Charge and spin orders in the t-U-V-J model: a slave-spin-1 approach

TL;DR

This work tackles the challenge of intertwined charge and spin orders in doped Mott insulators by introducing a spin-1 slave-spin formulation that decouples charge and spin into pseudo-spin (charge) and pseudo-fermion (spin) sectors. A self-consistent cluster mean-field approach is used to solve the two sectors, with a cluster DMRG treatment for the pseudo-spin part and an inhomogeneous MF treatment for the pseudo-fermion part, enabling stripe formation analyses across broad doping. The study reveals robust charge and spin stripe textures (CDW/SDW) and doped-Mott behavior controlled by $U$, $V$, $J$, and $\delta$, and demonstrates qualitative agreement with direct DMRG results while enabling larger system sizes. These findings provide a computationally economical framework to explore inhomogeneous orders in strongly correlated electrons and can inform future quantum-simulation platforms and finite-temperature extensions.

Abstract

Strongly-correlated fermion systems on a lattice have been a subject of intense focus in the field of condensed-matter physics. These systems are notoriously difficult to solve, even with state-of-the-art numerical methods, especially in regimes of parameters where degrees of freedom compete or cooperate at similar energy and length scales. Here, we introduce a spin-1 slave-particle technique to approximately treat the t-U-V-J fermionic model at arbitrary electron dopings in an economical manner. This formalism respectively maps the original charge and spin degrees of freedom into effective pseudo-spin and pseudo-fermion sectors, which are treated using a self-consistent cluster mean-field method. We study the phase diagram of the model under various conditions and report the appearance of charge and spin stripes within this formalism. These stripes are a consequence of the cluster mean-field treatment of the pseudo-particle sectors and have not been detected in previous slave-particle studies. The results obtained agree qualitatively well with what more reliable numerical methods capture.

Figures (17)

• Figure 1: Flow chart of the slave-particle spin-1 scheme to decouple the charge and spin degrees of freedom self-consistently. $t^k_{ij}$ ($J^k_{ij}$) represents the electron (spin) NN tunneling matrix at iteration $k$, renormalized appropriately in each sector. The matrices $t^k_{ij}$ and $J^k_{ij}$ inherit their spatial dependence from the correlation functions that renormalize the bare parameters $t$ and $J$. $U$ is the on-site interaction and $V$ the NN Coulomb interaction. The various correlation functions renormalize the microscopic couplings.
• Figure 2: Illustration of a $4\times 5$ cylindrical cluster embedded self-consistently in a mean-field environment. The black dashed lines represent the NN periodic hopping along $L_y$. The red sites and edges indicate the open sides of the cylindrical cluster. $t$ represents the NN hopping. $\mathbf{a}=\left(0,1\right)a$ and $\mathbf{b}=\left(1,0\right)a$ are the Bravais lattice vectors tiling the full square lattice, with the lattice spacing $a$ set to unity ($a=1$).
• Figure 3: Real-space signatures of the SDWs and CDWs as a function of doping at $J=0.2$ and $V=0$. Upper left panel: magnetization showing staggered AFM ordering in the pseudo-fermion sector (left inset) and pseudo-spin-1 modulations about the average magnetic moment showing CDWs (right inset)---the yellow (blue) color relates to surplus of holes (electrons) with respect to half-filling. The value of $\delta=1/16$ and $U=1.5$. Upper right panel: same as upper left panel, although for values of $\delta=1/8$ and $U=2$. Bottom left panel: pseudo-fermion magnetization $M$ along strips $l_x$ of 32 sites (dotted curves) and spin-1 projections $S^z$ (solid curves) for the same set of parameters as the panel above ($\delta=1/16$ and $U=1.5$). Bottom right panel: same data presented as that of the bottom left panel, although for parameters $\delta=1/8$ and $U=2$. For readability, the charge oscillations are amplified by a factor $10$ for both panels. We remind here that DMRG is only employed for the spin-1 XXZ transverse-field model that represents the charge sector.The calculations were carried out on a $4\times 32$ lattice with periodic boundary conditions along the longest side $L_y$.
• Figure 4: CDW ($\Delta_{\text{CDW}}$) and metallic ($\Phi$) order parameters vs $J$ and $U$ for various hole dopings. The value of the NN interaction $V=0$. Top panels: $\Delta_\text{CDW}$ for $\delta=0,1/16,1/8$, going from left to right. Bottom panels: $\Phi$ for $\delta=0,1/16,1/8$, going from left to right. The top (bottom) color bar shows the intensity of $\Delta_\text{CDW}$ ($\Phi$). The white regions display the parameter spaces where the slave-spin method fails to converge. The region between the diagonal dotted lines roughly encloses the region where phase separation occurs. The values of $\Phi$, when converging never reach numbers below $\sim 10^{-4}$. The calculations were carried out on a $4\times 32$ lattice with periodic boundary conditions along the longest side $L_y$. The black asterisks mark the data points shown in Fig. \ref{['fig:CDW_SDW']}, while the black arrows mark that shown in Fig. \ref{['fig:Phi_vs_J_U_zoom_J_0p1']}.
• Figure 5: wavelength of the CDW order parameter $\mathbf{Q}_{\text{CDW}}$ vs $U$ and $J$ for hole dopings $\delta=\{1/16,1/8\}$, and nearest-neighbor interaction $V=0$. The clusters sizes are $4\times32$. The color map shows the wavelength of the stripes in terms of amount of lattice sites, namely $2\pi\mathbf{Q}_{y;\text{CDW}}^{-1}$. A threshold of $\Delta_{\text{CDW}}\geq 1\times10^{-3}$ was put, and that explains why the white region widened when compared to Fig. \ref{['fig:Phi_vs_J_U']}. The black asterisks mark the data points shown in Fig. \ref{['fig:CDW_SDW']}.