From One to Two Dimensions: Magnetic Phases in Weakly Coupled Spin Ladders

TL;DR

This work addresses how weak interladder coupling shapes the magnetic phases of spin-1/2 Heisenberg ladders. By combining DMRG in the matrix-product-state formalism with a self-consistent mean-field treatment of interladder interactions, the authors map the zero-temperature phase diagram as a function of interladder coupling $J'$ and external field $h$ for different intraladder couplings $\gamma$, identifying spin-gapped, gapless (Luttinger-liquid-like), antiferromagnetic, and fully polarized phases. Key findings include an analytically known upper critical field $h_{c2} = 2J_{\parallel} + J_{\perp} + J'$, a field $h_m$ where $m^z=1/4$, and a critical interladder coupling $J'_c$ above which staggered order appears at zero field; the phase boundaries are largely geometric-configuration independent, with quantitative differences in staggered-order magnitude. The study further shows how magnetization curves $M(h)$ and their slope $\alpha$ can be used to extract Hamiltonian parameters from experiments, enabling material-specific parameter estimation for compounds like BPCB and terphenyls. Overall, the approach provides a practical, robust route to identify and characterize magnetically coupled ladder materials and to infer the underlying two-dimensional coupling from magnetization measurements.

Abstract

A large variety of materials can be approximately described by means of spin-1/2 Heisenberg ladders. Here, the Density Matrix Renormalization Group (DMRG) algorithm together with a previously established numerical self-consistent mean-field approximation is used to investigate the magnetic properties of spin ladders coupled in a second dimension. The full ground state phase diagram including spin-gapped, antiferromagnetic, ferrimagnetic and fully polarized phases is presented as a function of interladder and intraladder coupling and magnetic field. Measurement of the dependence of magnetization on applied magnetic field is shown to enable location of a material on the phase diagram and determination of the Hamiltonian parameters. These results provide a practical route toward identifying and characterizing magnetic materials composed of coupled spin ladders.

Figures (19)

• Figure 1: Depiction of spin ladder models. (a) Single spin ladder of length $L$ with coupling $J_{\perp}$ on rungs $i$ and $J_{\parallel}$ on legs $j$. (b) Coupled spin ladders with interladder coupling $J'$ are effectively a two-dimensional system with $J_{\parallel}$ along $x$-direction and alternating $J_{\perp}$ and $J'$ in $y$-direction. Each ladder $k$ is visualized by a red box. (c) Alternating interladder coupling $J'$ between two ladders $k=1,2$ on every second rung.
• Figure 2: (a) Schematic phase diagram of coupled spin ladders as a function of interladder coupling $J'$ (horizontal) and magnetic field $h$ (vertical) sketched for generic intraladder coupling $\gamma$. At small magnetic field below $h_{c_1}$ and interladder coupling $J' < J'_c$, the system is in a spin-gapped, decoupled ladder phase (gray). At fields larger than an upper critical field $h_{c_2}$, the system is fully spin polarized (pink). At intermediate fields the system possesses antiferromagnetic order (purple) except at $J'=0$ where one has decoupled Luttinger liquids (thick solid line). (b) Zero-field ($h = 0$) phase diagram as a function of rung coupling (horizontal) and interladder coupling (vertical) relative to leg coupling $J_{\parallel}$ shows symmetry between $J_{\perp}$ and $J'$. Black dot indicates square lattice Heisenberg model.
• Figure 3: Ground state phase diagram as a function of interladder coupling $J'$ (horizontal axis) and magnetic field $h$ (vertical axis) obtained with DMRG and self-consistent numerical mean-field theory for a spin ladder with $\gamma = 0.1$. (a) Magnetization along $z$ reveals the unpolarized spin-gapped phase and fully polarized phase. (b) Staggered magnetization along $x$ shows the boundaries of the antiferromagnetic phase (AFM).
• Figure 4: Magnetization data for a spin ladder with small intraladder coupling $\gamma = 0.1$ and mean-field coupling on every second rung. (a) Uniform magnetization along $z$ plotted against applied field $h$ for interladder couplings $J'$ shown in panel legend. For $J^\prime \leq 0.5$ the data show a spin gap ($m^z=0$ for a range of fields). By $J'=0.9$ the model is in a magnetic state with a non-zero susceptibility even as $h\rightarrow 0$. (b) Staggered magnetization along $x$ shows the boundaries of the antiferromagnetic phase. For $J' > J'_c$, the system has a nonzero staggered order even at zero magnetic field. (c) Expanded view of $m^z(h)$ from panel (a), showing weak dependence on $J'$ for small $J'$, with deviations becoming apparent for $J'=\gamma$. $J' = 0$ is data from a spin ladder without mean-field couplings. (d) Magnetization plotted against difference of field from critical value $h_{c_1}$. A region of linear scaling ($\delta = 1$) followed by a crossover to the square root scaling ($\delta = 1/2$) expected for an isolated chain is evident. (e) For larger $J' > J'_c$, the slope $\alpha = \dv{m^z(h)}{h}$ at $h\rightarrow 0$ increases with increasing $J'$. (f) The staggered order (lower plot) and its derivative (middle plot) can be inferred from the derivative of the magnetization (upper plot) and the inflection points (vertical lines). Normalized data in arbitrary units.
• Figure 5: Magnetization data for spin ladder with larger intraladder couplings (a)-(c) $\gamma = 1.5$ and (e)-(g) $\gamma = 3$ with mean-fields on every other rung. (a) Magnetization along $z$ for different interladder couplings between $J' = 0.01$ and $J' = 1$ indicates the presence of a spin gap for $J' \leq J'_c \approx 0.1$. (b) Staggered magnetization along $x$, with AFM order at zero field for $J' > J'_c$. Note that curve for $J' = 0.01$ shows peaks due to finite size effects discussed in the appendix. (c) The slope $\alpha$ of the magnetization increases with $J'$. (d) Magnetization along $z$ for a larger range of magnetic fields up to $h = 6$. (e)-(g) $m^z$, $m^x_a$, and $\alpha$, respectively, for $\gamma = 3$. As in panels (a) and (b), finite size effects are visible for the two smallest $J^\prime$-values. (h) $m^x_a$ for a larger range of magnetic fields for $\gamma = 1.5$ shows boundaries of staggered order, which is destroyed above $h_{c_2}$.