Order parameters and ground-state phase diagram of the interacting topological Su-Schrieffer-Heeger model with extended-range hoppings

Abstract: Topological insulators have attracted numerous attentions recent years, where the Su-Schrieffer-Heeger (SSH) model is one of the most studied models. While the interacting version of it has been explored recently, the interplay between interactions and long-range hoppings merit further investigations. In this work, we uncover a rich phase diagram of the interacting SSH model with extended-range hoppings, in which it consists of several topological phases, two novel superconducting-like (SC-like) phases and five distinct charge-density-wave (CDW) phases. We substantiate that the SC-like and two CDW phases are direct consequences of imbalanced interactions and extended-range hoppings. We derive the order parameters (OPs) for each of the phases and verify them in large-system simulations, finding consistency with the entanglement entropy and the fidelity in capturing the phase transitions. In contrast to the non-interacting case where the favored hoppings are unidirectional in the topological phases, the derived OPs suggest non-unidirectional hoppings are possible under the influence of interactions.

• The paper constructs interaction-aware, direction-symmetrized order parameters that identify most phases in the extended SSH model and agree with entanglement and fidelity diagnostics in exact diagonalization and DMRG.
• The phase diagram contains ten phases, including five charge-density waves, phase separation, topological phases strengthened by interactions, and period-four density waves enabled by extended hopping and imbalanced interactions.
• The paper identifies superconducting-like phases with algebraically decaying pair correlations, while highlighting severe finite-size effects and unresolved order parameters for the interacting W₁-like phase.

Model and scope

The paper studies the spinless interacting extended Su-Schrieffer-Heeger (ESSH) chain with intra-cell ($U$) and inter-cell ($V$) density-density interactions and next-nearest-neighbor hoppings $t_c$ (intra-cell, connecting $j,A$ to $j+1,B$) and $t_d$ (inter-cell, connecting $j,B$ to $j+2,A$), both of which preserve sublattice (chiral) symmetry. The Hamiltonian is

$$H = H_{\rm ESSH} + U\sum_j n_{j,A}n_{j,B} + V\sum_j n_{j,B}n_{j+1,A},$$

with the standard dimerization parametrization $t_a=-(1+\eta)$, $V$0 for benchmarking against the nearest-neighbor interacting SSH results of Yu et al. The non-interacting ESSH model supports winding-number phases $V$1 with $V$2 depending on which hopping dominates. The central methodological contribution is the application of an order-parameter (OP) construction scheme — based on extracting patterns from dominant Fock states of the ground state — to build explicit OPs for every phase in the interacting model, which had previously lacked well-established OPs.

Phases from limiting cases

By analyzing dominant Fock configurations in strong-coupling limits, the authors map out a schematic phase diagram containing ten phases. In the corners of the $V$3 plane: $V$4 yields CDW-2A ("1100" starting on A sites); $V$5 yields CDW-2B; $V$6 yields CDW-1A (the topologically trivial Mott insulator); and $V$7 yields phase separation (PS). Along the axes, $V$8 deepens the $V$9 phase while $t_c$0 deepens $t_c$1 — a notable result that interactions do not necessarily destroy topology but can enhance it, contrary to the common expectation that interactions destabilize SPT order.

Two additional phases, CDW-4A and CDW-4B (period-four density waves), arise specifically from the interplay between extended-range hoppings and imbalanced attractive interactions: for $t_c$2, strongly attractive $t_c$3 combined with vanishing $t_c$4 forces configurations compatible simultaneously with $t_c$5 pairing within cells and $t_c$6 bonding across cells, producing the "11110000" pattern (CDW-4A). These phases are absent in the nearest-neighbor interacting SSH model.

Refinement of the winding-number OPs

In the non-interacting case, the OP for $t_c$7 is a product of $t_c$8 unidirectional hopping operators, $t_c$9, with $j,A$0 growing linearly with system size. Interactions invalidate this construction: the sign of $j,A$1 biases the ground state toward Fock states in which individual hoppings point in different directions locally, so unidirectional products vanish even inside the correct phase. The fix is to symmetrize over all local Hermitian conjugations:

$j,A$2

a sum over the power set of hopping directions ($j,A$3 terms). Exact diagonalization at $j,A$4 shows that this refined OP attains comparable finite values throughout its own phase — including near $j,A$5 where the $j,A$6 character is strongest — whereas the naive non-interacting OP fails to reflect phase depth. A second consequence of interactions is that they localize particles, so the number of hoppings $j,A$7 must be truncated (fixed at $j,A$8 here) rather than scaled with system size.

The superconducting-like phase

For $j,A$9 and imbalanced attractive interactions, ED reveals a region of the phase diagram not captured by any CDW or winding-number OP. Analysis of the dominant Fock states shows configurations related by operators that move pairs of fermions, e.g. $j+1,B$0, while all odd-hopping operators vanish identically. This motivates defining a pairing operator $j+1,B$1 and its correlation function $j+1,B$2. DMRG simulations at $j+1,B$3 show that deep inside this SC-like phase, $j+1,B$4 decays as a power law (straight line on a log-log plot), consistent with quasi-long-range order permitted by the Mermin-Wagner theorem, whereas the decay deviates from power law upon crossing into the CDW-2B phase. The physical mechanism requires both extended hopping ($j+1,B$5) and interaction imbalance ($j+1,B$6); the sublattice index plays a role loosely analogous to spin. An analogous $j+1,B$7-like phase with pairing operator shifted by one site appears when $j+1,B$8 instead. To the authors' knowledge, such phases have not been reported in purely electronic SSH models without explicit Peierls electron-phonon coupling.

Large-system verification and remaining diagnostics

DMRG on $j+1,B$9 chains confirms that each refined OP dominates in its own phase, with OP crossings agreeing with transitions detected by half-block entanglement entropy and fidelity along six paths covering all identified phases. Under open boundary conditions, the CDW-2B, PS, and CDW-4A ground states are symmetry-broken (verified analytically via energy counting of boundary Fock states), so bare expectation values suffice there, while correlation functions are used under PBC. An apparent internal boundary in the PS phase at small even-$t_d$0 systems is shown to be a finite-size effect tied to whether $t_d$1 versus $t_d$2 energy contributions can be distinguished, which disappears for odd $t_d$3 and large systems.

Limitations and open questions

The paper is candid about one significant failure mode: the $t_d$4-like phase (and its conjectured counterpart $t_d$5-like phase) exhibits severe finite-size effects. In small systems $t_d$6 is negative, mimicking the $t_d$7 phase, but in large systems it becomes positive; fidelity fluctuates strongly, indicating ground-state degeneracy whose extent grows with system size without clear convergence. A trial OP constructed from the scheme does not behave cleanly, exposing a limitation of the Fock-state-based approach: it implicitly assumes small-system ground states are qualitatively representative of the thermodynamic limit, which fails here. The OP of the $t_d$8-like phase therefore remains undetermined, as do detailed characterizations of the SC-like and like-phases; the authors suggest topological markers as candidate diagnostics.

Conclusion

This work delivers a complete set of order parameters for the interacting ESSH model, yielding a ten-phase ground-state diagram including five distinct CDWs (two enabled only by the interplay of extended hoppings and imbalanced interactions), two SC-like phases with algebraically decaying pair correlations, and evidence that repulsive interactions can strengthen rather than destroy topological character. The refined, direction-symmetrized OPs are validated against entanglement entropy and fidelity in chains up to $t_d$9, while the unresolved $j,B$0-like phase delineates precisely where the underlying OP-construction methodology breaks down.