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, …
Laurens Vanderstraeten Tangent-space methods for MPS, part 2 44/47 Beyond
4) Finite density of excitations
study phase transitions as the condensation of quasiparticles
study finite-temperature properties with a thermally excited density of particles
study transport properties as particle diffusion (ballistic) processes
Laurens Vanderstraeten Tangent-space methods for MPS, part 2 45/47 Beyond
5) Finite systems and long-range interactions
real-time simulations of long-range interactions is very hard for TEBD, but straightforward for TDVP
Laurens Vanderstraeten Tangent-space methods for MPS, part 2 46/47 Beyond
6) Two-dimensions: PEPS
can we translate tangent-space methods to two dimensions?
resumming infinite sums in two dimensions?
canonical forms, effective parametrizations, … ?
see lecture by P. Corboz
Laurens Vanderstraeten Tangent-space methods for MPS, part 2 47/47