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arXiv:2605.05064

Interaction-controlled localization in one-dimensional chain: From edges to domain walls

Rahul Samanta, Sudin Ganguly, Santanu K. Maiti

Published May 6, 2026• 0 citations
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Abstract

Using Hartree-Fock mean-field approach, we study the role of on-site ($U$) and extended ($V$) Hubbard interactions on the existence and evolution of edge modes in a half-filled Su-Schrieffer-Heeger (SSH) chain. We analyze the energy spectrum, local probability amplitudes, and site-resolved charge and spin density profiles across topological, critical, and trivial hopping regimes. We find that the localization of bound states is controlled by the ratio $2V/U$, with edge spin-density-wave modes for $U>2V$ and mid-chain charge-density-wave domain walls for $U<2V$, independent of band topology. These results establish the correlation-driven origin of localized states in finite one-dimensional chains.

DOI
10.48550/arxiv.2605.05064

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Using Hartree-Fock mean-field approach, we study the role of on-site (UUU) and extended (VVV) Hubbard interactions on the existence and evolution of edge modes in a half-filled Su-Schrieffer-Heeger (SSH) chain. We analyze the energy spectrum, local probability amplitudes, and site-resolved charge and spin density profiles across topological, critical, and trivial hopping regimes. We find that the localization of bound states is controlled by the ratio 2V/U2V/U2V/U, with edge spin-density-wave modes for U>2VU>2VU>2V and mid-chain charge-density-wave domain walls for U<2VU<2VU<2V, independent of band topology. These results establish the correlation-driven origin of localized states in finite one-dimensional chains.
We investigate interacting Su-Schrieffer-Heeger (SSH) chains with two- and three-site unit cells using density matrix renormalization group (DMRG) simulations. By selecting appropriate filling fractions and sweeping across interaction strength Jz{J}_{z}Jz​ and dimerization $\ensuremath{\delta}$, we map out their phase diagrams and identify transition lines via entanglement entropy and magnetization measurements. In the two-site model, we observe the emergence of an interaction-induced antiferromagnetic intermediate phase between the topologically trivial and nontrivial regimes, as well as a critical region at negative Jz{J}_{z}Jz​ with suppressed magnetization and finite-size scaling of entanglement entropy. In contrast, the three-site model lacks an intermediate phase and exhibits asymmetric edge localization and antiferromagnetic ordering in both positive and negative Jz{J}_{z}Jz​ regimes. We further examine the response of edge states to Ising perturbations. In the two-site model, zero-energy edge modes are topologically protected and remain robust up to a finite interaction strength. However, in the three-site model, where the edge states reside at finite energy, this protection breaks down. Despite this, the edge-localized nature of these states survives in the form of polarized modes whose spatial profiles reflect the noninteracting limit.
We study two-particle states in a Su-Schrieffer-Heeger (SSH) chain with periodic boundary conditions and nearest-neighbor (NN) interactions. The system is mapped into a problem of a single particle in a two-dimensional (2D) SSH lattice with potential walls along specific edges. The 2D SSH model has a trivial Chern number but a non-trivial Zak's phase, the one-dimensional (1D) topological invariant, along specific directions of the lattice, which allow for the presence of topological edge states. Using center-of-mass and relative coordinates, we calculate the energy spectrum of these two-body states for strong interactions and find that, aside from the expected appearance of doublon bands, two extra in-gap bands are present. These are identified as bands of topological states localized at the edges of the internal coordinate, the relative distance between the two particles. As such, the topological states reported here are intrinsically many-body in what concerns their real space manifestation, having no counterpart in single-particle states derived from effective models. Finally, we compare the effect of Hubbard interactions with that of NN interactions to show how the presence of the topological bound states is specific to the latter case.
We address the effect of nearest-neighbor (NN) interactions on the topological properties of the Su-Schrieffer-Heeger (SSH) chain, with alternating hopping amplitudes t1{t}_{1}t1​ and t2{t}_{2}t2​. Both numerically and analytically, we show that the presence of interactions induces phase transitions between topologically different regimes. In the particular case of one-hole excitations in a half-filled SSH chain, the V/t2V/{t}_{2}V/t2​ versus t1/t2{t}_{1}/{t}_{2}t1​/t2​ phase diagram has topological phases at diagonal regions of the phase plane. The interaction acts in this case as a passivation potential. For general filling of the SSH chain, different eigensubspaces of the SSH Hamiltonian may be classified as topologically trivial and nontrivial. The two-hole case is studied in detail in the large interaction limit, and we show that a mapping can be constructed of the two-hole SSH eigensubspaces into one-particle states of a noninteracting one-dimensional (1D) tight-binding model, with interfaces between regions with different hopping constants and local potentials. The presence of edge states of topological origin in the equivalent chain can be readily identified, as well as their correspondence to the original two-hole states. Of these states only some, identified by us, are protected and, therefore, truly topological. Furthermore, we found that the presence of the NN interaction generates a state where two holes occupy two consecutive edge states. Such many-body states should also occur for arbitrary filling leading to the possibility of a macroscopic hole gathering at the surface (at consecutive edge states).
We consider a generalized Hubbard model with on-site interaction U, nearest-neighbor repulsion V, and nearest-neighbor hopping for spin \ensuremath{\sigma}, which depends on the sum of particles ${\mathrm{m}}_{\mathrm{\ensuremath{\sigma}}\ensuremath{-}|\mathrm{A}\mathrm{m}}$ with opposite spin in the two sites involved. The hopping matrix elements are denoted by tAA{\mathrm{t}}_{\mathrm{AA}}tAA​,tAB{\mathrm{t}}_{\mathrm{AB}}tAB​,tBB{\mathrm{t}}_{\mathrm{BB}}tBB​ for ${\mathrm{m}}_{\mathrm{\ensuremath{\sigma}}\ensuremath{-}|\mathrm{A}\mathrm{m}}$=0,1,2, respectively. For 0tAB{\mathrm{t}}_{\mathrm{AB}}tAB​tAA{\mathrm{t}}_{\mathrm{AA}}tAA​=tBB{\mathrm{t}}_{\mathrm{BB}}tBB​, we have determined the regions of parameters for which the ground-state (GS) of the one-dimensional system is a charge-density wave (CDW), a spin-density wave (SDW), or a metal (M), using Hartree-Fock, exact diagonalization of finite chains and quantum Monte Carlo. The results agree qualitatively with the exactly solvable limit of tAB{\mathrm{t}}_{\mathrm{AB}}tAB​=0. For 0tAB{\mathrm{t}}_{\mathrm{AB}}tAB​tAA{\mathrm{t}}_{\mathrm{AA}}tAA​=tBB{\mathrm{t}}_{\mathrm{BB}}tBB​, the GS is a M for sufficiently low values of U and V. In contrast, when tAA{\mathrm{t}}_{\mathrm{AA}}tAA​+tBB{\mathrm{t}}_{\mathrm{BB}}tBB​-2tAB{2\mathrm{t}}_{\mathrm{AB}}2tAB​=0, our results suggest that the GS is either a CDW or a SDW, with the boundary between them lying near the line U=2V.
The effect of Hubbard and Kondo interactions on the edge states in the half-filled Su-Schrieffer-Heeger chain of electrons is investigated by studying the behavior of charge quasiparticles using Kumar representation and the density matrix renormalization group method. For any finite dimerization of hopping, by increasing the Hubbard interaction, the edge states are found to transmigrate from the physical charge gap to a high energy gap through an intermediate phase without the edge states. The extent of this phase with no edge states shrinks smoothly upon increasing the dimerization. The transmigration of edge states from the charge gap to the high energy gap is also found to occur with Kondo interaction, but through an intermediate phase which itself changes from having no edge states for weak dimerization to having the edge states in the physical as well as the high energy gaps coexisting from moderate to strong dimerization.
Studying boundary excitations provides a powerful approach to probe correlations in topological phases. We propose that localized spins near the ends of a Su--Schrieffer--Heeger--Hubbard chain embedded in an insulating environment can be detected experimentally using scanning tunneling microscopy (STM) combined with electron-spin resonance. When the STM tip is in the contact regime, the tip--end-spin coupling realizes an effective Anderson impurity problem, giving rise to a Kondo peak at low bias. Spatially resolving the Kondo resonance width as the STM tip approaches the chain ends provides an indirect yet clear signature of these localized spins. To support this proposal, we use density-matrix renormalization group (DMRG) to calculate the spin gap and spin projection of end states for chains of various lengths and interaction strengths UUU at half filling. In the noninteracting limit (U=0U=0U=0), we derive simple analytical expressions that reproduce the numerical results for sufficiently long chains. We also discuss how the correlated phase of the isolated chain is characterized by boundary zeros in its single-particle Green's function, and briefly comment on their localization properties in relation to the boundary spins.