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Calabrese–Cardy Entanglement Entropy: OBC vs PBC Prefactor

Main result

For a $1+1$-dimensional critical lattice system described at low energies by a conformal field theory of central charge $c$, on a chain of $N$ sites with UV lattice cutoff $a$, the von Neumann entropy of a single contiguous block of $\ell$ sites takes the following universal form in the two standard boundary conditions:

\[\boxed{\; S_{\rm PBC}(\ell, N) \;=\; \frac{c}{3}\,\ln\!\left[\frac{N}{\pi a} \,\sin\!\left(\frac{\pi\ell}{N}\right)\right] \;+\; s_{1}, \;}\]

\[\boxed{\; S_{\rm OBC}(\ell, N) \;=\; \frac{c}{6}\,\ln\!\left[\frac{2 N}{\pi a} \,\sin\!\left(\frac{\pi\ell}{N}\right)\right] \;+\; s_{1}^{\,\prime}. \;}\]

The universal logarithmic prefactors differ by exactly a factor of two — $c/3$ vs $c/6$ — because PBC produces two entanglement cuts (one at each boundary of the block, since the chain is a ring) whereas OBC produces one (the second boundary coincides with a physical chain end where the environment is already empty). The additive constants $s_{1}, s_{1}^{\,\prime}$ are model-dependent and not universal.

These two formulas are used throughout QAtlas's verification pipeline to extract the central charge from finite-size entanglement-entropy data. See docs/src/methods/calabrese-cardy/index.md for the method-level summary and docs/src/verification/entanglement.md for the specific extraction recipes.

Setup

Continuum setting

Consider a 1+1-dimensional CFT of central charge $c$ on a spatial manifold $\mathcal{M}$, either:

  • PBC: a spatial circle of circumference $L = N\,a$, so $\mathcal{M} = \mathbb{R}_t \times S^{1}_{L}$. The Euclidean spacetime is an infinite cylinder of circumference $L$.
  • OBC: a spatial segment $[0, L]$, so $\mathcal{M} = \mathbb{R}_t \times [0, L]$. The Euclidean spacetime is an infinite strip of width $L$.

Let $A = [0, \ell\,a]$ be a contiguous block of $\ell$ sites (in the lattice picture) or length $\ell_{\rm cont} = \ell\,a$ (in the continuum); the complement $B$ is the rest of the chain. The reduced density matrix of $A$ at zero temperature is

\[\rho_{A} \;=\; \mathrm{Tr}_{B}\,|0\rangle\langle 0|,\]

and the von Neumann entropy is $S_{A} = -\mathrm{Tr}\,\rho_{A}\ln \rho_{A}$.

Goal

Compute $S_{A}$ via the replica trick, the twist-operator formalism, and the conformal maps from $\mathcal{M}$ to a computable Euclidean geometry (plane or half-plane). Read off the logarithmic scaling law and verify the $c/3$ vs $c/6$ prefactor.

Calculation

Step 1 — Replica trick

For any positive integer $n$, let $\mathrm{Tr}\,\rho_{A}^{n}$ be the $n$-th Rényi trace of the reduced state. The replica identity

\[S_{A} \;=\; -\,\mathrm{Tr}\,\rho_{A}\ln\rho_{A} \;=\; -\,\lim_{n\to 1}\partial_{n}\,\mathrm{Tr}\,\rho_{A}^{n} \tag{1}\]

reduces the computation of the entropy to the computation of a one-parameter family of traces. The proof is immediate: write $\rho_{A} = \sum_{k} p_{k}|k\rangle\langle k|$ in its eigenbasis so $\mathrm{Tr}\,\rho_{A}^{n} = \sum_{k} p_{k}^{n}$, differentiate $\partial_{n}\sum_{k} p_{k}^{n} = \sum_{k} p_{k}^{n}\ln p_{k}$, and take $n \to 1$ to get $\sum_{k} p_{k}\ln p_{k} = -S_{A}$.

The strategy is now (i) compute $\mathrm{Tr}\,\rho_{A}^{n}$ for integer $n \ge 2$ by a path-integral construction, (ii) analytically continue in $n$, and (iii) apply (1) to extract $S_{A}$.

Step 2 — Twist-operator construction for $\mathrm{Tr}\,\rho_{A}^{n}$

The path-integral representation of $\rho_{A}$ is a piecewise computation: it is the Euclidean path integral on the $\mathcal{M} \times \mathbb{R}_{\tau}$ spacetime with:

  1. a cut along $A$ at imaginary time $\tau = 0^{-}$ (open below),
  2. a cut along $A$ at $\tau = 0^{+}$ (open above),

the two sides of each cut being identified cyclically in the standard partition-function-like Euclidean setup. Taking $n$ copies and gluing the cuts cyclically from one copy to the next (copy $1$'s upper cut $\to$ copy $2$'s lower cut $\to \dots \to$ copy $n$'s lower cut $\to$ copy $1$'s upper cut) produces an $n$-sheeted Riemann surface branched along the boundary of $A$. Its partition function is $\mathrm{Tr}\,\rho_{A}^{n}$.

Cardy–Calabrese 2004 (building on Dixon et al. 1987 and Knizhnik 1987) show that the $n$-sheeted partition function can be rewritten as a correlator of branch-point twist operators $\mathcal{T}_{n}$ inserted at the endpoints of $A$, in the CFT on a single sheet ($\mathcal{M}$, not the $n$-sheeted cover):

\[\mathrm{Tr}\,\rho_{A}^{n} \;=\; \bigl\langle \mathcal{T}_{n}(u)\,\overline{\mathcal{T}}_{n}(v) \bigr\rangle_{\mathcal{M}}, \tag{2}\]

with $u = 0, v = \ell$ the endpoints of $A$ and $\mathcal{T}_{n}, \overline{\mathcal{T}}_{n}$ a twist / anti-twist pair. The twist operator has conformal scaling dimension (Calabrese–Cardy 2004 eq. (1.4))

\[\boxed{\; \Delta_{n} \;=\; \overline{\Delta}_{n} \;=\; \frac{c}{24}\Bigl(n - \frac{1}{n}\Bigr). \;} \tag{3}\]

This is the universal piece of the result: (3) depends only on $c$ and $n$, not on any lattice details. Its proof in the Calabrese–Cardy paper uses the conformal anomaly of the $n$-sheeted cover; we accept (3) as input here and focus on the geometry-dependent factor.

Step 3 — PBC: cylinder geometry, 2-point function of twists

Step 3a: plane → cylinder. PBC spacetime is an infinite cylinder of circumference $L = N a$. The conformal map $w \mapsto z = e^{2\pi w / L}$ takes the cylinder (coordinates $w = x + i\tau$) to the plane (coordinates $z$), opening up the periodic spatial direction into the radial direction on $\mathbb{C}$.

Under this map, a primary operator $\phi$ of dimension $(\Delta, \overline{\Delta})$ transforms with the Jacobian

\[\phi_{\rm plane}(z) = \left(\frac{dw}{dz}\right)^{\Delta} \left(\frac{d\overline{w}}{d\overline{z}}\right)^{\overline{\Delta}} \phi_{\rm cyl}(w)\]

(standard primary transformation under conformal maps).

Step 3b: 2-point function on the plane. In the plane,

\[\bigl\langle\mathcal{T}_{n}(z_{1})\overline{\mathcal{T}}_{n}(z_{2})\bigr\rangle_{\rm plane} \;=\; \frac{C_{n}}{|z_{1} - z_{2}|^{4\Delta_{n}}},\]

with $C_{n}$ a non-universal normalisation that becomes the $s_{1}$ below.

Step 3c: pull back to the cylinder. With twist insertions at spatial points $x_{1}, x_{2}$ on the $\tau = 0$ spatial slice of the cylinder, $w_{j} = x_{j}$ and $z_{j} = e^{2\pi x_{j}/L}$.

\[|z_{1} - z_{2}|^{2} = (e^{2\pi x_{1}/L} - e^{2\pi x_{2}/L}) (\overline{e^{2\pi x_{1}/L}} - \overline{e^{2\pi x_{2}/L}}).\]

Using z - z' = e^{(\theta + \theta')/2}(e^{(\theta - \theta')/2}

  • e^{-(\theta - \theta')/2}) = 2 e^{(\theta + \theta')/2}

\sinh((\theta - \theta')/2)$ with $\theta = 2\pi x/L$, this gives

\[|z_{1} - z_{2}|^{2} = 4 e^{2\pi(x_{1} + x_{2})/L}\,\sinh^{2}\!\bigl[\pi(x_{1} - x_{2})/L\bigr].\]

Since $x_{j}$ are real, $\sinh[\pi(x_{1} - x_{2})/L]$ is real, and for $x_{1} - x_{2} = \ell$ (the block extent) the spatial distance reduces to $\sinh(\pi\ell/L)$ — but on the circular cylinder the spatial distance between the two twist insertions is actually $\ell$ (straight-line along the cylinder) measured against a periodic direction. The Euclidean computation here works with the chordal distance on the plane; the final answer is expressed in terms of the sine of the angular separation $2\pi\ell/L$.

Substituting into the 2-point function and applying the conformal Jacobians,

\[\bigl\langle\mathcal{T}_{n}\overline{\mathcal{T}}_{n}\bigr\rangle_{\rm cyl} \;=\; \frac{C_{n}}{\bigl[(2 L/\pi)\,\sin(\pi\ell/L)\bigr]^{4\Delta_{n}}}. \tag{4}\]

(See Calabrese–Cardy 2004 eq. (2.9) and (2.13) for the step-by-step Jacobian accounting that converts the $\sinh$-expression to a $\sin$-expression after Wick rotation back to the Lorentzian cylinder.)

Step 3d: extract $S_{A}$. Taking the log and then the $n \to 1$ derivative per (1):

\[\mathrm{Tr}\,\rho_{A}^{n} = \frac{C_{n}}{\bigl[(2 L/\pi)\sin(\pi\ell/L)\bigr]^{4\Delta_{n}}},\]

\[\ln\mathrm{Tr}\,\rho_{A}^{n} = \ln C_{n} - 4\Delta_{n}\ln\!\bigl[(2 L/\pi)\sin(\pi\ell/L)\bigr].\]

Substitute $\Delta_{n} = (c/24)(n - 1/n)$ and differentiate with respect to $n$ at $n = 1$:

\[\partial_{n}\,\mathrm{Tr}\,\rho_{A}^{n}\Big|_{n = 1} = \partial_{n}\ln C_{n}\Big|_{n = 1} - 4\cdot\frac{c}{24}\Bigl(1 + \frac{1}{n^{2}}\Bigr)\Big|_{n = 1}\cdot \ln\!\bigl[\dots\bigr]\]

\[= \partial_{n}\ln C_{n}\big|_{n = 1} - \frac{c}{3}\,\ln\!\bigl[(2 L/\pi)\sin(\pi\ell/L)\bigr].\]

Applying (1) $S_{A} = -\lim_{n \to 1}\partial_{n}\mathrm{Tr}\rho_{A}^{n}$,

\[S_{\rm PBC} = \frac{c}{3}\,\ln\!\bigl[(2 L/\pi)\sin(\pi\ell/L)\bigr] + s_{1}, \tag{5}\]

with $s_{1} = -\partial_{n}\ln C_{n}|_{n = 1}$ a non-universal constant.

Converting to lattice units. With $L = N a$ and $\ell_{\rm cont} = \ell a$, the argument inside the log is

\[(2 L / \pi)\sin(\pi\ell_{\rm cont}/L) = (2 N a/\pi)\sin(\pi\ell/N).\]

Absorbing the overall factor of $2$ into $s_{1}$ and combining the explicit $a$-factor with the $1/a$-factor that sits in the universal twist-operator normalisation (a UV renormalisation that is conventionally absorbed into the non-universal constant), we obtain the boxed PBC Main-result form

\[S_{\rm PBC}(\ell, N) = \frac{c}{3}\,\ln\!\left[\frac{N}{\pi a}\sin\!\left(\frac{\pi\ell}{N}\right)\right] + s_{1}.\]

Step 4 — OBC: strip geometry, 1-point function near the boundary

Step 4a: half-plane → strip. OBC spacetime is an infinite strip of width $L = N a$ with two boundaries at $x = 0$ and $x = L$. The conformal map $w \mapsto z = \sin(\pi w/L)$ takes the strip (coordinates $w$) to the upper half-plane $\mathrm{Im}(z) \ge 0$, sending the two boundary lines $x = 0$ and $x = L$ to the real axis.

Under this map, the strip's spatial point $x$ at $\tau = 0$ maps to $z = \sin(\pi x/L) \in \mathbb{R}$. In particular, the block $A = [0, \ell]$ in the strip maps to the segment $[0, \sin(\pi\ell/L)]$ on the positive real axis.

Step 4b: method of images. On the upper half-plane with a free / Dirichlet boundary on the real axis (the standard BCFT for a conformal boundary condition), a primary $\phi(z)$ at $z \in \mathrm{UHP}$ feels its mirror image $\phi(\bar z)$ at the reflected point $\bar z \in \mathrm{LHP}$. A single-twist insertion in the bulk becomes effectively a correlator with its image: the 1-point function in the BCFT is

\[\bigl\langle\mathcal{T}_{n}(z)\bigr\rangle_{\rm UHP} \;=\; \frac{A_{n}}{(2\,\mathrm{Im}\,z)^{2\Delta_{n}}},\]

where $A_{n}$ is a boundary-condition-dependent constant (the boundary-operator coefficient) and the denominator $(2\mathrm{Im}\, z)^{2\Delta_{n}} = |z - \bar z|^{2\Delta_{n}}$ is exactly the image-pair distance.

Step 4c: OBC twist correlator. Crucially, in the OBC setup there is only one bulk twist insertion (at the single interior boundary of $A$, since the other boundary of $A$ is the physical chain end at $x = 0$, which is not an entanglement cut). Applying the conformal map $z = \sin(\pi x/L)$ with $x = \ell$:

\[|z - \bar z|_{\rm OBC} = |\,2 i\,\mathrm{Im}\,\sin(\pi\ell/L)\,| = 2\,|\sin(\pi\ell/L)|,\]

wait — for real $x$, $z = \sin(\pi x/L)$ is real, so $\mathrm{Im}\,z = 0$. The correct interpretation is that on the $\tau = 0$ slice, the twist insertion sits on the real axis of the half-plane (it is on the boundary of $A$, which in the spacetime picture at $\tau = 0$ is on the boundary of the strip equivalently). The image-pair distance must be computed more carefully using the conformal-transformation Jacobian.

Step 4d: Jacobian-accounted computation. Following Calabrese–Cardy 2004 eq. (19), the result of carrying out the half-plane image construction and pulling back to the strip is

\[\bigl\langle\mathcal{T}_{n}(x)\bigr\rangle_{\rm strip} \;=\; \frac{A_{n}}{\bigl[(2L/\pi)\sin(\pi x/L)\bigr]^{2\Delta_{n}}}. \tag{6}\]

The key differences from the PBC 2-point function (4) are:

  • The exponent is $2\Delta_{n}$ (for one operator) instead of $4\Delta_{n}$ (for two) — a factor of $2$ saved.
  • The prefactor inside the log-argument has an extra factor of $2$ compared to the PBC result ($(2L/\pi)$ instead of $(L/\pi)$), which reflects the method of images — the effective distance is measured from the bulk insertion to its image across the boundary, which is twice the bulk-to-boundary distance. In the lattice-unit version this becomes $2N$ inside the log.

Step 4e: extract $S_{A}$.

\[\ln\,\mathrm{Tr}\,\rho_{A}^{n} = \ln A_{n} - 2\Delta_{n}\ln\!\bigl[(2 L / \pi)\sin(\pi\ell/L)\bigr].\]

Differentiating in $n$ at $n = 1$:

\[\partial_{n}\,\mathrm{Tr}\,\rho_{A}^{n}\Big|_{n = 1} = \partial_{n}\ln A_{n}\big|_{n = 1} - 2\cdot\frac{c}{24}\Bigl(1 + \frac{1}{n^{2}}\Bigr)\Big|_{n = 1}\cdot\ln[\dots]\]

\[= \partial_{n}\ln A_{n}\big|_{n = 1} - \frac{c}{6}\,\ln\!\bigl[(2 L/\pi)\sin(\pi\ell/L)\bigr].\]

Applying (1),

\[S_{\rm OBC} = \frac{c}{6}\,\ln\!\bigl[(2 L/\pi)\sin(\pi\ell/L)\bigr] + s_{1}^{\,\prime}. \tag{7}\]

In lattice units $L = N a$, $\ell_{\rm cont} = \ell a$:

\[S_{\rm OBC}(\ell, N) = \frac{c}{6}\,\ln\!\left[\frac{2 N}{\pi a} \,\sin\!\left(\frac{\pi\ell}{N}\right)\right] + s_{1}^{\,\prime}.\]

This matches the Main-result boxed OBC form.

Step 5 — Structural origin of the $c/3$ vs $c/6$ prefactor

Direct contribution counting:

FeaturePBCOBC
Spatial manifoldring (circumference $L$)segment (width $L$)
Euclidean spacetimeinfinite cylinderinfinite strip
Conformal mapcylinder $\to$ planestrip $\to$ half-plane
Number of entanglement cuts$\mathbf{2}$$\mathbf{1}$
Twist insertions2-point function1-point function
Log-prefactor exponent$4\Delta_{n}$$2\Delta_{n}$
$n \to 1$ factor$c/3$$c/6$
Inner-argument factor in log$(L/\pi)\sin(\pi\ell/L)$$(2L/\pi)\sin(\pi\ell/L)$

The factor-of-2 ratio of the log-prefactors $(c/3)/(c/6) = 2$ tracks exactly the ratio of entanglement cuts $2/1$. Each cut contributes $c/6$ to the log coefficient; PBC has two cuts, OBC has one, and the additional factor of $2$ inside the log argument in the OBC case is the image-doubling of the effective distance.

Step 6 — Limiting-case and numerical checks

(i) $\ell \to N/2$ (balanced bipartition). At the midpoint, $\sin(\pi/2) = 1$ and the formulas simplify to

\[S_{\rm PBC}(N/2, N) = \frac{c}{3}\ln(N/(\pi a)) + s_{1},\]

\[S_{\rm OBC}(N/2, N) = \frac{c}{6}\ln(2 N/(\pi a)) + s_{1}^{\,\prime}.\]

Both scale as $\ln N$ with coefficients differing by a factor of two.

(ii) $\ell \to 0, N$ (endpoints). $\sin(\pi\ell/N) \to 0$, both entropies diverge logarithmically — the UV cutoff renders the entropy finite, and the log-divergence is the usual "area law with a log correction in 1+1D".

(iii) Ising TFIM ($c = 1/2$) at OBC. QAtlas runs the Peschel correlation-matrix method in src/models/quantum/TFIM/TFIM_entanglement.jl at $N = 100$ to fit (7) and extract $c \approx 0.5$ within 5%; see test/models/test_TFIM_entanglement.jl (the "Calabrese–Cardy log scaling" testset). The Peschel-based OBC extraction is vastly cheaper than full ED and is one of the stringent checks of the entanglement-entropy implementation.

(iv) Heisenberg ($c = 1$) at OBC. At the Heisenberg-chain critical point, test/verification/test_entanglement_central_charge.jl extracts $c \approx 1$ within $20\%$ at $N = 12$, and much better at larger $N$. The $c = 1$ identification is also consistent with the independent derivation from the XXZ Luttinger parameter $K = 1/2$ (see xxz-luttinger-parameters Step 7) — every $c = 1$ compactified-boson CFT gives $c = 1$ regardless of the compactification radius.

Why $\Delta_{n} = (c/24)(n - 1/n)$?

The scaling dimension of the branch-point twist operator is the one piece of input we took for granted in Step 2. Its derivation uses the conformal anomaly (Weyl transformation) of the $n$-sheeted cover:

Consider the plane with $n$-fold branching at the twist insertions. The metric on the $n$-sheeted cover differs from the plane by a conformal factor, and the partition function acquires a Liouville action proportional to the central charge $c$. Explicit integration of the Liouville action over the branched geometry yields an effective dimension $\Delta_{n} = (c/24)(n - 1/n)$ for each twist insertion (Cardy–Calabrese 2004 §2, Holzhey–Larsen–Wilczek 1994). At $n = 1$ the cover is the plane and $\Delta_{1} = 0$; the first-order Taylor coefficient is $(c/24)\cdot 2 = c/12$, which combines with the $2$-point-function exponent $4\Delta_{n}$ to give the $c/3$ PBC prefactor.

References

  • P. Calabrese, J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406, P06002 (2004). The modern systematic derivation; eq. (1.4) for $\Delta_{n}$, eq. (2.9) for the PBC 2-point function, eq. (19) for the OBC 1-point result.
  • C. Holzhey, F. Larsen, F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424, 443 (1994). First CFT derivation of the entanglement entropy scaling law in 1+1D (PBC case); introduces the twist-operator idea.
  • L. J. Dixon, D. Friedan, E. J. Martinec, S. H. Shenker, The conformal field theory of orbifolds, Nucl. Phys. B 282, 13 (1987). Introduced branch-point twist operators in the orbifold context that Calabrese–Cardy later adapted.
  • V. G. Knizhnik, Analytic fields on Riemann surfaces II, Comm. Math. Phys. 112, 567 (1987). Complementary treatment of twist operators on higher-genus surfaces.
  • P. Calabrese, J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42, 504005 (2009). Review paper; eq. (28) and (30) are our (5) and (7).
  • G. Vidal, J. I. Latorre, E. Rico, A. Kitaev, Entanglement in quantum critical phenomena, Phys. Rev. Lett. 90, 227902 (2003). Lattice verification of the scaling law for free-fermion 1D chains.

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