Entanglement Entropy in Free QFTs

• Entanglement entropy calculation is a method to quantify quantum correlations between subsystems in free quantum field theories through analytical and numerical techniques.
• It employs both Euclidean replica and real-time correlator methods to compute von Neumann and Renyi entropies, ensuring consistency across analytic and lattice frameworks.
• The approach incorporates geometric effects and scaling laws, revealing universal corrections in various dimensions and offering insights for quantum gravity and condensed matter research.

Entanglement entropy calculation quantifies quantum correlations between spatial regions or subsystems and is a central tool in quantum field theory, statistical mechanics, quantum gravity, and condensed matter. Modern research provides robust methodologies for computing entanglement entropy in free fields, interacts with lattice-continuum constructions, extends to black hole and holographic setups, and adapts to gauge and topologically ordered systems. This summary focuses on the rigorous, technical formulations for calculating entanglement entropy, primarily in free quantum field theories, highlighting analytic, numerical, and lattice methodologies, and their connections to scaling, geometry, and universal structures.

1. General Approaches: Euclidean Replica and Real-Time Correlator Methods

Two complementary formalisms express the entanglement entropy for free fields:

• Euclidean (Replica) Approach:

The reduced density matrix for a region $V$ is constructed by tracing out the degrees of freedom of the complement. The core computational tool is the R\'enyi entropy:

$S_n(V) = -\frac{1}{n-1} \log [\text{tr}(\rho_V^n)]$

The von Neumann entropy is recovered as $n \to 1$. The replica trick involves an $n$-sheeted Riemann manifold with sheets sewn at the entangling boundary, enabling the computation of $\text{tr}(\rho_V^n)$ as the ratio of partition functions on branched covers. For free bosons, the problem can be diagonalized in replica space, leading to multivalued field configurations subject to twisted boundary conditions:

$$\phi_k(x,0^+) = e^{i\frac{2\pi k}{n}}\phi_k(x,0^-) \quad (x\in V)$$

The replica sum can often be recast as a contour integral, e.g.,

$$S(V) = -2 \int_1^\infty d\lambda \frac{\log Z[-\lambda]}{(\lambda+1)^2}$$

for scalar fields.

• Real-Time (Correlator) Approach:

For Gaussian states, the reduced density matrix is entirely determined by two-point functions. For bosons with variables $\phi_i,\pi_i$, one computes

$$X_{ij} = \langle \phi_i \phi_j \rangle,\quad P_{ij} = \langle \pi_i \pi_j \rangle$$

and defines $C = \sqrt{X P}$. The entanglement entropy is

$$S(V) = \operatorname{tr}\left[(C+1/2)\log(C+1/2) - (C-1/2)\log(C-1/2)\right]$$

For fermions, $C_{ij} = \langle \psi_i \psi^\dagger_j \rangle$ ($0 \leq C \leq 1$) gives

$$S(V) = -\operatorname{tr}[(1-C)\log(1-C) + C \log C]$$

These formulas are widely used for lattice numerical calculations and for analytical evaluations in translationally invariant systems.

2. Explicit Results in Low and Higher Dimensions

• One Spatial Dimension (1+1D):

For massless free fields and a single interval of length $L$, the entropy exhibits universal scaling:

$$S(L) \sim \frac{c}{3}\log\left(\frac{L}{\epsilon}\right) + \text{const}$$

with $c$ the central charge and $\epsilon$ a UV cutoff. The so-called entropic $c$-function,

$c(L) = L \frac{dS(L)}{dL}$

is constant in conformal field theory ($c/3$), decaying at large $L$ for massive fields with subleading exponential corrections. For free scalars, the partition function in the presence of a branch cut can be linked to solutions of a Painlevé-type nonlinear ODE. For Dirac fields, bosonization maps the calculation to vertex operator correlators in sine-Gordon models.

• Two Spatial Dimensions (2+1D):

For regions with non-smooth boundaries (e.g., polygons), the entropy includes a universal logarithmic correction:

$$S(V) = (\text{area law}) + g_0[\partial V] \log\epsilon + S_0(V)$$

where $g_0(V) = \sum_{v_i\in \text{vertices}} s(x_i)$ is a sum over opening angles $x_i$ at the boundary with function $s(x)$. Explicit computation involves analysis of Green's functions on geometries with angular sectors.

• Higher Dimensions ($d>2$):

Universal terms can be extracted by dimensional reduction: considering a region $Z = B \times X$ (with $B$ large, $X$ compact), Fourier transforming in the large directions, and integrating over the resulting lower-dimensional entropic functions. The entropy structurally is

$$S(Z) \sim \frac{\text{Area}}{\epsilon^{d-1}} + (\text{subleading universal terms})$$

For four-dimensional CFTs, the universal logarithmic correction is controlled by the conformal anomaly coefficients and extrinsic curvature tensors of the entangling surface:

$$S_{\log} \sim s \log\epsilon, ~~ s = \frac{a}{720\pi} \int_{\partial V} (k_i^{\mu\nu} k_{i,\nu\mu} - (k_i^{\mu\mu})^2) + \ldots$$

with $a$, $c$ the central charges and $k_i^{\mu\nu}$ the second fundamental form of $\partial V$.

3. Analytic and Numerical Techniques

• Painlevé Analysis:

In 1+1D, exact formulas for the entropy as a function of mass and interval length are determined via nonlinear ODEs derived from the analytic structure of partition functions with branch cuts.

• Spectral and Heat Kernel Techniques:

In higher dimensions, spectral analysis and heat kernel expansion methods are crucial, particularly for extracting universal (logarithmic) corrections tied to the conformal anomaly.

• Lattice Simulations:

The real-time correlator method enables direct numerical computation of $S(V)$ in free boson and fermion theories, where the system is discretized and the relevant correlators are constructed; the entanglement spectrum and entropy are then efficiently calculable as matrix functions.

• Bosonization and Twisted Boundary Conditions:

For 1+1D Dirac fermions, bosonization translates the entropy problem to evaluating two-point functions of twist fields (vertex operators), facilitating analytic and numerical treatments.

• Dimensional Reduction for Higher $d$:

Calculations in higher dimensions can often be mapped onto effective lower-dimensional problems, in particular, integrating over towers of lower-dimensional massive modes.

4. Universal Scaling and Geometric Effects

The scaling structure of entanglement entropy reveals rich universal content:

• For regions with smooth boundaries in $d$-dimensions, the leading divergence is governed by the boundary area (the "area law"), with a subleading universal logarithmic term ("corner/cusp contribution") in even $d$.
• For example, polygons in 2+1D show that corner angles dominate the log term, with the universal $s(x)$ computable via solutions to ODEs derived from Green's function methods.
• In higher dimensions, universal terms include integrals over extrinsic and intrinsic curvatures of the boundary. These geometric invariants are tightly related to the structure of conformal anomalies, with the explicit dependence given in the logarithmic coefficient expressions.
• Non-smooth boundaries introduce additional universal logarithmic contributions, fully specified by the local geometry (e.g., local opening angles at vertices).

5. Comparison of Methods and Applications

• Equivalence of Replicated and Correlator Approaches:

The Euclidean replica and real-time correlator methods are shown to give consistent results for free fields, both theoretically and numerically.

• Utility for Lattice Calculations:

The correlator approach is extensively used for lattice simulations, enabling non-perturbative investigations, high-precision numerics, and direct comparison to analytic results.

• Physical Significance:

Explicit calculations in 1+1D exhibit the central charge in entropic scaling; in higher dimensions, calculations reveal the effect of geometry on leading and subleading entropy terms; such results inform understanding of quantum phase transitions, RG flows, and the physical meaning of universal coefficients.

• Mutual Information:

While entanglement entropy itself is UV-divergent (area-law divergence), mutual information between well-separated regions is finite and decays as a power law or exponentially (for massive fields), thus providing a cutoff-independent probe of entanglement structure.

6. Key Results and Conclusions

• Both replica and correlator approaches are rigorous and effective for free quantum field theories, including both analytic and numerical techniques.
• In 1+1D, exact scaling laws, Painlevé-type ODEs, and bosonization provide a highly precise picture of entanglement entropy and its universal content.
• In higher dimensions, geometric and anomaly considerations determine the structure of subleading and universal (logarithmic) corrections, and these are accessible both analytically (via heat kernel and spectral theory) and numerically.
• Universal coefficients (e.g., the central charge, conformal anomaly coefficients, or the function $s(x)$ for corners) encapsulate fundamental physical information about the underlying QFT, RG flows, and phase structure.
• Calculated entropies exhibit robust correspondence to mutual information and other information-theoretic measures, providing an organizing principle for interpreting quantum correlations in field theory.

Table: Summary of Main Formulas for Entanglement Entropy Calculation in Free QFTs

Dimension/Setting	Main Formula/Method	Universal Term(s)
1+1D, interval of length $L$	$S(L)\sim\frac{c}{3}\log(L/\epsilon)$	Central charge $c$
2+1D, region with corners	$S(V)=a\,R/\epsilon + \sum_v s(x_v)\log\epsilon+S_0$	$s(x)$ from angle at vertex
Higher $d$, smooth boundary	$$S(V) = (\text{Area})/\epsilon^{d-1} + s\log\epsilon +\ldots$$	$s$ from anomaly and geometry
Free bosons, region $V$	$$S(V) = \operatorname{tr}\left[(C+1/2)\log(C+1/2) - (C-1/2)\log(C-1/2)\right]$$	See above by dimension
Free fermions, region $V$	$$S(V) = -\operatorname{tr}[(1-C)\log(1-C) + C\log C]$$	See above by dimension

These methodologies, formulas, and their underlying technical foundations now form the basis for rigorous, quantitative studies of quantum information content in free quantum field theories, their lattice implementations, and their extensions to interacting cases and gravitational backgrounds (0905.2562).