Large Isolated Stripes on Short 18-leg $t$-$J$ Cylinders

Abstract

Spin-charge stripes belong to the most prominent low-temperature orders besides superconductivity in high-temperature superconductors. This phase is particularly challenging to study numerically due to finite-size effects. By investigating the formation of long, isolated stripes, we offer a perspective complementary to typical finite-doping phase diagrams. We use the density-matrix renormalization group algorithm to extract the ground states of an 18-leg cylindrical strip geometry, making the diameter significantly wider than in previous works. This approach allows us to map out the range of possible stripe filling fractions on the electron versus hole-doped side. We find good agreement with established results, suggesting that the spread of filling fractions observed in the literature is governed by the physics of a single stripe. Taking a microscopic look at stripe formation, we reveal two separate regimes - a high-filling regime captured by a simplified squeezed-space model and a low-filling regime characterized by the structure of individual pairs of dopants. Thereby, we trace back the phenomenology of the striped phase to its microscopic constituents and highlight the different challenges for observing the two regimes in quantum simulation experiments.

Figures (7)

• Figure 1: Stripe Formation: (a) Sketch of the cylindrical system geometry. A stripe is identified with the formation of a spin domain wall across the system, indicated by the sign of the cross-lattice spin correlations $C^{(L_x)}_{\hat{S}^z}$. (b) $t' \: - \: \nu$ diagram of stripe formation. Individual ground-state calculations are marked by points, and color-coded by $C^{(L_x)}_{\hat{S}^z}$. On the electron-doped side ($t'>0$), we find a narrow filling region featuring negative correlations centered around $\nu=1$. At lower values of $t'$, the domain where a stripe is formed extends to significantly lower fillings -- down to $\nu \sim 1/3$ on the hole-doped side ($t'<0$). For comparison, the black stars mark stripe fillings stabilized in the numerical literature Qin2020-AbsenceSuperconductivityPurebJiang2021-GroundstatePhaseDiagramXu2024-CoexistenceSuperconductivityPartiallyJiang2024-GroundstatePhaseDiagramChen2025-GlobalPhaseDiagramLu2024-EmergentSuperconductivityCompetingXu2022-StripesSpindensityWavesShen2024-GroundStateElectrondopedaZheng2017-StripeOrderUnderdopedRoth2025-SuperconductivityTwodimensionalHubbardaJiang2025-CompetitionChargedensitywaveSuperconductingaGu2025-SolvingHubbardModelLiu2025-AccurateSimulationHubbarda (see the Appendix for details). The grey shaded line at $t'=0$ indicates the near-degenerate filling fractions determined by B.-X. Zheng et al. Zheng2017-StripeOrderUnderdoped. As a guide to the eye, the black hatched region visualizes the range of stabilized fillings, which is well-reflected by the single-stripe results. In addition, the orange star indicates the stripe filling $\nu \approx 1/2$ observed in the LSCO cuprate compounds Cheong1991-IncommensurateMagneticFluctuationsTranquada1995-EvidenceStripeCorrelations.
• Figure 2: Microscopic Structure: High-probability product states revealing the charge and spin structure of the $\nu = 1/2$ states on $4 \times 18$ cylindrical strips at different values of $t'$. The spins are colored according to the staggered magnetization in order to visualize the formation of a spin domain wall. (a) At $t' = -0.2$, the doped holes are mostly separate and form a spin domain wall, which we associate with the stripe state. (b) At $t' = 0.0$, the dopants form a stripe with an internal pairing structure: A spin domain wall is formed, but the dopants combine into tightly bound pairs. (c) At $t' = 0.2$, there is no stripe at $\nu = 1/2$. The dopants form tightly bound pairs in an uninterrupted AFM background.
• Figure 3: Spin Correlations: (a) Cross-lattice spin correlations $C^{(L_x)}_{\hat{S}^z}$ averaged over all $1 \leq y \leq L_y$. Negative correlations indicate the formation of a spin domain wall. The lines show the correlations in the ground state of an $4 \times 18$ system at different values of $t'$, scanning the stripe filling fraction $\nu$. The dashed lines show the predictions of a simplified squeezed-space model. (b) Squeezed space picture for the formation of a spin domain wall: Focussing on a fixed-$y$ cut through the system, a single hole is expected to lead to a spin-domain wall (center) while an even number of holes does not disturb the AFM (top, bottom).
• Figure 4: MPS Mapping: Different ways to path the one-dimensional MPS (blue line) through the system's sites (black circles) are advantageous depending on the system's dimensionality and boundary conditions. Blue numbers index the sites in the MPS, black numbers label the coordinate $x$ (a-c), or $y$ (d). In our work, we use mapping (d) which allows us to study short, wide cylinders. (a) Mapping typical for one-dimensional systems. This mapping is optimal for OBCs, where the physical interaction range is the same as that of the MPS. Periodic boundary conditions would introduce a (physical) nearest-neighbor bond of MPS range $L - 1$ making this mapping unsuitable for PBCs. (b) MPS mapping for one-dimensional periodic systems Wilke2023-SymmetryprotectedBoseEinsteinCondensation. Compared to (a), the MPS range of most nearest-neighbor bonds is doubled to $2$. Even with PBCs, there are no nearest-neighbor bonds with MPS ranges greater than $2$, making calculations possible. (c) Zig-zag mapping typical for two-dimensional systems. Nearest-neighbor interactions in $y$-direction are also nearest-neighbor in the MPS. NN interactions in the $x$-direction have MPS range $L_y$ making calculations exponentially costly in the cylinder width. PBCs in the $y$-direction add bonds of range $L_y - 1$, so cylindrical boundary conditions add little complexity compared to open ones. The combination of these features makes cylinders with $L_y < L_x$ a popular choice for DMRG studies. (d) Combining (b) and (c) leads to an MPS mapping suited for two-dimensional systems with fully periodic boundary conditions or wide cylinders. Comparable to (c), nearest-neighbor bonds in the $x$-direction have MPS range 1, while most nearest-neighbor bonds in $y$-direction have range $2 \: L_x$ -- combining (c) with the doubling encountered in (b). This further restricts $L_x$ but enables studying two-dimensional systems with a long periodic direction using MPS.
• Figure 5: DMRG Convergence: (a-c) Tracking the correlations $C^{(L_x)}$, presented in Figs. \ref{['fig_filling_t_prime_diagram']} and \ref{['fig_szszs']}, with increasing bond dimension $m$. The curves converge at $m \approx 4 000$. (d, c) Dopant densities for parameters $t' = -0.2$ and $\nu = 4/9$, summed along the $x$ ($y$) direction, plotted along the $y$ ($x$) direction. $\langle \hat{n}^h_y \rangle$ becomes flat at $m \approx 4 000$.