Finite-entanglement and finite-size extrapolation

A finite chain of length L quantizes momenta and lets its boundaries influence the state. An infinite uniform MPS can avoid those ends but still acquire a finite correlation length because its transfer operator has a discrete spectrum. With a unique leading eigenvalue and a subleading eigenvalue of smaller magnitude, the ratio determines exponential decay of connected correlations in channels that overlap the corresponding modes. Other symmetries can suppress a mode’s contribution to a particular observable.

Normalize the leading transfer eigenvalue to one when defining that length. If a relevant eigenvalue has unit magnitude as well, the simple finite-correlation-length formula needs modification, as in noninjective or ordered representations. At a critical point, a finite-D MPS generally introduces an entanglement-induced cutoff even though the true correlation length diverges. Increasing L at fixed insufficient D can therefore leave observables unchanged for a numerical reason. Larger physical size alone does not remove finite-entanglement bias.

For a gapped local system well beyond its physical correlation length, boundary corrections to bulk observables can often be exponentially small, although surface contributions to energy per site can remain algebraic for open boundaries. At a critical point, finite-size corrections typically involve powers controlled by scaling dimensions and boundary conditions, sometimes with logarithms. Choosing a polynomial in inverse size because it fits three points is not a derivation. State the proposed asymptotic form and test its stability as the smallest sizes are removed.

Finite-entanglement scaling uses how an optimized tensor family approaches a critical state as D increases. Its induced correlation length is a more physical scaling variable than assuming D itself is a length. For suitable one-dimensional conformal critical points, half-chain entropy grows as central charge divided by six times the logarithm of correlation length, plus a nonuniversal constant. This statement assumes the relevant infinite-chain pure-state scaling regime and natural logs. Different geometries, boundaries, and logarithmic corrections require modified formulas.

At each selected size, converge local solver tolerance, sweep count, and bond dimension for the target observable before interpreting its size dependence. For real-time data, also converge time step and identify a time window stable under increased D. For PEPS, check environment contraction separately. Monte Carlo adds sampling error, autocorrelation, and possible sign bias. Numerical parameters should be varied one at a time or through a documented grid, so compensating errors do not create a misleading plateau.

Energy errors, correlation errors, and spectral errors need not share the same leading fit. Compare alternative justified asymptotic forms and report sensitivity to fitting windows. A finite time record limits frequency resolution, while analytic continuation from imaginary time adds ill-conditioning beyond ordinary cutoff extrapolation. Neither a sharper plot nor a smaller fit residual proves resolution of a physical narrow feature. A bisection lab provides a contrast: its bracket certifies an error for one scalar root, whereas an empirical many-body extrapolation needs additional physical and numerical evidence.