Parameter dependence of entanglement spectra in quantum field theories

In this paper, we explore the characteristics of reduced density matrix spectra in quantum field theories. Previous studies mainly focus on the function <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi mathvariant="script">P</a:mi><a:mo stretchy="false">(</a:mo><a:mi>λ</a:mi><a:mo stretchy="false">)</a:mo><a:mo>≔</a:mo><a:msub><a:mrow><a:mo>∑</a:mo></a:mrow><a:mrow><a:mi>i</a:mi></a:mrow></a:msub><a:mi>δ</a:mi><a:mo stretchy="false">(</a:mo><a:mi>λ</a:mi><a:mo>−</a:mo><a:msub><a:mrow><a:mi>λ</a:mi></a:mrow><a:mrow><a:mi>i</a:mi></a:mrow></a:msub><a:mo stretchy="false">)</a:mo></a:mrow></a:math>, where <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:msub><h:mi>λ</h:mi><h:mi>i</h:mi></h:msub></h:math> denote the eigenvalues of the reduced density matrix. We introduce a series of functions designed to capture the parameter dependencies of these spectra. These functions encompass information regarding the derivatives of eigenvalues concerning the parameters, notably including the function <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mrow><j:msub><j:mrow><j:mi mathvariant="script">P</j:mi></j:mrow><j:mrow><j:msub><j:mrow><j:mi>α</j:mi></j:mrow><j:mrow><j:mi>J</j:mi></j:mrow></j:msub></j:mrow></j:msub><j:mo stretchy="false">(</j:mo><j:mi>λ</j:mi><j:mo stretchy="false">)</j:mo><j:mo>≔</j:mo><j:msub><j:mrow><j:mo>∑</j:mo></j:mrow><j:mrow><j:mi>i</j:mi></j:mrow></j:msub><j:mfrac><j:mrow><j:mo stretchy="false">∂</j:mo><j:msub><j:mrow><j:mi>λ</j:mi></j:mrow><j:mrow><j:mi>i</j:mi></j:mrow></j:msub></j:mrow><j:mrow><j:mo stretchy="false">∂</j:mo><j:msub><j:mrow><j:mi>α</j:mi></j:mrow><j:mrow><j:mi>J</j:mi></j:mrow></j:msub></j:mrow></j:mfrac><j:mi>δ</j:mi><j:mo stretchy="false">(</j:mo><j:mi>λ</j:mi><j:mo>−</j:mo><j:msub><j:mrow><j:mi>λ</j:mi></j:mrow><j:mrow><j:mi>i</j:mi></j:mrow></j:msub><j:mo stretchy="false">)</j:mo></j:mrow></j:math>, where <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi>α</s:mi><s:mi>J</s:mi></s:msub></s:math> denotes the specific parameter. Computation of these functions is achievable through the utilization of Rényi entropy. Intriguingly, we uncover compelling relationships among these functions and demonstrate their utility in constructing the eigenvalues of reduced density matrices for select cases. We perform computations of these functions across several illustrative examples. Especially, we conducted a detailed study of the variations of <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi mathvariant="script">P</u:mi><u:mo stretchy="false">(</u:mo><u:mi>λ</u:mi><u:mo stretchy="false">)</u:mo></u:math> and <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:msub><z:mi mathvariant="script">P</z:mi><z:msub><z:mi>α</z:mi><z:mi>J</z:mi></z:msub></z:msub><z:mo stretchy="false">(</z:mo><z:mi>λ</z:mi><z:mo stretchy="false">)</z:mo></z:math> under general perturbation, elucidating their physical implications. In the context of holographic theory, we ascertain that the zero point of the function <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:msub><eb:mi mathvariant="script">P</eb:mi><eb:msub><eb:mi>α</eb:mi><eb:mi>J</eb:mi></eb:msub></eb:msub><eb:mo stretchy="false">(</eb:mo><eb:mi>λ</eb:mi><eb:mo stretchy="false">)</eb:mo></eb:math> possesses universality, determined as <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" display="inline"><jb:mrow><jb:msub><jb:mrow><jb:mi>λ</jb:mi></jb:mrow><jb:mrow><jb:mn>0</jb:mn></jb:mrow></jb:msub><jb:mo>=</jb:mo><jb:msup><jb:mrow><jb:mi>e</jb:mi></jb:mrow><jb:mrow><jb:mo>−</jb:mo><jb:mi>S</jb:mi></jb:mrow></jb:msup></jb:mrow></jb:math>, where <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" display="inline"><lb:mi>S</lb:mi></lb:math> denotes the entanglement entropy of the reduced density matrix. Furthermore, we exhibit potential applications of these functions in analyzing the properties of entanglement entropy. Published by the American Physical Society 2024