• View
• Edit
• Rename
• History
• Print
• Backlinks

MpExcitationAnsatz

Calculate excited states using the MPS excitation ansatz [1].

Synopsis

mp-excitation-ansatz [options] <psi> [psi-right]

Options

--help

-H, --Hamiltonian

-k, --momentum

--latticeucsize

--ky

--kyfix

--alpha

-n, --numeigen

-q, --quantumnumber

-r, --rotate

--string

-o, --output

--digits

--tol

--gmrestol

--unityepsilon

--seed

--random

--streaming

--no-streaming

--quiet

-v, --verbose

Description

This tool calculates the excited states for a given ground state using the MPS excitation ansatz {$$|\Phi_k(B)\rangle = \sum_n \mathrm{e}^{\mathrm{i}kn} \cdots A_L^{s_{n-2}} A_L^{s_{n-1}} B^{s_n} A_R^{s_{n+1}} A_R^{s_{n+2}} \cdots$$} where {$A_L$} and {$A_R$} are the left- and right-orthogonal forms respectively of the ground state. The lowest-energy states of this form are found for a specified momentum {$k$} by using a numerical eigensolver on the effective Hamiltonian for this state as the matrix {$B$} is varied.

We may also calculate topologically non-trivial excitations where the {$A_L$} and {$A_R$} matrices are taken from two different ground states, which can be done by specifying the optional command line option psi-right.

For 2D systems on a finite-width cylinder, we can calculate the {$y$}-momentum along the circumference of the cylinder by specifying the cylinder translation operator using the --string option. We can then target excited states which are simultaneously eigenstates of the {$y$}-momentum operator by penalising states with the wrong {$y$}-momentum [2]: this can be enabled by further specifying the --ky option.

Examples

As an example, we can calculate the Haldane gap, as was done in the original MPS excitation ansatz paper [1].

First, we find the ground state of the spin-1 antiferromagnetic Heisenberg model:

We can then calculate the energy gap at {$k = \pi$} by

We can calculate the dispersion relation for a range of {$k$} with the following command, using a larger error tolerance to speed up the calculation (in this example, the finite bond dimension of the ground state will have the most significant impact on the error anyway)

Notes

• There is some ambiguity to the meaning of the momentum when using a wavefunction or a lattice with a multi-site unit cell. The first Brillouin zone (FBZ) is taken to range from 0 to 2*pi*LatticeUnitCellSize/WavefunctionUnitCellSize. Here, LatticeUnitCellSize defaults to the number of sites in the lattice file’s unit cell, however, it can be manually overridden with --latticeucsize. The wavefunction and lattice unit cell sizes, as well as the FBZ range, are printed in the preamble of the output.
• When calculating the {$y$}-momentum with a different psi-right specified, the resulting values of ky will possibly be shifted by a constant, since the eigenvectors of the left/right transfer matrices with {$T_y$} have an arbitrary phase factor which needs to be fixed (see the notes on mp-ibc-overlap). If this happens, the workaround is to use the --kyfix option to shift the values of ky by subtracting the specified value.
• When calculating the {$y$}-momentum, if there are two or more excitations with the same energy, then the eigenvectors returned by ARPACK may not be eigenvectors of {$T_y$} and so their expectation values will be some linear combination of the {$y$}-momentum eigenvalues. This can be avoided by targeting a specific {$y$}-momentum by using the --ky option.

References