Signatures of correlated magnetic phases in the two-spin density matrix

Experiments with quantum gas microscopes have started to explore the antiferromagnetic phase of the two-dimensional Fermi-Hubbard model and effects of doping with holes away from half filling. In this work we show how direct measurements of the system averaged two-spin density matrix and its full counting statistics can be used to identify different correlated magnetic phases with or without long-range order. We discuss examples of phases which are potentially realized in the Hubbard model close to half filling, including antiferromagnetically ordered insulators and metals, as well as insulating spin liquids and metals with topological order. For these candidate states we predict the doping- and temperature dependence of local correlators, which can be directly measured in current experiments.

Figure 1

The two-spin reduced density matrix ${\rho }^{\mathrm{S}}$, measurable in ultracold atom experiments, and its full counting statistics (FCS) can be used to distinguish between symmetric and symmetry-broken phases in the Fermi-Hubbard model. The ground state on the square lattice at half filling has AFM order, which leads to nonvanishing singlet-triplet matrix elements $\langle t\left|{\rho }^{\mathrm{S}}\right|s\rangle \ne 0$ as well as a broad distribution of the triplet matrix element in the FCS, even if the correlation length is finite. Below half filling the precise nature of the ground state is still under debate, with doped quantum spin liquids as one possible scenario. These give rise to a $\mathrm{SU}\left(2\right)$ invariant two-spin reduced density matrix with vanishing singlet-triplet matrix elements, as well as a sharp delta-function distribution of the triplet amplitude.

Figure 2

Fingerprints for the spontaneous breaking of $\mathrm{SU}\left(2\right)$ invariance in the shot-to-shot FCS of the system-averaged reduced two-spin density matrix ${\rho }^{\mathrm{S}}$, see Eq. (8). When the $\mathrm{SU}\left(2\right)$ symmetry is spontaneously broken, the order parameter points in a different direction in every shot. This results in a broad distribution function of some of the matrix elements of ${\rho }^S$. (a) We use spin-wave theory to calculate the distribution of the triplet matrix element $\langle t\left|{\rho }^{\mathrm{S}}\right|t\rangle$ in (a) for an infinite Heisenberg AFM at half filling. For a $\mathrm{SU}\left(2\right)$ invariant quantum spin liquid the distribution function in an infinite system becomes a delta peak. (b) The distribution of the singlet matrix element $\langle s\left|{\rho }^{\mathrm{S}}\right|s\rangle$ is a delta peak in an infinite system even when the $\mathrm{SU}\left(2\right)$ symmetry is broken, because the singlet state $|s\rangle$ itself is $\mathrm{SU}\left(2\right)$ invariant.

Figure 3

Doping dependence of local correlations in the AFM metal. Panel (a) of this figure shows the short-range singlet, triplet, and ferromagnetic amplitudes ($p_s, p_t$, and $p_f$) as a function of doping $p$. We show here data for three temperatures, namely $T=0.025J$ (bold line), $T=0.125J$ (bold dashed), and $T=0.175J$ (dotted). The amplitudes decrease with increasing temperature due to larger thermal fluctuations. The triplet and ferromagnetic amplitudes approach each other when doping the system continuously from $p=0.01$ to $p=0.2$. The singlet amplitude $p_s$ remains larger than the other two amplitudes at large doping, by an amount determined by $c_{p\gtrsim 0.1}\approx 0.07$ in Eq. (12). The order parameter $\langle S_i^z\rangle$ in panel (b) indicates that the $\mathrm{SU}\left(2\right)$ symmetry is restored at a critical doping of $p\approx 0.06$ for the temperature $T=0.125J$.

Figure 4

Full counting statistics of local correlations in the AFM metal. We show the FCS of the system-averaged triplet matrix element $\langle t\left|{\rho }^S\right|t\rangle$ of the two-spin density matrix. The left panel (a) shows the FCS distribution at a fixed temperature $T=0.125J$ for different doping levels $p$. It narrows as function of $p$ and turns into a sharp peak at the critical doping $p\approx 0.05$, where the SU(2) symmetry is restored. The sharp peak at a value around 0.23 determines the parameter $c$ in the density matrix in Eq. (12), which takes the value $c_{p\approx 0.05}\approx 0.07$. The right panel (b) shows a similar behavior of the probability distribution as function of temperature at fixed doping, where thermal fluctuations restore the SU(2) symmetry and lead to a narrow peak around a value $\langle t\left|{\rho }^S\right|t\rangle =0.23$.

Figure 5

Doping dependence of local correlations in the FL* phase. We show the doping dependence of the short-range singlet, triplet, and ferromagnetic amplitudes ($p_s, p_t$, and $p_f$) in (a), for various temperatures $[T=0.05J$ (bold line), $T=0.1J$ (bold dashed), and $T=0.125J$ (dotted)] at $t=4J$. In (b) the doping dependence of spin-correlation functions and the parameter $c$, as defined in Eq. (12), is shown for the same parameters. The singlet amplitude in (a) decreases for larger doping and temperature due to the presence of holes, respectively thermal fluctuations. Since the state is SU(2) symmetric, triplet and ferromagnetic amplitude are identical. The magnitude of short-range spin correlations decreases by the presence of holes that destroy the AFM background ordering.

Figure 6

(a) Entries of the reduced two-spin density matrix in a $4\times 4$ system doped with a single hole, as a function of $J/t$. We used ED simulations for the ground state with total momentum $\mathit{k}=(\pi /2,\pi /2)$ and with total spin $S^z=1/2$. We compare our numerically exact results (symbols) to predictions by the geometric string theory (solid lines); for details see Ref. [54], where the ratio $J/t$ tunes the length of the string of displaced spins. The reduced two-spin density matrix of the doped system closely resembles the two-spin density matrix in an undoped but frustrated quantum magnet: In (b) we calculate ${\rho }^{\mathrm{S}}$ for a frustrated $J_1-J_2$ model on a periodic $4\times 4$ lattice with $S^z=0$, with diagonal next-nearest-neighbor couplings $J_2$. Locally the two systems become indistinguishable, and we find that larger values of $t/J$ correspond to larger values of $J_2/J_1$.

Figure 7

Panel (a) shows the singlet $p_{\text{s}}$, triplet $p_{\text{t}}$, and ferromagnetic $p_{\text{f}}$ matrix elements as a function of the external staggered field $h_z$. Increasing the staggered field drives the state closer to a classical Néel configuration. Panel (b) shows the full counting statistics of the triplet amplitude for different values of $h_z$. A strong external staggered magnetic field suppresses quantum fluctuations and we observe a continuous distribution of the triplet amplitude in the range $0\le \langle t\left|{\rho }^S\right|t\rangle \le 0.5$. Since quantum fluctuation locally disturbs the antiparallel alignment, the distribution has an onset at a finite value for a small external staggered field.