Entanglement Entropy in One-Dimensional Conformal Field Theory

Entanglement entropy quantifies how much information is shared between two parts of a quantum system. For a wavefunction describing many degrees of freedom, the Hilbert space can be partitioned into a region A and its complement B. The reduced density matrix for A, obtained by tracing over B, captures the statistics of measurements confined to A. The von Neumann entropy $S=-\mathrm{Tr}\left({\rho }_A\ln {\rho }_A\right)$ measures how mixed this reduced state is and thus how entangled A is with B. In a one-dimensional conformal field theory (CFT), this quantity can often be computed analytically thanks to the theory’s rich symmetry structure. The calculator on this page implements a widely used formula for the entanglement entropy of a single interval of length L in an infinite line, incorporating the central charge c that characterizes the CFT.

The ground-state entanglement entropy for an interval of length L embedded in an otherwise infinite system with ultraviolet cutoff a is given by the celebrated expression derived using the replica trick and conformal mapping techniques: $S=\frac{c}{3}ln\left(\frac{L}{a}\right)$. Here, the central charge c encodes the number of low-energy degrees of freedom; it is 1 for a free boson or free Dirac fermion and larger for theories with multiple components or additional symmetries. The logarithmic dependence on L reflects the scale invariance inherent in conformal theories: doubling the interval length increases the entanglement by a constant amount proportional to c. The cutoff a represents a microscopic scale below which the continuum description breaks down, such as a lattice spacing or a short-distance regulator introduced to render the entropy finite.

At nonzero temperature T, thermal fluctuations also contribute to entropy. A CFT at temperature T is conveniently described on a cylinder with circumference β = $\frac{\mathrm{ħc}}{k}$. In this case, the entanglement entropy of an interval embedded in an infinite thermal state becomes $S=\frac{c}{3}ln\left(\frac{\beta }{\mathrm{\pi a}}\right)sinh\left(\frac{\mathrm{\pi L}}{\beta }\right)$. The calculator evaluates this more general formula while allowing T = 0 to recover the ground-state result. Users provide c, L, a, and T; the script computes β from fundamental constants and returns the entropy. For T = 0, the hyperbolic sine reduces to its argument and the expression collapses to the logarithm form above.

Central charge acts as a fingerprint for universality classes of 1D quantum critical systems. For example, the Ising model at criticality has c = 1/2, a Luttinger liquid of interacting fermions has c = 1, and the tricritical Ising model has c = 7/10. The higher the central charge, the more rapidly entanglement grows with subsystem size. Because c appears only as a multiplicative factor, the calculator enables quick comparison of entanglement scaling across distinct models. Physically, higher c implies a larger density of low-energy excitations capable of straddling the boundary between A and B, thereby enhancing quantum correlations.

While the formula may appear elementary, its derivation intertwines several deep concepts. The replica trick introduces n copies of the system and computes the trace of the n-th power of the reduced density matrix, $\mathrm{Tr}({{\rho }_A}^n)$. This quantity corresponds to the partition function of the CFT on an n-sheeted Riemann surface. Conformal symmetry allows one to map this surface to simpler geometries where the partition function is easier to evaluate, after which an analytic continuation to n → 1 yields the entropy. This elegant method not only reproduces the logarithmic formula but also generalizes to multiple intervals, finite systems, or higher dimensions. However, explicit results often require careful analysis of branch points and twist operator correlation functions, topics explored extensively in the CFT literature.

To illustrate practical usage, suppose we study a spin chain near a quantum critical point described by a CFT with c = 1. Let the lattice spacing be a = 1 Å (1 × 10⁻¹⁰ m) and consider an interval of L = 1 μm (1 × 10⁻⁶ m). Setting T = 0, the calculator evaluates $S=\frac{1}{3}ln\left(\frac{10-6}{10-10}\right)$ ≈ 1.53 natural units (nats). Using logarithms base 2 would yield entropy in bits. Increasing L by an order of magnitude adds approximately 0.77 nats, demonstrating the slow but unbounded growth of entanglement in critical systems.

The table below lists entanglement entropies for several representative central charges and interval lengths at zero temperature with cutoff a = 1 nm. Such tabulated values provide intuition for how drastically c and L impact quantum correlations.

c	L (nm)	S (nats)
0.5	100	0.768
1	100	1.536
2	100	3.072
1	1000	2.303
2	1000	4.605

Entanglement entropy has become a cornerstone of modern condensed matter and quantum information research. In numerical simulations using density-matrix renormalization group (DMRG) or matrix product states, the scaling of entanglement with subsystem size reveals whether a system is gapped or critical. In holography, the Ryu–Takayanagi formula relates entanglement entropy in a boundary CFT to the area of minimal surfaces in a higher-dimensional gravitational bulk, hinting at deep connections between quantum information and spacetime geometry. Experimental advances in cold atoms and trapped ions have enabled direct measurement of second Rényi entropies, validating theoretical predictions and opening new avenues for exploring many-body entanglement.

Despite its power, the formula implemented here has limitations. It assumes an infinite line without boundaries; finite systems require replacing the logarithm with sinusoidal functions to account for reflections. It also neglects oscillatory corrections arising from lattice effects or marginal operators, which can modulate the entropy. For T > 0, the formula presumes thermal equilibrium; quenched or driven systems may exhibit different behavior. Nevertheless, within its domain of applicability, the equation captures universal features insensitive to microscopic details, making it invaluable for diagnosing criticality.

The calculator operates purely in your browser using double-precision arithmetic. It computes β = ħc/(k_B T) when T > 0, otherwise treating β as effectively infinite. The result is displayed in natural units (nats). If you prefer bits, divide the output by ln 2. Because the expression involves logarithms and hyperbolic functions, extreme values of L or T may exceed floating-point limits, so ensure inputs remain in realistic ranges (for example, 10⁻¹² m < a < L < 10³ m and 0 ≤ T < 10⁶ K). The simplicity of the interface conceals the profound physics underlying each computation: a single number encapsulates a web of correlations spread across your quantum system.

In summary, entanglement entropy offers a window into the nonclassical fabric of many-body states. By plugging a handful of parameters into this calculator, you can explore how conformal symmetry, thermal fluctuations, and geometric scales combine to produce measurable quantum correlations. The extensive discussion above aims to demystify the origin and significance of the formula while providing sufficient detail for deep dives into the literature. Whether you are a student grappling with the concept of central charge or a researcher testing analytic predictions against numerical data, this tool provides a convenient starting point for quantitative exploration.