tangent-space methods for matrix product states, part 1

Tangent-space methods for matrix product states, part 1

Laurens Vanderstraeten

University of Gent

Work in collaboration with people in Gent and Vienna: Frank Verstraete, Jutho Haegeman, Valentin Zauner, Damian Draxler


Overview of part 1

Tangent-space methods for MPS, part 1 2/33

Introduction: why tangent space methods?

MPS in the thermodynamic limit: canonical forms, Schmidt decomposition, etc.

Computing static properties

The MPS tangent space

Overview of part 1

What do we want to simulate?

Dynamical correlation functions

Real-time properties (quantum quenches, dynamical phase transitions, ...)

Transport properties

Laurens Vanderstraeten Tangent-space methods for MPS, part 1 3/33


Physical intuition behind dynamical properties

Dynamics are carried by quasiparticle excitations on top of a strongly-correlated background!



Physical intuition behind dynamical properties

Dynamics are carried by quasiparticle excitations on top of a strongly-correlated background!


In two-dimensional systems: quasiparticles can have anyonic statistics

State-of-the-art DMRG/MPS algorithms

Real-time evolution: TEBD

Dynamical correlation functions: apply local operator and evolve in time

Fourier transform


State-of-the-art DMRG/MPS algorithms

The physical intuition of low-energy dynamics in terms of quasiparticles is lost

Alternative: tangent-space methods

- Take an MPS in the thermodynamic limit

- Construct the tangent space on the MPS manifold

- This is the space that contains the low-energy dynamics

- real-time evolution

- elementary excitations



Overview of part 1






MPS in the thermodynamic limit

An MPS in the thermodynamic limit is represented as

the state only depends on the single tensor A

describes the bulk physics of a quantum ground state

is translation-invariant by construction

The norm is obtained by contracting with the complex conjugate:


bond dimension D

physical spins

Transfer matrix

The central object in all our calculations will be the transfer matrix, defined as

The transfer matrix has a number of properties:

- the largest eigenvalue is a positive number

rescale the tensor such that leading eigenvalue is one:

- the fixed points are positive definite matrices

normalize fixed points such that


Transfer matrix

This implies that

so that the norm of an MPS is given by

We ignore all boundary effects at infinity, because they have no effect on the physical properties!


Canonical forms

The representation of a state in terms of the MPS tensor is not unique, because

leaves the state invariant

We can use these gauge degrees of freedom to our advantage

canonical forms


Left-canonical form

The left-canonical form is obtained as

solution:


How do we find the matrix L ?

Left-canonical form

Analogously, we can find the right-canonical form

The left canonical form is determined up to a unitary gauge transformation


with

Mixed-canonical form

So that we can write the MPS as

Using left- and right canonical forms, we can introduce

Introduce the center matrix


translation-invariant state!

Schmidt decomposition

Take singular-value decomposition:

Since the center matrix is diagonal now, we can write this state as


Truncating an MPS

Take singular-value decomposition:

Truncate the singular values

BUT: not necessarily the optimal solution in the sense that

is minimized


smaller bond dimension

U and V are isometries


Overview of part 1






Compute expectation values

The expectation value of a physical observable is given by

The infinite products to the left and to the right are replaced by the left and right fixed points:

Left and right fixed points act as effective environments, representing an infinite part of the network


Compute correlation functions

A generic correlation function is given by

In between the two operators we have powers of the transfer matrix:


Compute correlation functions

disconnected part of the correlation function

exponentially decaying part

long-range order!

An MPS always has exponentially decaying correlations!


Structure factor

In experiments, one typically measures the Fourier transform of the connected correlation function

This is a sum of terms of the form

summing all terms...


BUT:

define regularized transfer matrix s.t.


Structure factor

So we can write the infinite number of terms as

The total expression for the structure factor is given by

How to compute efficiently?


Structure factor

We need to find

which is equivalent to solving the linear equation

Use iterative linear solvers for which only the action of the transfer matrix is needed


Computing the inverse directly would require operations


Overview of part 1






The tangent space

The class of matrix product states constitutes a non-linear manifold within Hilbert space

at every point we can define a tangent space


The tangent space

We can interpret a tangent vector as a local perturbation on a strongly-correlated background state

this perturbation is non-extensive

carries the notion of a quasi-particle

tangent space parametrizes the low-energy subspace on a given reference state


Gauge freedom in the tangent space

A transformation of the form

leaves the tangent vector invariant gauge degrees of freedom

We can introduce an effective parametrization for the B tensor

contains all parameters in the tangent vector


Effective parametrization

We determine the tensor such that

and

find as the null space of the matrix

orthonormalize

This parametrization implies


dimension:


Overlap with the MPS



Overlap between two tangent vectors


Parametrization of the tangent space

Overlap between two tangent vectors:


Only in this particular representation is the physical overlap between two tangent vectors equal to the Euclidean inner product of the parameters.

Differential geometry:

the fact that

introduces a non-trivial metric of the manifold

tangent-space methods for matrix product states, part 1

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