TY - JOUR

T1 - Geometry of variational methods

T2 - dynamics of closed quantum systems

AU - Hackl, Lucas

AU - Guaita, Tommaso

AU - Shi, Tao

AU - Haegeman, Jutho

AU - Demler, Eugene

AU - Cirac, J. Ignacio

N1 - 47+8 pages, 8 figures

PY - 2020

Y1 - 2020

N2 - We present a systematic geometric framework to study closed quantum systems based on suitably chosen variational families. For the purpose of (A) real time evolution, (B) excitation spectra, (C) spectral functions and (D) imaginary time evolution, we show how the geometric approach highlights the necessity to distinguish between two classes of manifolds: K\"ahler and non-K\"ahler. Traditional variational methods typically require the variational family to be a K\"ahler manifold, where multiplication by the imaginary unit preserves the tangent spaces. This covers the vast majority of cases studied in the literature. However, recently proposed classes of generalized Gaussian states make it necessary to also include the non-K\"ahler case, which has already been encountered occasionally. We illustrate our approach in detail with a range of concrete examples where the geometric structures of the considered manifolds are particularly relevant. These go from Gaussian states and group theoretic coherent states to generalized Gaussian states.

AB - We present a systematic geometric framework to study closed quantum systems based on suitably chosen variational families. For the purpose of (A) real time evolution, (B) excitation spectra, (C) spectral functions and (D) imaginary time evolution, we show how the geometric approach highlights the necessity to distinguish between two classes of manifolds: K\"ahler and non-K\"ahler. Traditional variational methods typically require the variational family to be a K\"ahler manifold, where multiplication by the imaginary unit preserves the tangent spaces. This covers the vast majority of cases studied in the literature. However, recently proposed classes of generalized Gaussian states make it necessary to also include the non-K\"ahler case, which has already been encountered occasionally. We illustrate our approach in detail with a range of concrete examples where the geometric structures of the considered manifolds are particularly relevant. These go from Gaussian states and group theoretic coherent states to generalized Gaussian states.

KW - quant-ph

KW - cond-mat.quant-gas

KW - cond-mat.str-el

KW - cond-mat.supr-con

U2 - 10.21468/SciPostPhys.9.4.048

DO - 10.21468/SciPostPhys.9.4.048

M3 - Journal article

VL - 9

JO - SciPost Physics

JF - SciPost Physics

SN - 2542-4653

M1 - 048

ER -