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arXiv:2111.13718 · Physical review. B./Physical review. B

Enhanced superconductivity and various edge modes in modulated t−J chains

Yong-Feng Yang, Jing Chen, Chen Cheng, Hong‐Gang Luo

Published Apr 14, 2022• 3 citations
arXivHTMLPDF

Abstract

We numerically investigate the ground state of the extended $t\text{\ensuremath{-}}J$ Hamiltonian with periodic local modulations in one dimension by using the density-matrix renormalization group method. Examining charge and spin excitation gaps, as well as the pair binding energy, with extrapolated results to the thermodynamic limit, we obtain a rich ground-state phase diagram consisting of the metallic state, the superconducting state, the phase separation, and insulating states at commensurate fillings. Compared to the homogeneous one-dimensional (1D) $t\text{\ensuremath{-}}J$ model, the superconductivity is greatly enhanced and stabilized by the flat-band structure. This superconducting state in periodically modulated chains shares similar properties with ladder systems: significant negative pair binding energy occurs, and the singlet-pairing correlation function dominates with the algebraic decay while the single-particle Green's function and spin correlation function decay exponentially. On the other hand, the superlattice structure leads to nontrivial topological nature in insulating states, characterized by different integer Chern numbers at different fillings. Due to the interplay among the topology, the interaction, and the 1D confinement, gapless edge modes show strong spin-charge separation and in different regions can relate to different collective modes, which are the charge of a single fermion, the magnon, and the singlet pair. We also find two interaction-driven topological transitions: (i) at particle filling $\ensuremath{\rho}=\frac{1}{2}$, the low-energy edge excitations change from the magnon to singlet pair, accompanied with pair formation in bulk; and (ii) at $\ensuremath{\rho}=\frac{3}{4}$, while the gapless edge mode is always the charge of a single fermion, there is a gap-closing point and a $\ensuremath{\pi}$-phase shift in the quasiparticle spectrum.

DOI
10.1103/physrevb.105.165123

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We numerically investigate the ground state of the extended t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J Hamiltonian with periodic local modulations in one dimension by using the density-matrix renormalization group method. Examining charge and spin excitation gaps, as well as the pair binding energy, with extrapolated results to the thermodynamic limit, we obtain a rich ground-state phase diagram consisting of the metallic state, the superconducting state, the phase separation, and insulating states at commensurate fillings. Compared to the homogeneous one-dimensional (1D) t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model, the superconductivity is greatly enhanced and stabilized by the flat-band structure. This superconducting state in periodically modulated chains shares similar properties with ladder systems: significant negative pair binding energy occurs, and the singlet-pairing correlation function dominates with the algebraic decay while the single-particle Green's function and spin correlation function decay exponentially. On the other hand, the superlattice structure leads to nontrivial topological nature in insulating states, characterized by different integer Chern numbers at different fillings. Due to the interplay among the topology, the interaction, and the 1D confinement, gapless edge modes show strong spin-charge separation and in different regions can relate to different collective modes, which are the charge of a single fermion, the magnon, and the singlet pair. We also find two interaction-driven topological transitions: (i) at particle filling \ensuremathρ=12\ensuremath{\rho}=\frac{1}{2}\ensuremathρ=21​, the low-energy edge excitations change from the magnon to singlet pair, accompanied with pair formation in bulk; and (ii) at \ensuremathρ=34\ensuremath{\rho}=\frac{3}{4}\ensuremathρ=43​, while the gapless edge mode is always the charge of a single fermion, there is a gap-closing point and a \ensuremathπ\ensuremath{\pi}\ensuremathπ-phase shift in the quasiparticle spectrum.
The t-J Hamiltonian is one of the cornerstones in the theoretical study of strongly correlated copper-oxide based materials. Using the density-matrix renormalization group method we obtain the phase diagram of the one-dimensional t-J chain in the presence of a periodic hopping modulation, as a prototype of coupled-segment models. While in the uniform 1D t-J model the near half-filling superconducting state dominates only at unphysically large values of the exchange coupling constant J/t>3; we show that a small hopping and exchange modulation very strongly reduces the critical coupling to be as low as J/t∼1/3--well within the physical regime. The phase diagram as a function of the electron filling also exhibits metallic, insulating line phases and regions of phase separation. We suggest that a superconducting state is easily stabilized if t-J segments creating local spin-singlet pairing are coupled to each other--another example is the ladder system.
Majorana zero modes (MZM-s) occurring at the edges of a one-dimensional (1D), ppp-wave, spinless superconductor, in the absence of fluctuations of the phase of the superconducting order parameter, are quintessential examples of topologically protected zero-energy modes occurring at the edges of 1D symmetry-protected topological phases. In this work, we numerically investigate the fate of the topological phase in the presence of phase fluctuations using the density matrix renormalization group (DMRG) technique. To that end, we consider a one-dimensional array of MZM-s on mesoscopic superconducting islands at zero temperature. Cooper-pair and MZM-assisted single-electron tunneling, together with finite charging energy of the mesoscopic islands, give rise to a rich phase diagram of this model. We show that the system can be in either a Mott-insulating phase, a Luttinger liquid (LL) phase of Cooper pairs, or a second gapless phase. In contrast to the LL of Cooper pairs, this second phase is characterized by nonlocal string correlation functions which decay algebraically due to gapless charge-eee excitations. The three phases are separated from each other by phase transitions of either Kosterlitz-Thouless or Ising type. Using a Jordan-Wigner transformation, we map the system to a generalized Bose-Hubbard model with two types of hopping and use DMRG to analyze the different phases and the phase transitions.
We investigate topological phase transitions driven by interaction and identify a topological Mott insulator state in one-dimensional fermionic optical superlattices through a numerical density-matrix renormalization-group method. Remarkably, the low-energy edge excitations change from spin-1/2 fermionic single-particle modes to spin-1 bosonic collective modes across the phase transition. Due to spin-charge separation, the low-energy theory is governed by an effective spin superexchange model, whereas the charge degree of freedom is fully gapped out. Such topological Mott state can be characterized by a spin Chern number and gapless magnon modes protected by a finite spin gap. The proposed experimental setup is simple and may pave the way for the experimental observation of exotic topological Mott states.
In the first part of the present Thesis, the ground state (GS) properties of the quasi-one-dimensional AB₂ Hubbard model are investigated taking the effects of charge and spin quantum fluctuations on equal footing. Using a functional integral approach, combined with a perturbative expansion in the strong-coupling regime, we obtain the Lagrangian density associated with the charge (Grassmann fields) and spin [SU(2) gauge fields] degrees of freedom. In the strong-coupling regime, we derive a perturbative low-energy theory suitable to describe the ferrimagnetic phase at half filling and the phases in the hole-doped regime. At half filling, a perturbative spin-wave analysis allows us to find the GS energy, sublattice magnetizations, and total spin per unit cell in the Lieb ferrimagnetic GS of the effective quantum Heisenberg model, in very good agreement with previous results. In the challenging hole doping regime away from half filling, we derive the corresponding t-J Hamiltonian. Under the assumption that charge and spin quantum correlations are decoupled, the evolution of the second-order spin-wave modes in the doped regime unveils the occurrence of spatially modulated spin structures and the emergence of phase separation in the presence of resonatingvalence-bond (RVB) states. We also calculate the doping-dependent GS energy and total spin per unit cell, including both Zeeman and orbital contributions, in which case it is shown that the spiral ferrimagnetic order collapses at a critical hole concentration. Notably, our analytical results in the doped regime are in very good agreement with density matrix renormalization group (DMRG) studies, where our assumption of spin-charge decoupling is numerically supported by the formation of charge-density waves in anti-phase with the modulation of the magnetic structure. In the second part, motivated by analogy with photonic lattices, we examine the edge states of a one-dimensional trimer lattice in the phases with and without inversion symmetry protection. In contrast to the Su-Schrieffer-Heeger model, we show that the edge states in the inversion-symmetry broken phase of the trimer model turn out to be chiral, i.e., instead of appearing in pairs localized at opposite edges they can appear at a single edge. Interestingly, these chiral edge states remain robust to large amounts of disorder. In addition, we use the Zak phase to characterize the emergence of degenerate edge states in the inversion-symmetric phase of the trimer model. Furthermore, we capture the essentials of the whole family of trimers through a mapping onto the commensurate off-diagonal Aubry-André-Harper model, which allows us to establish a direct connection between chiral edge modes in the two models, including the calculation of Chern numbers.We thus suggest that the chiral edge modes of the trimer lattice have a topological origin inherited from this effective mapping. Also, we find a nontrivial connection between the topological phase transition point in the trimer lattice and the one in its associated two-dimensional parent system, in agreement with results in the context of Thouless pumping in photonic lattices.
We study signatures of superconductivity in a two-leg ``checkerboard'' Hubbard ladder model, defined as a one-dimensional (period 2) array of square plaquettes with an intraplaquette hopping ttt and interplaquette hopping ${t}^{\ensuremath{'}}$, using the density-matrix renormalization-group method. The highest pairing scale (characterized by the spin gap or the pair binding energy, extrapolated to the thermodynamic limit) is found for doping levels close to half-filling, $U\ensuremath{\approx}6t$ and ${t}^{\ensuremath{'}}/t\ensuremath{\approx}0.6$. Other forms of modulated hopping parameters, with periods of either one or three lattice constants, are also found to enhance pairing relative to the uniform two-leg ladder, although to a lesser degree. A calculation of the phase stiffness of the ladder reveals that in the regime with the strongest pairing, the energy scale associated with phase ordering is comparable to the pairing scale.
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.