  Mean-field theories have proven to be efficient tools for exploring various
phases of matter, complementing alternative methods that are more precise but
also more computationally demanding. Conventional mean-field theories often
fall short in capturing quantum fluctuations, which restricts their
applicability to systems characterized by strong quantum fluctuations. In this
article, we propose a novel mean-field theory, density-matrix mean-field theory
(DMMFT). DMMFT constructs effective Hamiltonians, incorporating quantum
environments shaped by entanglements quantified by the reduced density
matrices. Therefore, it offers a systematic and unbiased approach to account
for effects of fluctuations and entanglements in quantum ordered phases. As
demonstrative examples, we show that DMMFT can not only quantitatively evaluate
the renormalization of order parameters induced by quantum fluctuations but can
even detect the topological order of quantum phases. Additionally, we discuss
the extensions of DMMFT for systems at finite temperatures and those with
disorders. Our work provides a novel and efficient approach to explore phases
exhibiting unconventional quantum orders, which can be particularly beneficial
for investigating frustrated spin systems in high spatial dimensions.
