Parton theory of angle-resolved photoemission spectroscopy spectra in antiferromagnetic Mott insulators

Angle-resolved photoemission spectroscopy (ARPES) has revealed peculiar properties of mobile dopants in correlated antiferromagnets (AFMs). But, describing them theoretically, even in simplified toy models, remains a challenge. Here, we study ARPES spectra of a single mobile hole in the $t-J$ model. Recent progress in the microscopic description of mobile dopants allows us to use a geometric decoupling of spin and charge fluctuations at strong couplings, from which we conjecture a one-to-one relation of the one-dopant spectral function and the spectrum of a constituting spinon in the undoped parent AFM. We thoroughly test this hypothesis for a single hole doped into a two-dimensional Heisenberg AFM by comparing our semianalytical predictions to previous quantum Monte Carlo results and our large-scale time-dependent matrix product state calculations of the spectral function. Our conclusion is supported by a microscopic trial wave function describing spinon-chargon bound states, which captures the momentum and $t/J$ dependence of the quasiparticle residue. From our conjecture we speculate that ARPES measurements in the pseudogap phase of cuprates may directly reveal the Dirac-fermion nature of the constituting spinons. Specifically, we demonstrate that our trial wave function provides a microscopic explanation for the sudden drop of spectral weight around the nodal point associated with the formation of Fermi arcs, assuming that additional frustration suppresses long-range AFM ordering. We benchmark our results by studying the crossover from two to one dimension, where spinons and chargons are confined and deconfined, respectively.

Figure 1

Magnetic polaron spectra and their unified description. (a) We perform td-MPS simulations of single-hole spectra in the $t-J$ model on $4\times 40$ cylinders of different geometry. At strong couplings $t\gg J$, here $t=3J$, a strong suppression of spectral weight is observed at $(\pi ,\pi )$ at low-to-intermediate energies. Details of the td-MPS calculations are provided in Appendix pp1. The spectrum is obtained along cuts in the Brillouin zone, calculated for different coverings of the cylinder by MPSs (both indicated in the top row). The dashed lines indicate the dispersion relations of the lowest two peaks (determined as local maxima of the spectrum), which we interpret as the ground and first vibrational states of the magnetic polaron. (b) At strong couplings, magnetic polarons can be understood as mesonlike spinon-chargon pairs connected by geometric strings. The spectral function $A(\omega ,\mathit{k})$ of a hole can be approximated by a convolution of the spinon and chargon contributions $A_{\mathrm{s}}(\omega ,\mathit{k})$ and $A_{\mathrm{c}}\left(\omega \right)$, respectively, where the center-of-mass momentum of the meson is carried by the heavy spinon. (c) The optimized slave-particle mean-field theory of fermionic spinons [44, 45] yields the correct shape of the magnetic polaron dispersion ${\omega }_{\mathrm{s}}\left(\mathit{k}\right)$, with a minimum at the nodal point $(\pi /2,\pi /2)$ and low-energy states along the edge $\partial \mathrm{MBZ}$ of the magnetic Brillouin zone. The contribution to the spectral weight $Z_{\mathrm{s}}^{\mathrm{MF}}\left(\mathit{k}\right)$ predicted by the mean-field spinon ansatz is indicated by the color plot at the bottom. It features a sharp drop at the nodal point $(\pi /2,\pi /2)$, which may be the root of the missing spectral weight on the backside of the Fermi arcs observed in cuprates [39].

Figure 2

Scaling of the first vibrational excitation. At strong coupling $t>J$, we analyze the energy $\mathrm{\Delta }E$ between the lowest two pronounced peaks in the spectrum. Our td-MPS results (red) on $4\times L_x$ cylinders are compared to quantum Monte Carlo calculations by Mishchenko et al. (blue, data extracted from Ref. [29]) and the effective geometric string approach (gray). A fit $\mathrm{\Delta }E/t=a{(J/t)}^{2/3}+b$ to our td-MPS data yields $a=2.51$ and $b=2\times {10}^{-3}$. All data are for the ground state at the nodal point $\mathit{k}=(\pi /2,\pi /2)$. See Appendix pp1 for a discussion how the peaks are extracted from our numerically obtained spectra. Finite-size effects in our td-MPS calculations are expected to be weak, but quantitative estimates of their size are difficult.

Figure 3

Ground-state quasiparticle weight and parton contributions. (a) The quasiparticle weight $Z(\pi /2,\pi /2)$ at the nodal point is shown as a function of $J/t$. We find that earlier numerical Monte Carlo studies by Brunner et al. [28] and Mishchenko et al. [29] predict values close to the bare chargon, or string, contribution $Z_{\mathrm{c}}={\left|{\psi }_{\mathrm{\Sigma }=0}^{\mathrm{FSA}}\right|}^2$ expected from the geometric string approach (solid gray line). This is confirmed by our DMRG simulations on cylinders with $L_r$ legs: we used bond dimensions $\chi =500$ ($\chi =600$) for $L_r=4$ ($L_r=6$). (b) We plot $Z_{(\pi /2,\pi /2)}/Z_{\mathrm{c}}$ as a function of $J/t$. The data show only weak dependence on $J/t$, indicating that $Z_{\mathrm{c}}(J/t)$ captures the main $J/t$ dependence of the quasiparticle weight; note that for $J/t\lesssim 0.2$ ($t/J\gtrsim 5$) finite-size effects start to become more sizable in the DMRG. The insets show the same data plotted over $t/J$.

Figure 4

Momentum dependence of the quasiparticle weight. We calculate the quasiparticle residue $Z\left(\mathit{k}\right)$, normalized by $Z_{\max }={max}_{\mathit{k}}Z\left(\mathit{k}\right)$, from the trial wave function (12) along a cut $(0,0)-(\pi ,0)-(\pi ,\pi )-(0,0)$ in a periodic $12\times 12$ system. Parameters are $t=3J$ and we used the optimized mean-field parameters $B_{\mathrm{st}}/J_{\mathrm{eff}}=0.44$ and $\mathrm{\Phi }=0.4\pi$. The solid green line is a guide to the eye. We compare our results to the bare mean-field prediction (solid blue line) and results from our td-MPS calculations (red dots). See Appendix pp1 for how $Z\left(\mathit{k}\right)$ is numerically extracted from td-MPS.

Figure 5

Dependence on $t/J$. Using the trial wave function (12) with ${\psi }_{\mathrm{\Sigma }}={\psi }_{\mathrm{\Sigma }}^{\mathrm{FSA}}$ we calculate the quasiparticle weight $Z\left(\mathit{k}\right)$ along the diagonal cut $(0,0)-(\pi ,\pi )$ for different values of $t/J$. We set $B_{\mathrm{st}}/J_{\mathrm{eff}}=0.44$ and $\mathrm{\Phi }=0.4\pi$ in a $12\times 12$ system; solid lines are guides to the eye.

Figure 6

Mean-field spinon contribution to the quasiparticle residue. $Z_{\mathrm{s}}^{\mathrm{MF}}\left(\mathit{k}\right)$ from Eq. (17) is shown in the following limiting cases: (a) $B_{\mathrm{st}}/J_{\mathrm{eff}}\to \infty$, (b) $B_{\mathrm{st}}=\mathrm{\Phi }=0$, (c) $B_{\mathrm{st}}=0$ and $\mathrm{\Phi }=0.4\pi$. The color bar is indicated on the right.

Figure 7

One-hole spectral function in the dimensional crossover. (a) For varying anisotropy $\alpha =t_y/t_x$ (indicated in top row) of the hopping elements, and ${\alpha }^2=J_y/J_x$ of the superexchange couplings, we use td-MPS to calculate the spectral function. We consider cylinders of length $L_x=40$ along $x$, with circumference $L_y=4$ along the periodic $y$ direction; $t_x/J_x=3$ is fixed. The MPS is wrapped around the cylinder along diagonals, which allows us to calculate diagonal cuts: $k_y=k_x+k_y^{\left(0\right)}$ with $k_y^{\left(0\right)}=\pi , \pi /2$, 0 (cuts 1,2,3, see inset below left panel). (b) Predictions for the spinon contribution $Z_{\mathrm{s}}$ to the spectral weight (color map) and dispersion from fermionic mean-field theory of spinons, as described in the text. Mean-field parameters are taken from Ref. [83]. The delta-function peaks are represented by broadened lines with integrated weight equal to $Z_{\mathrm{s}}\left(\mathit{k}\right)$. In (a) and (b) the locations of dispersion minima in the low-energy region (blue boxes) are indicated by gray arrows.

Figure 8

Quasiparticle weight $Z\left(k\right)$ for a Dirac quantum spin-liquid scenario, along the diagonal cut from (0,0) to $(\pi ,\pi )$. (a) When the long-range AFM order in the trial wave function, controlled by $B_{\mathrm{st}}/J_{\mathrm{eff}}$, is reduced, a sudden drop develops around the edge of the MBZ at $(\pi /2,\pi /2)$. Such behavior can be expected if a Dirac quantum spin liquid emerges upon frustrating spin-exchange interactions in the $t-J$ model. This scenario can be described qualitatively by the mean-field theory of fermionic spinons. We used $t/J=3$ and $\mathrm{\Phi }=0.4\pi$ in our trial wave function; at each momentum data points are slightly offset horizontally for better visibility. (b) For $B_{\mathrm{st}}=0$ and a large range of $t/J$ we observe a sharp drop of spectral weight around the nodal point. In (a) and (b) the solid lines are guides to the eye only.

Figure 9

One-hole spectral function. As in Fig. 1 of the main text, we use td-MPS methods to calculate the single-hole spectrum in the $t-J$ model on a $4\times 40$ cylinder. In (a) we used $t/J=1$ and in (b) we set $t/J=5$.

Figure 10

Convergence of td-MPS data. We check the convergence of our td-MPS calculations with the bond dimension $\chi$ in the real time and space correlation function $C_{\mathbf{i},\mathbf{j}}\left(t\right)$ for the latest time (8 in units of $1/J$) considered for $t/J=3$.

Figure 11

Linear prediction and Gaussian envelope. After the spatial Fourier transform, we use linear prediction to enhance the time window (orange dashed). Before Fourier transforming in time, the time-dependent data are multiplied with a Gaussian envelope of width ${\sigma }_{\omega }=0.25J$ (green dashed-dotted). Here, as an example, we show the corresponding time traces for $k_x=0, k_y=\pi /2$ at $t/J=3$.

Figure 12

First vibrational peak. We show the frequency cut of the one-hole spectral function at the nodal point $\mathit{k}=(\pi /2,\pi /2)$, for the same parameters as in Fig. 1 of the main text. The extracted positions of the ground state (first peak) and vibrationally excited (second peak) magnetic polaron are indicated by dashed lines.

Figure 13

Dependence of DMRG data on parameters. We check how our DMRG calculations of the ground-state quasiparticle weight $Z$ change when the bond dimension $\chi$ and the circumference $L_r$ of the cylinder are changed. The inset shows the difference $Z_{\mathrm{gs}}(L_r,\chi )-Z_{\mathrm{gs}}(L_r=4,\chi =500)$; note the overall scale ${10}^{-3}$.

Figure 14

Spinon-chargon repulsion. (a) We consider a model where the charge fluctuations are described by string states, corresponding to the sites of a Bethe lattice. (b) Assuming a central-symmetric potential on the Bethe lattice, the one-particle hopping problem on the Bethe lattice, describing the fluctuating string, can be mapped to a semi-infinite one-dimensional chain with the indicated hopping amplitudes and a linear potential along the chain. (c) The problem in (b) can be related to an infinite one-dimensional problem with a mirror symmetry around the origin and reduced hopping matrix elements $\sqrt{2}t$ to the central site, as compared to $\sqrt{3}t$ elsewhere. As described in the text, this inhomogeneous tunneling gives rise to a strong (of order $t$) microscopic spinon-chargon repulsion.

Figure 15

Chargon contribution the quasiparticle weight. $Z_{\mathrm{c}}(J/t)$ is calculated from the model in Eq. (B3), based on the frozen spin approximation (FSA), and shown in a double-logarithmic plot. In the strong coupling regime $J\ll t$ the asymptotic power law $Z_{\mathrm{c}}(J/t)\propto J/t$ is obtained as a result of spinon-chargon repulsion.

Figure 16

Size dependence of the quasiparticle weight. $Z\left(k\right)$ is shown along the diagonal cut from (0,0) to $(\pi ,\pi )$. We evaluated the spinon-chargon trial wave function for different system sizes (see legend). No significant finite-size dependence can be observed beyond $6\times 6$. We set $t/J=3, B_{\mathrm{st}}/J_{\mathrm{eff}}=0.44$, and $\mathrm{\Phi }=0.4\pi$. Solid lines are guides to the eye only. Some data points are slightly offset horizontally for better visibility; for $6\times 6$ only a selection of $\mathit{k}$ points are shown.

Figure 17

Dependence of the quasiparticle weight on string tension and staggered magnetic field. (a) We calculate the quasiparticle residue $Z(\pi /2,\pi /2)$ from the trial wave function for different string length distributions. The latter are determined by the linear string tension $dE/d\ell ={\lambda }_{dE/d\ell }2J(C_{{\mathit{e}}_x+{\mathit{e}}_y}-C_{{\mathit{e}}_x})$ used in the FSA ansatz; large ${\lambda }_{dE/d\ell }\gg 1$ corresponds to short strings, and vice versa. We use parameters $B_{\mathrm{st}}/J_{\mathrm{eff}}=0.44$ and $\mathrm{\Phi }=0.4\pi$ in a $12\times 12$ system. For $t/J=3$ the variational energy is minimized for ${\lambda }_{dE/d\ell }$ between ${10}^0$ to ${10}^1$ (see Ref. [35]). (b) The quasiparticle weight is calculated for different values of the staggered magnetic field $B_{\mathrm{st}}/J_{\mathrm{eff}}$ in the trial wave function (12). For large staggered fields, the chargon, or string, contribution $Z_{\mathrm{c}}$ is approached. For weak staggered fields a strong momentum dependence is observed. We set $t=3J, \mathrm{\Phi }=0.4\pi$, and worked in a $12\times 12$ system.