Ground-state phase diagram of the one-dimensional $t$-$J_{s}$-$J_{\tau}$ model at quarter filling

Quantum fluctuations are important features of microscopic systems, which give rise to plenty of interesting phenomena. In condensed matter physics, spin fluctuations play an important role in realizing various quantum states, such as spin liquids and superconductivity. One of the minimal theoretical models containing both spin and charge degrees of freedom is the $t$-$J$ model. The model was originally proposed to describe high-$T_{\mbox{\scriptsize c}}$ superconductivity [1], and its one-dimensional model has been studied to understand the fundamental properties of strongly correlated systems. Although this model contains only the kinetic energy term and the exchange energy term, various quantum states including a spin-gap state are realized [2], and it is interesting to investigate whether new quantum states appear when we include additional interactions existing in more realistic systems. One simple extension is the inclusion of repulsive interaction $V$ between the neighboring electrons. It has been reported that the repulsive interaction $V$ stabilizes the spin-gap phase at quarter filling [3], but other new states have not yet been obtained. Another approach to extend the $t$-$J$ model is to add new degrees of freedom of electrons.

Praseodymium contained in cage-shaped composites, such as $\mathrm{PrTi_{2}Al_{20}}$, has a non-Kramers doublet as the crystal-field ground state [4, 5]. The theoretical model of such materials is the two-channel Kondo lattice model (TCKLM)[6, 7], which has multiple degrees of freedom associated with the non-Kramers doublets. As one of the simplest models of interacting electron systems consisting of multiple degrees of freedom, we propose the “$t$-$J_{s}$-$J_{\tau}$ model,” which is not only an extension of the $t$-$J$ model but also an effective model of the TCKLM in the strong-coupling region.

In this paper, we study the ground-state properties of the model by the density matrix renormalization group (DMRG) method and investigate the effect of the channel degree of freedom on the ground state. The obtained results show that the spin-gap state is stabilized by weak channel fluctuations, while strong channel fluctuations lead to the transition to the insulator.

The model we study here is the following $t$-$J_{s}$-$J_{\tau}$ model:

	$\displaystyle H_{tJJ}$	$\displaystyle=-t\sum_{i\alpha\sigma}\left(a^{\dagger}_{i\sigma}b_{i\alpha}b^{\dagger}_{i+1,\alpha}a_{i+1,\sigma}+\mathrm{H.c.}\right)$
		$\displaystyle+J_{s}\sum_{i}{\bm{S}}_{i}\cdot{\bm{S}}_{i+1}+J_{\tau}\sum_{i}{\bm{\tau}}_{i}\cdot{\bm{\tau}}_{i+1}+V\sum_{i}n_{i}n_{i+1},$		(1)

where $a^{\dagger}_{i\sigma}$ and $b^{\dagger}_{i\alpha}$ are the creation operators of particles and “holes” at the $i$th site with spin $\sigma$ and channel $\alpha$, respectively, and the empty and double occupancies of $a^{\dagger}_{i\sigma}$ and $b^{\dagger}_{i\alpha}$ are inhibited:

$\displaystyle\sum_{\sigma}a^{\dagger}_{i\sigma}a_{i\sigma}+\sum_{\alpha}b^{\dagger}_{i\alpha}b_{i\alpha}=n_{i}+\sum_{\alpha}b^{\dagger}_{i\alpha}b_{i\alpha}=1,$

(2)

where $n_{i}=\sum_{\sigma}a_{i\sigma}^{\dagger}a_{i\sigma}$ is the number operator of the particles. The spin and channel-pseudospin operators are defined as ${\bm{S}}_{i}=\frac{1}{2}\sum_{\sigma\sigma^{\prime}}a^{\dagger}_{i\sigma}{\bm{\sigma}}_{\sigma\sigma^{\prime}}a_{i\sigma^{\prime}}$ and ${\bm{\tau}}_{i}=\frac{1}{2}\sum_{\alpha\alpha^{\prime}}b^{\dagger}_{i\alpha}{\bm{\sigma}}_{\alpha\alpha^{\prime}}b_{i\alpha^{\prime}}$, respectively.

This model is not only a generalization of the extended $t$-$J$ model but also derived from the TCKLM

	$\displaystyle H_{\mbox{\scriptsize TCKLM}}$	$\displaystyle=-\tilde{t}\sum_{i\sigma\alpha}\left(c^{\dagger}_{i\alpha\sigma}c_{i+1,\alpha\sigma}+\mathrm{H.c.}\right)$
		$\displaystyle~{}~{}+\frac{J}{2}\sum_{i\alpha\sigma\sigma^{\prime}}\tilde{{\bm{S}}}_{i}\cdot\left(c^{\dagger}_{i\alpha\sigma}{\bm{\sigma}}_{\sigma\sigma^{\prime}}c_{i\alpha\sigma^{\prime}}\right),$		(3)
	$\displaystyle\tilde{{\bm{S}}_{i}}$	$\displaystyle=\frac{1}{2}\sum_{\sigma\sigma^{\prime}}f^{\dagger}_{i\sigma}{\bm{\sigma}}_{\sigma\sigma^{\prime}}f_{i\sigma^{\prime}},$		(4)

as follows: assuming that the number of conduction electrons per local spins $n_{c}$ satisfies $1\leq n_{c}\leq 2$, the effective Hamiltonian in the strong-coupling region is given by the second-order perturbation of $1/J$ from the limit $J/\tilde{t}=\infty$. In this case, $a^{\dagger}$ and $b^{\dagger}$ are the composite particles defined as $a^{\dagger}_{i\sigma}=\frac{1}{\sqrt{6}}\left(2c_{i1\sigma}^{\dagger}c_{i2\sigma}^{\dagger}f_{i\bar{\sigma}}^{\dagger}-c_{i1\sigma}^{\dagger}c_{i2\bar{\sigma}}^{\dagger}f_{i\sigma}^{\dagger}-c_{i1\bar{\sigma}}^{\dagger}c_{i2\sigma}^{\dagger}f_{i\sigma}^{\dagger}\right)$ and $b^{\dagger}_{i\alpha}=\frac{1}{\sqrt{2}}\left(c_{i\alpha\uparrow}^{\dagger}f_{i\downarrow}^{\dagger}-c_{i\alpha\downarrow}^{\dagger}f_{i\uparrow}^{\dagger}\right)$ , respectively, as schematically represented in Fig. 1. The transfer integral $t$ in Eq. (1) is given as $t=3\tilde{t}/4$ and the effective interactions are

$\displaystyle J_{s}=\frac{1504t^{2}}{135J},\ J_{\tau}=\frac{64t^{2}}{9J},\ \ \ \frac{J_{\tau}}{J_{s}}\approx 0.64.$

(5)

The interactions $J_{s}$ and $J_{\tau}$ are the two largest terms obtained by the perturbation expansion, and the other ones, including next-nearest hopping, are ignored. The neglected long-range interactions are expected to suppress the phase separation caused by the above two interactions. Instead of treating all such terms explicitly, we consider the repulsion term $V$ to suppress the phase separation. Note that when $n_{c}=1$, which corresponds to the absence of the $a^{\dagger}$ particles ($n=0$), the model is reduced to the Heisenberg model of the channel degree of freedom [7].

In this study, we analyze the ground state of the Hamiltonian of Eq. (1) with an equal number of particles and holes at $n=1/2$ (quarter filling, $k_{F}=\pi/4$). This filling corresponds to $n_{c}=3/2$ in the TCKLM. Throughout this paper, we fix the nearest-neighbor interaction $V$ as $V/t=0.8$ and take the transfer integral $t$ as the unit of energy.

We use the DMRG method [8, 9] to analyze the ground states of the Hamiltonian of Eq. (1). In this method, the accuracy of the ground-state wave function is systematically controlled by the number of remaining states $m$. We increase $m$ up to 400 to see the convergence of the results, where the truncation error is less than $10^{-5}$. The system size is in the range of 128–192.

To suppresses the finite-size effect caused by the open boundary conditions used in the DMRG calculation, we apply the sine square deformation (SSD) [10, *cite:ssd_gendiar2] to the Hamiltonian. Since the SSD reproduces the bulk response to an external field [12], we use this property to obtain the excitation gap of the infinite system.

We first study the elementary excitations of the model to clarify how the interactions modify the low-energy properties of the system. We calculate the excitation gap for the charge ($\Delta_{n}$), spin ($\Delta_{s}$), and channel ($\Delta_{\tau}$) degrees of freedom in the parameter space of $J_{s}$-$J_{\tau}$. Figure 2 shows the excitation gaps obtained along the line defined by Eq. (5). With decreasing the parameter $J$ of the TCKLM (with increasing $J_{s}$ and $J_{\tau}$ of the $t$-$J_{s}$-$J_{\tau}$ model), the spin excitation gap first opens, and then, the charge and channel gaps open.

These successive transitions show the presence of the spin-gap phase. To further confirm the spin-gap phase, we systematically calculate the excitation gaps for various $J_{s}$ and $J_{\tau}$ and determine the ground-state phase diagram of the $t$-$J_{s}$-$J_{\tau}$ model. Figure 3 shows the obtained phase diagram consisting of five phases: metallic phase (no excitation gap), spin-gap phase (only the spin gap opens), channel-gap phase (only the channel gap opens), insulating phase (gap opens for all excitations), and phase separation. From the diagram, it is confirmed that the spin-gap phase is realized in the TCKLM between the metallic and insulating phases.

As shown in Fig. 3, the transition lines are symmetric with respect to the line of $J_{s}=J_{\tau}$. This arises from the invariance of the Hamiltonian of Eq. (1) and the particle filling $n=1/2$ under the transformation $(a^{\dagger}_{i\uparrow},a^{\dagger}_{i\downarrow},b^{\dagger}_{i1},b^{\dagger}_{i2})\rightarrow(b^{\dagger}_{i1},b^{\dagger}_{i2},a^{\dagger}_{i\uparrow},a^{\dagger}_{i\downarrow})$ with the exchange of $J_{s}$ and $J_{\tau}$, where the symmetry of the repulsive term $V$ is ensured by the condition of Eq. (2) ¹¹1 In the system under the open boundary condition, the repulsive term modifies the local chemical potential at both ends of the system. In this study, we use the SSD and remove such a chemical-potential difference. . Since this transformation exchanges the role of spin and channel degrees of freedom, the symmetric phase diagram is obtained. In the parameter sets we have studied, the direct transition between the metallic phase and the insulating phase occurs only on the line of $J_{s}=J_{\tau}$. This implies the existence of a quantum tetracritical point.

We have studied the ground states of the $t$-$J_{s}$-$J_{\tau}$ model, which is a minimal model consisting of multiple degrees of freedom. The low-energy excitations of the spin, charge and channel degrees of freedom have been calculated by the DMRG method with the SSD, and it was shown that the phase transition occurs from the metallic state to the spin-gap or channel-gap state when the exchange interactions exceed almost the bandwidth, roughly $\sqrt{J_{s}^{2}+J_{\tau}^{2}}\sim 4t$. For the symmetric case of $J_{s}=J_{\tau}$, however, the direct transition to the insulating state takes place. These results imply that weak channel fluctuations stabilize the spin-gap state of the $t$-$J$ model, while strong channel fluctuations lead to the transition to the insulating state which is characterized by the alternating product state of the spin and channel singlets.

This work was supported by JSPS KAKENHI Grant No. JP19K03708.