Q-deformed rainbows: a universal simulator of free entanglement spectra

Consider the two-spin Hamiltonian

$$\begin{align} \mathcal{H}_{\left(1\right)} = J_{1} \left(\sigma_{-1}^{x}\sigma_{1}^{x}+\sigma_{-1}^{y}\sigma_{1}^{y} \right) + h_{1} \left(\sigma_{-1}^{z} - \sigma_{1}^{z} \right). \end{align} \tag{ 1 }$$

The ground state of $\mathcal{H}$ is given by

$$\begin{align} \vert \psi_{1}\rangle = \frac{1}{\sqrt{\left[2\right]_{q_{1}}}}\left(q_{1}^{-\frac{1}{2}}\vert \uparrow\downarrow\rangle_{-1,1}-q_{1}^{\frac{1}{2}}\vert \downarrow\uparrow\rangle_{-1,1}\right), \end{align} \tag{ 2 }$$

and has ground state energy

$$\begin{align} E_{1} = -\left[2\right]_{q_{1}}J_{1}, \end{align} \tag{ 3 }$$

where

$$\begin{align} q_{1} = e^{\gamma_{1}}, \hspace{0.2cm} \sinh{\gamma_{1}} = \frac{h_{1}}{J_{1}}, \end{align} \tag{ 4 }$$

and $[x]_{q}$ is the so-called quantum dimension

$$\begin{align} \left[x\right]_{q} = \frac{q^{x}-q^{-x}}{q-q^{-1}}. \end{align} \tag{ 5 }$$

This ground state is the singlet of the quantum group $SU(2)_{q_{1}}$ [37]. Such q-deformed valence bonds have been considered in relation to a range of many-body models [38–41], including the anisotropic q-deformed generalization of the spin-1 AKLT chain as considered in [42, 43]. In the limit $h_{1}\rightarrow 0$ such that $q_{1}\rightarrow 1$, we recover the maximally entangled singlet state of the standard SU(2) Lie algebra. The degree of entanglement between this pair is directly related to the value of the deformation parameter, q₁, which is in turn directly related to our coupling and transverse field parameters via equation (4). To investigate this, we bipartition the system down the center of the chain into region A, and its complement B. The reduced density matrix of (2) is then determined for region A. The corresponding Rényi entropy of order α is given by

$$\begin{align} S_{A,1}^{\left(\alpha\right)} = \frac{1}{1-\alpha} \ln{\frac{1+q_{1}^{2\alpha}}{\left(1+q_{1}^{2}\right)^{\alpha}}}, \hspace{0.5cm} \alpha\gt 0, \alpha\neq 1 \end{align} \tag{ 6 }$$

and takes the maximum value $\ln{2}$ when $q_{1} = 1$ for all α as shown in figure 1. This expression for the Rényi entropy reflects a symmetry of our model as under the transformation $h_{1}\rightarrow -h_{1}$, such that $q_{1}\rightarrow\frac{1}{q_{1}}$, the value of the Rényi entropy of order α is unchanged.

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By considering the limit of $S_{A,1}^{(\alpha)}$ as $\alpha\rightarrow 1$ we obtain the expression for the von Neumann entanglement entropy of the pair

$$\begin{align} S_{A,1} = \ln{\left(1+q_{1}^{2}\right)}-\frac{q_{1}^{2}}{1+q_{1}^{2}} \ln{q_{1}^{2}}. \end{align} \tag{ 7 }$$

This entropy can be varied continuously to achieve all values in the maximal range $0\unicode{x2A7D} S_{A,1}\unicode{x2A7D} \ln{2}$, by varying $0\unicode{x2A7D} q_{1}\unicode{x2A7D} \infty$, or equivalently $-\infty\unicode{x2A7D} \frac{h_{1}}{J_{1}}\unicode{x2A7D} \infty$. We see that by varying the physical parameters of our model we can achieve all degrees of pairwise entanglement between the two spins.

The simple two-spin Hamiltonian presented above is the basis on which we construct our general model for a chain of any even number of spins. We now consider a chain of 2N spin-$\frac{1}{2}$ particles with the following Hamiltonian

$$\begin{align} \mathcal{H}_{\left(N\right)} &= \sum_{i = 1}^{N} h_{i} \left(\sigma_{-i}^{z} - \sigma_{i}^{z} \right) + J_{1} \left(\sigma_{-1}^{x}\sigma_{1}^{x}+\sigma_{-1}^{y}\sigma_{1}^{y} \right) \end{align}$$

$$\begin{align} &\quad+ \sum_{i = 2}^{N} J_{i}\left(\sigma_{-i}^{x}\sigma_{-\left(i-1\right)}^{x} + \sigma_{-i}^{y}\sigma_{-\left(i-1\right)}^{y} + \sigma_{i-1}^{x}\sigma_{i}^{x} + \sigma_{i-1}^{y}\sigma_{i}^{y} \right). \end{align} \tag{ 8 }$$

We have introduced the site labeling $\{-N, -(N-1), \ldots, -2, -1, 1, 2, \ldots, N-1, N\}$ such that sites −i and i are equidistant from a central bipartition of the chain, as shown in figure 2.

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In order to find the ground state of our model we have used the Real-Space Renormalization Group approach as first introduced by Ma and Dasgupta in [44] and later developed by Fisher with the application of the method to the Random Transverse Field Ising Chain [45]. This approach allows us to consider the ground state properties of random quantum chains by iteratively decimating the degrees of freedom with highest energy in order to derive an overall effective low-energy model. We start by first briefly reviewing this method with reference to the known results of the $h_{i}\rightarrow0$ limit of our model.

2.2.1. XX model Renormalization Group

In the limit, $h_{i} = 0$, our Hamiltonian (8) is equivalent to that of the inhomogeneous XX model acting on a chain of 2N spins

$$\begin{align} \mathcal{H}_{XX} = \sum_{i = 1}^{2N} J_{i} \left(\sigma_{i}^{x}\sigma_{i+1}^{x}+\sigma_{i}^{y}\sigma_{i+1}^{y} \right), \end{align} \tag{ 9 }$$

where we have re-adopted the standard site labeling $\{1, 2, \ldots, 2N-1, 2N\}$. Using the Renormalization Group (RG) approach for some random coupling profile, the highest energy term such that $J_{i} \gg J_{i-1}, J_{i+1}$, is identified and diagonalised independently of the rest of the chain. To zeroth-order in perturbation theory, the ground state of the system is then

$$\begin{align} \vert \psi\rangle = \vert \psi_{j\lt i}\rangle\otimes\vert \psi^{-}_{i}\rangle\otimes\vert \psi_{j\gt i}\rangle, \end{align} \tag{ 10 }$$

where $\vert \psi^{-}_{i}\rangle = \frac{1}{\sqrt{2}}(\vert \uparrow\downarrow\rangle_{i,i+1}-\vert \downarrow\uparrow\rangle_{i,i+1})$ is the maximally entangled singlet ground state of the two-site XX model and $\vert \psi_{j\lt i}\rangle$, $\vert \psi_{j\gt i}\rangle$ refer to the state of the spins to the left and right of the singlet respectively. To compute higher order corrections to the ground state of our system we consider the spins i and i + 1 to be 'frozen' into this singlet state. Then we employ perturbation theory to find the effect induced by quantum fluctuations on the neighboring spins, as shown in [12]. It is found that an effective coupling arises between sites i − 1 and i + 2 of strength

$$\begin{align} \tilde{J}_{i-1,i+2} = \frac{J_{i-1}J_{i+1}}{J_{i}}. \end{align} \tag{ 11 }$$

In this way the coupling between sites i and i + 1 is replaced by effective longer range interaction that captures the low-energy properties of the model. For a random coupling profile, successive iterations of this procedure yield a 'random singlet phase', as singlets form between the pairs of spins most strongly coupled after each decimation. In [12] Vitagliano, Riera and Latorre demonstrated how a coupling profile that decays exponentially away from the center of the chain produces a special form of ground state known as the 'concentric singlet phase'. This ground state is also known as the 'rainbow state', due to it is distinctive structure of a series of singlets symmetrically distributed around the center of the chain. For any given bipartition, the entanglement entropy is directly proportional to the number of singlets 'cut' by the bipartition. Thus, for such a coupling profile, the area law of entanglement entropy is maximally violated.

2.2.2.  Q-deformed model Renormalization Group

We now apply the Real-Space RG approach to the generalized model defined in (8). In the limit $J_{1}, h_{1} \gg J_{2}, h_{2}$, this yields the ground state

$$\begin{align} \vert \psi\rangle = \vert \psi_{i-1}\rangle\otimes\vert \psi_{1}\rangle\otimes\vert \psi_{i\gt 1}\rangle, \end{align} \tag{ 12 }$$

to zeroth-order in perturbation theory, where $\vert \psi_{1}\rangle$ is the q₁-deformed singlet as defined in equation (4). To compute corrections to the ground state at higher orders, second-order perturbation theory is used, as illustrated in figure 3(see also appendix A). We derive an effective Hamiltonian of the form (1) acting between sites −2 and 2 with a renormalized coupling

$$\begin{align} \tilde{J_{2}} = \frac{4J_{2}^{2}}{\left[2\right]_{q_{1}}^{2}J_{1}}, \end{align} \tag{ 13 }$$

and transverse field terms

$$\begin{align} \tilde{h}_{2} = h_{2} - \frac{2\left(q_{1}-\frac{1}{q_{1}}\right)J_{2}^{2}}{\left[2\right]_{q_{1}}^{2}J_{1}}. \end{align} \tag{ 14 }$$

In the case that $\tilde{J_{2}}, \tilde{h}_{2} \gg J_{3}, h_{3}$ this effective Hamiltonian can be diagonalised to yield an additional q₂-deformed singlet, $\vert \psi_{2}\rangle$, between sites −2 and 2, where $q_{2} = e^{\gamma_{2}}, ~\sinh{\gamma_{2}} = \frac{\tilde{h}_{2}}{\tilde{J_{2}}}$.

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If the couplings throughout the chain are selected such that $\tilde{J_{i}}, \tilde{h}_{i} \gg J_{i+1}, h_{i+1}$, then repeated iterations of this renormalization process will eventually yield the overall ground state

$$\begin{align} \vert \psi\rangle = \vert \psi_{1}\rangle\otimes\vert \psi_{2}\rangle\otimes\ldots\otimes\vert \psi_{N}\rangle, \end{align} \tag{ 15 }$$

where

$$\begin{align} \vert \psi_{i}\rangle = \frac{1}{\sqrt{\left[2\right]_{q_{i}}}}\left(q_{i}^{-\frac{1}{2}}\vert \uparrow\downarrow\rangle_{-i,i}-q_{i}^{\frac{1}{2}}\vert \downarrow\uparrow\rangle_{-i,i}\right), \end{align} \tag{ 16 }$$

and for i > 1

$$\begin{align} q_{i} = e^{\gamma_{i}} \text{with } \sinh{\gamma_{i}} = \frac{\tilde{h}_{i}}{\tilde{J_{i}}}. \end{align} \tag{ 17 }$$

The renormalized coupling and transverse field parameters for the effective Hamiltonian between spins −i and i are given by the recursive expressions

$$\begin{align} \tilde{J_{i}} = \frac{4J_{i}^{2}}{\left[2\right]_{q_{i-1}}^{2}\tilde{J}_{i-1}}, \end{align} \tag{ 18 }$$

$$\begin{align} \tilde{h}_{i} = h_{i} - \frac{2\left(q_{i-1}-\frac{1}{q_{i-1}}\right)J_{i}^{2}}{\left[2\right]_{q_{i-1}}^{2}\tilde{J}_{i-1}}. \end{align} \tag{ 19 }$$

From equations (18) and (19), we see that the expressions for $\tilde{J_{i}}$ and $\tilde{h}_{i}$ are dependent on all previous $\tilde{J}_{j\lt i}$, $\tilde{h}_{j\lt i}$. By fixing all previous i − 1 values, it is always possible to vary the associated physical parameters J_i and h_i such as to achieve any $1\unicode{x2A7D} q_{i}\unicode{x2A7D} \infty$. In this way, we will show that the deformation of each q-singlet can be individually tuned to achieve any degree of pairwise entanglement between a given pair of spins.

Using the reduced density matrix tensor product decomposition we derive the form of the Rényi entropy of order α of the ground state (15) across a central bipartition

$$\begin{align} S_{A}^{\left(\alpha\right)} = \frac{1}{1-\alpha} \sum_{i = 1}^{N}\ln{\frac{1+q_{i}^{2\alpha}}{\left(1+q_{i}^{2}\right)^{\alpha}}}, \hspace{0.3cm} \alpha\gt 0, \alpha\neq 1. \end{align} \tag{ 22 }$$

In the limit $\alpha \rightarrow 1$ we obtain an expression for the von Neumann entropy of the ground state

$$\begin{align} S_{A} = -\sum_{i = 1}^{N} \left[ \ln{\left(1+q_{i}^{2}\right)}-\frac{q_{i}^{2}}{1+q_{i}^{2}} \ln{q_{i}^{2}} \right]. \end{align} \tag{ 23 }$$

In section 2.1 we found the von Neumann entropy, $S_{A,1}$, of a single pair of spins as a function of the deformation parameter q₁, as given by (6). By extending this definition to that of the von Neumann entropy of the state $\vert \psi_{i}\rangle$ between spins −i and i

$$\begin{align} S_{A,i} = \ln{\left(1+q_{i}^{2}\right)}-\frac{q_{i}^{2}}{1+q_{i}^{2}} \ln{q_{i}^{2}}, \end{align} \tag{ 24 }$$

it is clear that the total von Neumann entropy is a sum of the individual von Neumann entropies of each concentric pair of spins on the chain. This is also true for the Rényi entropy, and is a natural consequence of the tensor product form of the reduced density matrix. By independently varying each q_i , we can therefore achieve all degrees of entanglement in the allowed maximal range $0\unicode{x2A7D} S_{A}\unicode{x2A7D} N\ln{2}$.

The entanglement spectrum was introduced by Li and Haldane [46] as an alternative entanglement measure that aimed to capture a complete representation of the entanglement between two subsystems [47]. The values of the spectrum, E_i , are related to the eigenvalues of the reduced density matrix, λ_i , via

$$\begin{align} \lambda_{i} = e^{-E_{i}}. \end{align} \tag{ 25 }$$

This exponential relationship means that the dominant quantum correlations depend predominantly on the 'lowest' part of the entanglement spectrum.

The entanglement spectrum reflects many of the physical properties of the system [48–52] and serves as a fingerprint of topological order [53–55]. For any non-interacting model, Wick's theorem shows that the spectrum can be constructed from a set of single-particle entanglement energies as

$$\begin{align} E^{f}_{j}\left(\epsilon\right) = E_{0} + \sum_{i = 1}^{N} n_{i}\left(j\right) \epsilon_{i}, \end{align} \tag{ 26 }$$

where E₀ is a normalization constant and each $n_{j} = \{0,1\}$ [56].

For our q-deformed rainbow, we find that

$$\begin{align} E_{0} = \sum_{i = 1}^{N} \ln{\left(1+q_{i}^{2}\right)}, \end{align} \tag{ 27 }$$

and

$$\begin{align} \epsilon_{i} = -\ln{q_{i}^{2}}. \end{align} \tag{ 28 }$$

Hence, the deformation parameters, q_i , of the q-deformed singlets directly determine the single-particle entanglement energies. As each q_i can take any value in the range $0\unicode{x2A7D} q_{i}\unicode{x2A7D}\infty$, each ε_i can be individually tuned to take any value $-\infty\unicode{x2A7D}\epsilon_{i}\unicode{x2A7D}\infty$.

By combining (17) and (28) we derive the simple relationship

$$\begin{align} \gamma_{i} = -\frac{\epsilon_{i}}{2}. \end{align} \tag{ 29 }$$

This in turn yields an expression for the required ratio of the renormalized parameters for a given pair in order to produce a specific desired single-particle entanglement energy

$$\begin{align} \frac{\tilde{h}_{i}}{\tilde{J_{i}}} = -\sinh{\left(\frac{\epsilon_{i}}{2}\right)}. \end{align} \tag{ 30 }$$

As a result, each single particle energy of the entanglement spectrum can be directly obtained by appropriately tuning a single effective magnetic field. In appendix B we expand these expressions to derive closed forms for the required ratio of the physical coupling parameters. In figure 4 the dependence of ε₂ on h₂ for fixed $J_{1},h_{1}$ and J₂ is illustrated. For the shown range, any desired ε₂ can be simulated by simply reading off the corresponding value of h₂. In this way, by fixing all previous i − 1 single-particle entanglement energies, ε_i can be tuned to any desired value by appropriately varying h_i .

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In section 2.1, we noted that a symmetry of our two-site Hamiltonian (1) results in the preservation of the von Neumann entropy, $S_{A,1}$, under the transformation $h_{1}\rightarrow -h_{1}$. Here, we will show that although $S_{A,2}$ possesses a similar symmetry under the transformation $\tilde{h}_{2}\rightarrow-\tilde{h}_{2}$, one of these values will yield a significantly higher fidelity than the other corresponding to the choice of sign of $\frac{h_{1}}{J_{1}}$.

In figure 5 we plot the fidelity between the q-deformed rainbow and the exact ground state of (8) as a function of the entanglement entropy between sites −2 and 2 for a range of constant values of h₁. For each curve $J_{1}, h_{1}$ and J₂ are fixed such that $S_{A,2}$ is a function of h₂. We see that for any desired value of $S_{A,2}$, for example $S_{A,2} = 0.5$ as indicated by the vertical gray line, the two intersections with each curve indicate two values of h₂ that correspond to the same degree of entanglement, but with a distinct difference in fidelity. As described these two solutions arise due to the natural symmetry of the entanglement entropy about $\tilde{h}_{2} = 0$, for which the degree of entanglement between sites −2 and 2 is maximized. To find the physical transverse field value $h_2^\text{max}$ that maximizes $S_{A,2}$ we substitute $\tilde{h}_{2} = 0$ into equation (14) to obtain

$$\begin{align} h_{2}^\text{max} = \frac{2\left(q_{1}-\frac{1}{q_{1}}\right)J_{2}^{2}}{\left[2\right]_{q_{1}}^{2}J_{1}}. \end{align} \tag{ 31 }$$

The symmetry of the entanglement entropy $S_{A,2}$ about $h_{2}^\text{max}$ is shown in figure 6(a) for the case $J_{1} = h_{1} = 1$, $J_{2} = 0.1$. By mapping these values onto the plot of fidelity with $S_{A,2}$ as shown in figure 6(b), we see that we have a 'high fidelity branch' corresponding to $h_{2}\unicode{x2A7E} h_{2}^\text{max}$ ($\tilde{h}_{2}\unicode{x2A7E} 0$) and a 'low fidelity branch' for $h_{2}\unicode{x2A7D} h_{2}^\text{max}$ ($\tilde{h}_{2}\unicode{x2A7D} 0$). For any desired value of $S_{A,2}$, the fidelity is clearly maximized by choosing the appropriate value of $h_{2} \unicode{x2A7E} h_{2}^\text{max}$. In contrast, if the sign of h₁ is reversed while J₁ remains fixed, as shown in figures 6(c) and (d), the opposite is true, and the fidelity is maximized by selecting the value of h₂ from the branch $h_{2}\unicode{x2A7D} h_{2}^\text{max}$. In this way, for fixed couplings J₁ and J₂, the magnitude and direction of the applied magnetic field terms h₁ and h₂ can be selected such as to maximize the accuracy of our model.

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Our system has a symmetry with respect to which pair of spins i and −i is used to tune a certain single particle entanglement energy ε_k . We can use this freedom, in conjunction with the optimization procedure of the previous subsection, to optimize the overall fidelity of our chain simulator.

Consider the case in which we want to use our model to generate a given pair of two-site von Neumann enanglement entropies, for example, $S_{A,i} = 0.2$ and $S_{A,j} = 0.6$. Our simulator has the freedom in the choice $i = 1, j = 2$ or $i = 2, j = 1$. In figure 7 we plot the two curves corresponding to $S_{A,1} = 0.2$, and $S_{A,1} = 0.6$ for $J_{1} = 1$, $J_{2} = 0.1$ and h₁ tuned accordingly. The analysis of the previous subsection has been applied to restrict to the appropriate high fidelity branch $h_{2} \unicode{x2A7E} h_{2}^{\text{max}}$. From figure 7 we observe that our model can be used to achieve $S_{A,1} = 0.2, S_{A,2} = 0.6$ with a significantly higher fidelity than $S_{A,1} = 0.6, S_{A,2} = 0.2$. The importance of the ordering of the degrees of entanglement is further illustrated in figure 8 with a direct comparison of the fidelity with the degree of entanglement between one pair when the other is maximally entangled. For all values of $S_{A,i}$, it is observed that the fidelity is maximized when the maximally entangled state lies between sites −2 and 2.

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In appendix C an exact relationship between our chosen parameters and the fidelity is given for the case N = 2. For any desired set of single-particle energies $\epsilon_{i}, \epsilon_{j}$ or entropies $S_{A,i}, S_{A,j}$ the corresponding field and coupling terms can be substituted into equation (C9) in order to directly determine the ordering that will optimize the fidelity.

In the previous two subsections we have shown that appropriate choices of the values of the transverse field parameters, and ordering of the degree of entanglement allow us to optimize the fidelity and thereby the accuracy of our model for the case N = 2. Although the simplest non-trivial implementation of our model, these techniques derived for the four-spin chain can also be used to optimize the fidelity of our simulator on larger chains. For example, in section 5.2 we will introduce an example of a non-uniform spectrum that our model can simulate known as the 'prime number spectrum', in which the single-particle entanglement energies for a chain with N pairs are the logarithms of the first N prime numbers. Performing a numerical optimization procedure for this spectrum on 10 sites with a coupling profile of the form $J_{i} = 10^{-\frac{i(i-1)}{2}}$ we find that the fidelity is maximized when $\{\epsilon\} = \{\ln{3}, \ln{11}, \ln{7}, \ln{5}, \ln{2}\}$. Choosing our coupling and transverse field terms such as to produce the single-particle energies in this order ensures that the ground state of (8) matches our q-deformed ground state with a fidelity $\gt0.988$. To highlight the importance of this choice we note that in contrast, using our simulator to produce the same spectrum with alternate ordering $\{\epsilon\} = \{\ln{11}, \ln{2}, \ln{3}, \ln{5}, \ln{7}\}$, yields a fidelity of only 0.878 to three decimal places.

Thus, our analysis for the case N = 2 also aids in the application of our simulator to larger chains. For larger chains however, there are additional features that we must discuss. We start by noting that, although our model is theoretically exact up to the thermodynamic limit, in practice, the balance between the rapid decay of our coupling and transverse field parameters ensuring the validity of our real-space RG approach and the finite precision of computational methods, only allows us to numerically simulate small system sizes. Such restrictions were discussed in the original work on the rainbow state model [12]. To perform error analysis on the system for larger chains we must therefore choose a coupling profile and single-particle entanglement energy spectrum known to yield a high fidelity. Here we will consider the effect of random errors on the simulation of the optimized prime number spectrum on 8 sites, $\{\epsilon\} = \{\ln{3}, \ln{7},\ln{5},\ln{2} \}$ for the coupling profile $J_{i} = 10^{-\frac{i(i-1)}{2}}$. This spectrum can be achieved with a fidelity $\gt 0.983$. For some fixed Δ we generate a profile of random errors $\{\delta_{i}\} \in [-\Delta,\Delta ]$, such that our parameters $J_{i}\rightarrow J_{i} + \delta_{i}$ and $h_{i}\rightarrow h_{i} + \delta_{i}$. For these altered parameters, equations (18), (19), (28) and (30) can be used to find the new entanglement spectrum and q-deformation parameters. These values can then be used to find a modified form of the q-deformed rainbow ground state. By taking the overlap between this state and the q-deformed rainbow for our desired entanglement profile we calculate a value for the fidelity. For each Δ this process is repeated 10 000 times and an average value for the fidelity calculated. The results for a selected range of Δ are shown in figure 9. We observe that as mentioned above, this drop in fidelity with the increasing size in error is related to the size of the weakest parameters in our model. As the size of the couplings and transverse fields increase from left to right we observe a drop in the fidelity each time the size of the error exceeds the magnitude of one of these parameters. Naturally, as our parameters become smaller with an increasing number of sites on our chain, the accuracy of our model becomes increasingly sensitive to errors.

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In the rainbow state model, the ground state is a tensor product of concentric maximally entangled singlets, or in the language of our model, $q_{i} = 1$, for all i. Here, we show how the parameters of our model can be chosen such that all deformation parameters take the same value, $q_{i} = q$, for some chosen q in the allowed range $0\lt q\lt\infty$. In this way, each concentric pair on our chain shares the same degree of pairwise entanglement, and all single-particle entanglement energies are equal.

We have defined $q_{1} = e^{\sinh^{-1}\left(\frac{h_{1}}{J_{1}}\right)}$ and $q_{i\gt 1} = e^{\sinh^{-1}\left(\frac{\tilde{h}_{i}}{\tilde{J_{i}}}\right)}$, such that the condition $q_{1} = q_{i} \implies \frac{h_{1}}{J_{1}} = \frac{\tilde{h}_{i}}{\tilde{J_{i}}}$. Re-arranging equation (4) and setting $q_{1} = q$ yields

$$\begin{align} \frac{h_{1}}{J_{1}} = \frac{1}{2}\left(q-\frac{1}{q} \right), \end{align} \tag{ 32 }$$

for some desired q > 0.

For all other pairs of sites the relations for the normalized couplings must be used. For example, by dividing equation (14) by equation (13) and equating with (32) we obtain

$$\begin{align} h_{2} = \frac{2J_{2}^{2}}{h_{1}} \frac{\left(1-q^{2}\right)^{2}}{\left(1+q^{2}\right)^{2}}. \end{align} \tag{ 33 }$$

For any fixed value of J₂ this relation can be easily implemented to find the required transverse field parameter to produce some desired q > 0.

In the same way, the ratio of equations (18) and (19) can be equated with (32) in order to obtain the general formula

$$\begin{align} h_{i} = \frac{4J_{i}^{2}}{h_{i-1}} \frac{\left(1-q^{2}\right)^{2}}{\left(1+q^{2}\right)^{2}}, \hspace{0.2cm} i\gt 2 \end{align} \tag{ 34 }$$

By iterating through and systematically determining each successive value of the required transverse field for some fixed coupling profile, these relations allow us to produce a one-dimensional chain in which each concentric pair shares the same degree of pairwise entanglement.

In figure 10 the variation of the fidelity with system size is plotted for a range of values of q. For each fixed number of sites on the chain, the fidelity increases with the value of q, corresponding to a better simulation of chains with a lower degree of entanglement between the pairs. The case with lowest fidelity corresponds to the reproduction of the rainbow state with q = 1 such that $h_{i} = 0$ for all i. For each fixed value of q, as the number of sites on the chain increases the fidelity drops off, rapidly for the increase from 2 to 3 pairs, then more slowly up to 5 pairs.

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Prime numbers play an important role in number theory. The Fundamental Theorem of Arithmetics [58] states that every natural number greater than one can be uniquely factorized as a product of prime numbers

$$\begin{align} N = 2^{n_{2}}3^{n_{3}}\ldots p^{n_{p}}\ldots, \end{align} \tag{ 35 }$$

where p is a prime and n_p counts the number of times that p appears in the factorization of N. In this way, prime numbers can be thought of as the building blocks of all natural numbers.

Let us introduce the Möbius function, $\mu(n)$:

$$\begin{align} \mu\left(n\right) = \begin{cases} 1, & n = 1, \\ \left(-1\right)^{r}, & n = p_{1}\ldots p_{r}, \\ 0, & \exists p, p^{2}\vert n, \end{cases} \end{align} \tag{ 36 }$$

where p are prime numbers. The symbol $p^{2}\vert n$ means that p² divides n. A square free integer is an integer whose factorization into products of primes does not contain any square of a prime numbers. $\mu(n)$ is therefore non-vanishing only on square free integers with value +1 if it contains an even number of primes and −1 if it contains an odd number of primes. In this way, the function µ is a sort of Fermi statistics if one thinks of the primes as being fermions [23].

Let us now consider an entanglement spectrum of the form

$$\begin{align} \lambda_{k} = \frac{A_{F}\lvert \mu\left(k\right)\rvert}{k^{s}}, \hspace{0.2cm} A_{F}, s\gt 0, \hspace{0.1cm} k = 1, \ldots ,\infty. \end{align} \tag{ 37 }$$

The normalization of the eigenvalues implies that

$$\begin{align} 1 = \sum_{k = 1}^{\infty} \lambda_{k} = A_{F} \prod_{p} \left(1+p^{-s}\right) = A_{F} \frac{\zeta\left(s\right)}{\zeta\left(2s\right)}, \end{align} \tag{ 38 }$$

where we have used the Euler product formula

$$\begin{align} \zeta\left(s\right) = \prod_{p} \frac{1}{1-p^{-s}}, \hspace{0.2cm} \text{Re}\left(s\right)\gt 1. \end{align} \tag{ 39 }$$

Using equation (25), the entanglement energies for this spectrum are given by

$$\begin{align} E_{k} = -\ln{\lambda_{k}} = -\ln{A_{F}} + s\ln{k}. \end{align} \tag{ 40 }$$

Where k is any square free integer. We equate this expression with that of the spectrum of a free fermionic system

$$\begin{align} -\ln{A_{F}} + s\ln{k} = E_{0} + \sum_{i = 1}^{N} n_{i}\left(k\right) \epsilon_{i}. \end{align} \tag{ 41 }$$

If k is a square free integer then from the Fundamental Theorem of Arithmetics one has

$$\begin{align} k = 2^{n_{2}}3^{n_{3}}\ldots p^{n_{p}}\ldots, \hspace{0.3cm} n_{i} = 0,1, \end{align} \tag{ 42 }$$

such that taking the logarithm of (42) yields

$$\begin{align} \ln{k} = \sum_{p:prime} n_{p} \ln{p}, \hspace{0.3cm} n_{p} = 0,1. \end{align} \tag{ 43 }$$

Hence, equation (41) is solved by

$$\begin{align} E_{0} &= -\ln{A_{F}} = \frac{\zeta\left(s\right)}{\zeta\left(2s\right)}, \end{align} \tag{ 44 }$$

$$\begin{align} \epsilon_{p}& = s\ln{p}. \end{align} \tag{ 45 }$$

The parameter s can be thought of as an entanglement temperature since it is common to all eigenenergies. The relation (45) has also been considered in [22, 23] with $\ln{p}$ being the single-particle energies of the primon gas. The partition function of this gas is related to the Riemann zeta function, $\zeta(s)$. In recent work, a prime number eigenvalue spectrum has also been experimentally realized by application of holographic optical traps [59], in agreement with previous theoretical results [60].

This example spectrum (45), motivated by the interesting parallel between the decomposition of integers in terms of prime numbers and the decomposition of free-system entanglement spectra in terms of single-particle energies, provides an example of a non-uniform entanglement spectrum that can be reproduced using our q-deformed model. To implement this spectrum for some finite number of pairs with fixed coupling profile $\{J_{i}\}$, the required values of the set $\{h_{i}\}$, can simply be read off from equations (B1)–(B3) in appendix B.