a tangent vector but the right-hand side points out of the manifold! Time-dependent variational principle (TDVP): project time evolution onto the tangent space Laurens Vanderstraeten Tangent-space methods for MPS, part 2 17/47 Time evolution in the MPS manifold In order to obtain the projection onto the manifold, we need the overlap with a general tangent vector Again, we compute the contributions where the Hamiltonian is to the left or to the right Laurens Vanderstraeten Tangent-space methods for MPS, part 2 18/47 Time evolution in the MPS manifold In order to obtain the projection onto the manifold, we need the overlap with a general tangent vector We can find the projection onto the tangent space by leaving the indices open Laurens Vanderstraeten Tangent-space methods for MPS, part 2 19/47 Time evolution in the MPS manifold BUT we have to take care of effective parametrization of the tangent space Laurens Vanderstraeten Tangent-space methods for MPS, part 2 20/47 The TDVP equations Linear Schrödinger equation in full Hilbert space is transformed into a highly non-linear differential equation for the MPS tensor We can integrate the TDVP equations by applying a first-order Euler scheme or we can use implicit/explicit schemes, higher-order Runge-Kutta, ... Laurens Vanderstraeten Tangent-space methods for MPS, part 2 21/47 Imaginary-time evolution Take the Schrödinger equation in imaginary time which results in a ground-state projection for infinite times Suppose we integrate the TDVP equation in imaginary time with a first-order Euler scheme: This is exactly the same update rule as the steepest descent method: Imaginary-time evolution corresponds to a steepest-descent optimization scheme using the tangent-space gradient! Laurens Vanderstraeten Tangent-space methods for MPS, part 2 22/47 Overview of part 2 Recap Variational optimization of ground states Time-dependent variational principle Elementary excitations Beyond Laurens Vanderstraeten Tangent-space methods for MPS, part 2 23/47 Elementary excitations We want to target isolated branches in the excitation spectrum of generic spin chains we can think of these excitations as quasiparticles on a strongly-correlated background state Tangent space ansatz for elementary excitations with momentum “boosted” tangent vector Laurens Vanderstraeten Tangent-space methods for MPS, part 2 24/47 Elementary excitations We can use the same effective parametrization Laurens Vanderstraeten Tangent-space methods for MPS, part 2 25/47 Computing expectation values summing three sums ... Same as before: operator to the left and to the right of both tensors Laurens Vanderstraeten Tangent-space methods for MPS, part 2 26/47 Computing expectation values summing three sums ... Same as before: ket-tensor to the left and to the right of bra-tensor and operator Laurens Vanderstraeten Tangent-space methods for MPS, part 2 27/47 Computing expectation values Same as before: operator and ket-tensor together to the left and right and Laurens Vanderstraeten Tangent-space methods for MPS, part 2 28/47 Computing expectation values Putting everything together Laurens Vanderstraeten Tangent-space methods for MPS, part 2 29/47 Finding the optimal tensors How do we obtain the tensors such that they describe the different excitations in a given momentum sector? Minimizing the energy orthogonal to the ground state Solve eigenvalue equation iteratively! Laurens Vanderstraeten Tangent-space methods for MPS, part 2 30/47 Eigenvalue equation Effective norm matrix effective norm matrix is unit matrix Generalized eigenvalue problem reduces to ordinary one: Laurens Vanderstraeten Tangent-space methods for MPS, part 2 31/47 Eigenvalue equation Effective energy matrix We only need the action of the effective energy matrix on a given input vector X Laurens Vanderstraeten Tangent-space methods for MPS, part 2 32/47 Eigenvalue equation Step 1: compute B tensor Step 2: compute all terms of the form Step 3: transform back to an output X vector Laurens Vanderstraeten Tangent-space methods for MPS, part 2 33/47 Eigenvalue equation Step 2: compute all terms of the form We do the usual contractions Laurens Vanderstraeten Tangent-space methods for MPS, part 2 34/47 Eigenvalue equation Step 2: compute all terms of the form We do the usual contractions Laurens Vanderstraeten Tangent-space methods for MPS, part 2 35/47 Eigenvalue equation Step 2: compute all terms of the form We do the usual contractions Laurens Vanderstraeten Tangent-space methods for MPS, part 2 36/47 Eigenvalue equation Putting everything together Laurens Vanderstraeten Tangent-space methods for MPS, part 2 37/47 Finding the optimal tensors How do we obtain the tensors such that they describe the different excitations in a given momentum sector? Minimizing the energy orthogonal to the ground state Solve eigenvalue equation iteratively! different eigenvalues and -vectors correspond to different quasiparticle excitations higher-energy solutions correspond to states in the continuum (not the right ansatz!) Laurens Vanderstraeten Tangent-space methods for MPS, part 2 38/47 Elementary excitations Connect these variational excitations to dynamical correlation functions Take only quasiparticle states in the sum so that we obtain a set of delta peaks spectral weights Laurens Vanderstraeten Tangent-space methods for MPS, part 2 39/47 Dynamical correlation functions We take the, by now, familiar steps for computing Laurens Vanderstraeten Tangent-space methods for MPS, part 2 40/47 Overview of part 2 Recap Variational optimization of ground states Time-dependent variational principle Elementary excitations Beyond Laurens Vanderstraeten Tangent-space methods for MPS, part 2 41/47 Beyond 1) Inverse-free versions of tangent-space algorithms remember effective parametrization requires inversion of left and right fixed points small Schmidt coefficients imply bad conditioning! alternative: mixed canonical form for a tangent vector (see lecture notes) Laurens Vanderstraeten Tangent-space methods for MPS, part 2 42/47 Beyond 2) Tangent-space methods for matrix product operators suppose we want to find the fixed point of a matrix product operator in the thermodynamic limit the fixed point can be approximated by an MPS algorithms for finding fixed points are inspired by TDVP (see lecture notes) Laurens Vanderstraeten Tangent-space methods for MPS, part 2 43/47 Beyond 3) Multi-particle excitations what about higher excitations? we can build up the whole spectrum starting from the isolated branches BUT the wave functions of these “scattering states” are non-trivial Multi-particle excitations require solving the two-particle scattering problem compute two-particle S matrix, bound state formation,