Complex Phase Diagram of Doped XXZ Ladder: Localization and Pairing

Introduction.—As one of the simplest models describing the doped Mott insulator, the $t$-$J$ model has attracted intense attention due to its potential to characterize systematically a complex phase diagram composed of a long-range Néel state, pseudogap and superconducting (SC) states, etc., as a function of doping Lee et al. (2006). Such a model can be accurately investigated by density matrix renormalization group (DMRG) method White (1992) in quasi one-dimensional (1D) square lattice cases (i.e., the “ladders”), which does exhibit quasi-1D SC behavior Poilblanc et al. (1995); Siller et al. (2002); Jiang et al. (2018, 2019a, 2019b) known as the Luther-Emery (LE) state, characterized by an exponential decay of spin correlations and power-law decays of SC and charge density wave (CDW) correlations Luther and Emery (1974); Jiang et al. (2019a); Seidel and Lee (2005). As a matter of fact, an intrinsic pairing force has been clearly identified even for two doped holes in the two- and four-leg $t$-$J$ laddersZhu et al. (2014); Sun et al. (2019). Here the half-filling isotropic spin background in the ladder is already a “spin liquid” without long-range Néel order Dagotto and Rice (1996), which would seem to support the resonance valence bond (RVB) idea of Anderson for the SC origin out of a pure repulsive interactionAnderson (1987); Baskaran et al. (1987).

However, given the same “RVB” or gapped spin background, the LE ground state can be replaced by a Luttinger-liquid-like state with the exponent very close to that of the free Fermi gas (FG) Jiang et al. (2019a) at finite doping, if the hopping term is modified to result in the so-called $\sigma\cdot$$t$-$J$ ladders Zhu et al. (2013); Zhu and Weng (2015). Such a dramatic contrast suggests that the mutual interplay between the spin and charge degrees of freedom, instead of the pure “RVB” itself, may play a decisive role in the LE phase. Here the $\sigma\cdot$$t$-$J$ model is equivalent to a $t$-$J$ type model with a ferromagnetic (FM) in-plane spin superexchange Jiang et al. (2019a). Moreover, although a long-range Néel ordered state would naturally occur in the two-dimensional (2D) case Auerbach (2012), such an order may be still realized in the spin ladder case by reducing the in-plane (transverse) spin superexchange coupling close to the Ising limit Narushima et al. (1995); Roy et al. (2017); Li et al. (2017). Thus, by systematically studying a doped XXZ model on a two-leg ladder as a prototypical Mott insulator, one may gain a full understanding of the mutual spin-charge interaction with utilizing the precise numerical tool-DMRG White (1992, 1993); Schollwöck (2005), which can in turn shed important light on the realistic 2D physics Anderson (1987) related to the high-$T_{c}$ cuprate Bednorz and Müller (1986); Keimer et al. (2015).

In this paper, we present a complex phase diagram obtained by DMRG in the lightly doped two-leg XXZ ladder via tuning the superexchange anisotropy at a given doping concentration. Near the Ising limit, a coexisting phase of charge localization (CL) and long-range spin density wave (SDW) (i.e., CL/SDW I) is identified. By reducing the anisotropy, the charge pairing and a distinct SDW with doubled wavelength emerges in a new phase (i.e., LPP/SDW II). Here LPP refers to a so-called lower pseudogap phase where the pairing is present but the system is short of SC phase coherence Weng and Qi (2006). Such a phase still remains insulating, concomitant with a classical SDW order in which the transverse (Goldstone-like) mode is gapped. Only when the long-range SDW order truly disappears at a weaker anisotropy, does the SC correlation finally become quasi-long-ranged as characterized by the Luther-Emery liquid which has been already well-established in the isotropic (t-J) limit Jiang et al. (2018, 2019a). On the contrary, in the in-plane FM XXZ regime, no pairing between the holes is ever found, where a FG-like phase emerges after the CL/SDW I vanishes. A schematic phase diagram for such a lightly doped XXZ model is summarized in Fig. 1(a).

Model and Method.—The doped XXZ model on a two-leg square ladder is given by

Here, $H_{t}$ describes the hole hopping process

where $\langle ij\rangle$ denotes the nearest-neighbor sites, with the no-double-occupancy constraint always enforced such that only a single electron can occupy each site: $\hat{n}_{i}\leq 1$. The spin superexchange term is given by

where $J$ is the strength of the $z$-component coupling, while $\Gamma$ denotes the anisotropy in the strength of the in-plane or transverse components. We shall fix $t/J=3$ and tune $\Gamma$ to study the evolution of the ground state at a doping concentration $\delta=N^{h}/N$ (typically at $\delta=1/12$ while other $\delta$’s have been also examined). Here $N^{h}$ is the number of holes and $N$ is the number of the lattice sites. Several limiting cases of the Hamiltonian (1) have been investigated previously. At the isotropic $\Gamma=1$ limit, the ground state is an LE liquid with quasi-1D SC Troyer et al. (1996); Hayward and Poilblanc (1996); Müller and Rice (1998); Jiang et al. (2019a) and it dramatically reduces to a Luttinger liquid behavior close to the free Fermi gas limit at $\Gamma=-1$ Jiang et al. (2019a). At the Ising limit of $\Gamma=0$, the doped holes repel each other with a vanished inverse compressibility which indicates the charge localization Weisse et al. (2006).

We employ DMRG to determine the ground state of the Hamiltonian (1) with system size $N=2\times N_{x}$ up to $N_{x}=192$. We shall keep up to 12000 number of states in each DMRG block to reduce the truncation error smaller than $2\times 10^{-9}$ and perform typically about 30-200 sweeps to achieve a reliable convergence. The high-performance matrix product state algorithm library GraceQ/MPS2 Sun and Peng is used to perform the simulations. Some detailed scaling analyses are also presented in Appendix.

Correlation functions.—Using the optimized DMRG ground state, the following physical quantities are investigated. The rung-averaged charge density along the ladder ($x$) direction is defined by $n(x)$, whose charge modulation behavior will be fitted by

where $Q_{\mathrm{CDW}}$ denotes the CDW wavevector and $A_{\mathrm{CDW}}$ the amplitude, which is scaled by $A_{\mathrm{CDW}}\sim N_{x}^{-K_{c}/2}$ with an exponent $K_{c}$. The rung-averaged equal-time single-particle Green’s function is defined as

where $x_{0}$ labelled the reference rung. An exponentially decayed $G_{\sigma}(r)\sim e^{-r/\xi_{G}}$ is characterized by the correlation length $\xi_{G}$, while a power-law-decayed $G_{\sigma}\sim r^{-K_{G}}$ is fitted by a Luttinger exponent $K_{G}$. The rung-averaged pair-pair correlation is given by

where $\hat{\Delta}^{\dagger}_{\alpha}(x,y)=\frac{1}{\sqrt{2}}\left[\hat{c}^{\dagger}_{(x,y),\uparrow}\hat{c}^{\dagger}_{(x,y)+\alpha,\downarrow}-\hat{c}^{\dagger}_{(x,y),\downarrow}\hat{c}^{\dagger}_{(x,y)+\alpha,\uparrow}\right]$ is the spin-singlet pair-field creation operator and $\alpha,\beta=\hat{x},\hat{y}$ defines the pairing bond orientation. The quasi-1D SC or non-SC state is characterized Jiang et al. (2018, 2019a) by $\Phi(r)\sim r^{-K_{sc}}$ or $\Phi(r)\sim e^{-r/\xi_{sc}}$, respectively.

Besides the above three quantities involving the charge degrees of freedom, the underlying spin degrees of freedom are described by the following quantities. The $z$-component spin structure factor is given by

The $z(x)$-component spin-spin correlations are defined by

Their asymptotic behavior will be either captured by the correlation length scale $\xi_{F^{z(x)}}$ or a Luttinger exponent $K_{F^{z(x)}}$ depending on the short-range or quasi-long-range order, respectively.

Two distinct insulating phases with SDW orders.—Let us first focus on the phase diagram of Fig. 1(a) at $\Gamma>0$ (top row) and examine the two sides of the critical point at $\Gamma_{c1}$. As shown in the top row of Fig. 1(b), the $S^{z}$ structure factor clearly indicates that $\Gamma_{c1}$ separates two SDW ordered states with the wavevector $q_{x}$ peaked at $\pi\pm 2\pi\delta$ (SDW I) and $\pi\pm\pi\delta$ (SDW II) with the “incommensurate” wavelength $\lambda_{\mathrm{SDW}}=1/\delta$ and $2/\delta$, respectively.

The first- and second-order derivatives of the ground state energy in Fig. 2(a) show a singularity at $\Gamma_{c1}$, suggesting a “second-order-like” phase transition. The wavelength of the CDW shown in Fig. 2(b) further demonstrates a jump at $\Gamma_{c1}$, which is consistent with that of the SDWs by $\lambda_{\mathrm{CDW}}=\lambda_{\mathrm{SDW}}/2$ (cf. Fig. 1(b) and Appendix ). The doubled CDW wavelength at $\Gamma>\Gamma_{c1}$ seems to indicate the pairing of the doped holes as opposed to the unpaired case at $\Gamma<\Gamma_{c1}$. Here we further examine the binding energy between the doped holes as defined by

where $E_{0}(x)$ is the ground state energy with $x$ holes. As shown in Fig. 2(c), $E_{b}$ is always positive and extrapolated to vanishing at $\Gamma<\Gamma_{c1}$, but is clearly saturated to a negative value at $\Gamma>\Gamma_{c1}$.

However, the finite binding energy of holes does not necessarily imply a (quasi) long-range pair-pair correlation, which is absent on both sides of $\Gamma_{c1}$ as presented by the exponential decays in Figs. 3(d) and (e). The corresponding single-particle Green’s function is also exponentially decaying as shown in Figs. 3(a) and (b). As far as the charge is concerned, two distinct phases separated by $\Gamma_{c1}$ seem all insulating, which is distinguished by the presence (absence) of local charge pairing. In Fig. 1(a), these two SDW phases are thus further denoted by CL and LPP.

One may further compare the transverse spin-spin correlation on either side of $\Gamma_{c1}$ in Figs. 3(g) and (h), respectively. The quantum nature of the magnetic order in SDW I is supported by the power-law behavior in Fig. 3(g), namely, the presence of the Goldstone-like mode in the transverse components. Note that the disorder-free “auto-localization” in the Ising ($\Gamma=0$) limit has been already pointed out in early analytical work Bulaevski et al. (1968), which should continuously persist in the regime of SDW I. But in the SDW II, the transverse spin-spin correlation becomes exponentially decaying in Fig. 3(h) as if the Goldstone-like mode be gapped via some novel “Higgs mechanism” concomitant with the pairing. It also indicates that LPP/SDW II resembles a classical magnetically ordered phase without the rigidity from the quantum protection of the gapless transverse mode.

Phase coherence achieved in the “spin liquid” regime: Superconducting at $\Gamma>\Gamma_{c2}$ vs. normal Fermi liquid at $\Gamma<\Gamma_{c3}<0$.—As illustrated in Fig. 1(a), the quasi long-range pair-pairing will be established at $\Gamma>\Gamma_{c2}$ as shown in Fig. 3(f). Simultaneously the SDW order disappears [cf. Figs. 1(b) and 3(i)]. And the corresponding $G(r)$ decays exponentially [cf. Fig. 3(c)]. This phase can be continuously connected to the well-studied LE phase in the isotropic limit at $\Gamma=1$ Jiang et al. (2019a). In other words, the quasi-1D SC coherence can be only realized with the disappearance of the magnetic order even though the local pairing is already present in the SDW II phase. Additionally, the establishment of the SC coherence at $\Gamma>\Gamma_{c2}$ can also be characterized by the qualitative change, from the exponential to the power-law decay, in the response to an inserted magnetic flux under periodic boundary condition Seidel and Lee (2005); Zhu et al. (2014) which is presented in the Appendix as a cross check.

By comparison, at $\Gamma<0$, the SDW I order terminates at $\Gamma_{c3}<0$ as shown in Figs. 1(a) and (b). However, instead of pairing, the doped holes will form a coherent Fermi-gas-like ground state at $\Gamma<\Gamma_{c3}<0$, with the Luttinger exponent $K_{G}\sim 1$, as shown in Appendix . It is emphasized that here the spin background also becomes a short-range “spin liquid” state as in the LE case at $\Gamma>\Gamma_{c2}$ since $H_{\mathrm{XXZ}}$ at $\Gamma<0$ can be exactly mapped to $H_{\mathrm{XXZ}}$ at $\Gamma>0$ by a unitary transformation Jiang et al. (2019a). However, the hopping term $H_{t}$ is mapped to the hopping term of the so-called $\sigma\cdot$$t$-$J$ defined by $H_{\sigma\cdot t}=-t\sum_{\langle ij\rangle,\sigma}\sigma\hat{c}^{\dagger}_{i\sigma}\hat{c}_{j\sigma}+\mathrm{h.c.}$ Jiang et al. (2019a). Consequently the main difference between $\Gamma<0$ and $\Gamma>0$ is precisely that between the $t$-$J$ and $\sigma\cdot$$t$-$J$ models Zhu et al. (2014); Sun et al. (2019), which can be solely attributed to the presence/absence of the phase string sign structure Sheng et al. (1996); Weng et al. (1997); Wu et al. (2008) as to be further elaborated below.

Discussion.—Four distinct phases have been identified in the present lightly doped XXZ model. In the Ising limit $|\Gamma|\rightarrow 0$, the spins of the system form an SDW I order, where the doped holes are always self-localized by the so-called $S^{z}$-string Bulaevski et al. (1968) (red-colored spins) as schematically illustrated in Fig. 4(b), which are displaced by the hopping of holes to result in string-like mismatches of the FM pattern along the hole path. It cannot be “repaired” via transverse spin flips until $|\Gamma|$ is sufficiently large beyond the regime of $\Gamma_{3c}<\Gamma<\Gamma_{1c}$. At $\Gamma>\Gamma_{1c}$, the $S^{z}$-string does get erased via spin flips, but in general a sequence of signs (phase string) Sheng et al. (1996); Weng et al. (1997); Wu et al. (2008) will be acquired by the holes [Fig. 4(c)], which is simply due to the fact that there is also a transverse $S^{\pm}$-string, similarly created by the hopping term $H_{t}$, which cannot be simultaneously “repaired” with the $S^{z}$-string through $H_{\mathrm{XXZ}}$. Any quantum spin zero-point motion can thus result in severe destructive interference effect due to Berry-phase-like phase strings associated with the holes. The only way to further eliminate such unescapable many-body quantum frustration (at least at low doping) is for holes to pair Zhu et al. (2014); Sun et al. (2019) up in SDW II and LE phases at $\Gamma>\Gamma_{1c}$. In sharp contrast, at $\Gamma<0$, the model is equivalent to the $\sigma\cdot$$t$-$J$ model, which does not generate any transverse spin string defect (phase string) and thus once $S^{z}$-string is erased at $\Gamma<\Gamma_{3c}<0$, the doped holes will simply behave like the free Landau’s quasiparticles as illustrated in Fig. 4(d), but there is no more incentive for holes to further pair up.

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