$Q$-Deformed Rainbows: a Universal Simulator of Free Entanglement Spectra

Consider the two-spins Hamiltonian

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

and has ground state energy

where

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

This ground state is the singlet of the quantum group $SU(2)_{q_{1}}$ [34]. Such $q$-deformed valence bonds have been considered in relation to a range of quantum many-body models [35, 36, 37, 38], including the anisotropic $q$-deformed generalization of the spin-$1$ AKLT chain as considered in [39, 40]. 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, $q_{1}$, 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 centre of the chain into region $A$, and its complement $B$. The reduced density matrix of (2) is then determined for region $A$. The corresponding Renyi entropy of order $\alpha$ is given by

and takes the maximum value $\ln{2}$ when $q_{1}=1$ for all $\alpha$ as shown in Figure 1. This expression for the Renyi 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 Renyi entropy of order $\alpha$ is unchanged.

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

This entropy can be varied continuously to achieve all values in the maximal range $0\leq S_{A,1}\leq\ln{2}$, by varying $0\leq q_{1}\leq\infty$, or equivalently $-\infty\leq\frac{h_{1}}{J_{1}}\leq\infty$. 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

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

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 [41] and later developed by Fisher with the application of the method to the Random Transverse Field Ising Chain [42]. 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 $h_{i}\rightarrow 0$ limit of our model.

Using the reduced density matrix tensor product decomposition we derive the form of the Renyi entropy of order $\alpha$ of the ground state (15) across a central bipartition

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

In Section II.1 we found the von Neumann entropy, $S_{A,1}$, of a single pair of spins as a function of the deformation parameter $q_{1}$, as given by (6). By extending this definition to that of the von Neumann entropy of the state $\ket{\psi_{i}}$ between spins $-i$ and $i$

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 Renyi 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 $0\leq S_{A}\leq N\ln{2}$.

The entanglement spectrum was introduced by Li and Haldane [43] as an alternative entanglement measure that aimed to capture a complete representation of the entanglement between two subsystems [44]. The values of the spectrum, $E_{i}$, are related to the eigenvalues of the reduced density matrix, $\lambda_{i}$, via

The 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 [45, 46, 47, 48, 49] and serves as a fingerprint of topological order [50, 51, 52]. For any non-interacting model, Wick’s theorem shows that the spectrum can be constructed from a set of single-particle entanglement energies as

where $E_{0}$ is a normalization constant and each $n_{j}=\{0,1\}$ [53].

For our $q$-deformed rainbow, we find that

and

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 $0\leq q_{i}\leq\infty$, each $\epsilon_{i}$ can be individually tuned to take any value $-\infty\leq\epsilon_{i}\leq\infty$.

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

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

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 $\epsilon_{2}$ on $h_{2}$ for fixed $J_{1},h_{1}$ and $J_{2}$ is illustrated. For the shown range, any desired $\epsilon_{2}$ can be simulated by simply reading off the corresponding value of $h_{2}$. In this way, by fixing all previous $i-1$ single-particle entanglement energies, $\epsilon_{i}$ can be tuned to any desired value by appropriately varying $h_{i}$.

In Section II.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 $h_{1}$. For each curve $J_{1},h_{1}$ and $J_{2}$ are fixed such that $S_{A,2}$ is a function of $h_{2}$. We see that for any desired value of $S_{A,2}$, for example $S_{A,2}=0.5$ as indicated by the vertical grey line, the two intersections with each curve indicate two values of $h_{2}$ 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 the value of $h_{2}$ yielding maximal entanglement between sites $-2$ and $2$. To find the value $h_{2}^{\text{max}}$ that maximises $S_{A,2}$ we set $\tilde{h}_{2}=0$ in equation (14) and obtain

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 $h_{2}\geq h_{2}^{\text{max}}$ and a ‘low fidelity branch’ for $h_{2}\leq h_{2}^{\text{max}}$. For any desired value of $S_{A,2}$, the fidelity is clearly maximised by choosing the appropriate value of $h_{2}\geq h_{2}^{\text{max}}$. In contrast, if $\frac{h_{1}}{J_{1}}$ is negative as shown in Figure 6(c) and (d), the opposite is true, and the fidelity is maximised by selecting the value of $h_{2}$ from the branch $h_{2}\leq h_{2}^{\text{max}}$. In this way, for fixed couplings $J_{1}$ and $J_{2}$, the direction of the magnetic fields applied to sites $i=-1,1$ dictate the magnitude and direction of the magnetic field that should be applied to sites $i=-2,2$ in order to maximise the accuracy of our model.

Our system has a symmetry with respect to which pair $i,-i$ of spins is used to tune a certain single particle entanglement energy $\epsilon_{k}$. We can use this freedom, in conjunction with the optimisation procedure of the previous Subsection to optimise the overall fidelity of our chain simulator. To proceed, we adopt the appropriately restricted range of $h_{2}\geq h_{2}^{\text{max}}$ that maximises the fidelity of two pairs of spins. We then consider the variation of the fidelity with $h_{2}$ for a range of fixed values of $h_{1}$ as shown in Figure 7(a). Here $\frac{h_{1}}{J_{1}}>0$ for each curve such that we have selected the values $h_{2}\geq h_{2}^{\text{max}}$. We first observe that for each fixed value of $h_{1}$, as $S_{A,2}$ increases, the fidelity decreases. This result is more notable for low values of $h_{1}$, such that the combination of parameters with lowest fidelity corresponds to the reproduction of the rainbow state with $h_{1}=h_{2}=0$. Hence, our model is best at accurately producing lower degrees of entanglement between the concentric pairs of sites.

For fixed $J_{1}$, a larger magnitude of $h_{1}$ coresponds to a lower value of $S_{A,1}$. Figure 7(a) therefore also shows that as the value of $h_{1}$ increases and $S_{A,1}$ decreases, the fidelity with which any desired $S_{A,2}$ can be achieved increases. 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 $i=2,j=1$. We will employ this freedom to choose the combination that maximises the fidelity. In Figure 7(b) we plot the two curves corresponding to $S_{A,1}=0.2$, and $S_{A,1}=0.6$. It is clear from this plot that the fidelity is always maximum by choosing the parameters such that $S_{A,1}=0.2,S_{A,2}=0.6$. In general it is always true that the fidelity is maximised by ordering the pairs such that $S_{A,i}\leq S_{A,i+1}$. Figure 8 further illustrates this 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 $S_{A,i}$, we observe that the fidelity is maximised when the maximally entangled state lies between sites $-2$ and $2$.

By combining equations (24) and (28), we see that the condition $S_{A,i}\leq S_{A,i+1}$ is equivalent to $\absolutevalue{\epsilon_{i}}\geq\absolutevalue{\epsilon_{i+1}}$. For some desired set of single-particle entanglement energies, $\{\epsilon_{i}\}$, our model allows for complete freedom in assigning which pair of sites corresponds to a given energy. In implementing our model we therefore choose to tune the parameters such that the magnitude of the single-particle energy generated by sites $-i$ and $i$ decreases with increasing $i$ in order to optimise the fidelity.

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 $0<q<\infty$. 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 $q_{i>1}=e^{\sinh^{-1}\left(\frac{\tilde{h}_{i}}{\tilde{J_{i}}}\right)}$, 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

for some desired $q>0$.

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

For any fixed value of $J_{2}$ 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

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 Figures 9(a) and (b) it is shown how all values of $1\leq q\leq 10$ and the corresponding entanglement entropies for the case $N=4$ can be produced with a very high level of fidelity. For just $J_{1}=1,J_{2}=0.01$ the error is of the order of $10^{-4}$.

Prime numbers play an important role in number theory. The Fundamental Theorem of Arithmetics [55] states that every natural number greater than one can be uniquely factorised as a product of prime numbers

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

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

where $p$ are prime numbers. The symbol $p^{2}|n$ means that $p^{2}$ 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 and its value is $+1$ if it contains an even number of primes and $-1$ if it contains an odd number of primes. In this way, the function $\mu$ is a sort of Fermi statistics if we think of the primes as being fermions.

Let us now consider an entanglement spectrum of the form

The normalization of the eigenvalues implies that

where we have used the Euler product formula

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

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

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

such that taking the logarithm of (42) yields

Hence, equation (41) is solved by

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 [56, 57] 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 realised by application of holographic optical traps [58], in agreement with previous theoretical results [59].

In Section IV.2, we saw that the highest ground state fidelity is achieved by fixing our parameters $\{h_{i}\}$ and $\{J_{i}\}$ such that $\absolutevalue{\epsilon_{i}}\geq\absolutevalue{\epsilon_{i+1}}$. Thus, in order for our model to most accurately reproduce this prime number spectrum, we choose $\epsilon_{i}=s\ln{p_{i}}$ such that $p_{i}>p_{i+1}$. The required values of the set $\{h_{i}\}$, for some fixed coupling profile $\{J_{i}\}$, can then be simply read off from equations (67), (68) and (69) in the Appendices.