Calculating the Luttinger liquid parameter for an interacting Kitaev chain quantum simulator

TL;DR

The paper tackles quantum simulation of the interacting Kitaev chain using a one-dimensional array of dipolar-coupled spin centers in a solid-state host. It introduces a magnetic-field configuration that enforces a $Z_2$ symmetry and yields a mapping to the Kitaev chain with tunable parameters $t$, $\Delta$, $U$, and $\mu$ as functions of geometry $\theta$ and fields $h^x$, $h^z$. The authors map out a phase diagram with antiferromagnetic, floating Luttinger-liquid phases, and a $Z_2$ symmetry-breaking phase, identifying PT and BKT transitions; they develop multiple methods to extract the Luttinger parameter $K$ from ground-state data, including Friedel oscillations and crosscap overlaps, aided by an energy-spectrum analysis to select nearly quantized $U(1)$ charge states. They show that entanglement entropy and central charge are unreliable for locating BKT transitions in floating regimes, whereas $K$ serves as a practical diagnostic, advancing solid-state quantum simulators for 1D strongly correlated physics. The work points toward experimental pathways using engineered spin-center couplings to realize scalable platforms at room temperature and beyond, enabling exploration of exotic quantum phases and topological edge modes.

Abstract

In this work, we introduce a solid-state platform for building quantum simulators using implanted spin centers in solid-state materials. We build upon the proposal for an $S=1$ chain of spin centers coupled through the magnetic dipole-dipole interaction and subjected to an external magnetic field as a quantum simulator for critical floating phases. We introduce another magnetic field and map the system to the interacting Kitaev chain. This setup, tunable through the applied fields and the orientation of the spin centers within the crystal, exhibits a variety of rich quantum behavior which notably includes floating phases, a $Z_2$ symmetry-breaking phase, and lines of both Berezinskii-Kosterlitz-Thouless (BKT) and Pokrovsky-Talapov transitions. Furthermore, we employ several novel methods to calculate the Luttinger liquid parameter in our model with incommensurate correlations. We find that these methods provide a route to identify BKT transitions with less computational resources than utilizing entanglement entropy and central charge.

Figures (7)

• Figure 1: Subplot (a) depicts a phase diagram showing the von Neumann entanglement entropy for the dipolar-coupled spin center chain portrayed in subplots (d) and (e). The angle $\theta$ defines the orientation of the chain relative to the spin centers' $\hat{z}$ symmetry axes, and $\Gamma_z$ is an effective magnetic field. The low entanglement entropy region between the red dashed lines is an AFM phase, while the red dashed lines are PT transitions. The purple dashed lines show BKT transitions, and the green region between the PT and BKT lines are critical floating phases. The orange dashed line is known analytically as the disorder line, and the high entanglement entropy region between the BKT and disorder lines is a $Z_2$ symmetry-breaking phase. The low entanglement entropy region for $\Gamma_z$ greater than the disorder line is a disordered phase. Along the cut where $\theta=0$, the $SU(2)$ symmetric Heisenberg point is the BKT point at $\Gamma_z=0$, there is a PT point at $\Gamma_z=4$, and between these points is a pure Luttinger liquid phase. The solid brown line is where the magnetization $m^z=1/16$. The phase diagram is reflected over $\Gamma_z=0$, $\theta=0$, and $\theta=\pi/2$. Subplots (b) and (c) show order throughout the phase diagram by plotting the wave vector $k^z_p$ that describes oscillations in the correlations, and the Fourier transform $\widetilde{S}^z_{k_p}$ at wave vector $k^z_p$ of the local expectation values $\langle S^z_i\rangle$, respectively.
• Figure 2: Hamiltonian parameters changing with the angle $\theta$ [see Fig. \ref{['fig:phaseDiagram']}(e)]. The parameters are shown for the spin representation [see Eq. (\ref{['eq:rotatedHam3']})] and the spinless fermionic representation [see Eq. (\ref{['eq:KitaevChain']})].
• Figure 3: Subplots (a) and (c) show entanglement entropy and central charge for a cut across the phase diagram where $\theta=\pi/2$. PT and BKT transitions are marked by black dashed lines at $\Gamma_z=1.38$ and $\Gamma_z=3.00$, respectively. Furthermore, the disorder line is marked by the black dashed line labeled DL at $\Gamma_z=5.477$. The phases are AFM to the left of the PT transition, floating between the PT and BKT transitions, $Z_2$ symmetry-breaking between the BKT transition and disorder line, and disordered to the right of the disorder line. For readability, subplot (c) doesn't show data points where $c>2$. Subplots (b) and (d) show entanglement entropy and central charge for a cut across the phase diagram where $m^z=1/16$ [see solid brown line in Fig. \ref{['fig:phaseDiagram']}(a)]. This line begins in the pure Luttinger liquid phase at $\theta=0$ and crosses a BKT transition when $\theta = 0.26\pi$. Both of these points are marked by black dashed lines. The $Z_2$ symmetry-breaking phase is on either side of the point that belongs to the pure Luttinger liquid phase, and the floating phases exist on the right side of the BKT transition.
• Figure 4: Subplot (a) shows an example of a fit of the spin density profile used to extract the Luttinger liquid parameter $K$ from Friedel oscillations, when $N=512$, $\theta=0.45\pi$, and $\Gamma_z=2.906$. Subplot (b) shows a polynomial fit of $K$ vs $\Gamma_z$ when $N=512$ and $\theta=0.45\pi$. The BKT transition occurs when $K=1/2$. Subplot (c) shows the magnetization $m^z$ for various system sizes and for $\Gamma_z$ near the PT transition when $\theta=0.45\pi$. For these system sizes, all PT transitions fall within the range $1.31\leq \Gamma_z \leq 1.32$. Subplot (d) shows the procedure when $\theta=0.45\pi$ to locate the PT and BKT transitions in the thermodynamic limit. They are each fit to an inverse polynomial of $N$ and extrapolated to $1/N=0$. Subplot (e) shows the $11$ points where we located PT and BKT transitions in the thermodynamic limit. Each set of transitions is fit to a Fourier series to find an equation that describes the entire PT and BKT lines.
• Figure 5: Example of tensor contractions used to efficiently calculate the crosscap overlap with a spin-$1/2$ ground state matrix product state with $N=8$.