I didn't come here to tell you how this is going to end. I came here to tell you how it's going to begin.
Tools /
MpEaMomentsCalculate (mixed) expectation values of EA wavefunctions with triangular or string operators. Synopsis
Options
Show help message.
Calculate the expectation value of this triangular operator (if unspecified, use wavefunction attribute Hamiltonian of psi ).
Calculate the expectation value of this string operator.
Calculate the overlap instead of an expectation value.
Use this value of the momentum instead of the one in the file, in units of {$\pi$}.
Use this value for the lattice unit cell size (defaults to the wavefunction attribute LatticeUnitCellSize of psi , or 1 otherwise).
(Triangular operators) Calculate expectation values of the operator for each power from 1 up to this value.
Calculate the moments (default, unless cumulants is specified).
Print the full moment polynomials.
Calculate the cumulants.
Show the result in Cartesian coordinates (equivalent to real imag ).
Show the result in polar coordinates (equivalent to mag arg ).
Display the real part of the result.
Display the imaginary part of the result.
Display the magnitude of the result.
Display the argument (angle) of the result.
Display the argument in radians instead of degrees.
Scale the results for a wavefunction with this unit cell size (otherwise, use the wavefunction’s unit cell size).
Force setting the degree of the MPO.
Don’t print column headings.
The tolerance to use for the linear solver algorithm (default 1e15 ).
The tolerance for checking transfer matrix eigenvalues with magnitude near unity (default 1e12 ).
Subtract this value from the phase (in degrees, or radians if radians is specfied).
Perform the calculation in a righttoleft fashion, rather than a lefttoright one. (This will usually return the same results, useful for debugging.)
Show all columns of the fixedpoint solutions (this will usually produce a lot of output, and is pretty much only useful for debugging).
Increase verbosity.
DescriptionThis tool calculates expectation values of an EA wavefunction with respect to a triangular or product MPO.
This is done using a similar method as If we have some {$n$}th order operator (such as the Hamiltonian taken to power {$n$}), the normalised expectation value will given by the {$n$}th moment {$\mu_n$}, which will be written as the complete Bell polynomial of the cumulants {$\kappa_m$} for {$m = 1, \ldots, n$}, each of which has the linear form in system size {$$\kappa_m = \kappa^E_m L + \kappa^\Delta_m.$$} Here, the linear component {$\kappa^E_m$} will be equal to the cumulant for the boundary infinite wavefunction, while the constant part {$\kappa^\Delta_m$} is a ‘correction’ due to the excitation: we call {$\kappa^E_m$} and {$\kappa^\Delta_m$} the boundary and excitation cumulants respectively. Hence, the {$n$}th moment will be an degree{$n$} polynomial, where the degree {$\geq 1$} components will be a mixture of the boundary and excitation cumulants, while the constant component will be the complete Bell polynomial of {$\kappa^\Delta_m$} for {$m = 1, \ldots n$}, and so we will call this the excitation moment {$\mu^\Delta_n$}. We can calculate the {$n$}th excitation cumulant {$\kappa^\Delta_n$} recursively by calculating each {$\mu^\Delta_n$} and using the identity {$$\kappa^\Delta_n = \mu^\Delta_n  \sum_{m=1}^{n1} \binom{n1}{m1} \kappa^\Delta_m \mu^\Delta_{nm}.$$} By default, the tool displays the excitation moments only: you can view the excitation cumulants using We can also calculate mixed expectation values by specifying the positional argument Notes
