Absence of logarithmic enhancement in the entanglement scaling of free fermions on folded cubes

The $d$-cube or hypercube graph $H(d,2)$ has a set of vertices $X_{d}$ given by the binary strings of length $d$,

$$X_{d}=\{v=(v_{1},v_{2},\dots,v_{d})\ |\ v_{i}\in\{0,1\}\}.$$

(4)

An edge connects two vertices $v$ and $v^{\prime}$ if there is a unique position $i$ such that they differ, i.e. $v_{i}\neq v_{i}^{\prime}$. We illustrate a $3$- and $4$-cube in Fig. 1. For an arbitrary pair of vertices $v$ and $v^{\prime}$, their relative distance is given by the Hamming distance,

$$\partial(v,v^{\prime})=|\{i\in\{1,2,\dots,d\}\ |\ v_{i}\neq v_{i}^{\prime}\}|.$$

(5)

The ground-state entanglement properties of free fermions defined on such lattices have been investigated, and results regarding bipartite entanglement and multipartite information have been obtained in [16, 17].

Folded $d$-cubes, denoted $\square_{d}$, are obtained by taking the antipodal quotient of a $d$-cube, i.e. by merging the pairs of vertices at distance $d$ in $H(d,2)$ [27]. The vertices of $\square_{d}$ are the equivalence classes in $X_{d}/\sim$, where the relation $\sim$ is defined by

$$v\sim v^{\prime}\iff\partial(v,v^{\prime})\in\{0,d\}.$$

(6)

Two classes $[v]$ and $[v^{\prime}]$ are connected by an edge in $\square_{d}$ if their respective representative strings $v$ and $v^{\prime}$ are at a Hamming distance $1$ or $d-1$ in $H(d,2)$. The graph $\square_{d}$ is also obtained by taking the vertices $X_{d-1}=X_{d}/\sim$ of a $(d-1)$-cube and connecting by an edge those at Hamming distance $1$ or $d-1$. The set $E$ of edges of $\square_{d}$ is thus

$$E=\{(v,v^{\prime})\in X_{d-1}\times X_{d-1}\ |\ \partial(v,v^{\prime})\in\{1,d-1\}\},$$

(7)

and the distance between any two vertices in $\square_{d}$ is given by

$$\text{dist}(v,v^{\prime})=\text{min}\{\partial(v,v^{\prime}),d-\partial(v,v^{\prime})\}.$$

(8)

We illustrate a folded $4$-cube in Fig. 2.

Each vertex $v=(v_{1},\dots,v_{d-1})\in X_{d-1}$ can be associated with a vector in $\mathbb{C}^{2^{d-1}}$ in the following way,

$$\ket{v}=\ket{v_{1}}\otimes\ket{v_{2}}\otimes\dots\otimes\ket{v_{d-1}},$$

(9)

where $\ket{0}=(1,0)^{t}$ and $\ket{1}=(0,1)^{t}$. In this basis, the adjacency matrix $\mathcal{A}$ of $\square_{d}$ is given by

$$\mathcal{A}=\left(\sum_{n=1}^{d-1}\underbrace{I\otimes I\otimes...\otimes I}_{n-1\text{ times}}\otimes\ \sigma_{x}\otimes\underbrace{I\otimes...\otimes I}_{d-1-n\text{ times}}\right)+\underbrace{\ \sigma_{x}\otimes\ \sigma_{x}\otimes...\otimes\ \sigma_{x}}_{d-1\text{ times}},$$

(10)

where $I$ is the $2\times 2$ identity matrix and $\sigma_{x}$ is a Pauli matrix. Indeed, one can check that

$$\bra{v}\mathcal{A}\ket{v^{\prime}}=\left\{\begin{array}[]{ll}1&\mbox{if }(v,v^{\prime})\in E\\ 0&\mbox{otherwise. }\end{array}\right.$$

(11)

Let us note that the distance function (8) implies that both folded cubes $\square_{2d}$ and $\square_{2d+1}$ have diameter $d$. In the following, we restrict ourselves to the case $\square_{2d+1}$.

For each vertex $v\in X_{2d}$, we define a pair of fermionic creation and annihilation operators $c_{v}^{\dagger}$ and $c_{v}$ that verify the canonical anti-commutation relations

$$\{c_{v},c_{v^{\prime}}\}=\{c_{v}^{\dagger},c_{v^{\prime}}^{\dagger}\}=0,\quad\{c_{v},c_{v^{\prime}}^{\dagger}\}=\delta_{vv^{\prime}}.$$

(12)

A nearest-neighbor Hamiltonian on the graph $\square_{2d+1}$ is then defined as

$$\begin{split}\mathcal{H}&:=\sum_{(v,v^{\prime})\in E}c_{v}^{\dagger}c_{v^{\prime}}=\bm{c}^{\dagger}\mathcal{A}\bm{c},\end{split}$$

(13)

where $\mathcal{A}$ is the adjacency matrix of $\square_{2d+1}$, and

$$\bm{c}=\sum_{v\in X_{2d}}c_{v}\ket{v},\quad\bm{c}^{\dagger}=\sum_{v\in X_{2d}}c_{v}^{\dagger}\bra{v}.$$

(14)

The Hamiltonian $\mathcal{H}$ can be diagonalized using the eigenvectors of the adjacency matrix $\mathcal{A}$. These are given by the $2d$-fold tensor product of the eigenvectors $\ket{\pm}$ of $\sigma_{x}$,

$$\ket{\pm}=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\ \pm 1\end{pmatrix},\quad\sigma_{x}\ket{\pm}=\pm\ket{\pm}.$$

(15)

The eigenvectors of $\mathcal{A}$ are denoted $\ket{\theta_{k},\ell}$, where $k$ indicates the number of vector $\ket{+}$ in the tensor product and $\ell\in\{1,2,\dots,\binom{2d}{k}\}$ is a label for the degeneracy,

$$\mathcal{A}\ket{\theta_{k},\ell}=\theta_{k}\ket{\theta_{k},\ell},\quad\theta_{k}=2k-2d+(-1)^{k}.$$

(16)

The degeneracy of the model is not entirely captured by the index $\ell$ since we also have $\theta_{2k}=\theta_{2k+1}$ for all $k\in\{0,\dots,d-1\}$. In terms of $\theta_{k}$ and the eigenvectors $\ket{\theta_{k},\ell}$, the free-fermion Hamiltonian $\mathcal{H}$ can be rewritten as

$$\mathcal{H}=\sum_{k=0}^{2d}\sum_{\ell=1}^{\binom{2d}{k}}\theta_{k}\hat{c}_{k\ell}^{\dagger}\hat{c}_{k\ell},$$

(17)

where $\hat{c}_{k\ell}$ and $\hat{c}_{k\ell}^{\dagger}$ are fermionic creation and annihilation operators which verify the same canonical anti-commutation relations as $c_{v}$, $c_{v}^{\dagger}$, and are defined by

$$\hat{c}_{k\ell}=\sum_{v\in X_{2d}}\bra{\theta_{k},\ell}\ket{v}c_{v},\quad\hat{c}_{k\ell}^{\dagger}=\sum_{v\in X_{2d}}\bra{v}\ket{\theta_{k},\ell}c_{v}^{\dagger}.$$

(18)

Let $|0\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}$ be the vacuum state which is annihilated by all operators $c_{v}$,

$${c}_{v}|0\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}=0,\quad\forall\ v\in X_{2d}.$$

(19)

The ground state $|\Psi_{0}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}$ of $\mathcal{H}$ is obtained by acting on the vacuum with all creation operators $\hat{c}^{\dagger}_{k\ell}$ associated with negative energy excitation, i.e. $\theta_{k}<0$. By filling up the Fermi sea, we have

$$|\Psi_{0}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}=\left(\prod_{k=0}^{2K+1}\prod_{\ell=1}^{\binom{2d}{k}}\hat{c}^{\dagger}_{k\ell}\right)|0\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}},$$

(20)

where $K$ is the integer such that $\theta_{2K}=\theta_{2K+1}<0$ and $\theta_{2K+2}\geqslant 0$. A direct computation shows that the two-point correlation matrix in this state is

$$\hat{C}_{vv^{\prime}}=\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}\Psi_{0}|c_{v}^{\dagger}c_{v^{\prime}}|\Psi_{0}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}=\sum_{k=0}^{2K+1}\sum_{\ell=1}^{\binom{2d}{k}}\bra{v}\ket{\theta_{k},\ell}\bra{\theta_{k},\ell}\ket{v^{\prime}}.$$

(21)

For any distance $n:=\text{dist}(v,v^{\prime})$, a combinatorial argument allows to express the correlation function as

$$\hat{C}_{vv^{\prime}}=\frac{1}{2^{2d}}\sum_{k=0}^{N}\binom{2d-n}{k}{}_{2}F_{1}\biggl{[}\genfrac{.}{.}{0.0pt}{}{-n\mathchar 44\relax\mkern 6.0mu-k}{2d-n-k+1};-1\biggr{]}+\frac{1}{2^{2d}}\sum_{k=N+1}^{2K+1}\binom{n}{2d-k}{}_{2}F_{1}\biggl{[}\genfrac{.}{.}{0.0pt}{}{n-2d\mathchar 44\relax\mkern 6.0mu\ k-2d}{n+k-2d+1};-1\biggr{]},$$

(22)

where $N=\text{min}(2d-n,2K+1)$ and ${}_{2}F_{1}$ is Gauss hypergeometric function. Let us also note that the correlation matrix $\hat{C}$ whose entries are $\hat{C}_{vv^{\prime}}$ can be expressed as a sum of projectors $E_{2k}+E_{2k+1}$ onto eigenspaces of $\mathcal{A}$,

$$\hat{C}=\sum_{k=0}^{K}\left(E_{2k}+E_{2k+1}\right),\quad E_{k}=\sum_{\ell=1}^{\binom{2d}{k}}\ket{\theta_{k},\ell}\bra{\theta_{k},\ell}.$$

(23)

For a given bipartition $A\cup B$ of a quantum many-body system in a pure state $|\Psi_{0}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}$, the entanglement entropy is defined as the von Neumann entropy of the reduced density matrix $\rho_{A}$,

$$S_{A}=-\text{tr}_{A}\left(\rho_{A}\ln{\rho_{A}}\right),\quad\rho_{A}=\text{tr}_{B}(|\Psi_{0}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}\Psi_{0}|).$$

(24)

We are interested in computing this quantity for free fermions on folded cubes $\square_{2d+1}$, in their ground state $|\Psi_{0}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}$ defined in (20). Since this state is a Slater determinant, the reduced density matrix $\rho_{A}$ is a Gaussian operator,

$$\rho_{A}=\frac{1}{\mathcal{K}}e^{-\mathcal{H}_{\text{ent}}},\quad\mathcal{K}=\text{tr}_{A}\left(e^{-\mathcal{H}_{\text{ent}}}\right),$$

(25)

where the entanglement Hamiltonian $\mathcal{H}_{\text{ent}}$ is quadratic in fermionic operators associated to degrees of freedom in region $A$,

$$\mathcal{H}_{\text{ent}}=\sum_{v,v^{\prime}\in A}h_{vv^{\prime}}c_{v}^{\dagger}c_{v^{\prime}}.$$

(26)

Owning to the quadratic nature of the free-fermion Hamiltonian, one can relate the matrix $h$ in the definition of the entanglement Hamiltonian (26) to the the restriction of the correlation matrix $\hat{C}$ to region $A$ [28],

$$h=\ln\left(\frac{1-C}{C}\right),$$

(27)

where $C$ is referred to as the truncated correlation matrix. It is defined as

$$C=\pi_{A}\hat{C}\pi_{A},\quad\pi_{A}=\sum_{v\in A}\ket{v}\bra{v},$$

(28)

with $\pi_{A}$ the projector in $\mathbb{C}^{2d}$ onto the vector space associated to sites in the region $A$. It follows that the entanglement entropy is given in terms of the eigenvalues $\lambda_{\ell}$ of the matrix $C$,

$$S_{A}=-\sum_{\ell}(\lambda_{\ell}\ln{\lambda_{\ell}}+(1-\lambda_{\ell})\ln{(1-\lambda_{\ell})}).$$

(29)

We shall restrict ourselves to the case where $A\subset X_{2d}$ is composed of the first $L$ neighborhoods of a given vertex $v_{0}\in X_{2d}$, i.e.

$$\pi_{A}=\sum_{i=0}^{L}E_{i}^{*},\quad E_{i}^{*}=\sum_{\begin{subarray}{c}v\in X_{2d}\text{ s.t.}\\ \text{dist}(v,v_{0})=i\end{subarray}}\ket{v}\bra{v}.$$

(30)

Such subsystems are balls centered on an arbitrary point, and are natural generalizations of single intervals in one-dimensional systems. Moreover, the symmetry of region $A$ shall allow us to perform exact calculations through the use of dimensional reduction.

Since folded cubes are vertex-transitive, we can choose $v_{0}=(0,0,\dots,0)$ without loss of generality. The number of sites in the $i$-th neighborhood of $v_{0}$ is given by

$$\text{tr}(E_{i}^{*})=\binom{2d}{i}+\binom{2d}{2d+1-i},$$

(31)

and the computation of $S_{A}$ thus reduces to the diagonalization of a square matrix $C$ of dimension $|A|=\text{tr}(\pi_{A})$,

$$\text{tr}(\pi_{A})=1+\sum_{i=1}^{L}\left(\binom{2d}{i}+\binom{2d}{2d+1-i}\right).$$

(32)

The computation of $S_{A}$ can be further simplified by an explicit block-diagonalization of $C$. Indeed, the truncated correlation matrix is part of the matrix algebra $\mathcal{T}$ generated by the projectors $E_{k}$ onto eigenspaces of the adjacency matrix and the projectors $E_{i}^{*}$ onto neighborhoods of the vertex $v_{0}=(0,0,\dots,0)$,

$$C=\sum_{i,j\leqslant L}\sum_{k\leqslant K}E_{i}^{*}\left(E_{2k}+E_{2k+1}\right)E_{j}^{*}.$$

(33)

Folded cubes are distance-regular graphs that satisfy the $Q$-polynomial property. Consequently, the algebra $\mathcal{T}$, known as the Terwilliger algebra of folded cubes, possesses interesting properties [20, 21, 22]. It is semi-simple and equivalent to the algebra generated by the adjacency matrix $\mathcal{A}$ and the dual adjacency matrix $\mathcal{A}^{*}$,

$$\mathcal{T}=\langle\mathcal{A},\mathcal{A}^{*}\rangle,$$

(34)

where $\mathcal{A}^{*}$ is the diagonal matrix whose entries are given by

$$\bra{v}\mathcal{A}^{*}\ket{v}=2^{2d}\bra{v}\left(E_{0}+E_{1}\right)\ket{v_{0}}=(-1)^{\text{wt}(v)}(2d+1-2\text{wt}(v)),$$

(35)

where the weight function wt counts the number of $1$ in a binary sequence,

$$\text{wt}(v)=\sum_{i=1}^{2d}v_{i}.$$

(36)

The dual adjacency matrix can be expressed in the basis (9) in terms of Pauli matrices as

$$\mathcal{A}^{*}=\left(\sum_{n=1}^{2d}\underbrace{\sigma_{z}\otimes\sigma_{z}\otimes...\otimes\sigma_{z}}_{n-1\text{ times}}\otimes\ I\otimes\underbrace{\sigma_{z}\otimes...\otimes\sigma_{z}}_{2d-n\text{ times}}\right)+\underbrace{\ \sigma_{z}\otimes\ \sigma_{z}\otimes...\otimes\ \sigma_{z}}_{2d\text{ times}}.$$

(37)

Using the expressions (10) and (37), one can show that the matrices

$$K_{1}=(-1)^{d}\mathcal{A}/2,\quad K_{2}=\{\mathcal{A},\mathcal{A}^{*}\}/4,\quad K_{3}=(-1)^{d}\mathcal{A}^{*}/2$$

(38)

verify the following defining relations of the algebra $\mathfrak{so}(3)_{-1}$,

$$\{K_{1},K_{2}\}=K_{3},\quad\{K_{2},K_{3}\}=K_{1},\quad\{K_{3},K_{1}\}=K_{2}.$$

(39)

The algebra $\mathfrak{so}(3)_{-1}$ amounts to the anti-commutator version of the Lie algebra $\mathfrak{so}(3)$, and it has he following Casimir operator,

$$\bm{K}^{2}=K_{1}^{2}+K_{2}^{2}+K_{3}^{2},\quad[\bm{K}^{2},K_{i}]=0.$$

(40)

From this identification, one gets that the vector space $\mathbb{C}^{2^{2d}}$ onto which the adjacency matrices $\mathcal{A}$, $\mathcal{A}^{*}$ and truncated correlation matrix $C$ act is a $\mathfrak{so}(3)_{-1}$-module. Using the standard representation theory of $\mathfrak{so}(3)_{-1}$ [29], this module can be decomposed into its irreducible components $\mathcal{V}_{j,r}$:

$$\mathbb{C}^{2^{2d}}=\bigoplus_{j=0}^{d}\bigoplus_{r=1}^{D_{j}}\mathcal{V}_{j,r},\quad D_{j}=\left\{\begin{array}[]{ll}\frac{2j+1}{2d+1}\binom{2d+1}{d-j}+\frac{2j+3}{2d+1}\binom{2d+1}{d-j-1}&\mbox{if }j\in\{0,1,2,\dots,d-1\},\\ 1&\mbox{if }j=d,\end{array}\right.$$

(41)

where $\mathcal{V}_{j,r}$ is a subspace of $\mathbb{C}^{2^{2d}}$ spanned by vectors $\ket{j,r,n}_{3}$,

$$\mathcal{V}_{j,r}=\text{span}\{\ket{j,r,n}_{3}\ |\ n\in\{0,1,\dots,j\}\}.$$

(42)

The matrices $K_{1}$ and $K_{3}$ act on these vectors respectively as tridiagonal and diagonal matrices,

$$\begin{split}K_{1}\ket{j,r,n}_{3}&=\sqrt{\frac{(j+n+2)(j-n)}{4}}\ket{j,r,n+1}_{3}+\delta_{n,0}\left(\frac{j+1}{2}\right)\ket{j,r,n}_{3}\\ &+(1-\delta_{n,0})\sqrt{\frac{(j+n+1)(j+1-n)}{4}}\ket{j,r,n-1}_{3},\\[8.5359pt] K_{3}\ket{j,r,n}_{3}&=(-1)^{n}\left(n+\frac{1}{2}\right)\ket{j,r,n}_{3}.\end{split}$$

(43)

The Casimir $\bm{K}^{2}$ also acts on $\mathcal{V}_{j,r}$ as a multiple of the identity,

$$\bm{K}^{2}\ket{j,r,n}_{3}=((j+1)^{2}-1/4)\ket{j,r,n}_{3}.$$

(44)

The matrix $K_{3}$ being diagonal in this basis, one finds that the projectors $E_{i}^{*}$ have a simple action,

$$E_{i}^{*}\ket{j,r,n}_{3}=\delta_{n,d-i}\ket{j,r,n}_{3}.$$

(45)

The representation theory of $\mathfrak{so}(3)_{-1}$ further guaranties the existence of an alternative basis for the modules $\mathcal{V}_{j,r}$, such that the roles of $K_{1}$ and $K_{3}$ are inverted. In other words, we have

$$\mathcal{V}_{j,r}=\text{span}\{\ket{j,r,k}_{1}\ |\ k\in\{0,1,\dots,j\}\},$$

(46)

where the action of $K_{1}$ and $K_{3}$ on $\ket{j,r,k}_{1}$ is given by

$$\begin{split}K_{3}\ket{j,r,k}_{1}&=\sqrt{\frac{(j+k+2)(j-k)}{4}}\ket{j,r,k+1}_{1}+\delta_{k,0}\left(\frac{j+1}{2}\right)\ket{j,r,k}_{1}\\ &+(1-\delta_{k,0})\sqrt{\frac{(j+k+1)(j+1-k)}{4}}\ket{j,r,k-1}_{1},\\[8.5359pt] K_{1}\ket{j,r,k}_{1}&=(-1)^{k}\left(k+\frac{1}{2}\right)\ket{j,r,k}_{1}.\end{split}$$

(47)

In this second basis, one finds a simple action of the projectors $E_{2k}+E_{2k+1}$ onto eigenspaces of $K_{1}$,

$$\left(E_{2k}+E_{2k+1}\right)\ket{j,r,k^{\prime}}_{1}=\left(\delta_{2k-d,k^{\prime}}+\delta_{d-2k-1,k^{\prime}}\right)\ket{j,r,k^{\prime}}_{1}.$$

(48)

The overlaps $Q_{k,n}:=\prescript{}{1}{\bra{j,r,k}\ket{j,r,n}}_{3}$ between these two bases of the submodule $\mathcal{V}_{j,r}$ can be computed explicitly. Indeed, equating the action of the Hermitian operator $K_{3}$ on the left and on the right in $\prescript{}{1}{\bra{j,r,k}K_{3}\ket{j,r,n}}_{3}$ and using (43) and (47) yield the three term recurrence relation

$$\begin{split}(-1)^{n}(n+1/2)Q_{n,k}&=\sqrt{\frac{(j+k+2)(j-k)}{4}}Q_{n,k+1}+\delta_{k,0}\left(\frac{j+1}{2}\right)Q_{n,k}\\ &+(1-\delta_{k,0})\sqrt{\frac{(j+k+1)(j+1-k)}{4}}Q_{n,k-1}.\end{split}$$

(49)

This recurrence is solved by anti-Krawtchouk polynomials $\hat{P}_{n}(x_{k})$ evaluated on the grid $x_{k}=(-1)^{k}(k+1/2)$ and modulated by appropriate weights $\Omega_{k}$ and normalisation functions $\Phi_{n}$,

$$Q_{k,n}=\sqrt{\frac{\Omega_{k}}{\Phi_{n}}}\hat{P}_{n}(x_{k}).$$

(50)

Explicit expressions for these functions are provided in App. A.

The overlaps $Q_{k,n}$ provide analytical expressions for the entries of the truncated correlation matrix in the basis of vectors $\ket{j,r,n}_{3}$,

$$\prescript{}{3}{\bra{j,r,n}C\ket{j^{\prime},r^{\prime},n^{\prime}}}_{3}=\delta_{j,j^{\prime}}\delta_{r,r^{\prime}}\left(\sum_{k=\left\lceil{\frac{d-j-1}{2}}\right\rceil}^{\text{min}\{K,\left\lceil{d/2}\right\rceil-1\}}Q_{d-2k-1,n}Q_{d-2k-1,n^{\prime}}+\sum_{k=\left\lceil{d/2}\right\rceil}^{\text{min}\{K,\left\lfloor{\frac{d+j}{2}}\right\rfloor\}}Q_{2k-d,n}Q_{2k-d,n^{\prime}}\right).$$

(51)

In this basis, the truncated correlation matrix $C$ exhibits a block-diagonal structure, where each submatrix $C|_{\mathcal{V}_{j,r}}$ depends solely on the value of $j$ and is independent of $r$. This property arises from the isomorphism between the submodules $\mathcal{V}_{j,r}$ corresponding to different values of $r$. Exploiting this block-diagonalization, we derive the following formula for the entanglement entropy $S_{A}$,

$$S_{A}=\sum_{j=0}^{d}D_{j}S(j),$$

(52)

where the terms $S(j)$ are given by

$$S(j)=-\sum_{\ell}(\lambda_{j,\ell}\ln\lambda_{j,\ell}+(1-\lambda_{j,\ell})\ln(1-\lambda_{j,\ell}))$$

(53)

and $\lambda_{j,\ell}$ are the eigenvalues of the submatrix $C|_{\mathcal{V}_{j,r}}$ restricted to the irreducible subspace $\mathcal{V}_{j,r}$ and with entries given by (51).

The coefficients $S(j)$ in (52) for the entanglement entropy $S_{A}$ can be interpreted as the entanglement entropy of inhomogeneous one-dimensional free-fermion systems. Indeed, in terms of the fermion operators

$$b_{j,r,n}=\sum_{v\in X_{2d}}\prescript{}{3}{\bra{j,r,n}\ket{v}}c_{v},\quad b_{j,r,n}^{\dagger}=\sum_{v\in X_{2d}}\bra{v}\ket{j,r,n}_{3}c_{v}^{\dagger},$$

(54)

one can rewrite the Hamiltonian $\mathcal{H}$ in (13) as a sum of Hamiltonians $\mathcal{H}_{j,r}$ acting on independent degrees of freedom,

$$\mathcal{H}=\sum_{j=0}^{d}\sum_{r=1}^{D_{j}}\mathcal{H}_{j,r}$$

(55)

where $\mathcal{H}_{j,r}$ describes free fermions on an inhomogeneous chain of length $j+1$, governed by the Hamiltonian

$$\mathcal{H}_{j,r}=(j+1)b_{j,r,0}^{\dagger}b_{j,r,0}+\sum_{n=0}^{j-1}\sqrt{(j+n+2)(j-n)}(b_{j,r,n+1}^{\dagger}b_{j,r,n}+b_{j,r,n}^{\dagger}b_{j,r,n+1}).$$

(56)

Since these Hamiltonians can be diagonalized in terms of anti-Krawtchouk polynomials, these are referred to as anti-Krawtchouk chains. From the point of view of these chains, the ground state $|\Psi_{0}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}$ is expressed as the tensor product of ground states $|\Psi_{0}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}_{j,r}$ of each chain in the decomposition (55),

$$|\Psi_{0}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}=\bigotimes_{j=0}^{d}\bigotimes_{r=1}^{D_{j}}|\Psi_{0}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}_{j,r},$$

(57)

and the correlation matrix decomposes as

$$\hat{C}=\bigoplus_{j=0}^{d}\bigoplus_{r=1}^{D_{j}}\hat{C}|_{\mathcal{V}_{j,r}}$$

(58)

where $\hat{C}|_{\mathcal{V}_{j,r}}$ is the correlation matrix in the ground state of $\mathcal{H}_{j,r}$. Given the simple action (45) of $E_{i}^{*}$ on each module $\mathcal{V}_{j,r}$, we also have that

$$\pi_{A}=\bigoplus_{j=0}^{d}\bigoplus_{r=1}^{D_{j}}\pi_{A}|_{\mathcal{V}_{j,r}}$$

(59)

where $\pi_{A}|_{\mathcal{V}_{j,r}}$ is the projector onto the last $L-d+j+1$ sites of the anti-Krawtchouk chain $j,r$,

$$\pi_{A}|_{\mathcal{V}_{j,r}}=\sum_{n=d-L}^{j}\ket{j,r,n}_{3}\prescript{}{3}{\bra{j,r,n}}.$$

(60)

Using (58) and (60), one recovers a block-diagonalization of the truncated correlation matrix $C$,

$${C}=\bigoplus_{j=0}^{d}\bigoplus_{r=1}^{D_{j}}\pi_{A}\hat{C}\pi_{A}|_{\mathcal{V}_{j,r}}=\bigoplus_{j=0}^{d}\bigoplus_{r=1}^{D_{j}}{C}|_{\mathcal{V}_{j,r}}$$

(61)

where $C|_{\mathcal{V}_{j,r}}$ is now interpreted as the truncated correlation matrix associated to the last $L-d+j+1$ sites of the ground state of $\mathcal{H}_{j,r}$. The coefficient $S(j)$ in (52) thus corresponds to the entanglement entropy contribution coming from the intersection of the region $A$ and the chain associated to the module $\mathcal{V}_{j,r}$.

In the following, we investigate the scaling of $S(j)$ as a function of $j$. The entanglement properties of inhomogeneous free-fermion chains solved by orthogonal polynomials have been intensely studied recently [30, 31, 32, 33, 34, 35].

We fix the ratios $\kappa:=K/d$ and $\xi:=(d-L)/(j+1)$ and use (53) to compute $S(j)$ via exact numerical diagonalization of the chopped correlation matrix (51). Physically, the ratio $\kappa$ corresponds to the filling fraction, whereas $\xi$ is (one minus) the aspect ratio, namely the ratio between the size $j+1$ of the chain, and the length $d-L$ of the complement of the intersection between $A$ and the chain. We present our numerical results for chains at half-filling, $\kappa=1/2$, in Fig. 3. We find a scaling of the form

$$S(j)=\frac{1}{6}\ln(j)+a_{1}(\kappa,\xi)+o(1)$$

(62)

in the limit of large $j$. This corresponds to a logarithmic violation of the area law in one dimension [6], which is typical for one-dimensional free-fermion models described by an underlying CFT with central charge $c=1$. The scaling (62) thus suggests that the anti-Krawtchouk chain is described by a CFT in curved space [36] with $c=1$, similarly to the Krawtchouk chain [32, 34].

The presence of oscillations in Fig. 3 can be attributed to sub-leading terms that have not been fully characterized in the present analysis. Similar oscillations have been observed in Krawtchouk chains and a conjecture regarding the sub-leading terms was proposed in [34].

For a fixed filling ratio $\kappa$, a fixed ratio $\Delta:=L/d$ and a fixed subsystem size $\ell=L-d+j+1$ in the chain, the entanglement entropy rather converges at large diameter $d$ to a constant value,

$$S(d-L-1+\ell)=a_{2}(\kappa,\Delta,\ell)+o(1)\leqslant\ell\ln(2).$$

(63)

Here, the bound on the entanglement entropy is determined by its value for a maximally entangled state given a subsystem of size $\ell$. Numerical analysis verifies that the magnitude of $a_{2}(\kappa,\Delta,\ell)$ remains close to $\ln(2)$ even for $\ell\neq 1$, indicating as expected that most of the entanglement originates from a highly entangled state at the boundary, see Fig. 4.

Let us fix the ratio $\Delta:=L/d$, where $L$ is the diameter of the region $A$ in the folded cube and $d$ is the diameter of the graph. We shall now consider how the entanglement entropy scales in the limit of large diameter $d$ at half-filling $K=d/2$. From numerical tests and the scaling given by (62) and (63), we find that the behavior of $S(j)$ is captured by

$$S(j)\sim\left\{\begin{array}[]{ll}\frac{1}{6}\ln{(d)}+O(1)&\mbox{if }1-\Delta<j/d,\\[5.69046pt] 0&\mbox{if }1-\Delta>j/d,\\[5.69046pt] a_{2}(\kappa,\Delta,\ell)&\mbox{if }1-\Delta\approx j/d.\end{array}\right.$$

(64)

In the first situation, $1-\Delta<j/d$, the scaling behavior follows equation (62). For $1-\Delta>j/d$, the contribution $S(j)$ arises from anti-Krawtchouk chains that do not intersect with the region $A$, resulting in a zero contribution. In the third regime, $1-\Delta\approx j/d$, the chains have a small intersection $\ell\ll d$ with the region $A$ compared to their size of $j+1$; the region $A$ in these chains is predominantly composed of a highly entangled site on the boundary, leading to a contribution of $a_{2}(\kappa,\Delta,\ell)\leqslant\ell\ln(2)$ to $S(j)$.