Adaptive Pseudoboson Density-Matrix Renormalization Group for Dilute 2D Systems

Abstract

Simulating strongly correlated systems in two dimensions is notoriously challenging due to rapid entanglement growth and frustration. Here, we introduce the adaptive projected-purified pseudoboson density-matrix renormalization group (A3P-DMRG) tailored to explore the ground states of dilute lattice models. The method compresses cluster Hilbert spaces by retaining only the most probable low-occupation Fock states, identified via probabilistic bounds and refined through a self-consistent mean-field basis optimization. We demonstrate that A3P-DMRG is advantageous in low-filling and weak-coupling regimes for large system sizes where conventional DMRG struggles. This establishes the method as a versatile tool for studying dilute quantum many-body systems relevant to ultra-cold atom quantum simulators, photonic lattices, Moiré materials and quantum chemistry.

Figures (6)

• Figure 1: Influence of cluster fluctuations on the effective many-body basis(a) Square lattice of hard-core bosons (yellow spheres). The gray box indicates a cluster of $W$ sites. The inset plot illustrates the probability distribution of the cluster-particle number for ground states with varying particle-fluctuations $\delta N_j$ and fixed average particle number $\bar{N_j}$. (b) Highest relevant occupation number $N_j^{\mathrm{max}}$ obtained from the Bennett bound (bars) and from exact calculations (crosses) for a cluster of size $W=10$ as a function of $\delta N_j$ and for $\bar{N_j}=1/2$. The discarded weight is fixed at $\Delta = 10^{-5}$. (c) Decay of Fock-state weights for the three different ground states with varying values of $\delta N$.
• Figure 2: Schematic overview of the adaptive pseudobosonic ansatz(a) Snake-like mapping of the hard-core bosonic onto the two-dimensional lattice. (b) A cluster of $W$ hard-core bosons is mapped onto a single pseudoboson with local dimension $M = 2^W$. The exponential suppression of relevant Fock-configurations enables truncation to an effective dimension $m$. (c) The broken $\mathrm{U}(1)$ symmetry is restored by projecting the purified pseudoboson states onto the physical submanifold satisfying $\tilde{\Sigma}_{P} + \tilde{\Sigma}_{B} = m$. (d) Improved pseudoboson basis of dimension $\bar{m}$ via self-consistent mean-field optimization scheme.
• Figure 3: Correlation between cluster particle-number fluctuations and ground-state accuracy(a) Average cluster particle-number fluctuation $\bar{\delta} N$ versus $J/J_{\perp}$ and $V_{\perp}/J_{\perp}$. (b) Relative ground-state energy error $\Delta E/E_0$ of the for the same parameters. Results are shown for an $L=W=8$ square lattice with open boundary conditions and $V/J_{\perp}=0$. The effective basis dimension is fixed at $m=93$ ($N\leq3$). (c) Fock-basis state spectrum of the ground state for two special points (yellow and green star) in the $J/J_{\perp}-V_{\perp}/J_{\perp}$ plane.
• Figure 4: Optimization of the effective pseudoboson basis(a) Effective basis dimension $m$ versus particle-number fluctuation $\delta N$ for the Bennett bound, , and approaches ($W=10$). (b) Cost function $D_F$ (blue line, left axis) and effective basis dimension $m$ (black line, right axis) versus mean-field coupling $\Phi$. (c) Relative ground-state error versus effective basis dimension $m$. The dashed line marks the approximate minimum of $D_F$. (d) Spectrum of the averaged cluster reduced density matrix in the pseudoboson basis for the optimal and full bases. Results are for an $L=W=8$ square lattice with open boundaries, $N=4$, and $V_{\perp}/J_{\perp}=V/J_{\perp}=0$, with discarded weight $\Delta=10^{-5}$.
• Figure 5: Runtime scaling of with hard-core boson and optimized pseudoboson bases Wall time $T$ of runs as a function of system length $L$ and number of threads (logical cores) for the hard-core boson basis (red plane) and the optimized pseudoboson basis (blue plane). All calculations were performed on the same computing node for a square lattice with open boundary conditions, density $n=1/L$, weak inter-cluster coupling $J/J_{\perp}=0.1$, vanishing interactions $V=V_{\perp}=0$, and flux per plaquette $\alpha=1/4$. The staging parameters (number of sweeps, bond dimension, and convergence criteria) were kept identical for both bases. The secondary axis in the $x$–$y$ plane indicates the effective basis dimension $\bar{m}(L)$ as a function of system size.