Entanglement detection in quantum materials with competing orders

Abstract

We investigate entanglement detection in quantum materials through criteria based on the simultaneous suppression of collective matter excitations. Unlike other detection schemes, these criteria can be applied to continuous and unbounded variables. By considering a system of interacting dipoles on a lattice, we show the detection of collective entanglement arising from two different physical mechanisms, namely, the ferroelectric ordering and the dressing of matter degrees of freedom by light. In the latter case, the detection shows the formation of a collective entangled phase not directly related to spontaneous symmetry breaking. These results open a new perspective for the entanglement characterization of competing orders in quantum materials, and have direct application to quantum paraelectrics with large polariton splittings.

Figures (8)

• Figure 1: Top: (Left) Schematic representation of a slab of quantum paraelectric of thickness $d$ in a planar cavity of length $L$. (Right) cubic lattice of dipoles described by pairs of conjugate variables with nearest neighbor ferroelectric coupling $(-J)$ and collective coupling with light in the cavity parametrized by the effective charge $q=Ze$, see Eq. \ref{['eq:Hgeneral']}. Bottom: Entanglement phase diagram in the $Z-J$ plane at $T=0$. The colored regions indicates the detection regions for entanglement between the dipoles induced, respectively, by the ferroelectric coupling (darkcyan) and the collective light-matter coupling (orange). The dot and the dashed line indicate the quantum phase transition between the paraelectric (PE) and ferroelectric (FE) phases.
• Figure 2: Entanglement detection in the purely ferroelectric model. Top panels: Harmonic potential, $k=0$. The hatching in panels (a) and (b) highlights the instability of the harmonic model for $J>1/6.$ (a) Squeezing parameter as a function of ${\bf q}=q(1,1,1)$. For illustration purposes, $\zeta_{{\bf q}}$ is cut-off between $-1$ and $1$. (b) Entanglement witnesses corresponding to the sets $(\mathbb{X}_{\boldsymbol{\pi}},\mathbb{P}_{\boldsymbol{0}})$ (solid lines) and $(\mathbb{X}_{\boldsymbol{0}},\mathbb{P}_{\boldsymbol{\pi}})$ (dashed lines) for increasing temperature from $T_{\rm cold} \simeq 5.8~\rm K$ (blue) to $T_{\rm hot} \simeq 17.4~\rm K$ (red). The horizontal dashed line indicates the bound. Bottom panels: Anharmonic potential, $k^2=0.05$. (c) Mode frequencies for ${\bf q}=\boldsymbol{0}$ (circles) and ${\bf q}=\boldsymbol{\pi}$ (diamonds) at $T=0$ across the ferroelectric QCP. Solid lines indicates the corresponding values for $k=0$. (d) Phase diagram in the $J-T$ plane. The dot indicates the QCP and the magenta line indicates the the FE-PE phase boundary. The open circles bounded area indicates the entanglement detection region.
• Figure 3: Entanglement induced by the light-matter interaction $Z$. In all panels $T=0$. (a) Position (red) and momentum (blue) spectral functions for $J=0.05$ and $Z=4.0$, compared to the $Z=0$ case (thin black lines). The dashed line marks the fundamental cavity frequency. (b) Position (circles) and momentum (diamonds )fluctuations at ${\bf q}=\boldsymbol{0}$ (open symbols) and ${\bf q}=\boldsymbol{\pi}$ (solid symbols) as a function of $Z$, and fixed $J=0.05$. (c) and (d) Entanglement witnesses ${\rm EW}(\mathbb{X_{{\bf q}=\boldsymbol{\pi}}},\mathbb{P}_{{\bf q}=\boldsymbol{0}})$ (darkcyan diamonds) and ${\rm EW}(\mathbb{X_{{\bf q}=\boldsymbol{0}}},\mathbb{P}_{{\bf q}=\boldsymbol{\pi}})$ (orange circles) as a function of $Z$ and fixed $J=0.05$, panel (c), and as a function of $J$ and fixed $Z=15$ [panel (d)]. The color code matches the different detection regions in the entanglement phase diagram of Fig. \ref{['fig:fig0']}.
• Figure 4: Left panels: Linear response dynamics of the position operator for ${\bf q}=\boldsymbol{=}0$ (top) and ${\bf q}=\boldsymbol{\pi}$ (bottom) and increasing value of $J$. Right panels: Position response functions obtained by Fourier transform of the time signals.
• Figure 5: Softening of the ferroelectric mode at different temperatures and as a function of $J$.