Microscopic spinon-chargon theory of magnetic polarons in the $t\text{−}J$ model

The interplay of spin and charge degrees of freedom, introduced by doping mobile holes into a Mott insulator with strong antiferromagnetic (AFM) correlations, is at the heart of strongly correlated matter such as high-$T_c$ cuprate superconductors. Here, we capture this interplay in the strong coupling regime and propose a trial wave function of mobile holes in an AFM. Our method provides a microscopic justification for a class of theories which describe doped holes moving in an AFM environment as mesonlike bound states of spinons and chargons. We discuss a model of such bound states from the perspective of geometric strings, which describe a fluctuating lattice geometry introduced by the fast motion of the chargon, relative to the spinon. This is demonstrated to give rise to short-range hidden string order, signatures of which have recently been revealed by ultracold atom experiments at elevated temperatures. We present evidence for such short-range hidden string correlations also at zero temperature by performing numerical density-matrix renormalization-group simulations. To test our microscopic approach, we calculate the ground-state energy and dispersion relation of a hole in an AFM, as well as the magnetic polaron radius, and obtain good quantitative agreement with advanced numerical simulations at strong coupling. We discuss extensions of our analysis to systems without long-range AFM order to systems with short-range magnetic correlations.

Figure 1

Mesonlike spinon-chargon bound states and short-range hidden string order. A single hole in a 2D AFM forms a mesonlike bound state of a spinon and a chargon, similar to quark-antiquark pairs forming mesons in high-energy physics. (a) In the $t\text{−}J$ model, a spinon can be bound to a chargon by a geometric string of displaced spins, as illustrated in (b) (note the indices of spins along the string on the right). Geometric string states $\mathrm{\Sigma }$ can be defined in arbitrary spin backgrounds, where they provide an adequate approximate basis in the presence of strong local AFM correlations. (c) Signatures of geometric strings can be found in individual Fock configurations by analyzing the difference to a classical Néel pattern. (d) We use the matrix product state formalism (MPS) and the DMRG algorithm to generate snapshots of the $T=0$ ground state of the $t\text{−}J$ model with a single hole, similar to the recent measurements using ultracold fermions [29]. In (d) we show the distribution function of the length $\ell$ of stringlike patterns emanating from the hole. A striking difference is observed between a localized and a mobile hole (MPS, indicated by symbols connected with dashed lines), for $\ell =0$ in particular. Mobile holes are described quantitatively by the geometric string theory [frozen-spin approximation (FSA), shaded ribbons indicating statistical errors] which is based on the string length distributions $p_{\ell }^{\mathrm{FSA}}$ (see inset). Since the string patterns illustrated in (c) are sensitive to quantum fluctuations in the spin background, the FSA distribution from the inset in (d) differs quantitatively from the results in the main panel. Quantum fluctuations are also responsible for the even-odd effect observed most clearly for a localized hole.

Figure 2

Dressing cloud of magnetic polarons. Using the MPS formalism and DMRG we calculate local spin correlation functions $C_n^z\left(d\right)=\langle {\widehat{n}}_{{\mathit{r}}_h}^h{\widehat{S}}_{{\mathit{r}}_2}^z{\widehat{S}}_{{\mathit{r}}_1}^z\rangle /\langle {\widehat{n}}_{{\mathit{r}}_h}^h\rangle$, where $n=1$ ($n=2$) denotes nearest-neighbor (diagonal next-nearest-neighbor) configurations of ${\mathit{r}}_1$, ${\mathit{r}}_2$, as a function of the bond-center distance $d=|({\mathit{r}}_1+{\mathit{r}}_2)/2-{\mathit{r}}_h|$ of the two spins from the hole. At strong coupling, the distortion of the magnetic spin environment around the hole follows a shape which is described well by the geometric string approach (FSA), except for some additional features between $d\approx 1.5$ and $2.5$ captured only by DMRG. Calculations are performed for one hole and $S_{\mathrm{tot}}^z=\frac{1}{2}$ as described in Sec. 1c.

Figure 3

Magnetic polaron radius. We calculate the size of the magnetic polaron as a function of $t/J$: (a) by determining the average length $\langle \ell \rangle$ of the stringlike objects revealed in individual snapshots, and (b) by fitting the dependence of local spin correlations $C_n\left(d\right)$ on the bond-center distance by a function of the form $C_n^{\infty }+a{e}^{-d/R_{\mathrm{mp}}}$ and interpreting the fit parameter $R_{\mathrm{mp}}$ as the magnetic polaron radius. For small values of $t$, the fit to $C_2^z\left(d\right)$ is not meaningful and we do not provide any data points in this regime. MPS and FSA are defined in the figure caption of Fig. 1.

Figure 4

Frozen-spin approximation. In the approximate FSA basis we only allow processes where the motion of the chargon displaces the surrounding spins without changing their quantum states. As a result, nearest-neighbor correlations $C_{{\mathit{e}}_y}$ (red) [next-nearest-neighbor correlations $C_{{\mathit{e}}_x+{\mathit{e}}_y}$ ( yellow), respectively] in the frozen-spin background (a) contribute to next-nearest-neighbor correlators (nearest-neighbor correlators, respectively) measured in states with longer string lengths (b). This leads to the approximately linear string tension, Eq. (10), binding spinons to chargons.

Figure 5

Dispersion relation of a single hole in an AFM. We consider cuts through the Brillouin zone along the path sketched in (a), starting at (0,0). (a) The shape of the mean-field spinon dispersion [Eq. (13)] is shown at $B_{\mathrm{st}}=0.44J_{\mathrm{eff}}$ for $\mathrm{\Phi }=0.4\pi$ (left) and $\mathrm{\Phi }=0,\pi$ (right). (b) We calculate the magnetic polaron dispersion for $t=3J$ in a $12\times 12$ system from the meson trial wave function [Eq. (14)] (string-VMC) and compare it to predictions by the analytical FSA theory combined with a simplified tight-binding expression for the spinon dispersion [Eq. (18)]. (c) As expected from the strong coupling spinon-chargon picture, only the part of the variational energy associated with spin-exchange terms $\langle {\widehat{\mathcal{H}}}_J\rangle$ depends on ${\mathit{k}}_{\mathrm{MP}}$ (top), whereas $\langle {\widehat{\mathcal{H}}}_t\rangle$ is nondispersive (bottom).

Figure 6

Simplified tight-binding description of the spinon dispersion. In a classical Néel state, a spinon at site ${\mathit{j}}_1^s$ (left) can hop to another site ${\mathit{j}}_2^s$ by spin-exchange processes on the bonds indicated by ellipses. The spin exchange on the bond indicated by the red ellipse leads to the spinon configuration shown on the right, where ${\mathit{j}}_2^s={\mathit{j}}_1^s+2{\mathit{e}}_x$ and the string length changes by two units, ${\ell }_{{\mathrm{\Sigma }}_2}={\ell }_{{\mathrm{\Sigma }}_1}+2$. Similarly, spinons at ${\mathit{j}}_2^s={\mathit{j}}_1^s-2{\mathit{e}}_x$, $\pm 2{\mathit{e}}_y$, $+{\mathit{e}}_x\pm {\mathit{e}}_y$, and $-{\mathit{e}}_x\pm {\mathit{e}}_y$ can be reached, with ${\ell }_{{\mathrm{\Sigma }}_2}={\ell }_{{\mathrm{\Sigma }}_1}\pm 2$.

Figure 7

Bandwidth $W$ of a single hole in an AFM. We compare our variational results (string-VMC) from Eq. (14), valid at strong couplings, to quantum Monte Carlo simulations by Brunner et al. [16], ED studies by Dagotto et al. [10] and Leung and Gooding [15], a variational wave function by Sachdev [8], and spin-wave calculations by Martinez et al. [11] and by Liu and Manousakis [12]. The overall shape of $W(J/t)$ is well captured by the effective FSA theory [Eq. (18)]. The string-VMC calculations are performed at $B_{\mathrm{st}}=0.44J_{\mathrm{eff}}$, $\mathrm{\Phi }=0.4\pi$ in a $12\times 12$ system.

Figure 8

Ground-state energy of a hole in an AFM. We compare our variational result from the meson trial wave function in Eq. (14) (string-VMC) to quantum Monte Carlo calculations by Mishchenko et al. [17] (QMC) and semianalytical predictions by the tight-binding FSA theory from Eq. (18). In the shaded region, defined by $J<0.05t$, the ground state is expected to be a Nagaoka polaron [18, 40]. The string-VMC calculations are performed at $B_{\mathrm{st}}=0.44J_{\mathrm{eff}}$, $\mathrm{\Phi }=0.4\pi$ in a $12\times 12$ system at ${\mathit{k}}_{\mathrm{MP}}=(\pi /2,\pi /2)$.

Figure 9

Optimization of the trial wave function. We change the average length of the geometric string in the spinon-chargon wave function (14), $\langle \ell \rangle ={\sum }_{\mathrm{\Sigma }}{\left|{\psi }_{\mathrm{\Sigma }}\right|}^2{\ell }_{\mathrm{\Sigma }}$ shown in the inset, by rescaling the linear string tension in Eq. (10) with a factor ${\lambda }_{dE/d\ell }$. The resulting variational energy $\langle {\widehat{\mathcal{H}}}_{t\text{−}J}\rangle -E_{0\mathrm{h}}$, in units of $J$, is calculated as a function of ${\lambda }_{dE/d\ell }$ for the parameter $t=2J$. The string-VMC calculations are performed at $B_{\mathrm{st}}=0.44J_{\mathrm{eff}}$, $\mathrm{\Phi }=0.4\pi$ in a $12\times 12$ system at ${\mathit{k}}_{\mathrm{MP}}=(\pi /2,\pi /2)$. Lines are guides to the eye.

Figure 10

Magnetic polaron cloud. Using the trial wave function (string-VMC) in Eq. (14) we calculate the local spin correlations $C_n\left(d\right)=\langle {\widehat{n}}_{{\mathit{r}}_h}^h{\widehat{\mathit{S}}}_{{\mathit{r}}_1}·{\widehat{\mathit{S}}}_{{\mathit{r}}_2}\rangle /\langle {\widehat{n}}_{{\mathit{r}}_h}^h\rangle$, where $d=|({\mathit{r}}_1+{\mathit{r}}_2)/2-{\mathit{r}}_h|$ is the bond-center distance between ${\mathit{r}}_1$ and ${\mathit{r}}_2$ and $n=1$ ($n=2$) corresponds to nearest- (next-nearest-) neighbor spin correlations. (a) For $t=3J$ we compare our string-VMC result $C_n\left(d\right)/3$ to $C_n^z\left(d\right)$ obtained from the same DMRG simulations as in Fig. 2: MPS corresponds to the full solution in the case of a mobile hole and the FSA predictions are generated from snapshots of the undoped Heisenberg AFM. (b) We calculate the momentum dependence of $C_n\left(d\right)$ from the trial wave function at $t=2J$. The string-VMC calculations in (a) and (b) are performed at $B_{\mathrm{st}}=0.44J_{\mathrm{eff}}$, $\mathrm{\Phi }=0.4\pi$ in a $14\times 14$ system; in (a) $\mathit{k}=(\pi /2,\pi /2)$.

Figure 11

Comparison of the different theoretical approaches. We calculate the local spin correlations $C_n\left(d\right)/3$ and $C_n^z\left(d\right)$ for $n=1,2$ in the vicinity of a mobile dopant, using different theoretical approaches: (a) the trial wave function (string-VMC) from Eq. (14), (b) by numerical DMRG simulations (MPS), and (c) using the FSA and starting from snapshots for the undoped Heisenberg model generated by DMRG. The string-VMC calculations in (a) are performed at $B_{\mathrm{st}}=0.44J_{\mathrm{eff}}$, $\mathrm{\Phi }=0.4\pi$ in a $14\times 14$ system at $\mathit{k}=(\pi /2,\pi /2)$.