Forestalled Phase Separation as the Precursor to Stripe Order

Technical Summary: Forestalled Phase Separation as the Precursor to Stripe Order

Problem Statement
The nature of the strange-metal and pseudogap regimes in high-temperature cuprate superconductors remains a central puzzle in condensed matter physics. These regimes emerge at finite temperatures above the stripe-ordered and superconducting phases in the phase diagram of the two-dimensional Fermi-Hubbard model (FHM). While theoretical scenarios have long linked phase separation (PS)—the coexistence of hole-rich metallic regions and hole-poor antiferromagnetic (AFM) domains—to charge-density modulations and superconductivity, the precise mechanism governing the transition from disordered metals to ordered stripes is debated. Specifically, it is unclear whether the system undergoes a true macroscopic phase separation or if a precursor state involving charge clustering exists before the onset of static stripe order.

Methodology
The authors employ modern tensor network techniques to simulate the finite-temperature FHM on a square lattice with nearest-neighbor hopping ($t=1$) and on-site Coulomb repulsion ($U=10$). They utilize two complementary approaches to address different aspects of the problem:

1. Infinite Projected Entangled Pair States (iPEPS): Used in the grand-canonical ensemble to simulate the thermodynamic limit on an infinite square lattice. This method, employing the purification technique and the Neighborhood Tensor Update (NTU) algorithm, allows for the calculation of charge susceptibility ($\chi_{charge} = \partial n / \partial \mu$) without finite-size effects.
2. Minimally Entangled Typical Thermal States (METTS): Applied to finite cylinders (widths $W=4, 6$; lengths $L \le 40$) in the canonical ensemble. This method generates typical thermal states via imaginary-time evolution of product states, enabling the direct visualization of spatial charge inhomogeneities and the statistical analysis of hole clusters.

Key Results

• Charge Susceptibility Maximum: iPEPS simulations reveal a distinct maximum in the charge susceptibility at temperatures above the stripe ordering transition, centered around a filling of $n \approx 0.9$. As temperature decreases, the system transitions from a smooth high-temperature behavior to a regime characterized by a finite, non-divergent peak in $\chi_{charge}$. This behavior is distinct from the divergent susceptibility associated with true ground-state phase separation.
• Forestalled Phase Separation: The peak in susceptibility is attributed to the formation of fluctuating charge clusters. METTS snapshots visualize these as "hole-rich" regions surrounded by AFM domains. Unlike genuine phase separation where domains would macroscopically segregate, these clusters fluctuate in size and do not lead to a divergent susceptibility. The authors term this state "forestalled phase separation."
• Cluster Statistics: Statistical analysis of METTS snapshots shows that at intermediate temperatures, hole clusters grow in size with a stepwise aggregation pattern (near-integer hole increments), driven by the AFM background which penalizes extended domain walls. The mean cluster size $\bar{m}$ increases as temperature drops, reaching a broad maximum in the "forestalled PS" window.
• Transition to Stripe Order: Upon further cooling (below $T \approx 0.06$), the fluctuating clusters condense into a static stripe order. This is marked by a shift in the magnetic structure factor from the Néel vector $(\pi, \pi)$ to the stripe vector $(7\pi/8, \pi)$ and the locking of the mean cluster size to a specific value ($\bar{m} \approx 12$ for $W=4$). The charge structure factor $S_c(k)$ confirms this transition, showing a low-momentum peak shifting inward with system size at intermediate $T$ (indicative of long-wavelength fluctuations) and stabilizing at a specific wavevector $k=(\pi/4, 0)$ at low $T$.

Significance and Claims
The paper claims to identify a crucial intermediate regime in the FHM phase diagram that acts as a precursor to stripe order. The primary contribution is the characterization of "forestalled phase separation," a state where charge carriers cluster into fluctuating domains reminiscent of phase separation but are prevented from macroscopic segregation by the eventual onset of stripe order.

The authors argue that this clustering is driven by antiferromagnetic correlations, which favor half-filled regions and effectively attract holes into clusters. This mechanism suggests that the pseudogap and strange-metal regimes may be understood through the interplay of AFM order and dynamic charge clustering, rather than a simple crossover to a Fermi liquid or a direct transition to static stripes. The work provides numerical evidence that the "Widom line" (a line of susceptibility maxima extending from a quantum critical point) corresponds to this clustering window, organizing the phase space between the disordered metal and the stripe-ordered phase. The findings align with experimental observations of precursor charge density wave fluctuations in cuprates and offer a theoretical framework for understanding the thermodynamic crossover in the pseudogap regime.