Mott transition and electronic excitation of the Gutzwiller wavefunction

The $t$-$J$ model is defined by the following Hamiltonian:

where ${\tilde{c}}_{i,\sigma}$ denotes the annihilation operator of an electron with spin $\sigma$ at site $i$ under the constraint of no double occupancy, and $\langle i,j\rangle$ means that sites $i$ and $j$ are nearest neighbors. Here, $n_{i,\sigma}$ and ${\bm{S}}_{i}$ denote the number operator with spin $\sigma$ and the spin operator at site $i$, respectively, and $n_{i}=\sum_{\sigma}n_{i,\sigma}$. In this paper, we consider the $t$-$J$ models on a chain ($t_{i,j}=t$, $J_{i,j}=J$), planar square lattice ($t_{i,j}=t$, $J_{i,j}=J$), ladder ($t_{i,j}=t$ and $J_{i,j}=J$ in the legs; $t_{i,j}=t_{\perp}$ and $J_{i,j}=J_{\perp}$ in the rungs), and bilayer square lattice ($t_{i,j}=t$ and $J_{i,j}=J$ in the layers; $t_{i,j}=t_{\perp}$ and $J_{i,j}=J_{\perp}$ between the layers).

Hereafter, the numbers of sites and electrons are denoted by $N_{\rm s}$ and $N_{\rm e}$, respectively. The electron density and doping concentration are defined as $n=N_{\rm e}/N_{\rm s}$ and $\delta=1-n$, respectively. At half filling, $n=1$ and $\delta=0$. For a ladder and bilayer, the momentum in the interchain or interlayer direction is denoted by $k_{\perp}$. The momenta on a ladder and bilayer are represented as $(k_{x},k_{\perp})$ and $(k_{x},k_{y},k_{\perp})$, respectively. The shorthand notations ${\bm{0}}$ and ${\bm{\pi}}$ are used for $(0,0)$ and $(\pi,\pi)$, respectively. As a compact notation, $k_{x}$ and $k_{y}$ are sometimes denoted by $k_{1}$ and $k_{2}$, respectively.

In this paper, the numerical results for $J/t=0.5$ on a chain and plane; $J/t=0.25$, $t_{\perp}/t=2$, and $J_{\perp}/t=1$ on a ladder; and $J/t=0.25$, $t_{\perp}/t=4$, and $J_{\perp}/t=4$ on a bilayer with $t>0$ are presented. The calculations were performed under periodic boundary conditions on clusters of $N_{\rm s}=120$ for the chain and ladder, $N_{\rm s}=400$ for the plane, and $N_{\rm s}=200$ for the bilayer. Typically, several millions of samples were generated following several hundreds of sweeps in the Monte Carlo calculations.

In this paper, the ground state is assumed to be the Gutzwiller wavefunction $|\Phi\rangle$, defined as GutzwillerWF

where $c^{\dagger}_{{\bm{k}},\sigma}$ denotes the creation operator of an electron with momentum ${\bm{k}}$ and spin $\sigma$, and $|0\rangle$ represents the vacuum. Here, ${\rm P}_{\rm d}$ denotes the projection operator that forbids double occupancy. The excitation energy $\varepsilon({\bm{k}})$ and spectral weight $W({\bm{k}})$ of the electron-addition excited state ${\tilde{c}}^{\dagger}_{{\bm{k}},\sigma}|\Phi\rangle$ averaged with respect to spin are obtained as follows:

where ${\tilde{c}}_{{\bm{k}},\sigma}$ denotes $c_{{\bm{k}},\sigma}$ with the constraint of no double occupancy. The expectation value of an operator ${\cal O}$ by $|\Phi\rangle$ can be evaluated as the sample average of weight $w_{i}$ for configuration $|i\rangle$ generated with probability $p_{i}$ using a Monte Carlo method Ceperley ; GrosVMC :

where

It should be noted that ${\tilde{c}}^{\dagger}_{{\bm{k}},\sigma}$ and ${\rm P_{d}}$ commute:

because ${\tilde{c}}^{\dagger}_{i,\sigma}{\rm P}_{\rm d}|\alpha\rangle_{i}={\rm P}_{\rm d}c^{\dagger}_{i,\sigma}|\alpha\rangle_{i}$, where $c^{\dagger}_{i,\sigma}$ and $|\alpha\rangle_{i}$ denote the creation operator of an electron with spin $\sigma$ and a state (0, $\uparrow$, $\downarrow$, or $\uparrow\downarrow$) at site $i$, respectively. Thus, the first term on the right-hand side of Eq. (3) can be calculated as the energy of the Gutzwiller wavefunction where an electron with momentum ${\bm{k}}$ and spin $\sigma$ is added to the Fermi sea prior to projection. This can significantly reduce the computational complexity of the electron-addition energy. On the other hand, ${\tilde{c}}_{{\bm{k}},\sigma}{\rm P_{d}}|{\rm FS}\rangle\neq{\rm P_{d}}c_{{\bm{k}},\sigma}|{\rm FS}\rangle$ in general.

The chemical potential $\mu$ at $N_{\rm e}=m$ can be calculated as follows:

where $E_{m\pm 1}$ denotes the ground-state energy at $N_{\rm e}=m\pm 1$ for $\mu=0$. In this paper, because the ground state is assumed to be expressed as the Gutzwiller wavefunction, $\mu$ can be calculated using the lowest energies of the Gutzwiller wavefunction with an electron added to ($E_{m+1}$) and removed from ($E_{m-1}$) the Fermi sea at $N_{\rm e}=m$ prior to projection. Similarly, the Fermi momentum ${\bm{k}}_{\rm F}$ can be determined as the momentum where the dispersion relation of the Gutzwiller wavefunction with an electron added or removed prior to projection crosses the Fermi level.

The spectral function is defined as

where $\varepsilon_{l}$ denotes the excitation energy of the normalized eigenstate $|l\rangle$ from the ground state $|{\rm GS}\rangle$. In this paper, the single-mode approximation is employed, where the electron-addition spectral function [$A({\bm{k}},\omega)$ for $\omega>0$] is approximated as

If the excitation is essentially represented by a dominant mode, the single-mode approximation can capture the essential excitation feature. It has been shown that the electron-addition excitation ($\omega>0$) can be effectively represented by a single mode in the $t$-$J$ models near half filling (at each $k_{\perp}$ for the ladder and bilayer) [Figs. 1(c), 1(f), 1(i), and 1(l)] Kohno1DtJ ; Kohno2DtJ ; KohnoDIS . However, if the spectral weight is spread over a wide range of $\omega$ at each ${\bm{k}}$, as observed in the electron-removal excitation ($\omega<0$) in the $t$-$J$ and Hubbard models [Figs. 1(c), 1(f), and 1(i)] KohnoRPP ; Kohno1DHub ; Kohno2DHub ; Kohno1DtJ ; Kohno2DtJ ; KohnoDIS ; KohnoHubLadder ; KohnoAF ; KohnoSpin , the single-mode approximation exhibits a single peak at the weighted mean value of $\omega$. The mode for $\omega<0$ in this approximation can exhibit an excitation gap even if the true excitation is gapless. Thus, in this paper, we only consider the electron-addition excitation ($\omega>0$), which exhibits a significant characteristic toward the Mott transition.

At zero electron density ($n=0$), the spectral function for an added electron is the same as that in a noninteracting system [Figs. 1(a), 1(d), 1(g), and 1(j)], because no other electrons exist. As the electron density increases (i.e., the chemical potential is increased), the Fermi level moves into the (lower) band and the spectral function in the single-mode approximation [$A^{\rm s}({\bm{k}},\omega)$; Eq. (11)] for $\omega>0$ becomes that indicated in Figs. 1(b), 1(e), 1(h), and 1(k).

The validity of the results is confirmed by their comparison with the results obtained using the non-Abelian dynamical density-matrix renormalization-group (DDMRG) method for the chain [Fig. 1(c)] Kohno1DtJ and ladder [Fig. 1(f)] KohnoDIS , those obtained using the cluster perturbation theory (CPT) for the plane [Fig. 1(i)] Kohno2DtJ , and those of the effective theory near half filling for $t_{\perp}\gg t$ and $J_{\perp}\gg J$ for the bilayer [Fig. 1(l)] KohnoDIS . In the effective theory KohnoDIS , the dispersion relation at $k_{\perp}=\pi$ for $\omega>0$ is obtained as follows:

and that of the other modes can be expressed as

on the ladder ($d=1$) and bilayer ($d=2$), where the $x$ and $y$ components of ${\bm{k}}_{\rm F}$ are denoted by $k_{{\rm F}1}$ and $k_{{\rm F}2}$, respectively. In Fig. 1(l), the spectral weight at each ${\bm{k}}$ is approximated as $1.5\delta$ for $\omega>0$ at $k_{\perp}=\pi$, $0.5-\delta$ for $\omega<0$ at $k_{\perp}=\pi$, and 0.5 at $k_{\perp}=0$. The contributions from the continua (multiparticle processes) are neglected.

In the following sections, the changes in the dispersion relation and spectral weight with the electron density are discussed.

At $n=0$, because the added electron behaves as a noninteracting electron, the dispersion relation can be expressed as

on the chain ($d=1$) and plane ($d=2$), and as

on the ladder ($d=1$) and bilayer ($d=2$), where the Fermi level is set to the bottom of the (lower) band [Figs. 1(a), 1(d), 1(g), and 1(j)].

As illustrated in Figs. 2(a)–2(f), the dispersion relation of the electron-addition excitation [$\omega=\varepsilon({\bm{k}})$] changes continuously as the electron density increases from $n=0$ [Eqs. (14) and (15)]. To clarify the electron-density dependence, Figs. 3(a), 3(d), 3(g), and 3(j) display the characteristic energies: $\varepsilon(\pi)$ on the chain, $\varepsilon({\bm{\pi}})$ on the plane, $\Delta\varepsilon$ and $\varepsilon_{\rm c}$ on the ladder and bilayer. Here, $\Delta\varepsilon$ and $\varepsilon_{\rm c}$ denote the bandwidth and band center, respectively, which are defined as

where ${\bm{Q}}_{\rm min}$ and ${\bm{Q}}_{\rm max}$ represent $0$ and $\pi$ on the ladder, and ${\bm{0}}$ and ${\bm{\pi}}$ on the bilayer.

For the chain and plane, the dispersion relation continues to disperse even in the limit of $n\rightarrow 1$ [Figs. 2(a), 2(d), 3(a), and 3(d)]. This implies that the mode for $\omega>0$, which is continuously deformed from that of a noninteracting electron at $n=0$, does not become flat toward the Mott transition, in contrast to the conventional single-particle picture of the effective-mass divergence.

For the ladder and bilayer, the dispersion relation at $k_{\perp}=0$ shrinks to $\omega\rightarrow 0$ at $k_{x}=\pi$ and $(k_{x},k_{y})={\bm{\pi}}$, respectively, in the limit of $n\rightarrow 1$ [Figs. 2(b) and 2(e)], whereas the dispersion relation at $k_{\perp}=\pi$ continues to disperse [Figs. 2(c), 2(f), 3(g), and 3(j)]. Although these features are similar to those of the transition from a metal to a band insulator, the spectral weight at $k_{\perp}=\pi$ gradually disappears toward the Mott transition, in contrast to the conventional band picture, as shown in Sec. III.3.

To clarify the nature of the electron-addition excitation in the limit of $n\rightarrow 1$, we consider the excitation at half filling ($n=1$), where the $t$-$J$ model is reduced to the Heisenberg model. For the chain and plane, the dominant spin excitation exhibits the following spin-wave dispersion relation desCloizeaux ; AndersonSW :

on the chain and

for $k_{x}=k_{y}=k$ on the square lattice, where the spin-wave velocities of the Heisenberg models on the chain and square lattice have been obtained as $v_{\rm 1D}=\pi J/2$ [desCloizeaux, ] and $v_{\rm 2D}=1.18(2)\sqrt{2}J$ [Singh, ], respectively. As illustrated in Figs. 3(a) and 3(d), $\varepsilon(\pi)$ and $\varepsilon({\bm{\pi}})$ in the limit of $n\rightarrow 1$ reasonably well approach $e_{\rm 1D}(\pi/2)$ and $e_{\rm 2D}({\bm{\pi}}/2)$ (open red diamonds) on the chain and plane, respectively.

For the ladder and bilayer, the dispersion relation of the spin excitation at $k_{\perp}=\pi$ for $J_{\perp}\gg J$ can effectively be expressed as

on the ladder ($d=1$) and bilayer ($d=2$) KohnoDIS . As indicated in Figs. 3(g) and 3(j), $\Delta\varepsilon$ (solid purple triangles) and $\varepsilon_{\rm c}$ (solid blue circles) in the limit of $n\rightarrow 1$ are reduced to the bandwidth $\Delta e$ (open red triangles) and band center $e_{\rm c}$ (open red circles) of the spin excitation at half filling, respectively, which are defined as

The above results are consistent with the general relationship between the dispersion relation of the spin excitation in a Mott insulator ($N_{\rm e}=N_{\rm s}$) and that of the electron-addition excitation in the small-doping limit ($N_{\rm e}=N_{\rm s}-1$) shown in Ref. [KohnoDIS, ]: spin-excited states in a Mott insulator can emerge in the electron-addition spectrum outside the Fermi surface, exhibiting the magnetic dispersion relation shifted by ${\bm{k}}_{\rm F}$ in the small-doping limit KohnoRPP ; Kohno1DHub ; Kohno2DHub ; Kohno1DtJ ; Kohno2DtJ ; KohnoAF ; KohnoSpin ; KohnoDIS ; KohnoHubLadder . By applying this relationship, the dispersion relation of the electron-addition excitation in the small-doping limit is expected to be

for $\pi/2<k_{x}<3\pi/2$ on the chain [${\bm{k}}_{\rm F}=\pi/2$; Eq. (18)],

for $\pi/2<k<3\pi/2$ along $k_{x}=k_{y}=k$ on the plane [${\bm{k}}_{\rm F}={\bm{\pi}}/2$; Eq. (19)], and Eq. (12) on the ladder at $k_{\perp}=\pi$ [$d=1$; ${\bm{k}}_{\rm F}=(\pi,0)$; Eq. (20)] and bilayer at $k_{\perp}=\pi$ [$d=2$; ${\bm{k}}_{\rm F}=({\bm{\pi}},0)$; Eq. (20)]. The results that are obtained simply by assuming that the ground state is the Gutzwiller wavefunction [Figs. 1(b), 1(e), 1(h), 1(k), 2(a), 2(c), 2(d), 2(f), 3(a), 3(d), 3(g), and 3(j)] agree reasonably well with this behavior [Eqs. (12), (23), and (24); Fig. 1(l)].

At $n=0$, $W({\bm{k}})=1$ because the electron-addition excitation is the same as that of a noninteracting system [Figs. 1(a), 1(d), 1(g), and 1(j)]. As the electron density increases, the spectral weight outside the Fermi surface gradually decreases, as illustrated in Figs. 2(g)–2(l). To clarify the electron-density dependence, Figs. 3(b), 3(e), 3(h), and 3(k) display the characteristic spectral weights: $W(\pi)$ on the chain, $W({\bm{\pi}})$ on the plane, $W(\pi,\pi)$ on the ladder, and $W({\bm{\pi}},\pi)$ on the bilayer. At $n=1$, $W({\bm{k}})=0$ because an electron cannot be added to the ground state with the constraint of no double occupancy. The Hubbard gap can be regarded as infinitely large.

The spectral weights at $k_{\perp}=0$ on the ladder and bilayer remain nonzero even in the limit of $n\rightarrow 1$ [Figs. 2(h) and 2(k)], as in the case of the transition from a metal to a band insulator. However, the spectral weights on the chain, plane, and ladder at $k_{\perp}=\pi$, as well as on the bilayer at $k_{\perp}=\pi$ gradually disappear toward the Mott transition ($n\rightarrow 1$) [Figs. 2(g), 2(i), 2(j), 2(l), 3(b), 3(e), 3(h), and 3(k)].

These results imply the following: For the chain and plane, the dispersing mode crossing the Fermi level (Sec. III.2), which is continuously deformed from that of a noninteracting electron at $n=0$, loses its spectral weight and gradually disappears toward the Mott transition without flattening of the dispersion relation KohnoRPP ; Kohno1DHub ; Kohno2DHub ; Kohno1DtJ ; Kohno2DtJ ; KohnoAF ; KohnoSpin . For the ladder and bilayer, the mode at $k_{\perp}=\pi$, which is continuously deformed from the noninteracting antibonding band ($k_{\perp}=\pi$) at $n=0$, persists as a dispersing mode in the metallic phase (Sec. III.2) but loses its spectral weight and gradually disappears as $n\rightarrow 1$ KohnoDIS ; KohnoHubLadder , contrary to the conventional band picture in which the number of bands is considered to be determined by the number of atomic orbitals in a unit cell AshcroftMermin and invariant with the electron density provided that symmetry breaking does not occur (neither emergence nor disappearance of a band is expected).

The momentum distribution function is defined as

which can be calculated as $n({\bm{k}})=1-n/2-W({\bm{k}})$ owing to the sum rule Stephan_nk . It has been established that $n({\bm{k}})$ of the Gutzwiller wavefunction exhibits a discontinuity at ${\bm{k}}_{\rm F}$ in the metallic phase [Figs. 2(g), 2(h), 2(j), and 2(k)] GutzwillerWF ; MetznerPRL ; MetznerPRB ; YokoyamaShiba1dGW ; Gros1dGW . The value of this discontinuity is called the quasiparticle weight, which is represented by $Z$ in this paper [Figs. 3(c), 3(f), 3(i), and 3(l)]. The volume inside the Fermi surface of the Gutzwiller wavefunction is the same as the noninteracting Fermi sea (cf. Luttinger’s theorem LuttingerTheorem ). The same volume as the noninteracting Fermi sea and $Z\neq 0$ in the metallic phase are usually identified as evidence of a Fermi liquid. In this sense, the Gutzwiller wavefunction can be regarded as a Fermi-liquid-like state.

As illustrated in Figs. 3(c) and 3(f), the quasiparticle weight $Z$ on the chain and plane decreases continuously to zero toward the Mott transition. The Brinkman-Rice picture is based on this behavior: $Z\rightarrow 0$ implies the divergence of the effective mass $m^{*}$, because $m^{*}\propto 1/Z$ in the Fermi-liquid theory, assuming that the renormalization of $m^{*}$ is only due to the $\omega$ dependence of the self-energy BrinkmanRice . If electronic excitation can essentially be represented by the single mode of the Fermi-liquid quasiparticle with $m^{*}\rightarrow\infty$, the Mott transition should be characterized by the flattening of the dispersion relation toward the Mott transition, as is widely believed according to the Brinkman-Rice picture.

In contrast, as discussed in Secs. III.1–III.3, the results on the chain and plane indicate that the mode crossing the Fermi level, which is continuously deformed from a noninteracting band at $n=0$, does not become flat toward the Mott transition, but loses its spectral weight for $\omega>0$, even if the ground state is assumed to be a Fermi-liquid-like state with the same volume as the noninteracting Fermi sea and a nonzero $Z$ that decreases continuously to zero toward the Mott transition.

The results exhibiting $Z\neq 0$ on the chain [Figs. 2(g) and 3(c)] are due to the Gutzwiller wavefunction MetznerPRL ; MetznerPRB . In a 1D system, the low-energy properties are generally described as a Tomonaga-Luttinger liquid HaldaneTLL ; TomonagaTLL ; LuttingerTLL ; MattisLiebTLL where $Z=0$ GiamarchiBook ; EsslerBook . Thus, the picture of $m^{*}\rightarrow\infty$ for the Fermi-liquid quasiparticle is generally inapplicable to 1D systems. Nevertheless, the gradual loss of the spectral weight from the dispersing mode that exhibits the momentum-shifted magnetic dispersion relation in the small-doping limit has been shown in 1D systems Kohno1DHub ; Kohno1DtJ as well as in 2D systems Kohno2DHub ; Kohno2DtJ . This implies that this characteristic is general and fundamental in the Mott transition, regardless of whether the ground state is like a Fermi liquid or not. That is, this characteristic is not highly sensitive to the ground-state properties or dimensionality, but would generally be robust in the Mott transition.

For the ladder and bilayer, $Z$ remains nonzero even in the limit of $n\rightarrow 1$ [Figs. 2(h), 2(k), 3(i), and 3(l)], as in the case of the transition from a metal to a band insulator. Nevertheless, similarly to the cases of the chain and plane, the dispersion relation of the antibonding band is deformed into the momentum-shifted magnetic dispersion relation in the small-doping limit [Figs. 2(c), 2(f), 3(g), and 3(j)]. Furthermore, the spectral weight at $k_{\perp}=\pi$ decreases continuously to zero toward the Mott transition [Figs. 2(i), 2(l), 3(h), and 3(k)] KohnoDIS ; KohnoHubLadder , contrary to the conventional band picture. These results also support the general and fundamental characteristic of the Mott transition.

The characteristic discussed in this paper can also be demonstrated using a model whose ground state is the Gutzwiller wavefunction. It is known that the ground state of the 1D supersymmetric $t$-$J$ model with $1/r^{2}$ interaction ($J/t=2$) is the Gutzwiller wavefunction SUSYGS . The electron-addition spectral function $A^{+}(k_{x},\omega)$ of this model has been obtained analytically SUSYAkw . According to the analytical expression of $A^{+}(k_{x},\omega)$ SUSYAkw , the dominant mode (upper edge of the continuum) that is continuously deformed from the noninteracting band at $n=0$ loses its spectral weight and gradually disappears toward the Mott transition. Its dispersion relation continues to disperse and becomes

for $k_{\rm F}<k_{x}<2\pi-k_{\rm F}$ in the small-doping limit (Fermi momentum $k_{\rm F}\rightarrow\pi/2$), where $e_{\rm HS}(k_{x})$ denotes the dispersion relation of the dominant mode of the spin excitation at half filling (the Haldane-Shastry model) HSSkw :

This clearly demonstrates that the characteristic discussed in this paper can appear even in a system whose ground state is a Fermi-liquid-like state exhibiting gradual disappearance of the quasiparticle weight toward the Mott transition [Fig. 3(c)].

In conventional single-particle pictures, an electronic quasiparticle or a hole is considered as a carrier and the Mott transition is considered to be characterized as one of the following two possibilities ImadaRMP : the divergence of the effective mass $m^{*}\rightarrow\infty$ or the disappearance of the carrier density $n_{\rm c}\rightarrow 0$. The former is based on the Fermi-liquid theory, where interaction effectively makes the electronic quasiparticle heavier. The latter is based on a band picture such as the mean-field approximation for the antiferromagnetic order OverhauserSDW or Hubbard’s decoupling approximation HubbardApprox , where holes in a doped Mott insulator can be regarded as carriers. Discussions on the Mott transition have mostly focused on which picture is more appropriate and intense controversies have arisen, particularly in relation to cuprate high-temperature superconductors ImadaRMP ; DagottoRMP . For the distinction between $m^{*}\rightarrow\infty$ and $n_{\rm c}\rightarrow 0$, the ground-state properties such as the quasiparticle weight $Z$, antiferromagnetic order, and sign of the Hall coefficient are important.

In contrast, the characteristic discussed in this paper is not highly sensitive to the ground-state properties Kohno1DHub ; Kohno2DHub ; Kohno1DtJ ; Kohno2DtJ ; KohnoDIS ; KohnoRPP ; KohnoSpin ; KohnoAF ; KohnoHubLadder . The quasiparticle weight $Z$ or the presence or absence of a spin gap or antiferromagnetic order in a Mott insulator is not significant. In fact, essentially the same characteristic of the Mott transition appears on the square lattice [Figs. 2(d), 2(j), 3(d), and 3(e)] and on the chain [Figs. 2(a), 2(g), 3(a), and 3(b)], although the Gutzwiller wavefunction on a square lattice exhibits an antiferromagnetic long-range order at half filling GWAForderLi ; GWAForderEnt , whereas that on a chain does not GWspinPRL ; GWspinPRB . Instead, the existence of spin excitation in the energy regime that is much lower than the charge gap in a Mott insulator is important for this characteristic. This spin–charge separation can be regarded as a defining factor of a Mott insulator KohnoHubLadder . In fact, in a band insulator, spin–charge separation does not occur; the lowest spin- and charge-excitation energies are the same as the band gap because the excitations are described in terms of electronic single particles KohnoRPP ; KohnoAF .

It should be noted that the spin–charge separation in the metallic phase of a 1D system means that excitations in the low-energy limit are described in terms of independent spin and charge excitations GiamarchiBook ; TakahashiBook ; EsslerBook , rather than electronic quasiparticles. The lowest excitation energies for the spin $\Delta_{\rm s}$ and charge $\Delta_{\rm c}$ of the order of $1/N_{\rm s}$ are different. In a Mott insulator, the spin–charge separation is more robust [$\Delta_{\rm s}\ll\Delta_{\rm c}=O(U)$ for Coulomb repulsion $U\gg t$; $\Delta_{\rm s}\ll\Delta_{\rm c}=\infty$ in the $t$-$J$ (Heisenberg) model] and general, regardless of the dimensionality.

Although there are physical quantities that can distinguish between an insulator and a metal, such as the Drude weight KohnDrudeWeight , the characterization of the Mott transition should reflect a general characteristic of a Mott insulator that can distinguish a Mott insulator from a band insulator KohnoHubLadder . The above-mentioned spin–charge separation in a Mott insulator provides such a characteristic. Because the characteristic of the Mott transition discussed in this paper reflects the spin–charge separation of a Mott insulator, it would be general regardless of the dimensionality Kohno1DHub ; Kohno2DHub ; Kohno1DtJ ; Kohno2DtJ ; KohnoDIS ; KohnoRPP ; KohnoSpin ; KohnoAF ; KohnoHubLadder . Although an antiferromagnetic order may be considered as important in the Mott transition, it is not essential to the Mott transition. This is because not only an antiferromagnetically ordered insulator, but also a spin liquid, which is an insulator exhibiting spin excitation (with or without a spin gap) in the energy regime that is much lower than the charge gap without a magnetic order, is usually regarded as a Mott insulator. The quasiparticle weight $Z$ in the metallic phase or structural instability is not essential to the Mott transition either, because the Mott transition can occur even on a chain with $Z=0$ or without being accompanied by lattice distortion.

The emergence of electronic states in the Hubbard gap upon doping a Mott insulator has been recognized since the early 1990s Eskes ; DagottoDOS , but the interpretations thereof have been controversial. The emergent states have been interpreted as part of the upper Hubbard band that is quickly shifted by doping SakaiImadaPRL ; SakaiImadaPRB , composite-particle states ImadaCofermionPRL ; ImadaCofermionPRB ; PhillipsRMP ; PhillipsRPP , and a spin-polaron shake-off band EderOhtaIPES ; EderOhta2DHub . In these interpretations, the mode of the emergent states is essentially separated by an energy gap from the mode around the Fermi level, even if the spin excitation of a Mott insulator is gapless SakaiImadaPRL ; SakaiImadaPRB ; ImadaCofermionPRL ; ImadaCofermionPRB ; PhillipsRMP ; PhillipsRPP ; EderOhtaIPES ; EderOhta2DHub . In contrast, another interpretation is that the emergent states are essentially the spin-excited states that exhibit the magnetic dispersion relation shifted by ${\bm{k}}_{\rm F}$ in the electronic spectrum Kohno1DHub ; Kohno2DHub ; Kohno1DtJ ; Kohno2DtJ ; KohnoDIS ; KohnoRPP ; KohnoSpin ; KohnoAF ; KohnoHubLadder . If the magnetic excitation of a Mott insulator is gapless, the mode of the emergent electronic states should also be gapless in the small-doping limit.

The behavior of the characteristic mode discussed in this paper can also be understood in the final interpretation above Kohno1DHub ; Kohno2DHub ; Kohno1DtJ ; Kohno2DtJ ; KohnoDIS ; KohnoRPP ; KohnoSpin ; KohnoAF ; KohnoHubLadder , even if a Fermi-liquid-like ground state is assumed [Figs. 1(b), 1(e), 1(h), 1(k), 3(a), 3(d), 3(g), and 3(j); Eqs. (12), (18), (19), (20), (23), and (24)]. When viewed from the low-electron-density side, this mode is continuously deformed from a noninteracting band at $n=0$, gradually losing its spectral weight toward the Mott transition [Figs. 2(a), 2(c), 2(d), 2(f), 2(g), 2(i), 2(j), 2(l), 3(a), 3(b), 3(d), 3(e), 3(g), 3(h), 3(j), and 3(k)], which implies that this mode also has the same origin as a noninteracting band at $n=0$.

The physical picture of this characteristic of the Mott transition has been described as follows Kohno1DHub ; Kohno2DHub ; Kohno1DtJ ; Kohno2DtJ ; KohnoDIS ; KohnoRPP ; KohnoSpin ; KohnoAF ; KohnoHubLadder : From the metallic side, the charge degrees of freedom freeze toward the Mott transition, while the electronic motion is preserved in the spin degrees of freedom. This picture has been derived based on the spectral feature indicating that an electronic mode representing an electronic particle with spin and charge gradually loses its identity (spectral weight) toward the Mott transition, while the dispersion relation is continuously reduced to the magnetic dispersion relation shifted by ${\bm{k}}_{\rm F}$ in the Mott transition. From the insulating side, spin excitation emerges as electronic excitation because the charge character is added by doping. This picture has been derived based on the spectral feature indicating that the electronic excitation in the small-doping limit exhibits the magnetic dispersion relation shifted by ${\bm{k}}_{\rm F}$ (a spin-excited state at ${\bm{k}}={\bm{p}}$ from the ground state with $N_{\rm e}=m$ can overlap with an electron-addition excited state at ${\bm{k}}={\bm{p}}+{\bm{k}}_{\rm F}$ from the ground state with $N_{\rm e}=m-1$ where an electron with ${\bm{k}}={\bm{k}}_{\rm F}$ on the Fermi surface is removed KohnoDIS ). The emergence in the electronic spectrum implies that the excitation not only has a spin character, but also gains a charge character owing to doping (the electronic excitation should have the same quantum numbers as an electron).

The above picture contrasts with conventional single-particle pictures: the electronic quasiparticle becomes extremely heavy and immobile when $m^{*}\rightarrow\infty$, and the number of mobile holes disappears (full filling of electrons) when $n_{\rm c}\rightarrow 0$. In these single-particle pictures, the decoupling of the spin and charge degrees of freedom toward the Mott transition is not considered. These pictures do not explain how a metallic state changes into a Mott insulating state that exhibits spin excitation in the energy regime that is much lower than the charge gap (the spin–charge separation characteristic of a Mott insulator).