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arXiv:2409.09344 · Physical review. B./Physical review. B

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

Junhao Zhang, Jie Hou, Jie Lou, Yan Chen

Published Apr 8, 2025• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.111.165110

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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\ensuremath-Jt\text{\ensuremath{-}}Jt\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\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J chain model with larger NNN values, operating within the same filling regime.
We examine the ground-state phase diagram of the ttt-JJJ model in one dimension by means of the density matrix renormalization group. This model is characterized by a rich phase diagram as a function of the exchange interaction JJJ and the density nnn, displaying Luttinger-liquid (LL) behavior both of repulsive and attractive (i.e., superconducting) natures, a spin-gap phase, and phase separation. The phase boundaries separating the repulsive from the attractive LL phase as JJJ is increased, and also the boundaries of the spin-gap region at low densities, and phase separation at even larger JJJ, are determined on the basis of correlation functions and energy gaps. In particular, we shed light on a contradiction between variational and renormalization-group (RG) results about the extent of the spin-gap phase, which results larger than the variational but smaller than the RG one. Furthermore, we show that the spin gap can reach a sizable value ( 0.1t~0.1t 0.1t) at low enough filling, such that preformed pairs should be observable at temperatures below these energy scales. No evidence for a phase with clustering of more than two particles is found on approaching phase separation.
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 $\ensuremath{\eta}$-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.
By using the bosonization approach and the renormalization group (RG) technique, we study the half-filled band one-dimensional t–U–J model with additional on-bond repulsion (W>0) in the weak-coupling regime. The presence of on-bond repulsion is responsible for realization of a metallic phase in the system, and the phase diagram is strongly controlled by the symmetry of the model. By analyzing the RG flow diagram and comparing order parameters, the phase boundaries are determined and the structure of the phase diagram is clarified. In the case of SU (2) ⊗ SU (2) symmetry, the phase diagram consists of a metallic phase characterized by a Luttinger liquid (LL) and two insulting phases characterized by the degenerate spin-density-wave (SDW) and the bond-charge-density-wave (BCDW). In the SU (2) ⊗ U(1)-symmetric case, the phase diagram contains two metallic phases: a LL and a Luther–Emery phase, and three insulating phases: the transverse SDW ( SDW ±), the longitudinal SDW ( SDW z) and the dimerized BCDW. The insulating charge-density-wave and bond-spin-density-wave (BSDW) phases are always suppressed in the ground state. In addition, the system show a long-ranged order in the BCDW and SDW z phases.
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.
The phase diagram of the two-leg $t\text{\ensuremath{-}}{J}_{z}$ ladder is explored, using the density-matrix renormalization group method. Results are obtained for energy gaps, electron density profiles, and correlation functions for the half filled and quarter filled cases. The effective Lagrangian velocity parameter ${v}_{\ensuremath{\rho}}$ is shown to vanish at half filling. The behavior of the one-hole gap in the Nagaoka limit is investigated, and found to disagree with theoretical predictions. A tentative phase diagram is presented, which is quite similar to the full $t\text{\ensuremath{-}}J$ ladder, but scaled up by a factor of about 2 in coupling. Near half filling a Luther-Emery phase is found, which may be expected to show superconducting correlations, while near quarter filling the system appears to be in a Tomonaga-Luttinger phase.
Inspired by the observation of a robust ddd-wave superconducting phase driven by tuning the next-nearest-neighbor (NNN) electron hopping in recent density matrix renormalization group (DMRG) studies of six- and eight-leg $t\text{\ensuremath{-}}J$ model, we systematically study the phase diagram of the two-leg $t\text{\ensuremath{-}}J$ ladder with the NNN couplings (including NNN hopping and spin interaction) in a large region of doping level, by means of the DMRG calculations. Upon doping from half filling, we identify the Luther-Emery liquid (LEL) phase, which can be distinguished as the pairing-dominant and charge density-dominant regime by comparing the Luttinger parameter ${K}_{\ensuremath{\rho}}$. With the growing NNN couplings, pairing correlations are enhanced and correspondingly ${K}_{\ensuremath{\rho}}$ increases, driving the system from the density-dominant to the pairing-dominant regime. In the Tomonaga-Luttinger liquid (TLL) phase in the larger doping region, we identify two TLL regimes with different features of charge density correlation. At the quarter filling (1/21/21/2 doping level), we find that the strong dimer orders of bond energy in the open system actually decay algebraically and thus do not indicate a spontaneous translational symmetry breaking. Our results show that in the LEL phase of the two-leg ladder, the NNN couplings seem to play the similar role as that on the wider $t\text{\ensuremath{-}}J$ cylinder, and studies on this more accessible system can be helpful toward understanding the emergence of the remarkable ddd-wave superconducting phase on the wider system.