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

Abstract

The behaviour of correlations across a bipartition is an indispensable tool in diagnosing quantum phases of matter. Here we present a spin chain with position-dependent XX couplings and magnetic fields, that can reproduce arbitrary structure of free fermion correlations across a bipartition. In particular, by choosing appropriately the strength of the magnetic fields we can obtain any single particle energies of the entanglement spectrum with high fidelity. The resulting ground state can be elegantly formulated in terms of $q$-deformed singlets. To demonstrate the versatility of our method we consider certain examples, such as a system with homogeneous correlations and a system with correlations that follow a prime number decomposition. Hence, our entanglement simulator can be easily employed for the generation of arbitrary entanglement spectra with possible applications in quantum technologies and condensed matter physics.

Figures (9)

• Figure 1: The variation of the Renyi entropy, $S_{A,1}^{(\alpha)}$ for states given in \ref{['eq:SU(2)_q singlet']} as a function of their deformation parameter $q_{1}$, for a range of fixed values of $\alpha$. The Renyi entropy takes maximal value $S_{A,1}^{(\alpha)}=\ln{2}$ when the deformation parameter $q_{1}=1,$ for all $\alpha$, and is unchanged under the transformation, $h_{1}\rightarrow -h_{1}$, such that $q_{1}\rightarrow\frac{1}{q_{1}}$, reflecting a symmetry of the Hamiltonian \ref{['eq:2-body']}.
• Figure 2: The $q$-deformed spin model for a chain of $2N$ sites. The blue lines represent the XX coupling terms, $J_{i}$, and the red arrows represent the magnitude and direction of the transverse magnetic field, $h_{i}$. The magnitude of both the coupling and transverse field are symmetric about the centre of the chain, with decreasing strength moving outwards.
• Figure 3: The Real-Space RG procedure. (a) Our model \ref{['eq:model']} acting on a chain of four spins. For $J_{1},h_{1}\gg J_{2},h_{2}$ perturbation theory yields a $q$-deformed singlet \ref{['eq:q1']} between the central two spins. (b) These spins are integrated out and an effective Hamiltonian of the form \ref{['eq:2-body']} is found to act between sites $-2$ and $2$ with renormalized coupling $\Tilde{J_{2}}$ and transverse field $\Tilde{h}_{2}$. (c) Diagonalization of this effective Hamiltonian yields the ground state $\ket{\psi}=\ket{\psi_{1}}\otimes\ket{\psi_{2}}$. The difference in colour of the bonds between the two pairs indicates the difference in correlations that can be achieved by appropriately tuning $J_{1}, J_{2}, h_{1}$ and $h_{2}$.
• Figure 4: The values of $h_{2}$ required to generate any $0\leq\epsilon_{2}\leq10$ for fixed $J_{1}=h_{1}=1$, $J_{2}=0.01$ ($\epsilon_{1}=-2\sinh^{-1}{(1)}$). Any desired value of $\epsilon_{2}$ in this range can be obtained by selecting the corresponding value of $h_{2}$.
• Figure 5: Variation of the ground state fidelity with the entanglement entropy of the outer pair for different fixed values of $h_{1}$ with $J_{1}=1$ and $J_{2}=0.1$. The grey line shows an example of a desired outer entanglement entropy, $S_{A,2}=0.5$. The two intersections with each curve for $h_{1}>0$ indicate two possible values of $h_{2}$ to generate the desired $S_{A,2}$ with a distinct difference in fidelity. This choice can be used to optimise the accuracy of our model.