Valence transition in topological Kondo insulator

TL;DR

This paper explores valence and topological phase transitions in a topological Kondo insulator using slave-boson analysis, revealing complex first-order and continuous transitions influenced by Coulomb interactions and temperature.

Contribution

It introduces a comprehensive phase diagram showing various valence, magnetic, and topological transitions driven by Coulomb correlation and temperature effects.

Findings

First-order valence transition near topological transition

Thermal valence transition from mixed valence to Kondo region

Shift from continuous to first-order antiferromagnetic transition

Abstract

We investigate the valence transition in three-dimensional topological Kondo insulator through slave-boson analysis of periodic Anderson model. By including the effect of intra-atomic Coulomb correlation $U_{fc}$ between conduction and local electrons, we find a first-order valence transition from Kondo region to mixed valence upon ascending of local level above a critical $U_{fc}$, and this valence transition usually occurs very close to or simultaneously with a topological transition. Near the parameter region of zero-temperature valence transition, rise of temperature can generate a thermal valence transition from mixed valence to Kondo region, accompanied by a first-order topological transition. Remarkably, above a critical $U_{fc}$ which is considerable smaller than that generating paramagnetic valence transition, the original continuous antiferromagnetic transition is shifted to first order one, at which a discontinuous valence shift takes place. Upon increased $U_{fc}$, the paramagnetic valence transition approaches then converges with the first-order antiferromagnetic transition, leaving an significant valence shift on the magnetic boundary. The continuous antiferromagnetic transition, first-order antiferromagnetic transition, paramagnetic valence transition and topological transitions are all summarized in a global phase diagram. Our proposed exotic transition processes can help to understand the thermal valence variation as well as the valence shift around the pressure-induced magnetic transition in topological Kondo insulator candidates and in other heavy-fermion systems.

Tables

Table 1. Table 1: Two sets of EHA used in this work

	$t_d$	$t_d^{\prime }$	$t_d^{\prime \prime }$	$t_f$	$t_f^{\prime }$	$t_f^{\prime \prime }$
EHA(I)	1	0.15	0	-0.2	-0.02	0
EHA(II)	1	-0.375	-0.375	-0.2	0.09	0.09

Equations

$$\mathcal{H}=$$

$$+$$

$$H_{\mathrm{MF}}=N[\lambda (b^2-1)-U_{fc}{n}_c{n}_f]+$$

$$\displaystyle\sum_{\mathbf{k},\alpha,\beta}(c^{\dagger}_{\mathbf{k}\alpha},f^{\dagger}_{\mathbf{k}\alpha})\left(\begin{array}[]{cc}\tilde{\epsilon}^{c}_{\mathbf{k}}\delta_{\alpha\beta}&\tilde{V}\mathbf{S}_{\mathbf{k}}\cdot\vec{\sigma}_{\alpha\beta}\\ \tilde{V}\mathbf{S}_{\mathbf{k}}\cdot\vec{\sigma}_{\alpha\beta}&\tilde{\epsilon}^{f}_{\mathbf{k}}\delta_{\alpha\beta}\end{array}\right)\left(\begin{array}[]{cc}c_{\mathbf{k}\beta}\\ f_{\mathbf{k}\beta}\end{array}\right),$$

$$n_t=\frac{2}{N}\sum _{\mathbf{k},\pm }\theta (-E_{\mathbf{k}}^{\pm }),$$

$$n_f=\frac{1}{N}\sum _{\mathbf{k},\pm }\theta (-E_{\mathbf{k}}^{\pm })[1\mp \frac{{\widetilde{ϵ}}_{\mathbf{k}}^c-{\widetilde{ϵ}}_{\mathbf{k}}^f}{\sqrt{({\widetilde{ϵ}}_{\mathbf{k}}^c-{\widetilde{ϵ}}_{\mathbf{k}}^f{)}^2+4V^2{b}^2{S}_{\mathbf{k}}^2}}],$$

$$\lambda =-\frac{1}{N}\sum _{\mathbf{k},\pm }\theta (-E_{\mathbf{k}}^{\pm })[{ϵ}_{\mathbf{k}}^f\pm \frac{2V^2{S}_{\mathbf{k}}^2-{ϵ}_{\mathbf{k}}^f({\widetilde{ϵ}}_{\mathbf{k}}^c-{\widetilde{ϵ}}_{\mathbf{k}}^f)}{\sqrt{({\widetilde{ϵ}}_{\mathbf{k}}^c-{\widetilde{ϵ}}_{\mathbf{k}}^f{)}^2+4V^2{b}^2{S}_{\mathbf{k}}^2}}],$$

$$\mathcal{H}=N(h{m}_f-\eta n_f-U_{fc}{n}_c{n}_f)+\sum _{\mathbf{k}}{\Psi }_{\mathbf{k}}^{†}{\mathbf{H}}_{\mathbf{k}}{\Psi }_{\mathbf{k}},$$

$$\displaystyle\mathbf{H}_{\mathbf{k}}=\left(\begin{array}[]{cc}\mathbf{H}^{d}_{\mathbf{k}}&\mathbf{V}_{\mathbf{k}}\\ \mathbf{V}^{+}_{\mathbf{k}}&\mathbf{H}^{f}_{\mathbf{k}}\end{array}\right),$$

$$\displaystyle\mathbf{V}_{\mathbf{k}}=V\left(\begin{array}[]{cccc}0&0&Z_{-}s_{z}&Z_{+}\Gamma_{\mathbf{k}}\\ 0&0&Z_{-}\Gamma^{*}_{\mathbf{k}}&-Z_{+}s_{z}\\ Z_{+}s_{z}&Z_{-}\Gamma_{\mathbf{k}}&0&0\\ Z_{+}\Gamma^{*}_{\mathbf{k}}&-Z_{-}s_{z}&0&0\end{array}\right)$$

$$\displaystyle\mathbf{H}^{d}_{\mathbf{k}}=\left(\begin{array}[]{cc}(t^{\prime}_{c}\gamma_{\mathbf{k}}+U_{fc}n_{f}-\mu)&u^{c}_{\mathbf{k}}\\ u^{c}_{\mathbf{k}}&(t^{\prime}_{c}\gamma_{\mathbf{k}}+U_{fc}n_{f}-\mu)\end{array}\right)\otimes I_{2},$$

$${\mathbf{H}}_{\mathbf{k}}^f=$$

$$F=$$

$$+\sum _{\mathbf{k},n,m}({\mathbf{H}}_{\mathbf{k}}{)}_{nm}⟨({\Psi }_{\mathbf{k}}^{†}{)}_n({\Psi }_{\mathbf{k}}{)}_m⟩,$$

$$⟨({\Psi }_{\mathbf{k}}^{†}{)}_n({\Psi }_{\mathbf{k}}{)}_m⟩=\sum _{i=1}^8({\mathrm{U}}_{\mathbf{k}}{)}_{ni}^{\ast }({\mathrm{U}}_{\mathbf{k}}{)}_{mi}{f}_{\mathbf{k}}^{(i)},$$

$$n_t=\frac{2}{N}\sum _{\mathbf{k},\pm }{f}_{\mathbf{k}}^{\pm },$$

$$n_f=\frac{1}{N}\sum _{\mathbf{k},\pm }{f}_{\mathbf{k}}^{\pm }[1\mp \frac{{\widetilde{ϵ}}_{\mathbf{k}}^c-{\widetilde{ϵ}}_{\mathbf{k}}^f}{\sqrt{({\widetilde{ϵ}}_{\mathbf{k}}^c-{\widetilde{ϵ}}_{\mathbf{k}}^f{)}^2+4V^2{Z}^2{S}_{\mathbf{k}}^2}}],$$

$$\eta =\frac{2Z}{N}\frac{\partial Z}{\partial n_f}\sum _{\mathbf{k},\pm }{f}_{\mathbf{k}}^{\pm }[{ϵ}_{\mathbf{k}}^f\pm \frac{2V^2{S}_{\mathbf{k}}^2-{ϵ}_{\mathbf{k}}^f({\widetilde{ϵ}}_{\mathbf{k}}^c-{\widetilde{ϵ}}_{\mathbf{k}}^f)}{\sqrt{({\widetilde{ϵ}}_{\mathbf{k}}^c-{\widetilde{ϵ}}_{\mathbf{k}}^f{)}^2+4V^2{Z}^2{S}_{\mathbf{k}}^2}}],$$

Full text

I Introduction

Since the proposal of topological intrinsic in some special Kondo insulators (KI) Dzero10 ; Dzero12 , renewed attention has been attracted on these old materials, now are known as ”topological Kondo insulators”(TKI) represented by SmB6 Tran12 ; Alexandrov13 ; Legner14 ; Werner14 ; Roy14 ; Kim14 ; Baruselli14 ; Li14 ; Xu14 ; Alexandrov15 ; Zheng18 ; Pirie18 ; Ying18 ; Broekey18 ; Ohtsubo19 ; Fuhrman19 . In these TKIs, the strong spin-orbit coupled hybridization between $d$ and $f$ electrons guarantees time-reversal symmetry (TRS) and generates a band inversion between $d$ and $f$ orbits at certain time-reversal-invariant momenta (TRIM), leading to a topologically protected state classified by $Z_{2}$ invariants Dzero10 ; Dzero12 ; Dzero13 ; Alexandrov13 ; Dzero16 . Via the spin- and angle-resolved photoemission spectroscopy (SARPES), the metallic surface states with Dirac points have been observed in SmB6, confirming its topological characteristic Xu14 ; Xu16 . A Variety of theoretical works have been carried out for TKI to reveal its rich topological phases, topological transitions, surface states, and magnetic transitions, through first-principle calculations as well as model calculations for periodic Anderson model (PAM) Tran12 ; Lu13 ; Legner14 ; Werner14 ; Roy14 ; Kim14 ; Baruselli14 ; Legner15 ; Yu15 ; Alexandrov15 .

Recently, the high-pressure studies of SmB6 demonstrate a magnetic transition at 6$\sim$8 GPa  Barla05 ; Derr06 ; Derr08 ; Paraskevas15 ; Emi16 ; Butch16 ; Zhou17 , around which the mean valence $\nu$ of Sm ion (Smν+) displays a quite rapid variation from 2.5 to saturated value 3 Zhou17 , in addition, the mean valence also increases as temperature rises Mizumaki09 ; Wu17 , similar to other TKI candidates such as YbB12 and pressured golden SmS Annese06 ; Li14-PRB ; Hagiwara17 . In the hole representation for the $f$ shell (in the filled 4$f^{6}$ base) of Sm atom, the valence $\nu$ is related to the mean occupation of $f$ holes $n_{f}$ by $\nu=2+n_{f}$ Derr06 ; Alexandrov13 , so the valence shift of Sm ion indicates a variation of $f$-occupation with pressure, from mixed valence (MV), to Kondo region (or local moment region) in which the $f$ electrons are nearly localized to create a magnetic order. In this context, the possible valence transition or valence crossover in TKI, as well as the relation to the magnetic transition deserve further theoretical investigation.

For Ce- and Yb- based heavy-fermion compounds, the first-order valence transition (FOVT) was observed decades ago in pressure studies Felner85 ; Mito03 ; Park06 , in which the valences of Ce and Yb ion increase abruptly at a critical temperature, and under enhanced pressure, FOVT can be suppressed and terminates at a critical end point (CEP) to become a valence crossover. The valence of Ce (Yb) is manipulated by the electron (hole) occupation number $n_{f}$ of the 4$f$ shell Watanabe09 , so the discontinuous valence jump indicates a first-order transition from MV (with small $n_{f}$) to Kondo region (nearly localized $f$ electrons with $n_{f}\sim 1$). Pressure can drive magnetic transitions in some heavy-fermion compounds, and it’s found that FOVT reaches the magnetic boundary (e.g. in YbInCu4 Immer97 ; Mito03 ), implying a strong interplay between FOVT and magnetic transition. The zero-temperature FOVT in heavy-fermion systems can be understood through PAM by considering the on-site Coulomb repulsion $U_{fc}$ between local $f$ and conducting $c$ electrons Goltsev01 ; Watanabe06 . Upon ascent of the energy level $\epsilon_{f}$ of $f$ orbit by strengthened pressure, the influence of $U_{fc}$ causes a much rapider ascent of the renormalized $f$ level (relative to the chemical potential), pours the electrons into conduction band then consequently drives a decrease of $f$ occupation number $n_{f}$, leading to the crossover behavior of the valence. Above a critical $U_{fc}$, the valence crossover is strengthened then finally changed into FOVT from Kondo region to MV, showing a abrupt fall of $n_{f}$ at a critical $\epsilon_{f}$ Watanabe08 ; Watanabe09 .

Beside $\epsilon_{f}$, pressure applied on heavy-fermion compounds can lead to variations of other model parameters such as the hybridization strength, which all can affect the FOVT Watanabe09 , so the experimentally observed FOVT is a combined outcome within model description. In YbInCu4, the enhanced pressure drives a FOVT from MV to Kondo region Mito03 , contrary to Ce-based systems, therefore the pressure-induced FOVT cannot be simply attributed to the ascent of $\epsilon_{f}$ Goltsev01 . By contrast, the FOVT from MV to Kondo region in Ce- and Yb- compounds by rising temperature is a purer effect thus can be interpreted more straightforwardly, but such thermal FOVT is still lacking of theoretical investigation. On the other hand, for SmB6 and TKI candidate golden SmS, near the pressure-driven magnetic boundary, the valence of Sm ion shows an active increase Zhou17 ; Barla04 ; Butch16 , similarly, at low temperatures, the pressure-induced magnetic transition in YbInCu4 holds simultaneously a FOVT Mito03 , therefore, the relation of FOVT or valence crossover in TKI (and also in other heavy-fermion systems) to the magnetic transition should be clarified in an unified framework. Particularly, in TKI, variation of model parameters can produce various topological phases and distinct transition processes among them, and can driven magnetic transition as well Li18 ; Li18-2 , so interest questions arise, as how the FOVT or valence crossover appears in TKI? what is the relation between FOVT, topological transitions and magnetic transition? can any unrevealed novel transition process takes place in TKI?

This work is devoted to answer above questions. We first study the zero-temperature valence transition in three-dimensional TKI, modeling by spin-orbit coupled PAM with $U_{fc}$ interactions. Similar to other typical KIs, we find the rise of $f$ energy level $\epsilon_{f}$ can induce a rapid decrease of $f$ electron number $n_{f}$, causing a valence crossover of $f$ orbit, then above a critical $U_{fc}$, the valence crossover is shifted to a FOVT from Kondo region to MV. We also find that the $U_{fc}$ interaction has an significant impact on the topological boundaries of TKI. Remarkably, FOVT generally takes place very close to a topological boundary, and with slightly greater $U_{fc}$ than the critical one, FOVT can simultaneously cause a first-order topological transition, which should be clarified more rigorously in future studies. We also propose a thermal FOVT of TKI from MV to Kondo region by rising temperature in a narrow parameter regime, simultaneously with a first-order topological transition. Furthermore, we find the effect of $U_{fc}$ interaction leads to a strong variation of $f$ valence near the continuous antiferromagnetic (AF) transition in TKI, and further enhancement of $U_{fc}$ can drive the AF transition from continuous one to first order one, meanwhile, the continuous valence variation is shifted to a FOVT at the AF boundary. In a narrow $U_{fc}$ window, we find a gradual approaching then convergence between paramagnetic (PM) FOVT and FOVT-associated first-order AF transition on $U_{fc}$-$\epsilon_{f}$ plane. In addition, such FOVT-associated first-order AF transition can also be generated by increased temperature in some parameter regime. The PM FOVT, two classes of AF transitions and the topological transitions are all summarized in a global phase diagram.

Our work provides new insight into the exotic transition precesses in TKI, and can be used to qualitatively understand the observed thermal valence variation as well as the valence change around the magnetic transition in SmB6, other TKIs and heavy-fermion systems.

II valence transition and topological transitions

We use the spin-1/2 half-filled PAM in cubic lattice with a spin-orbit coupled $c$-$f$ hybridization to describe TKI Dzero12 ; Legner14 . This model is adopted frequently in the literature and successfully reveals the topological aspects of TKI Werner14 ; Alexandrov15 ; Li18 ; Peters18 . Besides, the on-site $c$-$f$ Coulomb interaction $U_{fc}$ is the crucial driving force of FOVT in heavy-fermion systems thus should be included Goltsev01 ; Watanabe08 ; Watanabe09 . The model Hamiltonian of PAM reads:

[TABLE]

in which half-filling of total electrons $n_{t}=n_{c}+n_{f}=2$ is fixed by adjusting the chemical potential $\mu$, $\vec{\sigma}$ is three-dimensional vector formed by three Pauli matrices. We choose the electron hopping amplitude (EHA) up to next-next-nearest-neighbor, with the two sets of EHAs shown in Tab.1, both retaining an insulating bulk gap Li18 . The spin- and oriental- dependent hybridization between $c$ and $f$ electrons on neighboring sites linking by coordination vector $\vec{l}$ guarantees the TRS of above PAM. The on-site $f$-$f$ Coulomb repulsion $U$ is general large thus is set to infinite for simplicity Alexandrov13 ; Alexandrov15 ; Li18 ; Peters18 , appropriate for applying the standard slave-boson mean-field technique, through which we arrive at the effective Hamiltonian in momentum space as

[TABLE]

in which $\alpha$ and $\beta$ represent spin orientations, $\lambda$ is the lagrange multiplier, $b$ is the mean expectation value of slave bosons obeying the relation $n_{f}=1-b^{2}$. The effective hybridization is renormalized as $\tilde{V}=Vb$. $U_{fc}$ term in Eq. 1 has been decoupled via Hatree-Fock approximation. The renormalized $c$ and $f$ dispersions are $\tilde{\epsilon}^{c}_{\mathbf{k}}=\epsilon^{c}_{\mathbf{k}}+U_{fc}n_{f}-\mu$ and $\tilde{\epsilon}^{f}_{\mathbf{k}}=b^{2}\epsilon^{f}_{\mathbf{k}}+\epsilon_{f}+\lambda+U_{fc}n_{c}-\mu$, respectively, in which the tight-binding dispersions $\epsilon^{c}_{\mathbf{k}}$ and $\epsilon^{f}_{\mathbf{k}}$ are determined by their corresponding EHAs Li18 . $\mathbf{S}_{\mathbf{k}}=(\sin\mathbf{k}\cdot\mathbf{a}_{1},\sin\mathbf{k}\cdot\mathbf{a}_{2},\sin\mathbf{k}\cdot\mathbf{a}_{3})$, Alexandrov15 where $\mathbf{a}_{1}$, $\mathbf{a}_{2}$, $\mathbf{a}_{3}$ are the element vectors of cubic lattice.

The quasi-particle dispersions are derived by diagonalizing the Hamiltonian matrix in Eq.6 (in its modified form) as $E^{\pm}_{\mathbf{k}}=\frac{1}{2}[\tilde{\epsilon}^{c}_{\mathbf{k}}+\tilde{\epsilon}^{f}_{\mathbf{k}}\pm\sqrt{(\tilde{\epsilon}^{c}_{\mathbf{k}}-\tilde{\epsilon}^{f}_{\mathbf{k}})^{2}+4\tilde{V}^{2}S^{2}_{\mathbf{k}}}]$, both doubly degenerated, then the ground state energy is expressed by $E_{g}=N[\lambda(b^{2}-1)-U_{fc}n_{c}n_{f}]+2\sum_{\mathbf{k},\pm}\theta(-E^{\pm}_{\mathbf{k}})E^{\pm}_{\mathbf{k}}$. The mean-field parameters $\lambda$, $b$, and $\mu$ are then solved by saddle point approximation of $E_{g}$ via $\partial E_{g}/\partial\mu=-n_{t}$, $\partial E_{g}/\partial\lambda=0$ and $\partial E_{g}/\partial b=0$, to obtain the following set of equations

[TABLE]

where $\theta(-E^{\pm}_{\mathbf{k}})$ is an step function. The above equations should be solved by numerical iteration.

In SmB6, the itinerant 5d band and local 4f band both locate near the Fermi level Lu13 , causing the emergence of intra-atomic Coulomb repulsion $U_{fc}$. Similarly, in Ce and Yb systems, such interactions are also non-ignorable and play a crucial role in their valence transitions, although the magnitude of $U_{fc}$ may differs from each other Watanabe09 . In this sense, the pressure- or temperature-induced valence variation of SmB6 and other TKIs Mizumaki09 ; Zhou17 ; Wu17 ; Annese06 ; Li14-PRB ; Hagiwara17 should be understood on the basis of $U_{fc}$ interaction. On the other hand, the pressure applied on heavy-fermion systems enhances the hybridization strength $V$, and elevates the local $f$ level $\epsilon_{f}$ as well, therefore, to simulate the valence transition in TKI, we first test the valence change (embodied by variation of $n_{f}$) as a function of $\epsilon_{f}$ under various magnitudes of $U_{fc}$, then study the evolution of valence transition point with $V$.

The numerical results are illustrated in Fig.1 with EHA(II) and $V=1$. With weak $U_{fc}$, $n_{f}$ decreases gradually with ascending $\epsilon_{f}$, showing a valence crossover behavior, see Fig.1a. While $U_{fc}$ is enhanced, $n_{f}$ shows a stronger decline. At a critical value $U_{fc}=4.31$, $n_{f}$ drops sharply at $\epsilon_{f}=-2.89$, indicating the emergence of a quantum critical point (QCP), see Fig.1b. While $U_{fc}$ is further increased, $n_{f}$ shows an abrupt jump from $n_{f}\sim 1$ to $n_{f}\ll 1$ at a critical $\epsilon_{f}$, generating a FOVT from Kondo region in which local moments weakly hybridize with $c$ electrons, to MV region in which $c$ and $f$ electrons are strongly coupled, see Fig.1c-d. The FOVT is generated by strong $U_{fc}$ interaction which forces the $f$ electrons to pour into $c$ band when the effective $f$ level is suddenly lifted during the ascent of $\epsilon_{f}$.

Previous works have shown that the variation of $\epsilon_{f}$ or $V$ can produce successive topological transitions driven by closing and reopening of bulk insulating gap in TKI Tran12 ; Legner14 ; Li18-2 , and through present calculations, we find that these topological boundaries can be shifted sensitively by $U_{fc}$ towards lower $\epsilon_{f}$ direction, see the black dashed lines in Fig.1a-c and blue dashed lines in Fig.2. Remarkably, attributing to the rapid variation of bulk gap around the QCP of FOVT, the QCP usually locates very close to a certain topological transition, see Fig.1b. Upon further increased $U_{fc}$ from the QCP, the topological boundary first approaches then finally converges with FOVT, leaving a first-order topological transition characterized by discontinuous change of insulating gap. The unconventional first-order topological transition is a direct outcome of FOVT in TKI, and should be verified more rigorously in future work.

The QCP, FOVT and topological transitions are summarized on $U_{fc}$-$\epsilon_{f}$ plane from $U_{fc}$=[math] in Fig.2a. In Fig.2b, the phase diagram around the QCP is shown in enlarged pattern, in which the FOVT successively converges with the topological boundaries at two points, one at $U_{fc}$=$4.38$ and $\epsilon_{f}$=$-2.93$, the other at $U_{fc}$=$4.82$ and $\epsilon_{f}$=$-3.19$, separating the FOVT into three segments: (1) when 4.31$<U_{fc}<$4.38, FOVT occurs near conventional STI${}_{\bar{\Gamma}\bar{X}}$-WTI${}_{\bar{X}}$ transition ( the subscripts denote the Dirac points on the surface Brillouin zone Li18-2 ); (2) when 4.38$<U_{fc}<$4.82, FOVT occurs simultaneously with first-order STI${}_{\bar{\Gamma}\bar{X}}$-WTI${}_{\bar{X}}$ transition; (3) when $U_{fc}>$4.82, a first-order STI${}_{\bar{\Gamma}\bar{X}}$-STI${}_{\bar{M}}$ transition takes place at FOVT. Below the QCP, namely when $U_{fc}<$4.31, $U_{fc}$ is insufficient to produce FOVT, but rather a valence crossover between Kondo region and MV. Therefore, whether a heavy-fermion system undergoes a FOVT mainly depends on the strength of $U_{fc}$, e.g., in SmB6 and SmS, the pressure- and temperature- induced valence shift in PM phase seem like a crossover behavior Butch16 ; Mizumaki09 ; Zhou17 , which may imply a weak $U_{fc}$. It is also found that the critical $U_{fc}$ of QCP mainly depends on the hybridization strength $V$. Due to the enhanced coherence between localized and itinerant electrons upon the increase of $V$, stronger $U_{fc}$ is required to develop a QCP of valence transition, and the critical $\epsilon_{f}$ is pushed to deeper direction, see Fig.2c.

III thermal valence transitions

The ground-state FOVT discussed in above section may be experimentally realizable through external pressure, e.g. in Ce and Yb systems Felner85 ; Mito03 ; Park06 , however, the alteration of external factors will lead to a series of changes for model parameters in PAM which are hard to track, hence it is difficult to compare above theoretical results with experiments. Therefore, in this section, we turn to the possible valence transition caused by temperature variation in TKI, which can be more clearly interpreted theoretically. Such ”thermal valence transition” has been detected in Ce and Yb systems decades ago, in which the increase of temperature causes a FOVT from MV to Kondo state Felner85 ; Park06 ; Watanabe09 , but the mechanism and transition process of thermal valence transition are still lacking of theoretical verification. In SmB6 and TKI candidates such as golden SmS, YbB6 and YbB12, the valences of Sm and Yb both increase with temperature at ambient pressure, and in some systems, the valence even reaches the saturated value 3 Annese06 ; Li14-PRB ; Mizumaki09 ; Wu17 ; Butch16 ; Hagiwara17 , but it seems no discontinuous valence shift was observed in PM phase. We expect that the thermal valence behavior in TKI can be dramatically alternated under different magnitudes of pressure, since the pressure applied can affect the interaction strengths in heavy-fermion systems and lead to significant change of valence in the ground state Annese06 ; Butch16 ; Zhou17 ; Hagiwara17 .

At finite temperatures, the quasi-particle spectrums share the same formulas as those of the ground state, except that the mean-field parameters are now temperature-dependent, and are determined by the saddle-point equations derived from the free energy $F=N[\lambda(b^{2}-1)-U_{fc}n_{c}n_{f}]-2T\sum_{\mathbf{k},\pm}\ln(1+e^{-E^{\pm}_{\mathbf{k}}/T})$ by $\partial F/\partial\mu=-n_{t}$, $\partial F/\partial\lambda=0$ and $\partial F/\partial b=0$, giving rise to the equation set the same as the ground state one (Eq.7), except that now the step function $\theta(-E^{\pm}_{\mathbf{k}})$ should be replaced by Fermi distribution function $f^{\pm}_{\mathbf{k}}=1/(1+e^{E^{\pm}_{\mathbf{k}}/T})$ at non-zero temperatures.

The $T$-$\epsilon_{f}$ phase diagrams are given in Fig.3a and 3b, with two magnitudes of $U_{fc}=4.1$ and $4.4$, which are slightly smaller and greater than the QCP value $U_{fc}=4.31$, respectively. As temperature increases, $b$ is reduced gradually, but only when $\epsilon_{f}$ exceeds a certain value (-0.6 and -0.525 for $U_{fc}=4.1$, $4.4$, respectively), $b$ (consequently $\tilde{V}$) gradually vanishes at a critical temperature, results in an second-order insulator-metal transition from heavy-fermion MV insulating state to a decoupled metallic state (DM) in which $n_{f}=1$ with completely localized $f$ electrons. Surprisingly, when $\epsilon_{f}$ is further reduced, the MV-DM transition becomes first-order, indicating a thermal FOVT, at which $b$ and $n_{f}$ undergo an abrupt jump to 0 and 1, respectively, Such discontinuous thermal decoupling of heavy-fermion state and MV-DM thermal FOVT accompanied by an insulator-metal transition deserves further verification.

Remarkably, when $U_{fc}$ is greater than that of the QCP, we find that with $\epsilon_{f}$ slightly above that triggers the zero-temperature FOVT, there are two successive FOVTs with increasing temperature, first MV-to-Kondo FOVT, then Kondo-to-DM FOVT, see Fig.4, with the transition boundaries on $T$-$\epsilon_{f}$ plane shown in the inset of Fig.3b. Similar to zero-temperature case, the FOVT between MV and Kondo states also holds a first-order WTI${}_{\bar{X}}$-STI${}_{\bar{\Gamma}\bar{X}}$ topological transition. As $\epsilon_{f}$ rises, the Kondo region gradually shrinks, then two thermal FOVTs converge, leaving only a MV-DM transition. Moreover, with $\epsilon_{f}$ lower than that of zero-temperature FOVT, only Kondo-DM FOVT occurs with increased temperature. For $U_{fc}$ less than the QCP value, no thermal MV-Kondo FOVT appears, since it is insufficient to support zero-temperature FOVT between MV and Kondo states.

In Ce systems, pressure can suppress the thermal FOVT, forces it to terminate at a CEP, leading to a thermal valence crossover behavior Gschneidner78, this feature corresponds to a shift of thermal behavior from first-order to continuous one. In this sense, since pressure elevates $\epsilon_{f}$, the $\epsilon_{f}$-driven order change of thermal transitions may explain the appearance of CEP in Ce systems, which is marked by the red dots in Fig.3. In contrast to Ce systems, for SmB6 and TKI candidates SmS, YbB6 and YbB12, the Sm and Yb valences in PM phases seem to increase continuously with temperature Annese06 ; Li14-PRB ; Mizumaki09 ; Wu17 ; Butch16 ; Hagiwara17 , showing no clear signal of thermal FOVT. The absence of thermal FOVT in these TKIs may be ascribed to their weak $U_{fc}$ interactions or their high $f$ level $\epsilon_{f}$ which exceeds the CEP in Fig.3, leading to a continuous valence increase with temperature. Previous works suggested that magnetic field can shift the CEP of FOVT in Ce and Yb systems towards lower $U_{fc}$ and higher $\epsilon_{f}$ Watanabe09 ; Watanabe09 , similarly, we expect magnetic field can also shift the CEP of TKI to the right side on $T$-$\epsilon_{f}$ plane in Fig.3, and consequently recover a thermal FOVT in TKI under ambient pressure, which requires experimental verification.

IV valence variation around the magnetic transition

The high-pressure experiments of SmB6 display a first-order transition to magnetically ordered state at 6$\sim$8 Gpa Barla05 ; Derr06 , around which the Sm valence shows a rapid increase Zhou17 , afterwards, the Sm valence in the magnetic phase increases smoothly with pressure then approaches pure trivalent above 10 Gpa. For TKI candidate golden SmS Li14-PRB , the pressure-induced magnetic transition at about 2 Gpa also exhibits first-order character Barla04 , but around this transition, the valence increase of Sm in SmS is much sharper than in SmB6 Butch16 . In YbInCu4, with enhanced pressure, FOVT reaches the magnetic boundary at low temperatures Immer97 ; Mito03 , implying that the pressure-induced magnetic transition can hold a FOVT simultaneously. The active valence change at and around the magnetic transition indicates an intimate relation between these two transitions in TKI and in other heavy-fermion systems.

In order to study the valence variation around the magnetic transition in TKI, which is mostly likely an antiferromagnetic (AF) transition Peters18 ; Chang18 , we adopt Kotliar-Ruckenstein (K-R) slave-boson mean-filed method Kotliar86 ; Yang93 ; Sun93 ; Sun95 . This technique produces very close results of TKI to those of the conventional slave-boson approach, and has the advantage to include the magnetic order conveniently Li18 ; Li18-2 . The main feature of K-R representation for AF phase is that each $f$- creation and annihilation operator in $c$-$f$ hybridization and $f$-$f$ hopping is multiplied by a factor $Z_{+}$ or $Z_{-}$ with $Z_{\pm}=\sqrt{\frac{2(1-n_{f})}{2-n_{f}\mp m_{f}}}$, depending on the spin and sublattice of the $f$ operator, in which $m_{f}$ is the staggered magnetization of $f$ electrons. Secondly, $\epsilon_{f}$ is shifted by a parameter $\eta$. $Z_{\pm}$ and $\eta$ are analogue to $b$ and $\lambda$ in conventional slave-boson method, respectively. The K-R mean-field treatment leads to the following effective Hamiltonian

[TABLE]

in which the summation of $\mathbf{k}$ is restricted to the magnetic Brillouin zone (MBZ), $h$ is a staggered parameter associated with $m_{f}$, $\Psi^{\dagger}_{\mathbf{k}}$=$(d^{\dagger}_{\mathbf{k}A\uparrow}$,$d^{\dagger}_{\mathbf{k}A\downarrow}$, $d^{\dagger}_{\mathbf{k}B\uparrow}$,$d^{\dagger}_{\mathbf{k}B\downarrow}$, $f^{\dagger}_{\mathbf{k}A\uparrow}$,$f^{\dagger}_{\mathbf{k}A\downarrow}$, $f^{\dagger}_{\mathbf{k}B\uparrow}$,$f^{\dagger}_{\mathbf{k}B\downarrow})$ is eight-component creation operator for $c$ and $f$ electrons in sublattice and spin spaces, the Hamiltonian matrix is given by

[TABLE]

in which

[TABLE]

hybridizes $c$ and $f$ electrons on neighboring sites.

[TABLE]

[TABLE]

are the sublattice versions of tight-binding Hamiltonian for $c$ and $f$, respectively, in which the $f$ hopping amplitudes are renormalized by $Z_{\pm}$. $I_{2}$ is a two-order unit matrix, $\Gamma_{\mathbf{k}}=s_{x}-\textrm{i}s_{y}$, $\gamma_{\mathbf{k}}=-4(c_{x}c_{y}+c_{x}c_{z}+c_{y}c_{z})$, $e^{\pm}_{\mathbf{k}}=\epsilon_{f}+\eta+U_{fc}n_{c}-\mu\mp h+t^{\prime}_{f}Z^{2}_{\pm}\gamma_{\mathbf{k}}$, $u^{c(f)}_{\mathbf{k}}=t_{c(f)}\lambda_{\mathbf{k}}+t^{\prime\prime}_{c(f)}g_{\mathbf{k}}$ with $\lambda_{\mathbf{k}}=-2(c_{x}+c_{y}+c_{z})$ and $g_{\mathbf{k}}=-8c_{x}c_{y}c_{z}$, where we denote $c_{i}=\cos k_{i}$ and $s_{i}=\sin k_{i}$ ($i=x,y,z$) for simplicity.

The free energy can be expressed as the expectation value of the effective Hamiltonian by

[TABLE]

where subscripts $n$ and $m$ are row or column numbers. Use the unitary transformation matrix $\mathrm{U}_{\mathbf{k}}$ extracted from numerical diagonalization of $\mathbf{H}_{\mathbf{k}}$, we have

[TABLE]

where $f^{(i)}_{\mathbf{k}}=1/(1+e^{E^{(i)}_{\mathbf{k}}/T})$ is the Fermi distribution of quasi-particles, and equals the step function $\theta(-E^{(i)}_{\mathbf{k}})$ at zero temperature. The set of equations determining the parameters $n_{f}$, $m_{f}$, $h$, $\eta$, $\mu$ can be obtained through the zero point of the derivation of free energy (Eq.25) with respect to them, by using the formulas of matrix elements $(\mathbf{H}_{\mathbf{k}})_{nm}$ in Eq.11 Li18 .

In order to give a global phase diagram including both PM and AF phases through K-R solution, we also solve the K-R saddle-point equations for $n_{f}$, $\eta$ and $\mu$ in PM phase, which are similar to those in section II and III:

[TABLE]

in which $Z=\sqrt{\frac{2(1-n_{f})}{2-n_{f}}}$, $\tilde{\epsilon}^{f}_{\mathbf{k}}=Z^{2}\epsilon^{f}_{\mathbf{k}}+\epsilon_{f}+\eta+U_{fc}n_{c}-\mu$, $f^{\pm}_{\mathbf{k}}=1/(1+e^{E^{\pm}_{\mathbf{k}}/T})$ for finite temperatures and equals the step function $\theta(-E^{\pm}_{\mathbf{k}})$ at zero temperature, where $E^{\pm}_{\mathbf{k}}=\frac{1}{2}[\tilde{\epsilon}^{c}_{\mathbf{k}}+\tilde{\epsilon}^{f}_{\mathbf{k}}\pm\sqrt{(\tilde{\epsilon}^{c}_{\mathbf{k}}-\tilde{\epsilon}^{f}_{\mathbf{k}})^{2}+4V^{2}Z^{2}S^{2}_{\mathbf{k}}}]$

We first study the zero-temperature phase diagram on $U_{fc}$-$\epsilon_{f}$ plane. For convenience, we choose EHA(I) in Tab.1, set various values of $U_{fc}$ from zero, then calculate the corresponding evolution of $m_{f}$ and $n_{f}$ vs $\epsilon_{f}$ to locate the AF transitions. Above the AF boundaries, the evolution of PM phases is computed via Eq.27 to determine the FOVT and topological boundaries. The results are shown with increasing $U_{fc}$ from Fig.5a to 5f. For small $U_{fc}$, $m_{f}$ variation indicates an second-order magnetic transition, at which $n_{f}$ changes continuously with a kink; when $U_{fc}$ exceeds 2.92, we find a first-order magnetic transition at which $n_{f}$ shows an abrupt jump, indicating a FOVT-associated first-order AF transition; in the PM phase with $\epsilon_{f}$ above the first-order AF boundary, $n_{f}$ decreases rapidly with $\epsilon_{f}$, and when $U_{fc}>$3.906 gives rise to a FOVT at a critical $\epsilon_{f}$ above the first-order AF transition; with further increased $U_{fc}$, the two transitions approach each other and then converge, leaving a large $n_{f}$ jump at the first-order AF transition.

The resulting ground-state phase diagram on $U_{fc}$-$\epsilon_{f}$ plane is summarized in Fig.6, including the topological transitions in PM phase. In Fig.6a, it can be clearly seen that a critical point C1 at ($U_{fc}=2.92,\epsilon_{f}=-1.92$) separates the AF boundaries into continuous part and first-order part, which is below and above C1, respectively. The FOVT in PM phase is denoted by blue solid lines in Fig.6a, and the enlarged view near the QCP at ($U_{fc}=3.906,\epsilon_{f}=-2.348$) is demonstrated in Fig.6c, in which the relative position between FOVT and topological boundaries behaves similarly to Fig.2b. Here we should point out that in K-R solution, the critical $U_{fc}$ and $\epsilon_{f}$ for the QCP of FOVT in PM phase is slightly greater and lower than the conventional slave-boson solution, respectively, meanwhile the FOVT and topological transition processes in PM phase remain essential the same. The interplay between first-order AF transition and PM FOVT is clearly illustrated in Fig.6b: with increasing $U_{fc}$ from QCP, the two boundaries approach each other and finally converge at C2 ($U_{fc}=4.01,\epsilon_{f}=-2.45$), leading to an single first-order AF boundary accompanied by a FOVT with significant valence jump. In addition, the coexistence of FOVT-associated AF transition and PM FOVT on $\epsilon_{f}$ axis only occurs in a narrow $U_{fc}$ regime from 3.906 to 4.01. Therefore, the FOVT-associated first-order AF transitions are also classified into two classes: one with weak valence shift below C2, and the other with strong valence jump above C2. Consequently, the order of AF transition, as well as the ”strength” of first-order AF transition both depend on the magnitude of $U_{fc}$, in this sense, the weak first-order magnetic transitions in pressured SmB6 and SmS may be ascribed to their relatively small $d$-$f$ intra-atomic Coulomb repulsion as analysed in above sections Barla05 ; Derr06 ; Barla04 , while the remarkable valence change at the magnetic transition in YbInCu4 may be attributed to its stronger $U_{fc}$ Immer97 ; Mito03 . It is worth noting that the critical $U_{fc}=2.92$ of C1 which triggers a first-order AF transition is much smaller than that drives a FOVT in PM phase ($U_{fc}=3.906$ of QCP), explaining why Sm valence varies continuously with temperature at ambient pressure in SmB6 and SmS, meanwhile first-order AF transitions appear under high pressure.

Then we study the thermal phase transitions on $T$-$\epsilon_{f}$ plane, the results are displayed in Fig.7. For $U_{fc}$ smaller than the critical point C1, the zero-temperature AF transition with $\epsilon_{f}$ is continuous, while $\epsilon_{f}$ descends, the thermal AF transition is shifted to first-order with valence jump (see Fig.7a), implying that the critical point C1 separating these two types of AF transitions is temperature dependent and is pushed by increasing temperature towards lower $\epsilon_{f}$ and weaker $U_{fc}$. By comparison, for $U_{fc}$ stronger than zero-temperature C1, thermal AF transition remains first-order, because $U_{fc}$ keeps greater than C1 when temperature rises. For temperature above the AF transitions, $Z$ factor reduces with temperature, leading to the increase of $n_{f}$ then finally a decoupling of heavy-fermion state to DM state with $n_{f}=1$ at a critical temperature, giving rise to HF-DM transition. By contrast to AF transitions, the order of this HF-DM transition is only weakly temperature-dependent and mainly depends on $U_{fc}$: when $U_{fc}$ is less or larger than zero-temperature C1 in Fig.6a, HF-DM transition is continuous or first-order, respectively, and the latter coexists a FOVT. As $\epsilon_{f}$ descends, the AF boundary and HF-DM boundary gradually converge, leaving a first-order AF-DM transition with abrupt valence shift. The thermal first-order AF transition coexisting a FOVT deserves more rigorous theoretical and experimental examinations.

V conclusion and discussion

To summarize, we have proposed a series of exotic valence transition processes in TKI, driven by the effect of on-site Coulomb repulsion $U_{fc}$ between conduction and local electrons in the PAM. For the ground state, we located a QCP on the $U_{fc}$-$\epsilon_{f}$ plane, above which a FOVT boundary separates the Kondo state with MV state, and holds an abrupt valence jump upon ascending $\epsilon_{f}$. The QCP is very close to a topological transition, and the FOVT boundary gradually approaches then converges with the topological boundary, leading to a discontinuous topological transition. Near the parameter regime of the zero-temperature FOVT, increase in temperature can lead to a series of thermal FOVTs among the Kondo state, MV state and DM state. In addition, we found a CEP on $T$-$\epsilon_{f}$ plane which terminates the thermal FOVT from heavy-fermion state to DM state, and shift the thermal decoupling of heavy-fermion state into continuous one under higher $\epsilon_{f}$ above the CEP. Further descent of $\epsilon_{f}$ can generate an AF order in TKI, and above a critical $U_{fc}$ which is considerably smaller than that of the QCP which triggers a PM FOVT, the $\epsilon_{f}$-driven continuous AF transition is shifted to first-order, accompanied by a FOVT. Further increasing of $U_{fc}$ can push the first-order AF transition to approach then to join the PM FOVT boundary, giving rise to a first-order AF transition with large valence shift. The thermal AF transitions can also be classified into continuous one or first-order one, depending mainly on the magnitude of $U_{fc}$. The distinct valence-variation processes with temperature and pressure observed in TKI candidates SmB6, golden SmS and other heavy-fermion compounds, as well as their valence behaviors near the magnetic transitions can be qualitatively understood by our results in terms of their different magnitudes of $U_{fc}$. This work has used an simplified PAM with single conduction and $f$ band to describe TKI, actually in real cubic TKI systems such as SmB6, dominant hybridization channel involves spin-degenerated 5$d_{x^{2}-y^{2}}$, 5$d_{3z^{2}-r^{2}}$ orbits, and $\Gamma^{(1)}_{8}$, $\Gamma^{(2)}_{8}$ 4$f$ quartet, resulting in much complicated tight-binding and hybridization terms Takimoto11 ; Alexandrov13 ; Roy14 . Although the multiple parameters can all influence the detailed valence-variation process in TKI, we believe that the intra-atomic Coulomb interaction $U_{fc}$ between $d$ and $f$ orbits still provides a crucial factor controlling the valence transition, and the magnitudes of $U_{fc}$ in TKI candidates may be examined through first-principle simulations Lu13 .

Acknowledgements.

H. Li is supported by NSFC (No. 11764010) and Guangxi Natural Science Foundation (No. 2017GXNSFAA198169). Y. Liu and H. F. Song thank the Science Challenge Project (Grant No. TZ2018002). Y. Liu is also supported by Open Research Fund Program of the State Key Laboratory of Low-Dimensional Quantum Physics (No. KF201702) and the SPC-Lab Research Fund (No. XKFZ201605). Z. Y. Wang is supported by NSFC (No. 11564008). X. J. Zheng is supported by NSFC (No. 11704084) and Guangxi NSF (No. 2017GXNSFBA198115). X. Ming is supported by NSFC (No. 11864008) and Guangxi NSF (No. 2018GXNSFAA138185).