# Conformal Field Theory Ground States as Critical Points of an Entropy
Function

Ting-Chun Lin（林鼎鈞  Department of Physics, University of California at San
Diego, La Jolla, CA 92093, USA  Hon Hai Research Institute, Taipei, Taiwan
John McGreevy  Department of Physics, University of California at San Diego,
La Jolla, CA 92093, USA

###### Abstract

We derive an entropy formula satisfied by the ground states of 1+1D conformal
field theories. The formula implies that the ground state is the critical
point of an entropy function. We conjecture that this formula may serve as an
information-theoretic criterion for conformal field theories, which differs
from the conventional algebraic definition. In addition to these findings, we
use the same proof method to extract the six global conformal generators of
the conformal field theory from its ground state. We validate our results by
testing them on different critical lattice models with excellent agreement.

Quantum entanglement and quantum information have played important roles in
the study of quantum matter. For 2+1D gapped phases, this includes topological
entanglement entropy [1, 2], entanglement spectrum [3], and the recent work on
chiral central charge [4] and minimal total central charge [5]. For 1+1D
conformal field theories (CFTs), the entanglement entropy [6, 7, 8] of the
ground state is related to the central charge. These tools are useful to
distinguish quantum phases analytically and numerically.

More ambitiously, one may wonder if the reverse holds, namely, could there be
entanglement conditions that are satisfied and only satisfied by ground states
for certain quantum phases? One proposal is given by Shi, Kato, and Kim [9]
where they stated two conditions that are conjectured to be satisfied and only
satisfied by the ground states of topological quantum field theories111More
precisely, any state that satisfies the conditions is the ground state of a
Hamiltonian that belongs to a topological phase. And any topological phase has
a Hamiltonian whose ground state satisfies the conditions.. Remarkably, using
the two conditions, they are able to derive many known properties of 2+1D
topological orders. Another nice feature is that because the two conditions
only involve the entropies of local regions, the conditions can be checked
easily. This program is called entanglement bootstrap and can be applied to
other settings, including gapped domain walls [10] and higher dimensions [11].

Given the success of entanglement bootstrap for gapped topological orders, one
may wonder if a similar set of entropy conditions exists for gapless states.
In this work, we propose a new set of ultra-violet (UV)-independent entropy
conditions that apply to the ground states of 1+1D unitary CFTs. The
conditions additionally hold for 1+1D gapped phases at RG fixed points. In the
spirit of entanglement bootstrap, we conjecture that these conditions
characterize the ground states of 1+1D RG fixed points with Lorentz
symmetry.222Notice that in this statement we make a distinction between CFT
and RG fixed point. In fact we know examples of lattice wavefunctions that
satisfy Equation 1 that are not CFT ground states. Nevertheless, they are
renormalization group fixed points. To be clear, for the more specific case of
CFT, this paper shows that Equation 1 is a necessary condition, and we
conjecture that Equation 1 can also be viewed as a sufficient condition for RG
fixed points. Furthermore, the following argument suggests that demanding
Equation 1 for multiple choices of regions may be a sufficient condition for
CFT. The condition Equation 1 is a strong constraint; the fact that there are
solutions to Equation 1 is itself surprising. Notice that each vector equation
has roughly the same number of constraints as the number of variables in
$|\psi\rangle$, and therefore imposing the vector equation for multiple
choices of intervals is overconstrained. When the state has $\geq 6$
intervals, if we require the vector equation for every choice of three
contiguous intervals, there are at least $2$ independent vector equations,
generically. That means, in general, we do not expect any $|\psi\rangle$ that
satisfy every vector equation to exist. However, as we have shown, each CFT
ground state is a solution to the vector equations. This is suggesting that
these vector equations have intricate relations that are yet to be understood,
and it is possible that this relation between equations has something to do
with the Virasoro algebra.

Let $A,B,C$ be three consecutive intervals; see Figure 1. Our essential
observation is that the ground state of a 1+1D unitary CFT $|\psi\rangle$
satisfies

$K_{\Delta}\propto I\qquad\text{and}\qquad
K_{\Delta}|\psi\rangle\propto|\psi\rangle$ (1)

where

$\displaystyle K_{\Delta}$
$\displaystyle\coloneqq(K_{AB}+K_{BC})-\eta(K_{A}+K_{C})$
$\displaystyle\qquad-(1-\eta)(K_{B}+K_{ABC})$ (2)

and $K_{X}\coloneqq-\log\rho_{X}$555The reader might worry that
$-\log\rho_{X}$ is not well-defined when $\rho_{X}$ has zero eigenvalues.
Here, we note that the equation $K_{\Delta}|\psi\rangle\propto|\psi\rangle$
remains well-defined, because it projects out the problematic eigenvectors.
This point is discussed in further detail after the derivation of the main
results. is the entanglement Hamiltonian of the reduced density matrix
$\rho_{X}\coloneqq\operatorname{Tr}_{\bar{X}}|\psi\rangle\langle\psi|$, $I$ is
the identity operator, and $\eta$ is the cross ratio of the intervals.
Moreover, when the central charge $c$ is known, the proportionality constant
is given by

$K_{\Delta}=\frac{c}{3}h(\eta)\qquad\text{and}\qquad
K_{\Delta}|\psi\rangle=\frac{c}{3}h(\eta)|\psi\rangle$ (3)

where $h(\eta)\coloneqq-\eta\log\eta-(1-\eta)\log(1-\eta)$ is the binary
entropy function. Finally, we observe that Eq. (1) is the condition for
$|\psi\rangle$ to be a critical point of the following function:

$\displaystyle S_{\Delta}(|\psi\rangle)$
$\displaystyle\coloneqq(S_{AB}+S_{BC})-\eta(S_{A}+S_{C})$
$\displaystyle\qquad-(1-\eta)(S_{B}+S_{ABC})$ (4)

where $S_{X}\coloneqq S(\rho_{X})$ is the von Neumann entanglement entropy
between $X$ and $\bar{X}$. The function is nonnegative because it is a convex
combination of two nonnegative quantities, by weak monotonicity
$S_{AB}+S_{BC}-S_{A}-S_{C}\geq 0$ and strong subadditivity
$S_{AB}+S_{BC}-S_{B}-S_{ABC}\geq 0$ [13, 14]. All the statements above can be
extended to the ground state on a finite circle and the thermal state on an
infinite line.

The formulae discussed above were derived in continuum CFT. In many physical
realizations, CFT arises as an approximation to a lattice model. Importantly,
we show numerically that the equations hold approximately for various lattice
models and the error
$\Big{\lvert}K_{\Delta}|\psi\rangle-\frac{c}{3}h(\eta)|\psi\rangle\Big{\rvert}$
decays as the number of sites increases (as a power law).

(a)$x_{1}$$x_{2}$$x_{3}$$x_{4}$$A$$B$$C$$\eta=\frac{(x_{2}-x_{1})(x_{4}-x_{3})}{(x_{3}-x_{1})(x_{4}-x_{2})}$(b)$x_{1}$$x_{2}$$x_{3}$$x_{4}$$A$$B$$C$$\eta=\frac{\sin(\pi
x_{12}/L)\sin(\pi x_{34}/L)}{\sin(\pi x_{13}/L)\sin(\pi x_{24}/L)}$ Figure 1:
(a) Three consecutive intervals on an infinite system with the corresponding
cross ratio $\eta$. (b) Three consecutive intervals on a circle with
circumference $L$. Here, $x_{ij}\coloneqq x_{j}-x_{i}$ denotes the distance
between $x_{i}$ and $x_{j}$.

Therefore, after obtaining the basic formulas, we explore their implications
on states supported on a discrete lattice, by the following device: divide up
the spatial direction into a collection of line segments, and regard each
segment as a site of a lattice model with infinite-dimensional local Hilbert
space. We thereby identify new equations that could potentially hold for
lattice models with finite-dimensional local Hilbert space, which can then be
validated numerically. Specifically, we express the entanglement Hamiltonians
of intervals and the six global conformal generators as a combination of
1-site and 2-site entanglement Hamiltonians of the CFT ground state. This has
two important implications. First, when the state satisfies the entropy
formula, the six global conformal generators have the expected scaling
properties under real-space RG. This provides evidence that a state satisfying
the entropy formula enjoys conformal symmetry. Second, the result yields an
algorithm that can reconstruct the CFT local Hamiltonian from the CFT ground
state. This is remarkable since the parent Hamiltonian construction [15] that
applies to matrix product states cannot be applied to gapless phases due to
their long-range correlations. This algorithm offers an alternative method for
recovering the local Hamiltonians from states near RG fixed points, including
gapped and gapless phases. This result corroborates the general principle that
a single representative wavefunction contains all the universal data about a
state of matter.

We want to highlight an implicit theme of this work, which aims to go beyond
the traditional algebraic formulation of CFT and provide an alternative
analytic formulation. Typically, CFTs are defined using algebras with
equalities, which do not allow for the discussion of _approximate CFTs_.
However, many examples exist that we would like to categorize as approximate
CFTs, such as QFTs that are slight perturbations of CFTs or critical lattice
models that are CFTs in the IR limit. Because these models have different
Hilbert spaces, finding a criterion that applies to all cases is challenging.
To address this issue, we utilize the entanglement entropy and entanglement
Hamiltonian, which are agnostic to the Hilbert space. We propose to define
approximate CFTs as systems whose ground state approximately satisfies Eq. 1.
Further research is needed to determine the usefulness of this proposal.

##  I CFTs and their entanglement properties

Conformal field theories (CFTs) are field theories with scale invariance. Such
scale invariance often implies a larger invariance called the conformal
symmetries, which include transformations that locally look like rescalings.
These conformal symmetries appear naturally in many physical systems,
including the fixed points of the RG flow and the critical points of
statistical models.

For 1+1D CFT, the entanglement entropy [8] and the entanglement Hamiltonian
(EH) [16] of the ground state on an interval $[x_{1},x_{2}]$ are known to be

$S_{[x_{1},x_{2}]}=\frac{c}{3}\log\frac{x_{2}-x_{1}}{\epsilon}$ (5)

and

$K_{[x_{1},x_{2}]}=2\pi\int_{x_{1}}^{x_{2}}dx\frac{(x-x_{1})(x_{2}-x)}{x_{2}-x_{1}}T_{00}(x)+\textnormal{const.}$
(6)

where $c$ is the central charge, $\epsilon$ is the uniform UV cutoff, $T$ is
the stress-energy tensor, and const. is a number that depends on the UV cutoff
$\epsilon$. These two equations are the key to showing the main results. As we
will see, even though both equations suffer from UV divergences, the
combinations in Eq. (1) and Eq. (3) are free from UV divergences.

##  II Derivation of the main results

We first show Eq. (1) and Eq. (3) for the ground state of a 1+1D CFT on an
infinite system. Then we show the equivalence of Eq. (1) to the critical point
condition of $S_{\Delta}$ defined in Eq. (4). Let $A=[x_{1},x_{2}]$,
$B=[x_{2},x_{3}]$, and $C=[x_{3},x_{4}]$ be three consecutive intervals. For
convenience, we define the function
$f_{[x^{\prime},x^{\prime\prime}]}(x)=\frac{(x-x^{\prime})(x^{\prime\prime}-x)}{x^{\prime\prime}-x^{\prime}}1_{[x^{\prime},x^{\prime\prime}]}$
where $1_{[x^{\prime},x^{\prime\prime}]}$ is the indicator function. By Eq.
(6),

$\displaystyle(K_{AB}+K_{BC})-\eta(K_{A}+K_{C})-(1-\eta)(K_{B}+K_{ABC})$
$\displaystyle\hskip
15.00002pt=\int_{-\infty}^{\infty}dx\,\big{(}(f_{AB}+f_{BC})-\eta(f_{A}+f_{C})$
$\displaystyle\hskip
50.00008pt-(1-\eta)(f_{B}+f_{ABC})\big{)}T_{00}(x)+\textnormal{const.}$ (7)

A straightforward calculation shows
$(f_{AB}+f_{BC})-\eta(f_{A}+f_{C})-(1-\eta)(f_{B}+f_{ABC})=0$ from which
Equation 1 follows.

To obtain the ratio in Eq. (3), we multiply $\langle\psi|$ on both sides.
Because $\langle\psi|K_{A}|\psi\rangle=S_{A}$, the left hand side becomes
$(S_{AB}+S_{BC})-\eta(S_{A}+S_{C})-(1-\eta)(S_{B}+S_{ABC})$ which evaluates to
$\frac{c}{3}h(\eta)$ using Eq. (5).

Now we show that Eq. (1) is precisely the condition for vanishing variation of
$S_{\Delta}$ with respect to the state. Recall that $|\psi\rangle$ is a
critical point if the gradient is $0$ subject to $(\langle\psi|+\langle
d\psi|)(|\psi\rangle+|d\psi\rangle)=1$, i.e., $\langle
d\psi|\psi\rangle+\langle\psi|d\psi\rangle=0$. To compute the gradient of
$S_{\Delta}=(S_{AB}+S_{BC})-\eta(S_{A}+S_{C})-(1-\eta)(S_{B}+S_{ABC})$ we use
the first order derivative of the entanglement entropy:

$dS_{X}(|\psi\rangle)=\langle
d\psi|K_{X}|\psi\rangle+\langle\psi|K_{X}|d\psi\rangle$ (8)

as reviewed in the Supplementary Material. Therefore, the critical point
$|\psi\rangle$ has

$dS_{\Delta}=\langle
d\psi|K_{\Delta}|\psi\rangle+\langle\psi|K_{\Delta}|d\psi\rangle=0$ (9)

for all $|d\psi\rangle$ satisfying $\langle
d\psi|\psi\rangle+\langle\psi|d\psi\rangle=0$. Recall
$K_{\Delta}=(K_{AB}+K_{BC})-\eta(K_{A}+K_{C})-(1-\eta)(K_{B}+K_{ABC})$. This
is equivalent to $K_{\Delta}|\psi\rangle\propto|\psi\rangle$ in Eq. (1). Note
that the left equation in Eq. (1), $K_{\Delta}\propto I$, does not follow from
$K_{\Delta}|\psi\rangle\propto|\psi\rangle$ and the critical point condition
in general, but does follow under certain assumptions that will be explained
later in this section.

More generally, the result can be extended to the ground state on a finite
circle and the thermal state on an infinite line. In both cases, the
entanglement entropy and the entanglement Hamiltonian are known [8, 16], so
the proof strategy still works. The difference is to replace the cross-ratio
$\eta$ with the effective cross-ratio $\eta_{\text{eff}}$ where
$\eta^{L}_{\text{eff}}=\frac{\sin(\pi x_{12}/L)\sin(\pi x_{34}/L)}{\sin(\pi
x_{13}/L)\sin(\pi x_{24}/L)}$ for the ground state on a circle of length $L$
and $\eta^{\beta}_{\text{eff}}=\frac{\sinh(\pi x_{12}/\beta)\sinh(\pi
x_{34}/\beta)}{\sinh(\pi x_{13}/\beta)\sinh(\pi x_{24}/\beta)}$ for the
thermal state on an infinite line of inverse temperature $\beta$.

We now discuss the subtle relation between the operator equation
$K_{\Delta}=\frac{c}{3}h(\eta)$ and the vector equation
$K_{\Delta}|\psi\rangle=\frac{c}{3}h(\eta)|\psi\rangle$. We first present an
argument that shows their equivalence for quantum field theories, then discuss
the difference in their approximate versions. It is clear that the operator
equation implies the vector equation and typically the other way does not hold
for finite dimensional Hilbert spaces. One extreme example for finite
dimensional Hilbert spaces is that $K_{X}=-\log\rho_{X}$ could have
singularities when $\rho_{X}$ has zero eigenvalues, while $K_{X}|\psi\rangle$
is still well-defined with continuity, because $\lim_{\lambda\to
0}\lambda\log\lambda=0$. On the other hand, for quantum field theories, due to
the Reeh-Schlieder theorem [17], $|\psi\rangle\in\mathcal{H}_{ABCD}$ is
cyclic, which means $O_{D}|\psi\rangle$ is dense in $\mathcal{H}_{ABC}$ for
$O_{D}$ supported on $D$. Because $K_{\Delta}$ is only supported on
$\mathcal{H}_{ABC}$, $K_{\Delta}$ and $O_{D}$ commutes, and we have
$K_{\Delta}O_{D}|\psi\rangle=\frac{c}{3}h(\eta)O_{D}|\psi\rangle$. Since
$O_{D}|\psi\rangle$ is dense, this implies the operator equation
$K_{\Delta}=\frac{c}{3}h(\eta)$.

In the approximate case, one might hope that
$K_{\Delta}\approx\frac{c}{3}h(\eta)$ in the operator norm and
$K_{\Delta}|\psi\rangle\approx\frac{c}{3}h(\eta)|\psi\rangle$ in the vector
norm. Numerically, we observe that the approximate operator equation does not
hold, while the approximate vector equation holds. This suggests that the
vector equation is more valid because it is stable against perturbation.
Nevertheless, we suspect that the operator equation remains stable under a
different norm which requires further investigation.

##  III Local-to-global implications

In the last section, we established that $K_{\Delta}=\frac{c}{3}h(\eta)$ for
any consecutive intervals $A,B,C$. In this section, we apply this relation to
intervals with integer-valued endpoints which can be associated with a lattice
model. The benefit of having a lattice model is that it allows for concrete
numerical verifications. We demonstrate that several quantities can be
expressed as a sum of local operators. First, we prove that the EH of an
interval can be expressed as a sum of 1-site and 2-site EHs from Eq. (1). This
implies that the reduced density matrix on an arbitrary interval can be
reconstructed from the 2-site reduced density matrix.

We first introduce the notation. To simplify the presentation, we define the
reduced EH as $\widetilde{K}_{X}\coloneqq
K_{X}-\langle\psi|K_{X}|\psi\rangle$, which is the EH shifted by a constant so
that $\langle\psi|\widetilde{K}_{X}|\psi\rangle=0$. This removes the constant
term in Eq. (6), which implies that $\widetilde{K}_{\Delta}=0$. The reduced EH
of an interval $[a,b]$ is denoted as $\widetilde{K}_{[a,b]}$. In the following
discussion, we only consider intervals with $a,b\in\mathbb{Z}$, so that they
can be associated with a lattice model. The corresponding lattice model
regards each interval of length $1$, $[a,a+1]$, as a single site666Of course,
these ‘sites’ have infinite-dimensional Hilbert spaces, and this is just a
rewriting of the continuum CFT information. In what follows, we will use the
same relation in systems where the Hilbert space of each site is replaced by a
finite-dimensional Hilbert space.. Therefore, the interval $[a,b]$ corresponds
to $b-a$ sites. From now on, we refer to the reduced EH simply as EH.

We now show that any EH of an interval can be written as a sum of 1-site and
2-site EHs, using Eq. (1) recursively. We begin with the example of a 3-site
EH. Take $A=[0,1],B=[1,2],C=[2,3]$. The equivalent form of Eq. (1)
$\widetilde{K}_{\Delta}=0$ implies

$\widetilde{K}_{[0,2]}+\widetilde{K}_{[1,3]}-\frac{1}{4}\widetilde{K}_{[0,1]}-\frac{1}{4}\widetilde{K}_{[2,3]}-\frac{3}{4}\widetilde{K}_{[1,2]}-\frac{3}{4}\widetilde{K}_{[0,3]}=0.$
(10)

Therefore,

$\widetilde{K}_{[0,3]}=-\frac{1}{3}\widetilde{K}_{[0,1]}+\frac{4}{3}\widetilde{K}_{[0,2]}-\widetilde{K}_{[1,2]}+\frac{4}{3}\widetilde{K}_{[1,3]}-\frac{1}{3}\widetilde{K}_{[2,3]}.$
(11)

Similarly, for a 4-site EH, take $A=[0,1],B=[1,2],C=[2,4]$, and we have

$\displaystyle\widetilde{K}_{[0,4]}$
$\displaystyle=-\frac{1}{2}\widetilde{K}_{[0,1]}+\frac{3}{2}\widetilde{K}_{[0,2]}-\widetilde{K}_{[1,2]}+\frac{3}{2}\widetilde{K}_{[1,4]}-\frac{1}{2}\widetilde{K}_{[2,4]}$
$\displaystyle=-\frac{1}{2}\widetilde{K}_{[0,1]}+\frac{3}{2}\widetilde{K}_{[0,2]}-\frac{3}{2}\widetilde{K}_{[1,2]}+2\widetilde{K}_{[1,3]}$
$\displaystyle\qquad\qquad-\frac{3}{2}\widetilde{K}_{[2,3]}+\frac{3}{2}\widetilde{K}_{[2,4]}-\frac{1}{2}\widetilde{K}_{[3,4]}.$
(12)

More generally, for an $n$-site EH, take $A=[0,1],B=[1,2],C=[2,n]$, and we
have

$\displaystyle\widetilde{K}_{[0,n]}$
$\displaystyle=-\frac{n-2}{n}\widetilde{K}_{[0,1]}+\frac{2(n-1)}{n}\widetilde{K}_{[0,2]}-\widetilde{K}_{[1,2]}$
$\displaystyle\qquad\qquad+\frac{2(n-1)}{n}\widetilde{K}_{[1,n]}-\frac{n-2}{n}\widetilde{K}_{[2,n]}.$
(13)

By recursively expanding $\widetilde{K}_{[1,n]}$ and $\widetilde{K}_{[2,n]}$,
we obtain

$\widetilde{K}_{[0,n]}=\sum_{j=-\infty}^{\infty}f_{2}(j+1)\widetilde{K}_{[j,j+2]}+f_{1}(j+\frac{1}{2})\widetilde{K}_{[j,j+1]}$
(14)

where $f_{2}(x)=\frac{2x(n-x)}{n}1_{[0,n]}$,
$f_{1}(x)=\frac{-2x(n-x)+\frac{3}{2}}{n}1_{[0,n]}$, and $1_{[0,n]}$ is the
indicator function. Therefore, any EH can be written as a sum of 1-site and
2-site EHs, which means the reduced density matrix on arbitrary intervals can
be reconstructed from the 2-site reduced density matrix. This could be viewed
as a solution to the quantum marginal problem for CFTs where the Petz recovery
map does not apply.

######  Remark 1.

We note that the decomposition of EH into a sum of 1-site and 2-site EHs is
similar to Eq. (6) in field theory, where EH is a sum of local terms
$T_{00}(x)$. We also remark that there are different ways to decompose
$\widetilde{K}_{[0,n]}$, yet all of them lead to the same expression. This
nontrivial property implies certain consistency relations between
$\widetilde{K}_{\Delta}=0$ across different choices of $A,B,C$.

_The state contains the universal data—_ The same logic that leads to Eq. (1)
implies that the CFT Hamiltonian (and hence all of the CFT data) can be
extracted from the groundstate. In fact, the same is true of all 6 global
conformal generators $H,P,D,M_{10},C_{0},C_{1}$ which are the Hamiltonian,
momentum, dilatation, boost, and special conformal generators. We first find
their corresponding expressions for CFT ground states when Eq. (6) applies.
Because $H=\int_{-\infty}^{\infty}dx\,T_{00}(x)$,
$M_{10}=\int_{-\infty}^{\infty}dx\,xT_{00}(x)$,
$C_{0}=\int_{-\infty}^{\infty}dx\,x^{2}T_{00}(x)$, applying Eq. (6), we have

$\displaystyle H$
$\displaystyle=\frac{1}{\pi}\sum_{j=-\infty}^{\infty}\left(\widetilde{K}_{[j,j+2]}-\widetilde{K}_{[j,j+1]}\right)$
$\displaystyle=\frac{1}{\pi}\sum_{j=-\infty}^{\infty}\widetilde{K}^{\prime}_{[j,j+2]},$
(15) $\displaystyle M_{10}$
$\displaystyle=\frac{1}{2\pi}\sum_{j=-\infty}^{\infty}\left((2j+2)\widetilde{K}_{[j,j+2]}-(2j+1)\widetilde{K}_{[j,j+1]}\right)$
$\displaystyle=\frac{1}{\pi}\sum_{j=-\infty}^{\infty}(j+1)\widetilde{K}^{\prime}_{[j,j+2]},$
(16) $\displaystyle C_{0}$
$\displaystyle=\frac{1}{\pi}\sum_{j=-\infty}^{\infty}\left((j+1)^{2}\widetilde{K}_{[j,j+2]}-(j^{2}+j+1)\widetilde{K}_{[j,j+1]}\right)$
$\displaystyle=\frac{1}{\pi}\sum_{j=-\infty}^{\infty}\left((j+1)^{2}\widetilde{K}^{\prime}_{[j,j+2]}-\frac{1}{2}\widetilde{K}_{[j,j+1]}\right),$
(17)

where
$\widetilde{K}^{\prime}_{[j,j+2]}=\widetilde{K}_{[j,j+2]}-\frac{1}{2}\widetilde{K}_{[j,j+1]}-\frac{1}{2}\widetilde{K}_{[j+1,j+2]}$
is introduced to simplify the equations. Because $P={\bf i}[M_{10},H]$,
$D=\frac{{\bf i}}{2}[C_{0},H]$, $C_{1}={\bf i}[C_{0},M_{10}]$, we have

$\displaystyle P$ $\displaystyle=\frac{{\bf
i}}{\pi^{2}}\sum_{j=-\infty}^{\infty}\left[\widetilde{K}^{\prime}_{[j+1,j+3]},\widetilde{K}^{\prime}_{[j,j+2]}\right],$
(18) $\displaystyle D$ $\displaystyle=\frac{{\bf
i}}{\pi^{2}}\sum_{j=-\infty}^{\infty}\bigg{(}(j+\frac{3}{2})\left[\widetilde{K}^{\prime}_{[j+1,j+3]},\widetilde{K}^{\prime}_{[j,j+2]}\right]$
$\displaystyle\qquad\qquad-\frac{1}{4}\left[\widetilde{K}_{[j,j+1]},\widetilde{K}^{\prime}_{[j,j+2]}\right]$
$\displaystyle\qquad\qquad-\frac{1}{4}\left[\widetilde{K}_{[j+1,j+2]},\widetilde{K}^{\prime}_{[j,j+2]}\right]\bigg{)},$
(19) $\displaystyle C_{1}$ $\displaystyle=\frac{{\bf
i}}{\pi^{2}}\sum_{j=-\infty}^{\infty}\bigg{(}(j+1)(j+2)\left[\widetilde{K}^{\prime}_{[j+1,j+3]},\widetilde{K}^{\prime}_{[j,j+2]}\right]$
$\displaystyle\qquad\qquad-\frac{j+1}{2}\left[\widetilde{K}_{[j,j+1]},\widetilde{K}^{\prime}_{[j,j+2]}\right]$
$\displaystyle\qquad\qquad-\frac{j+1}{2}\left[\widetilde{K}_{[j+1,j+2]},\widetilde{K}^{\prime}_{[j,j+2]}\right]\bigg{)}.$
(20)

This constructs the 6 global conformal generators from 1-site and 2-site EHs.
We use the term reconstructed Hamiltonian to refer to the output of the
process of taking a state and constructing an associated Hamiltonian as in
(15). This state could be on a lattice or in the continuum.

We make two comments on the expression for $P$. First, a similar expression
was considered in [18, Sec II.C]. The idea is to define the momentum density
$p_{j}={\bf i}[h_{j},h_{j-1}]$ in terms of the Hamiltonian density $h_{j}$
where $H=\sum h_{j}$. One challenge in implementing this idea in practice is
that the Hamiltonian $H$ is only equal to the CFT Hamiltonian $H_{CFT}$ up to
a multiplicative factor. Consequently, the momentum density $p_{j}$ is
determined only up to a multiplicative factor. In our work, we fix this
multiplicative factor and provide a better theoretical understanding. Second,
there is a no-go theorem that prevents expressing the momentum operator as a
sum of local operators [19, Corollary 6.1] which seemingly contradicts our
expression above. There are several ways to reconcile this apparent
contradiction. One is to note that our expression holds exactly only for
lattice models with an infinite local Hilbert space dimension, whereas the no-
go theorem applies to models with a finite local Hilbert space dimension.
Another way to reconcile this is to recognize that, for models with a finite
local Hilbert space dimension, $e^{-{\bf i}P}$ is not precisely equal to the
lattice translation operator [20]. Instead, they are only approximately equal
at low energies, as we verify numerically below.

We further comment that there are additional commutation relations between the
generators which are not utilized in this work, such as $[M_{10},P]=-{\bf i}H$
and $[H,P]=0$. In the context of field theories, this phenomenon where higher
order commutator of EHs are related to linear combinations of EHs can be
understood from the operator product expansion (OPE) of $T_{zz}(t,x)$. First,
because EH is a sum of $T_{zz}(t=0,x)$ and $T_{\bar{z}\bar{z}}(t=0,x)$, the
commutator of EHs is a sum of the commutators of $T_{zz}(t=0,x)$ and
$T_{\bar{z}\bar{z}}(t=0,x)$. Second, the commutator of $T_{zz}(t=0,x)$ is
generated by the singular part of the OPE of $T_{zz}$. Finally, the singular
part of the OPE of $T_{zz}$ is generated by $T_{zz}$ and its descendent
$\partial_{z}T_{zz}$. The second and the final part can be summarized into the
following expression [21, Equation (B23)]

$\displaystyle[T_{zz}(0,x_{1}),T_{zz}(0,x_{2})]$ $\displaystyle=\frac{{\bf
i}\pi c}{6}\partial_{x_{1}}^{3}\delta(x_{1}-x_{2})$ $\displaystyle+4\pi{\bf
i}T_{zz}(0,x_{2})\partial_{x_{1}}\delta(x_{1}-x_{2})$ $\displaystyle-2\pi{\bf
i}\partial_{z}T_{zz}(0,x_{2})\delta(x_{1}-x_{2}).$ (21)

We leave these further relations between the higher order commutators of EHs
and the linear combinations of EHs for future study.

Having defined the 6 conformal generators using 1-site and 2-site EHs, we now
show that they have the expected scaling from real-space RG. We first study
the case of the Hamiltonian $H$. To perform real-space RG, we consider a new
system which blocks 2 sites in the original system into 1 site. We then
compare the two reconstructed Hamiltonians using Eq. (15), which are

$\displaystyle H_{1}$
$\displaystyle=\frac{1}{\pi}\sum_{j=-\infty}^{\infty}\widetilde{K}_{[j,j+2]}-\widetilde{K}_{[j,j+1]}$
(22) $\displaystyle H_{2}$
$\displaystyle=\frac{1}{\pi}\sum_{j=-\infty}^{\infty}\widetilde{K}_{[2j,2j+4]}-\widetilde{K}_{[2j,2j+2]}.$
(23)

Note that the state we used for the reconstructions is the same. The only
difference is the size of the block. By expanding the 4-site EH as a sum of
1-site and 2-site EH using Eq. (12), we have $H_{2}=2H_{1}$. Similarly, one
can define $P_{2}$ and show that $P_{2}=2P_{1}$, $D_{2}=D_{1}$,
$(M_{10})_{2}=(M_{10})_{1}$, $(C_{0})_{2}=\frac{1}{2}(C_{0})_{1}$, and
$(C_{1})_{2}=\frac{1}{2}(C_{1})_{1}$. These are precisely the scalings of the
generators for CFT under the transformation $x\to x/2$, $t\to t/2$.

It is perhaps not surprising that the scaling property holds because the
construction is motivated by the field theories. However, what is surprising
is that the derivation of the scaling property only uses the condition (1),
which is independent from the field theory description of CFTs. This
observation supports the conjecture that states satisfying Eq. (1) are the CFT
ground states.

##  IV Numerical tests

We now move on to numerics and verify that Eqs. (1), (3), (15) and (18) hold
approximately for critical lattice models. First, we test the validity of the
main Eqs. (1) and (3) on small sizes of four different critical lattice models
and on large sizes of free fermions. We find that these equations hold
approximately for all the models we consider and the error decreases as the
size increases. Then, we test the reconstruction of the Hamiltonian and
momentum operator Eqs. (15) and (18) on small sizes of the four models.

The critical lattice models we consider are the critical transverse field
Ising model, critical three-state Potts model, XX model, and Heisenberg
model777These models are all quantum critical and their universal properties
are described respectively by the following conformal field theories: Ising
model with $c=1/2$; Potts model with $c=4/5$; compact scalar CFT with radius
$1$ and $1/\sqrt{2}$. defined as:

$\displaystyle H_{\text{Ising}}$ $\displaystyle=\sum_{i}-Z_{i}-X_{i}X_{i+1}$
(24) $\displaystyle H_{\text{Potts}}$
$\displaystyle=\sum_{i}-\mathcal{Z}_{i}-\mathcal{Z}^{\dagger}_{i}-\mathcal{X}_{i}\mathcal{X}^{\dagger}_{i+1}-\mathcal{X}^{\dagger}_{i}\mathcal{X}_{i+1}$
(25) $\displaystyle H_{\text{XX}}$
$\displaystyle=\sum_{i}X_{i}X_{i+1}+Y_{i}Y_{i+1}$ (26) $\displaystyle
H_{\text{Heisenberg}}$
$\displaystyle=\sum_{i}X_{i}X_{i+1}+Y_{i}Y_{i+1}+Z_{i}Z_{i+1}$ (27)

where $X,Y,Z$ are the Pauli matrices and $\mathcal{X},\mathcal{Z}$ are the
qutrit Pauli matrices

$\mathcal{X}=\begin{pmatrix}0&1&0\\\ 0&0&1\\\
1&0&0\end{pmatrix},\quad\mathcal{Z}=\begin{pmatrix}1&0&0\\\ 0&\omega&0\\\
0&0&\omega^{2}\end{pmatrix},\quad\omega=e^{2\pi{\bf i}/3}$ (28)

Table 1 lists the errors associated with Eqs. (1) and (3). The error in Eq.
(1) is defined as the norm of the component in $K_{\Delta}|\psi\rangle$
orthogonal to $|\psi\rangle$,
$\Big{\lvert}K_{\Delta}|\psi\rangle-\langle\psi|K_{\Delta}|\psi\rangle|\psi\rangle\Big{\rvert}$,
which is also the standard deviation of $K_{\Delta}$, $\sqrt{\langle
K_{\Delta}^{2}\rangle-\langle K_{\Delta}\rangle^{2}}$. The error in Eq. (3) is
defined as the norm
$\Big{\lvert}K_{\Delta}|\psi\rangle-\frac{c}{3}h(\eta)|\psi\rangle\Big{\rvert}$.
We computed these errors for ground states on circles with circumferences
$L=4,8,12$, where $(A,B,C)$ have lengths $(1,1,1),(2,2,2),$ and $(3,3,3)$,
respectively. In all cases, $\eta=1/2$. We observe that except for the
accidental case where the XX model and Heisenberg model with circumference $4$
have $0$ errors, the error decreases as the system size increases. This result
is consistent with the intuition that the lattice model approximates the CFT
better in the IR as the system size increases.

$L$ |  Error in Eq. (1) |  Error in Eq. (3)
---|---|---
| 4 | 8 | 12 | 4 | 8 | 12
Ising Model | 0.0282 | 0.0090 | 0.0057 | 0.0288 | 0.0090 | 0.0057
Potts Model | 0.0422 |  |  | 0.0434 |  |
XX Model | 0 | 0.0399 | 0.0120 | 0.0486 | 0.0416 | 0.0125
Heisenberg | 0 | 0.0562 | 0.0279 | 0.0566 | 0.0583 | 0.0286
Table 1: The error in Eq. (1)
$\Big{\lvert}K_{\Delta}|\psi\rangle-\langle\psi|K_{\Delta}|\psi\rangle|\psi\rangle\Big{\rvert}$
and the error in Eq. (3)
$\Big{\lvert}K_{\Delta}|\psi\rangle-\frac{c}{3}h(\eta)|\psi\rangle\Big{\rvert}$
for ground states on circles with circumferences $L=4,8,12$.

Figure 2 shows the error in Eq. (1) for free fermions across various system
sizes. We again observe that the error decreases as the system size increases
which roughly scales as $1/L^{2}$. Simulating free fermions on a large system
size is feasible due to their lower complexity [22, 23].

Figure 2: The error in Eq. (1)
$\Big{\lvert}K_{\Delta}|\psi\rangle-\langle\psi|K_{\Delta}|\psi\rangle|\psi\rangle\Big{\rvert}$
for the ground state of the critical free fermions. The circle is partitioned
into four equal sized intervals, where the lengths of $A,B,C$ are $L/4$ and
has cross ratio $\eta=1/2$. The error roughly scales as $1/L^{2}$.

We now test the reconstructed Hamiltonian and momentum on a circle in Eqs. (3)
and (4) where

$\displaystyle H_{rec}$
$\displaystyle=\frac{\sin(2\pi/L)}{2L\sin^{2}(\pi/L)}\sum_{j=0}^{L-1}\widetilde{K}_{[j,j+2]}-\widetilde{K}_{[j,j+1]}$
(29) $\displaystyle P_{rec}$ $\displaystyle=\frac{{\bf
i}}{\pi^{2}}\sum_{j=0}^{L-1}\left[\widetilde{K}^{\prime}_{[j+1,j+3]},\widetilde{K}^{\prime}_{[j,j+2]}\right].$
(30) 888The analogous formula for the exact reconstructed momentum is
$\displaystyle P_{rec}$
$\displaystyle=\frac{\sin^{3}(2\pi/L)\pi}{8L\sin^{4}(\pi/L)}\frac{i}{\pi^{2}}\sum_{j=0}^{L-1}[\widetilde{K}_{[j+1,j+3]},\widetilde{K}_{[j,j+2]}]$
$\displaystyle\hskip
60.00009pt-\left(1-\frac{2\sin^{2}(\pi/L)}{\sin^{2}(2\pi/L)}\right)[\widetilde{K}_{[j,j+2]},\widetilde{K}_{[j,j+1]}]$
$\displaystyle\hskip
60.00009pt-\left(1-\frac{2\sin^{2}(\pi/L)}{\sin^{2}(2\pi/L)}\right)[\widetilde{K}_{[j+1,j+2]},\widetilde{K}_{[j,j+2]}]$

The factor $\frac{\sin(2\pi/L)}{2L\sin^{2}(\pi/L)}$ in $H_{rec}$ comes from
the coefficient obtained when expressing EH as an integral of $T_{00}(x)$. On
the other hand, the factor $\frac{{\bf i}}{\pi^{2}}$ in $P_{rec}$ is based
solely on the understanding at the infinite-size limit in Eq. (18). Despite
this, as we will see, the reconstructed Hamiltonian and momentum agree
excellently. 999Note that $H_{rec}$ is equal to $H_{CFT}$ since both have the
same expression in terms of $T$. On the contrary, because we put the system on
a circle $P_{rec}$ is only equal to $P_{CFT}$ in the thermodynamic limit.

Before presenting the result, we first remark that the field theory
Hamiltonian and momentum satisfy

$\displaystyle H_{CFT}\,|\Delta,s\rangle$
$\displaystyle=\frac{2\pi}{L}\left(\Delta-\frac{c}{12}\right)|\Delta,s\rangle$
(31) $\displaystyle P_{CFT}\,|\Delta,s\rangle$
$\displaystyle=\frac{2\pi}{L}s\,|\Delta,s\rangle,$ (32)

where $|\Delta,s\rangle$ is the image under the state-operator correspondence
of a scaling operator of dimension $\Delta$ and spin $s$. We emphasize that
$H_{rec}$ is equal to $H_{CFT}$ up to a constant shift so that $H_{rec}$ has
ground state energy $0$, i.e. $H_{rec}=H_{CFT}-E_{0}$ 101010The constant
$-\frac{2\pi}{L}\frac{c}{12}$ can be recovered by using the EHs of the ground
state on an infinite system instead of the EHs of the length $L$ finite
system, which we have tested for critical free fermions. However, we do not
discuss this aspect further because it is generally not possible to know the
EHs on an infinite system directly.. In particular, the multiplicative factor
is fixed, meaning that the scaling dimensions $\Delta$ can be obtained without
the rescaling required if one directly uses the spectrum of an arbitrary
critical Hamiltonian.

Figure 3 compares the spectra of the original Hamiltonian to the spectrum of
the reconstructed Hamiltonian at low energy. The reconstructed Hamiltonian
$H_{rec}$ is rescaled by $\frac{L}{2\pi}$ and the original Hamiltonian is
rescaled to fit $H_{rec}$. We observe the spectra have excellent agreement
even at the small system size where $L=4$.111111We emphasize that this is a
pleasant surprise: although we use field theory to derive our formula for the
reconstructed Hamiltonian, its form is numerically effective even on quite
small lattices. It would be interesting to understand why it works so well.
Locality clearly plays an important role. We note that the ideal Hamiltonian
reconstructed from the saddle point of the entropy function $S_{\Delta}$ and
the model Hamiltonians we study (Eqs. 24-27) both involve at most 2-site
operators. This means that the artificial model Hamiltonian can already be
close to the ideal reconstructed CFT Hamiltonian.

Figure 3: Spectra of the original Hamiltonian and the reconstructed
Hamiltonian, where the states are ordered by energy along the X-axis. Top
left: Ising model with $L=4$. Top right: Lowest $40$ eigenvalues for Ising
model with $L=12$. Bottom left: Heisenberg model with $L=4$. Bottom right:
Lowest $40$ eigenvalues for Heisenberg model with $L=12$.

Figure 4 compares the spectrum of ${\bf i}\log T$ to the spectrum of the
reconstructed momentum at low energy, where $T$ is the translation operator by
1 site. We test if $T\approx e^{-{\bf i}P_{rec}}$. Again both spectra are
rescaled by $\frac{L}{2\pi}$ and we expect both to take integer values. We
observe the spectra agree at low energy and the agreement improves as the
system size increases.

We provide three technical comments on this test. First, we note that in
lattice models, it is not generally true that $T=e^{-{\bf i}P_{CFT}}$, for
example, in the Heisenberg and XX models. This is because ${\bf i}\log T$ may
have a constant shift $s_{0}$, where ${\bf i}\log
T|\psi\rangle=(s_{0}+\frac{2\pi}{L}s)|\psi\rangle$. As an example, if we
instead define the transverse field Ising model as
$H_{\text{Ising}}=\sum_{i}-Z_{i}+X_{i}X_{i+1}$, then the new ground state is
obtained by applying $Z$ on the even sites of the usual ground state. This
results with a shift $s_{0}=\pi$. Sometimes this shift cannot be removed
easily due to anomaly [20]. This is why we did not show the reconstruction in
the case of Heisenberg and XX models for simplicity. Next, we remark on two
surprising properties of $P_{rec}$. One is that $\frac{L}{2\pi}P_{rec}$ is
close to taking integer values at low energy. The other is that it is often
the case that $-\pi\leq P_{rec}\leq\pi$. Both properties are expected when
knowing $T\approx e^{-{\bf i}P_{rec}}$. However, from Eq. (30) alone, it is
unclear why this happens; we leave this for future research. Finally, although
we lack a full analytic understanding of the factor $\frac{{\bf i}}{\pi^{2}}$
in $P_{rec}$, it works well numerically. The precise nature of this factor on
a finite system, and whether this is a correct choice or merely a coincidence,
is left for further exploration.

Figure 4: Spectra of ${\bf i}\log T$ and the reconstructed momentum operator
$P$, where $T$ is the translation operator by 1 site. The states are ordered
by energy along the X-axis. In the infinite-size limit, this order corresponds
to the order of the scaling dimension of the corresponding operator from the
state-operator correspondence. For the small system sizes shown here, these
orders are not yet the same. This is why the order changes in the figure when
the system size changes. Top left: Ising model with $L=8$. Bottom left: Ising
model with $L=12$. Top right: Potts model with $L=6$. Bottom right: Potts
model with $L=8$.

_Further directions—_ The further directions are sorted based on some
subjective measure of attractiveness.

(a) We show that CFT ground states satisfy the condition in Eq. (1). Motivated
by entanglement bootstrap, we ask whether the converse is also true, i.e. if a
state satisfying the condition can be interpreted as a CFT ground state.
Additionally, we would like the statement to be robust, useful even when the
state only satisfies the condition approximately.

(b) We proposed various formulas that hold exactly when the state is a CFT
ground state. However, for the ground state of critical lattice models, the
formula only holds approximately. What is the convergence behavior and the
finite size scaling?

(c) Relatedly, one may wonder if we allow combinations of not only 1-site and
2-site EH but also 3-site EH, which combination has the best approximation to
the Hamiltonian and the momentum. Furthermore, what if we include EH on larger
sites? Could this lead to a sequence of reconstructions that converges to the
actual operator? Finding a good approximation is crucial for obtaining a
better estimation of the OPE based on the method discussed in [24].

(d) Can we reproduce the key results in CFT directly using quantum
information? For example, can we show that all nontrivial states must have
$c\geq\frac{1}{2}$? Can we show the ground state energy is
$-\frac{2\pi}{L}\frac{c}{12}$ on a circle with circumference $L$? Can we show
that the reconstructed momentum operator should be approximately integer
valued at low energy and has a norm roughly below $\pi$?

(e) The appearance of $h(\eta)$ in Eq. (3) suggests that $\eta$ can be
interpreted as a probability. Could we find a physical meaning for this
observation?

(f) We provide an information theoretic criterion for CFT ground state. Can we
transform this criterion into an algorithm that searches for CFTs by screening
states that satisfy Eq. (1)? We believe the answer is yes and plan to explore
this aspect in an upcoming paper.

(g) We provide an entropy criterion that appears to describe ground states for
1+1D unitary CFTs. We suspect a similar formula exists for 1+1D chiral CFTs,
d+1D CFTs, and perhaps complex CFTs [25, 26].

(h) We show that CFT ground states are critical points of the function
$S_{\Delta}$, with a value proportional to the central charge. What is its
relation to the Zamolodchikov $c$-function [27] and the entropic $c$-function
[28]? For example, it is known that the second and third-order expansion of
the Zamolodchikov $c$-function near the CFT point contains the CFT data,
including the scaling dimensions and the OPEs. Does the same hold for the
entropy function we proposed?

(i) We showed that 1+1D phases at RG fixed points with Lorentz symmetry
satisfy Eq. (1). What happens to other RG fixed points without Lorentz
symmetry, such as those with dynamical critical exponent $z\neq 1$. In the
case of free fermion models with $z\in\mathbb{Z}^{>0}$, the formula continues
to hold. When $z$ is even, the ground state is simply a product state, and
when $z$ is odd, the ground state is same as the ground state for $z=1$, which
is a CFT.

##  V Acknowledgments

We thank Isaac Kim and Hao-Chung Cheng for suggesting a simple proof to
compute the first-order derivative of entanglement entropy in Eq. (8). We
thank Meng Cheng and Yijian Zhou for clarifying why in general $T\neq e^{-{\bf
i}P}$. We thank Bowen Shi and Daniel Ranard for helpful discussions. This work
was supported in part by funds provided by the U.S. Department of Energy
(D.O.E.) under the cooperative research agreement DE-SC0009919, and by the
Simons Collaboration on Ultra-Quantum Matter, which is a grant from the Simons
Foundation (652264, JM). TCL would like to thank the assistance of ChatGPT in
improving parts of the paper for clarity, including this sentence.

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##  Appendix A Supplemental Material

###  A.1 Proof of Eq. (8)

Let $|\psi\rangle\in\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ be a pure state. We
now show the first order derivative of entanglement entropy
$S_{A}(|\psi\rangle)$ claimed in Eq. (8). Because
$d\rho_{A}=\operatorname{Tr}_{B}(|\psi\rangle\langle
d\psi|)+\operatorname{Tr}_{B}(|d\psi\rangle\langle\psi|)$, it is sufficient to
show

$dS(\rho_{A})=\operatorname{Tr}_{A}((-\log\rho_{A})d\rho_{A}).$ (33)

Using product rule in calculus,

$dS(\rho)=d\operatorname{Tr}((-\log\rho)\rho)=\operatorname{Tr}((-\log\rho)d\rho)-\operatorname{Tr}(\rho\,d(\log\rho)).$
(34)

We suffice to show the second term on the RHS is $0$. This is done by showing

$\operatorname{Tr}(\rho\,d(\log\rho))=\operatorname{Tr}(d\rho)$ (35)

where $\operatorname{Tr}(d\rho)=d\operatorname{Tr}(\rho)=0$.

Because $\rho$ is hermitian, we can write $\rho=e^{M}$. Equivalently, we need
to show

$\operatorname{Tr}(e^{M}dM)=\operatorname{Tr}(de^{M}).$ (36)

This is obtained by an equation from the perturbation theory of path integrals

$de^{M}=\int_{0}^{1}dt\,e^{tM}dMe^{(1-t)M}.$ (37)

We sketch the proof of the final equation:

$\displaystyle de^{M}$
$\displaystyle=\lim_{n\to\infty}d\prod_{i=1}^{n}e^{M/n}$ (38)
$\displaystyle=\lim_{n\to\infty}\sum_{i=1}^{n}e^{\frac{i-1}{n}M}\,(de^{M/n})\,e^{\frac{n-i}{n}M}$
(39)
$\displaystyle=\lim_{n\to\infty}\sum_{i=1}^{n}e^{\frac{i-1}{n}M}\,(dM/n)\,e^{\frac{n-i}{n}M}$
(40) $\displaystyle=\int_{0}^{1}dte^{tM}dMe^{(1-t)M}$ (41)

where the third equality uses $e^{M/n}=I+M/n+O(1/n^{2})$. The higher order
term $O(1/n^{2})$ converges to $0$ as $n\to\infty$, because there are only $n$
terms.

More directly, the key step Eq. (35) can also be obtained through the
following equality [29, Lemma 3.4]

$d\log\rho=\int_{-\infty}^{\infty}dt\,\beta_{0}(t)\,\rho^{-\frac{1+{\bf
i}t}{2}}\,d\rho\,\rho^{-\frac{1-{\bf i}t}{2}}$ (42)

where $\beta_{0}(t)=\frac{\pi}{2}(\cosh(\pi t)+1)^{-1}$. When substituting
$d\log\rho$ in Eq. (35), we obtain

$\operatorname{Tr}(\rho\,d(\log\rho))=\int_{-\infty}^{\infty}dt\,\beta_{0}(t)\operatorname{Tr}(d\rho)=\operatorname{Tr}(d\rho)$
(43)

using $\int_{-\infty}^{\infty}dt\beta_{0}(t)=1$.

###  A.2 No other relations

Here we argue that relations of the form (1) or (3) exhaust linear relations
among single-interval entanglement Hamiltonians of CFT groundstates. What we
mean is that any other linear entropy formula derivable from (5) and (6) is
implied by our fixed-point equation (1) or (3).

The idea is the following. The observation is that each $K_{A}$ corresponds to
a piecewise quadratic function $\beta_{A}$. We wish to identify all linear
relations between these piecewise quadratic functions. For convenience, divide
up the line or circle into a set of atomic intervals (we’ll revisit this
below). Our relation shows that $\beta_{A}$ for any region can be decomposed
into a linear combination of one-site and two-site functions, meaning the
functions associated with a single atomic interval or two atomic intervals. So
it is sufficient to show that there is no linear dependence among one-site and
two-site functions. This is easy to check:
$a(x_{3}-x)(x-x_{1})+b(x_{3}-x)(x-x_{2})+c(x_{4}-x)(x-x_{2})=0$ for all
$x\in[x_{1},x_{2}]$ implies $a=b=c=0$.

If instead we wish to work in the continuum, suppose there is some relation
that is not of our form. Use the intervals appearing in the relation as the
atomic intervals and repeat the above argument.

We remark in passing that this question of other relations can be described as
a cohomology problem in functional analysis. Consider a formal vector space
$V$ spanned by states labelled by intervals on the line or circle $|A\rangle$,
with no relations. There is map $\phi$ from $V$ to the Hilbert space of
piecewise quadratic functions $\phi:|A\rangle\to\beta_{A}(x)$. The kernel of
the map $\phi$ is spanned by all relations that follow from (5) and (6). There
is map $\psi$ into $V$ determined by relations of the form (1) or (3). The
previous paragraph shows that there is no cohomology at $V$ – the kernel of
$\phi$ is the image of $\psi$. The fact noted above Remark 1 about the
associativity of decomposition of $\widetilde{K}_{[0,n]}$ implies that there
are relations among our relations, meaning that this chain complex can be
further extended to the left.