This paper provides an in-depth analysis of entanglement entropy (EE) in quantum field theory (QFT), with a particular focus on its computation using the replica trick and its applications to both conformal and non-conformal systems. Beginning with an introduction to the basics of QFT, the study explains how entanglement entropy quantifies the quantum correlations between subsystems in a pure state, represented by the von Neumann entropy of the reduced density matrix. The replica trick is employed to derive the entanglement entropy, involving path integrals over n-sheeted Riemann surfaces. The paper demonstrates the method’s utility in 1+1-dimensional conformal field theory (CFT), where the central charge governs universal properties of entropy. It further extends the analysis to massive field theories, finite systems, and topological phases, exploring how deviations from conformal symmetry affect entanglement. The work also delves into the AdS/CFT correspondence, showcasing how holographic techniques facilitate entanglement entropy calculations in higher-dimensional systems via the Ryu-Takayanagi formula. In addition, the study investigates entanglement entropy in quantum lattice systems, considering the effects of spatial and thermal fluctuations. Numerical methods are used to compute the scaling of entanglement entropy with interval length for varying system masses and temperatures, highlighting the influence of these parameters on entropy behavior. Fractal lattice structures are explored to uncover unique entropy scaling laws and self-similar entanglement patterns. This study advances the under-standing of EE in diverse quantum systems and establishes a foundation for exploring its role in quantum criticality and fractal geometries. The paper also explores multiscale entanglement entropy (MSE), offering a comprehensive framework for future research in quantum critical phenomena and non-equilibrium systems, shedding light on the complex relationship between quantum entanglement and thermal effects. This work serves as a foundational reference for future theoretical and computational studies in quantum entanglement
Pseudo entropy extends entanglement entropy from a single quantum state to a pair of nonorthogonal states and is generally complex. Taking advantage of Cauchy-Riemann equations, Kramers-Kronig relations, and analytic continuation in state parameters, we show how and to what extent real and imaginary parts of pseudo entropy can be derived from ordinary entanglement entropy. For families with holomorphic coefficients in finite-dimensional Hilbert spaces, the reduced transition matrix equals the ordinary reduced density matrix formula evaluated at complex parameters. Then a convergent Taylor series gives the real and imaginary parts of pseudo entropy from even and odd derivatives of entanglement entropy at the real midpoint. The matrix identity also gives formulae for excess pseudo entropy and Renyi entropies, and interpolation formulae for families with polynomial coefficients. We apply these results to boundary-state quenches and thermal states in conformal field theory, and to fermionic and bosonic Gaussian states and quenches. In conformal field theory, Kramers-Kronig relations give the first moment of the imaginary part in terms of the central charge and one-point functions for boundary states.
The bipartite entanglement entropy of a segment of length
l in
1+1-dimensional conformal field theories (CFT) follows the formula $S=\frac{c}{3}\mathrm{ln}l+\ensuremath{\gamma}$, where
c is the central charge of the CFT and $\ensuremath{\gamma}$ is a cutoff-dependent constant which diverges in the absence of an ultraviolet cutoff. According to this formula, systems with larger central charges have more bipartite entanglement entropy. Using quantum Fisher information (QFI), we argue that systems with bigger central charges not only have larger bipartite entanglement entropy, but also have more multipartite entanglement content. In particular, we argue that since a system with a smaller smallest scaling dimension has a larger QFI, the multipartite entanglement content of a CFT is dependent on the value of the smallest scaling dimension present in the spectrum of the system. We show that our argument seems to be consistent with some of the existing results regarding the von Neumann entropy, negativity, and localizable entanglement in
1+1 dimensions. Furthermore, we also argue that the QFI decays under renormalization group flow between two unitary CFTs. Finally, we also comment on nonconformal but scale-invariant systems.
We propose a new product entropy, defined as the Rényi (or von Neumann) entropy of a normalized product operator constructed from two density matrices. We establish a duality showing that the SVD entanglement entropy of a subsystem for two pure states is exactly equivalent to the product entropy of the complementary subsystem. This connection provides both a transparent physical interpretation in terms of the spectral diversity of the subsystem state product and a computationally efficient route that bypasses the reduced transition matrix. For low-lying eigenstates, we derive analytical expressions for the subsystem product entropy between the ground state and primary excitations in two-dimensional conformal field theories, explicitly verified against the critical Ising chain. In quantum quench dynamics, the quasiparticle picture yields time evolution in the scaling limit: following a global quench, the subsystem product entropy exhibits distinct sequences of thermalization and revivals, whereas under a local operator quench, it develops characteristic plateaus whose constant values are determined by the inserted operator. Extensive numerical calculations on the critical Ising chain confirm the analytical predictions with excellent accuracy.
We show the presence of universal features in the entanglement entropy of regularized boundary states for (1+1)D conformal field theories on a circle when the reduced density matrix is obtained by tracing over right- or left-moving modes. We derive a general formula for the left-right entanglement entropy in terms of the central charge and the modular S matrix of the theory. When the state is chosen to be an Ishibashi state, this measure of entanglement is shown to precisely reproduce the spatial entanglement entropy of a (2+1)D topological quantum field theory. We explicitly evaluate the left-right entanglement entropies for the Ising model, the tricritical Ising model and the su[over ^](2)_{k} Wess-Zumino-Witten model as examples.
We characterize the operator-theoretic origin of entanglement entropy in the GeoVac Coulomb lattice, a finite spectral graph realization of the hydrogenic angular momentum eigenbasis on
S3. Five results are presented. First, the von Neumann entropy of any single-particle reduced density matrix derived from a non-degenerate ground state of a purely one-body fermionic Hamiltonian on a finite lattice is identically zero; numerically, we confirm
Skin/Sfull∼10−14 for He at
nmax=2,3 in the angular momentum eigenbasis. All ground-state entanglement is generated by the two-body interaction
Vee, with the non-degeneracy qualifier essential: open-shell and dissociated-bond states escape the floor. Second, the holographic area-law scaling
Sn=klogAn with
An∝n4, asserted on one-particle grounds in earlier framework papers, is rigorously recovered as a pair-counting statement:
An=gn2, where
gn=2n2 is the shell degeneracy. The factor of four is the two-body signature. Third, in the Coulomb lattice the cusp concentrates as a single ``hot node'' on the
(1s,1s) pair-state of the
Vee graph; entanglement is co-located with cusp mass. Fourth, on the Bargmann–Segal harmonic oscillator lattice the ground-state entropy of any two-fermion closed-shell state interacting via any central two-body potential is identically zero, a consequence of Moshinsky–Talmi total-quanta conservation; this is a rigidity statement structurally parallel to the Fock and Bargmann–Segal theorems of Paper 24. Fifth, Coulomb-lattice block entropy obeys a universal scaling
SB=A(w~B/δB)γ, where
w~B is the dimensionless
Vee mass off-diagonal in the one-body eigenbasis and
δB the dimensionless first
H1 gap; the local exponent extrapolates to
γ∞≈1.96 in the joint
Z,nmax→∞ limit (Richardson on
nmax=2,3,4,5), slightly below the non-degenerate second-order Rayleigh–Schrödinger value of
2, with the residual gap attributed to multi-shell aggregation in the basis-extension limit. Entropy is slotted into the exchange-constant taxonomy as a projection-class observable: zero on the combinatorial graph, finite only after the metric embedding of
1/r12. GeoVac Paper 27. Part of the GeoVac research corpus (precision observations group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.