# Calculating the norm matrix to solve the A-body Schr\"odinger equation within a set of non-orthogonal many-body states

## TL;DR

This paper introduces a new, general method for calculating the norm matrix in many-body quantum physics, improving the solution of the A-body Schrödinger equation with non-orthogonal states, exemplified on Bogoliubov states.

## Contribution

It presents an alternative, broadly applicable method for computing the norm matrix in many-body problems involving non-orthogonal states, addressing longstanding phase ambiguity issues.

## Key findings

- Method successfully applied to Bogoliubov states
- Numerical tests on a toy model demonstrate effectiveness
- Provides a more general approach than previous solutions

## Abstract

There are efficient many-body methods, such as the (symmetry-restored) generator coordinate method in nuclear physics, that formulate the A-body Schr\"odinger equation within a set of nonorthogonal many-body states. Solving the corresponding secular equation requires the evaluation of the norm matrix and thus the capacity to compute its entries consistently and without any phase ambiguity. This is not always a trivial task, e.g. it remained a long-standing problem for methods based on general Bogoliubov product states. While a solution to this problem was found recently in Ref. [L. M. Robledo, Phys. Rev. C79, 021302 (2009)], the present work introduces an alternative method that can be generically applied to other classes of states of interest in many-body physics. The method is presently exemplified in the case of Bogoliubov states and numerically illustrated on the basis of a toy model.