Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2411.14724 · Physical Review A

Edge-edge correlations without edge states: η -clustering state as the ground state of the extended attractive SU(3) Hubbard chain

Hironobu Yoshida, Niclas Heinsdorf, Hosho Katsura

Published Jun 6, 2025• 0 citations
arXivHTMLPDF

Abstract

We explore the phase diagram of the extended attractive $\mathrm{SU}(3)$ Hubbard chain with two-body hopping and nearest-neighbor attraction at half-filling. In the large on-site attraction limit, we identify three different phases: phase separation (PS); Tomonaga-Luttinger liquid (TLL); and charge density wave (CDW). Our analysis reveals that the $\ensuremath{\eta}$-clustering state, a three-component generalization of the $\ensuremath{\eta}$-pairing state, becomes the ground state at the boundary between the PS and TLL phases. On an open chain, this state exhibits an edge-edge correlation, which we call boundary off-diagonal long-range order (bODLRO). Using the density matrix renormalization group (DMRG) method, we numerically study the phase diagram of the model with large but finite on-site interactions and find that the numerical results align with those obtained in the strong coupling limit.

DOI
10.1103/physreva.111.063307

Related papers

One-dimensional extended Hubbard model with soft-core potential

Thomas Botzung, Guido Pupillo, Pascal Simon +2 more

We investigate the $T=0$ phase diagram of a variant of the one-dimensional extended Hubbard model where particles interact via a finite-range soft-shoulder potential. Using Density Matrix Renormalization Group (DMRG) simulations, we evidence the appearance of Cluster Luttinger Liquid (CLL) phases, similarly to what first predicted in a hard-core bosonic chain [M. Mattioli, M. Dalmonte, W. Lechner, and G. Pupillo, Phys. Rev. Lett. 111, 165302]. As the interaction strength parameters change, we find different types of clusters, that encode the order of the ground state in a semi-classical approximation and give rise to different types of CLLs. Interestingly, we find that the conventional Tomonaga Luttinger Liquid (TLL) is separated by a critical line with a central charge $c=5/2$, along which the two (spin and charge) bosonic degrees of freedom (corresponding to $c=1$ each) combine in a supersymmetric way with an emergent fermionic excitation ($c=1/2$). We also demonstrate that there are no significant spin correlations.

arXiv:1909.12168•Sep 26, 2019•1 citation

Ground-state phase diagram of the quarter-filled interacting Su-Schrieffer-Heeger chain

Y. H. Wang, Yin Zhong

We map the ground-state phase diagram of the quarter-filled spinless Su-Schrieffer-Heeger chain with nearest-neighbor interactions by combining exact diagonalization (ED) and density-matrix renormalization group (DMRG) under twist-averaged boundary conditions to suppress shell effects. A unified scaling analysis of the equal-time charge structure factor ${S}_{\mathrm{cdw}}(q)$---including the size dependence of ${S}_{\mathrm{cdw}}({q}_{min})$, the extrapolation of ${S}_{\mathrm{cdw}}({q}_{min})/L$ versus $1/L$, the evolution of ${K}_{\mathrm{eff}}(L)=\frac{1}{2}L\phantom{\rule{0.16em}{0ex}}{S}_{\mathrm{cdw}}({q}_{min})$, and the power-law behavior of ${S}_{\mathrm{cdw}}(2{k}_{F})$---disentangles phase separation (PS), Luttinger liquids with pairing-dominated correlations ($K>1/2$), Luttinger liquids dominated by $2{k}_{F}$ bond-order-wave (BOW) correlations ($K<1/2$), and a long-range $2{k}_{F}$ charge-density-wave (CDW) phase. The phase structure, controlled by the interaction $V/t$ and dimerization $\ensuremath{\delta}t/t$, is as follows: PS for $V/t\ensuremath{\lesssim}\ensuremath{-}2$; a pairing-dominated Luttinger liquid for $\ensuremath{-}2<V/t<0$; for $V/t>0$ and moderate $|\ensuremath{\delta}t/t|$ the system realizes a $2{k}_{F}$-BOW-dominated Luttinger liquid; and for approximately $V/t\ensuremath{\gtrsim}1.5$ together with $|\ensuremath{\delta}t/t|\ensuremath{\lesssim}0.9$ it enters a long-range $2{k}_{F}$ CDW, whose domain expands toward smaller $|\ensuremath{\delta}t/t|$ with increasing $V/t$. Quantitative agreement between ED and DMRG across system sizes and boundary treatments underscores the robustness of this phase map and provides a benchmark for the competition between interactions and dimerization in one-dimensional dimerized chains.

Physical review. B./Physical review. B•Nov 21, 2025•1 citation

Quantum information analysis of the phase diagram of the half-filled extended Hubbard model

Cornelius Mund, Örs Legeza, R. M. Noack

We examine the phase diagram of the half-filled one-dimensional extended Hubbard model using quantum information entropies within the density-matrix renormalization group. It is well known that there is a charge-density-wave phase at large nearest-neighbor and a small on-site Coulomb repulsion and a spin-density wave at small nearest-neighbor and large on-site Coulomb repulsion. At intermediate Coulomb interaction strength, we find an additional narrow region of a bond-order phase between these two phases. The phase transition line for the transition out of the charge-density-wave phase changes from first order at strong coupling to second order in a parameter regime where all three phases are present. We present evidence that the additional phase-transition line between the spin-density-wave and bond-order phases is infinite order. While these results are in agreement with recent numerical work, our study provides an independent unbiased means of determining the phase boundaries by using quantum information analysis, yields values for the location of some of the phase boundaries that differ from those previously found, and provides insight into the limitations of numerical methods in determining phase boundaries, especially those of infinite-order transitions.

arXiv:0904.4673•Physical Review B•Jun 25, 2009•37 citations

Ground-state phase diagram of the SU(3) t − J chain

Junhao Zhang, Jie Hou, Jie Lou +1 more

Distinct from the $\text{SU}(2)$ case, the fermionic systems with $\text{SU}(N)$ symmetry are expected to exhibit novel physics, such as exotic singlet formation. Using the density matrix renormalization group technique, we obtain the ground-state phase diagram of the $\text{SU}(3)$ $t\text{\ensuremath{-}}J$ chain for density $n<1$. The ground-state phase diagram includes the Luttinger liquid, the extended Luther-Emery liquid characterized by a spin gap, and the phase separation state. We quantitatively assess the characteristics of the three phases by measuring spin gap, compressibility, various correlation functions, and structure factors. We further study the extended Luther-Emery liquid phase and discover molecular superfluid quasi-long-range order. The mechanism of the molecular superfluid is the combination of three $\text{SU}(3)$ fermions on sites that are not entirely connected. Accordingly, we can speculate the behavior of the $\text{SU}(N)$ $t\text{\ensuremath{-}}J$ chain model with larger $N$ values, operating within the same filling regime.

arXiv:2409.09344•Physical review. B./Physical review. B•Apr 8, 2025•1 citation

Ground-state instabilities in a Hubbard-type chain with particular correlated hopping at non-half-filling

Weiting Chen, Jun Zhang, Hanqin Ding

At non-half-filling, a one-dimensional interacting electronic model (t–U–R) is studied analytically. The particular correlated hopping keeps particle–hole symmetry of the Hubbard model. Physically, the model involved is equivalent to an extended Hubbard chain incorporating a two-body (X) and a three-body (X˜) site-bond interactions with X˜=2X. The analysis of weak-coupling theory leads to the ground-state phase diagram, which consists of three types of metallic phases, characterized by the charge- (CDW) and spin-density-wave (SDW) and singlet-superconductivity (SS) instabilities.

Results in Physics•Apr 24, 2023•5 citations

Ground State Phase Diagram of an Extended Hubbard Chain with Spin-Dependent Interaction

Er-Yin Lin, Hanqin Ding, Jun Zhang

A half-filled one-dimensional (1D) extended Hubbard model (U-V_⊥-V_∥) with on-site and nearest-neighbor spin-dependent interactions is studied analytically. The ground state phase diagram is obtained in the weak-coupling regime by performing the bosonization and renormalization group methods. The result shows that the nearest-neighbor spin-dependent attraction influences the low-energy physics of the 1D correlated electron system, and changes the topological structure of the ground state phase diagram of the U-V model.

Chinese Journal of Physics•Apr 1, 2015•0 citations
We explore the phase diagram of the extended attractive SU(3)\mathrm{SU}(3)SU(3) Hubbard chain with two-body hopping and nearest-neighbor attraction at half-filling. In the large on-site attraction limit, we identify three different phases: phase separation (PS); Tomonaga-Luttinger liquid (TLL); and charge density wave (CDW). Our analysis reveals that the \ensuremathη\ensuremath{\eta}\ensuremathη-clustering state, a three-component generalization of the \ensuremathη\ensuremath{\eta}\ensuremathη-pairing state, becomes the ground state at the boundary between the PS and TLL phases. On an open chain, this state exhibits an edge-edge correlation, which we call boundary off-diagonal long-range order (bODLRO). Using the density matrix renormalization group (DMRG) method, we numerically study the phase diagram of the model with large but finite on-site interactions and find that the numerical results align with those obtained in the strong coupling limit.
We investigate the T=0T=0T=0 phase diagram of a variant of the one-dimensional extended Hubbard model where particles interact via a finite-range soft-shoulder potential. Using Density Matrix Renormalization Group (DMRG) simulations, we evidence the appearance of Cluster Luttinger Liquid (CLL) phases, similarly to what first predicted in a hard-core bosonic chain [M. Mattioli, M. Dalmonte, W. Lechner, and G. Pupillo, Phys. Rev. Lett. 111, 165302]. As the interaction strength parameters change, we find different types of clusters, that encode the order of the ground state in a semi-classical approximation and give rise to different types of CLLs. Interestingly, we find that the conventional Tomonaga Luttinger Liquid (TLL) is separated by a critical line with a central charge c=5/2c=5/2c=5/2, along which the two (spin and charge) bosonic degrees of freedom (corresponding to c=1c=1c=1 each) combine in a supersymmetric way with an emergent fermionic excitation (c=1/2c=1/2c=1/2). We also demonstrate that there are no significant spin correlations.
We map the ground-state phase diagram of the quarter-filled spinless Su-Schrieffer-Heeger chain with nearest-neighbor interactions by combining exact diagonalization (ED) and density-matrix renormalization group (DMRG) under twist-averaged boundary conditions to suppress shell effects. A unified scaling analysis of the equal-time charge structure factor Scdw(q){S}_{\mathrm{cdw}}(q)Scdw​(q)---including the size dependence of Scdw(qmin){S}_{\mathrm{cdw}}({q}_{min})Scdw​(qmin​), the extrapolation of Scdw(qmin)/L{S}_{\mathrm{cdw}}({q}_{min})/LScdw​(qmin​)/L versus 1/L1/L1/L, the evolution of Keff(L)=12LScdw(qmin){K}_{\mathrm{eff}}(L)=\frac{1}{2}L\phantom{\rule{0.16em}{0ex}}{S}_{\mathrm{cdw}}({q}_{min})Keff​(L)=21​LScdw​(qmin​), and the power-law behavior of Scdw(2kF){S}_{\mathrm{cdw}}(2{k}_{F})Scdw​(2kF​)---disentangles phase separation (PS), Luttinger liquids with pairing-dominated correlations (K>1/2K>1/2K>1/2), Luttinger liquids dominated by 2kF2{k}_{F}2kF​ bond-order-wave (BOW) correlations (K<1/2K<1/2K<1/2), and a long-range 2kF2{k}_{F}2kF​ charge-density-wave (CDW) phase. The phase structure, controlled by the interaction V/tV/tV/t and dimerization $\ensuremath{\delta}t/t$, is as follows: PS for $V/t\ensuremath{\lesssim}\ensuremath{-}2$; a pairing-dominated Luttinger liquid for $\ensuremath{-}2<V/t<0$; for V/t>0V/t>0V/t>0 and moderate $|\ensuremath{\delta}t/t|$ the system realizes a 2kF2{k}_{F}2kF​-BOW-dominated Luttinger liquid; and for approximately $V/t\ensuremath{\gtrsim}1.5$ together with $|\ensuremath{\delta}t/t|\ensuremath{\lesssim}0.9$ it enters a long-range 2kF2{k}_{F}2kF​ CDW, whose domain expands toward smaller $|\ensuremath{\delta}t/t|$ with increasing V/tV/tV/t. Quantitative agreement between ED and DMRG across system sizes and boundary treatments underscores the robustness of this phase map and provides a benchmark for the competition between interactions and dimerization in one-dimensional dimerized chains.
We examine the phase diagram of the half-filled one-dimensional extended Hubbard model using quantum information entropies within the density-matrix renormalization group. It is well known that there is a charge-density-wave phase at large nearest-neighbor and a small on-site Coulomb repulsion and a spin-density wave at small nearest-neighbor and large on-site Coulomb repulsion. At intermediate Coulomb interaction strength, we find an additional narrow region of a bond-order phase between these two phases. The phase transition line for the transition out of the charge-density-wave phase changes from first order at strong coupling to second order in a parameter regime where all three phases are present. We present evidence that the additional phase-transition line between the spin-density-wave and bond-order phases is infinite order. While these results are in agreement with recent numerical work, our study provides an independent unbiased means of determining the phase boundaries by using quantum information analysis, yields values for the location of some of the phase boundaries that differ from those previously found, and provides insight into the limitations of numerical methods in determining phase boundaries, especially those of infinite-order transitions.
Distinct from the SU(2)\text{SU}(2)SU(2) case, the fermionic systems with SU(N)\text{SU}(N)SU(N) symmetry are expected to exhibit novel physics, such as exotic singlet formation. Using the density matrix renormalization group technique, we obtain the ground-state phase diagram of the SU(3)\text{SU}(3)SU(3) $t\text{\ensuremath{-}}J$ chain for density n<1n<1n<1. The ground-state phase diagram includes the Luttinger liquid, the extended Luther-Emery liquid characterized by a spin gap, and the phase separation state. We quantitatively assess the characteristics of the three phases by measuring spin gap, compressibility, various correlation functions, and structure factors. We further study the extended Luther-Emery liquid phase and discover molecular superfluid quasi-long-range order. The mechanism of the molecular superfluid is the combination of three SU(3)\text{SU}(3)SU(3) fermions on sites that are not entirely connected. Accordingly, we can speculate the behavior of the SU(N)\text{SU}(N)SU(N) $t\text{\ensuremath{-}}J$ chain model with larger NNN values, operating within the same filling regime.
At non-half-filling, a one-dimensional interacting electronic model (t–U–R) is studied analytically. The particular correlated hopping keeps particle–hole symmetry of the Hubbard model. Physically, the model involved is equivalent to an extended Hubbard chain incorporating a two-body (X) and a three-body (X˜) site-bond interactions with X˜=2X. The analysis of weak-coupling theory leads to the ground-state phase diagram, which consists of three types of metallic phases, characterized by the charge- (CDW) and spin-density-wave (SDW) and singlet-superconductivity (SS) instabilities.
A half-filled one-dimensional (1D) extended Hubbard model (U-V_⊥-V_∥) with on-site and nearest-neighbor spin-dependent interactions is studied analytically. The ground state phase diagram is obtained in the weak-coupling regime by performing the bosonization and renormalization group methods. The result shows that the nearest-neighbor spin-dependent attraction influences the low-energy physics of the 1D correlated electron system, and changes the topological structure of the ground state phase diagram of the U-V model.