Transfer Matrices and Excitations with Matrix Product States

V. Zauner, J. Haegeman, M.M. Rams, V. Stojevic, D. Draxler, M. Degroote, L. Vanderstraeten, N. Schuch and F. Verstraete

We investigate the relation between the dispersion relation of low energy excitations and static 
correlation functions in the ground state of local quantum many-body Hamiltonians in the context of 
tensor networks. As a central object in obtaining static correlations we relate the Matrix Product 
State Transfer Matrix (MPS-TM) to the Quantum Transfer Matrix (QTM) at zero temperature and show 
that it contains important information about the location and magnitude of the dispersion 
relation's minima. We elaborate on the peculiar structure of the MPS-TM's eigenspectrum and give 
several arguments as to how it affects the structure of the low energy spectrum of the system as 
well as the form of static correlation functions. We further derive a bound for the decay of 
momentum-resolved static correlation functions as a function of the dispersion relation. Lastly we 
establish a relation between the exact QTM and a finite bond dimension MPS approximation of the 
ground state using renormalization group arguments. We present supporting numerical data for 
one-dimensional lattice and continuum models as well as two-dimensional lattice models on a 
cylinder.