1D CFT Entanglement Entropy Calculator
What this calculator computes
This page estimates the von Neumann entanglement entropy of a single interval in a one-dimensional conformal field theory. In plain language, you pick a segment of a quantum system, call that segment the interval, and ask how strongly that segment is entangled with the rest of the line. In 1D CFT, the answer is unusually elegant: instead of depending on every microscopic detail, the leading behavior is controlled by a small set of quantities. The calculator asks for exactly those quantities and then evaluates the standard expression for an infinite system in either the ground state or a thermal state.
The four inputs correspond directly to the physics. The central charge c tells you how many effective low-energy degrees of freedom the theory carries. The interval length L is the size of the subsystem whose entropy you want. The UV cutoff a is the short-distance regulator that keeps the logarithm finite; physically, it stands in for the microscopic lattice spacing or other smallest resolved length. The temperature T lets you move from the zero-temperature result to the finite-temperature expression, where thermal correlations modify the pure logarithmic growth.
For the ground state of an infinite 1D CFT, the calculator uses the familiar formula
This expression immediately explains two key features of the output. First, the entropy grows only logarithmically with interval size, so doubling L does not double S. Second, the answer is only meaningful when L is larger than the cutoff a. If you make the interval smaller than or equal to the regulator, the logarithm becomes zero or negative in a way that is not physically appropriate for the continuum formula, so the calculator blocks that case.
When the temperature is nonzero, the page switches to the thermal interval formula
with inverse temperature
as implemented in the script on this page. That implementation restores SI constants and uses the speed of light for the characteristic velocity. In many field theory notes you will see units chosen so that ℏ = kB = v = 1; this calculator instead asks for L and a in meters and T in kelvin. If you are working with a condensed-matter realization whose low-energy velocity is not the speed of light, interpret the page result as the relativistic version of the formula or rescale your physical length scale accordingly.
How to choose the inputs without guessing
The central charge c is dimensionless. For a free compact boson or many simple benchmark CFTs, c = 1 is a common starting value. Minimal models have smaller rational values, while theories with multiple gapless modes can have larger central charge. Because the entropy is proportional to c, changing central charge rescales the final answer linearly. If you are comparing two theories with the same interval geometry and temperature, the ratio of their entanglement entropies is often largely the ratio of their central charges in this leading formula.
The interval length L should be entered as the physical size of the subsystem. The UV cutoff a should be the smallest meaningful distance in the same unit system, so meters here. What matters mathematically is the ratio L/a. A large ratio means there is a wide separation between the interval scale and the microscopic regulator, which is exactly the regime where the continuum expression is most useful. A ratio only slightly above one means you are probing the edge of the approximation, and the answer becomes very sensitive to microscopic details that the idealized CFT does not try to model.
The temperature T is entered in kelvin. If you want the ground-state result, set T = 0. That does not mean the calculator assumes your lab sample is literally at absolute zero; it means you want the zero-temperature formula, which is often appropriate when the thermal correlation length is much larger than the interval. As T rises, the quantity β shrinks, the hyperbolic sine term becomes more important, and the entropy crosses over from pure ground-state scaling to thermally influenced behavior.
If you are unsure whether your temperature is effectively low or high for a chosen interval, compare L with β. When L ≪ β, the finite-temperature formula reduces smoothly toward the ground-state logarithm. When L becomes comparable to or larger than β, thermal effects matter much more strongly. That is why this page shows β in the result table whenever T is nonzero: it helps you see the physically relevant comparison scale instead of only the final entropy value.
What the result means
The calculator reports entropy in nats, not bits, because the formulas use the natural logarithm. If you want bits, divide the reported entropy by ln 2. This is not a change in physics, only a change of logarithm base. A result of 2.30 nats, for example, is about 3.32 bits. The number is dimensionless even though you enter lengths and temperature, because those dimensional inputs only appear in unitless combinations such as L/a or πL/β.
When you interpret a computed value, ask three practical questions. First, is the sign and size plausible? Entanglement entropy from this formula should be finite and typically positive in the physical regime of interest. Second, is the scale separation credible? A huge answer obtained from a tiny cutoff may simply reflect that you have pushed the continuum approximation far beyond any realistic microscopic model. Third, do the dependencies move in the direction you expect? Increasing c should increase S. Increasing L at fixed a should increase S. Raising temperature can also increase the interval entropy once thermal effects are relevant.
The page also includes a copy button that summarizes the numerical scenario. That is useful when you are comparing several possible intervals or temperatures. Entanglement calculations are often most informative when used comparatively: one interval versus another, one theory versus another, or ground state versus thermal state. Saving a short summary makes those comparisons reproducible instead of relying on memory.
Worked examples
Here is a simple ground-state check. Suppose you choose c = 1, L = 10-6 m, a = 10-9 m, and T = 0. The ratio L/a = 1000, so the entropy is
S = (1/3) ln(1000) ≈ 2.302585 nats.
This is a good test case because the numbers are easy to inspect by hand. If the result panel shows approximately 2.302585 nats, the inputs are being interpreted consistently. If you then change only the central charge to c = 2, the entropy doubles to about 4.605170 nats. That linear scaling is one of the quickest sanity checks available on the page.
Now consider a finite-temperature example with c = 1, L = 2 × 10-6 m, a = 10-9 m, and T = 50 K. The script computes β = ℏc/(kBT), which is about 4.58 × 10-5 m in this implementation. Since L is smaller than but not absurdly smaller than β, the thermal correction is present but not dominant. You should get an entropy of roughly 2.53 nats, a little larger than the zero-temperature value for a comparable interval scale. The precise number matters less than the pattern: thermal physics shifts the answer through the sinh factor and the crossover length set by β.
One more instructive comparison is to keep c and a fixed while varying only L. Because the formula contains a logarithm, increasing the interval by a factor of ten adds a constant amount, not a multiplicative explosion. That is a very different dependence from an area-law estimate in higher dimensions or a simple linear extensive quantity. If you are new to entanglement calculations, this logarithmic growth is the main feature to remember: 1D critical systems behave differently from generic gapped systems.
Why the logarithm and the thermal sinh term appear
At zero temperature, the logarithm reflects scale invariance. A conformal field theory has no preferred length scale in the continuum description, so the interval size L can only appear relative to the cutoff a. That symmetry constraint is why the answer depends on ln(L/a) rather than on an arbitrary polynomial in L. The central charge then supplies the universal prefactor that distinguishes one CFT from another.
At finite temperature, Euclidean time becomes compact with circumference set by the inverse temperature. Conformal mapping turns the problem from the plane into a cylinder, and the logarithm of a simple distance becomes the logarithm of a hyperbolic sine. That is the origin of the sinh(πL/β) term. You do not need to reproduce the full derivation to use the calculator well, but it helps to know that the formula is not arbitrary. Every piece reflects a physical or geometric scale: the interval size, the short-distance cutoff, the inverse temperature, and the central charge.
At a more abstract level, any calculator can be viewed as an output function of a few inputs. The two MathML expressions below are retained because they capture that generic viewpoint, even though this page specializes it to entanglement entropy.
For this specific calculator, the abstract lesson is simple: the entropy is not a mysterious black box. It is a well-defined function of four inputs, and each input has a physically interpretable role. If changing a parameter gives a surprising trend, you can usually track the reason to one of those roles rather than assuming the calculator failed.
Assumptions, edge cases, and sanity checks
This page is designed for the standard single-interval formula on an infinite line. It does not include boundary entropies, finite-size circles, multiple disjoint intervals, Rényi index dependence, or model-specific subleading corrections. If your problem involves any of those ingredients, the calculator is still useful as a baseline estimate, but not as the final word.
The script also defends against numerical edge cases. If any field is nonnumeric, computation stops. If L or a is nonpositive, the formula is invalid. If L ≤ a, the page warns that the logarithm would be nonphysical for the continuum setup. At very large L or very high T, the hyperbolic sine can overflow in ordinary floating-point arithmetic, so the script asks you to reduce the input scale. That is not a conceptual failure of the formula; it is simply a reminder that numerical implementation has finite limits.
A final practical tip: if you are working in natural units from a paper, resist the temptation to paste those values into the form unchanged. This page expects meters and kelvin in the finite-temperature branch because the constants are restored explicitly. Convert first, then compute. If you only want a purely symbolic or unit-free result, set T = 0 and focus on the ratio L/a, which is the cleanest universal quantity in the ground-state formula.
Mini-game: Tune the entangling interval
This optional arcade mini-game turns the quasiparticle picture behind interval entanglement into something you can feel. Colored partner excitations separate across a one-dimensional chain. Your job is to drag the interval so that, at each scan pulse, you cut as many entangled pairs as possible with the interval boundary. Exactly one member of a pair inside the interval scores; both inside or both outside means that pair does not contribute to the cut. As the round heats up, thermal noise appears and the scan cadence speeds up, echoing the crossover from the ground-state formula to the finite-temperature expression. The game never changes the calculator result; it is simply a quick, replayable way to build intuition for what larger entropy means.
Why this mechanic fits the calculator: in the quasiparticle picture, entropy rises when your chosen interval cuts more entangled pairs. The on-screen interval is your stand-in for L, the minimum width marker represents the UV scale a, and the later thermal clutter stands in for finite-temperature effects related to β.