Stripes on a 6-Leg Hubbard Ladder

Abstract

While DMRG calculations find stripes on doped n-leg t-J ladders, little is known about the possible formation of stripes on n-leg Hubbard ladders. Here we report results for a 7×6 Hubbard model with 4 holes. We find that a stripe forms for values of U/t ranging from 6 to 20. For U/t ∼ 3 − 4, the system exhibits the domain wall feature of a stripe, but the hole density is very broadened. PACS numbers: PACSThe nature of the ground state of the two dimensional t-J model continues to be controversial, even if one restricts one's attention to numerical simulations. Density matrix renormalization group (DMRG) calculations [1] for n-leg t-J ladders (with n ranging from 2 to 8) with J/t in the physical region of interest for the cuprates, find that stripes are formed when the ladders are doped with holes [2]. In contrast, the Green's function Monte Carlo (GFMC) results of Sorella, et al., on square, periodic t-J lattices find a d x 2 −y 2 -wave superconducting ground state [3]. The aspect ratio of an n-leg lattice and the open boundary conditions used in the DMRG calculations may be sufficient to favor a striped state. Alternatively, the choice of the guiding trial wave function for the GFMC may bias the system towards a superconducting state [4]. It is clear that the t-J model is a delicately balanced system. Thus it is interesting to go back one step and study the underlying Hubbard model in which the onsite Coulomb interaction U is a parameter.Much less is known about the nature of the ground state of the 2D Hubbard model, which is computationally more demanding. Previous DMRG calculations[5] on 3-leg Hubbard ladders found that diagonal three-hole stripes with a linear filling density of unity formed when U/t was greater than 5. Early mean-field Hartree-Fock calculations for the 2D Hubbard model found that vertical stripes were favored for U/t less than of order 4 and diagonal stripes formed at larger values of U/t [6]. These Hartree-Fock stripes have a linear filling density of one and their width is set by a coherence length ξ 0 ∼ t/∆ 0 with ∆ 0 the gap of the half-filled system. As U increases, ξ 0 decreases, approaching a lattice spacing at larger values of U/t. Eventually, when U exceeds twice the bandwidth, the mean-field stripes disappear [7]. Note, however, that the Hartree-Fock treatments do not include pairing and that their energy per hole is of order 1t too high. Here, we extend the DMRG study of stripe formation in the Hubbard model and in particular examine the question of whether stripes form on a 6-leg Hubbard ladder and how their structure depends upon U/t. Our results represent the first reliable ground state results for a Hubbard cluster large enough to exhibit an unambiguous stripe. From this single cluster we are not able to make conclusions about the infinite 2D lattice; on the other hand, our results strongly suggest that stripes are low-lying states in the Hubbard model, as they are in the t-J model.The cluster geometry was chosen with some care. Stripe-like behavior involving only two holes c...

Ground-state phase diagram of the square lattice Hubbard model from density matrix embedding theory

“…The strongest SC order found in DMET roughly occurs in the same region as seen in earlier Green's function cluster calculations [41,47]. However, this region is not typically found to be superconducting in ground-state wave-function calculations using DMRG and AFQMC on finite lattices, even though such calculations achieve significantly higher energy accuracies than the Green's function cluster studies [25,32,81,82]. The significance of the DMET result is that the energy error bar in this region (e.g., 0.001t for U = 4, n = 0.8, t = −0.2) is comparable to or better than the accurate ground-state wave-function calculations, yet SC order is still seen.…”

“…In this region, a variety of spin-density [25,43,45,46,49,[83][84][85], charge-density [25,[86][87][88], pair-density wave [88][89][90][91], and stripe orders [30,32,51,52,85,[92][93][94][95] have been posited in both the Hubbard model and the simpler t-J model, with different types of orders seen in different simulation methods. These inhomogeneous phases are proposed to be relevant in the pseudogap physics [89,90,[96][97][98][99][100].…”

“…(We refer to DCA/CDMFT/VCA collectively as Green's function cluster theories.) These pioneering works have suggested rich phenomenology in the phase diagram including metallic, antiferromagnetic, and d-wave (and other kinds of) superconducting phases, a pseudogap regime, inhomogeneous orders such as stripes, and charge, spin, and pairdensity waves, as well as phase separation [6,19,20,24,25,[27][28][29]32,35,[39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58]. However, as different numerical methods have yielded different pictures of the ground-state phase diagram, a precise quantitative picture of the ground-state phase diagram has yet to emerge.…”

“…Away from these limits, however, approximations are necessary. Many numerical methods have been applied to the model at both finite and zero temperatures, including fixed-node, constrained path, determinantal, and variational quantum Monte Carlo (QMC) [19][20][21][22][23][24][25][26][27][28][29], density matrix renormalization group (DMRG) [30][31][32], dynamical cluster (DCA) [33,34], (cluster) dynamical mean-field theories (CDMFT) [35,36], and variational cluster approximations (VCA) [37,38]. (We refer to DCA/CDMFT/VCA collectively as Green's function cluster theories.)…”

Diagonalization- and Numerical Renormalization-Group-Based Methods for Interacting Quantum Systems

“…The DMRG has been used to study a number of multi-chain and two-dimensional systems, including the multichain Heisenberg model [97], the frustrated Heisenberg model in one dimension [82] and in two dimensions [93], the two-leg Hubbard ladder [98], CaV 4 O 9 [99], and the multichain and two-dimensional t-J model [100,101,102,103].…”