Mott Transition in the Two-Dimensional Hubbard Model

Spectral properties of the two-dimensional Hubbard model near the Mott transition are investigated by using cluster perturbation theory. The Mott transition is characterized by freezing of the charge degrees of freedom in a single-particle excitation that leads continuously to the magnetic excitation of the Mott insulator. Various anomalous spectral features observed in high-temperature superconductors are explained in a unified manner as properties near the Mott transition.

The metal-insulator transition due to Coulomb interactions between electrons is called the Mott transition. In the insulating phase, the charge excitation has an energy gap whereas the spin excitation is usually gapless [1]. On the other hand, a metal far away from a Mott insulator exhibits free-electron-like behaviors. Thus, the question of how the two limits can be reconciled at the Mott transition is a fundamental puzzle in condensed-matter physics. In particular, in relation to the anomalous properties of cuprate high-temperature (high-T c ) superconductors [2][3][4], the Mott transition in a two-dimensional (2D) system has been investigated from various viewpoints [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. However, its nature remains controversial.
This Letter shows how the single-particle excitation in a 2D metal changes into the spin-wave mode of the Mott insulator [1], through analyses of numerical results on the 2D Hubbard model according to the exactly known properties of a one-dimensional (1D) system [21]. In addition, anomalous spectral features observed in high-T c cuprates, such as the pseudogap, Fermi arc, flat band, dopinginduced states, spinon-like and holon-like branches, as well as kink and waterfall in the dispersion relation [2][3][4], are explained in a unified manner as properties of the 2D Hubbard model near the Mott transition. The 2D Hubbard model is defined by the following Hamiltonian: where c iσ and n iσ denote the annihilation and number operators of an electron with spin σ at site i, respectively, and i, j means that sites i and j are nearest neighbors on a square lattice. The doping concentration is denoted by δ. We consider the spectral function defined as A(k, ω) ≡ − 1 π ImG(k, ω), where G(k, ω) denotes the retarded single-particle Green function at momentum k and energy ω [7] of the 2D Hubbard model at zero temperature. We employ cluster perturbation theory (CPT) [20] to obtain G(k, ω), using (4 × 4)-site cluster Green functions calculated by exact diagonalization. The size of clusters treatable with exact diagonalization in CPT is larger than that in cellular dynamical mean-field theory [14]. We consider the properties of hole-doped systems with t > 0 for 0 ≤ k y ≤ k x ≤ π without loss of generality.
A characteristic feature of the Mott transition is the doping-induced spectral-weight transfer from the upper Hubbard band (UHB) to the lower Hubbard band (LHB) [4-6, 9-14, 21]. As shown in Figs. 1(i-k), the spectral weight transferred to the LHB for ω > 0 is primarily carried by the mode originating from the upper edge of the spinon-antiholon continuum. The energy of the mode at (π,π) [ǫ(π, π)] does not reach zero even in the δ → 0 limit [ Fig. 2(c)], and its spectral weight gradually disappears as δ → 0 [ Fig. 2(h)]: the mode remains dispersing with the spectral weight fading away as δ → 0 [ Fig. 1(p)]. This feature is the same as that in 1D [Figs. 2(d,h)] [21] and contrasts with that of a doped band insulator and with that of a Fermi liquid as the effective mass m * → ∞ where the effective bandwidth ǫ * (k) ∝ 1/m * [7].

(l)] (blue diamonds). Dotted green line indicates
√ 2v2D/t, where spin-wave velocity of the 2D Heisenberg model v2D ≈ 1.18 √ 2J [27] (J = 4t 2 /U ). Inset shows J/t dependence. (f) Same as (e) but for 1D. Solid blue curve shows ǫπ/t(= ǫ0/t) [21]. Dotted green line indicates v1D/t, where spin-wave velocity of the 1D Heisenberg model v1D = πJ/2 [29]. (g) ǫp/t (blue diamonds) and ǫ (π,0) /t (red circles) [ Fig. 1(l)]. Dotted green line indicates a fit in the large U/t regime, assuming ǫp, ǫ (π,0) ∝ J. Inset shows J/t dependence. (h) Spectral weight A for ω > 0 in the LHB. Blue squares with solid line show 2D results for U/t = 8. Red circles with dashed line denote 1D results for U/t = 8 taken from ref. [21]. Dotted green line indicates results for t = 0 [10]. Inset shows the Mott gap ∆ between the LHB and the upper Hubbard band at δ = 0. Blue diamonds show 2D results. Dotted red curve denotes 1D results [33]. [1,27]. The continuous evolution to the spin-wave mode is consistent with the scaling behavior of spin correlations [7,17,18]. In the extremely large U/t regime, the ferromagnetic fluctuation arising from Nagaoka ferromagnetism [28] could dominate the antiferromagnetic fluctuation near the Mott transition, an outcome that is beyond the scope of the present study. Note that the relationship to the spin-wave mode is essentially the same as that in 1D [21]: in 1D, the dispersion relations of the mode for ω > 0 in the LHB and the spinon mode reduce to ǫ 1D (k) = −v 1D cos k in the δ → 0 limit in the large U/t limit [ Figs. 2(a,f)] [21], where v 1D denotes the spin-wave velocity of the 1D Heisenberg model [29]. The modes in 1D lead continuously to the spin excitations at δ = 0 whose dominant mode shows E(k) = |ǫ 1D (k − π/2)| [29].
Properties near (π,0).− As shown in Figs. 1(j-l), the dominant mode near (π,0) near the Mott transition is located below ω = 0, whose dispersion relation is anomalously flat. This anomalously flat dispersion relation has been found in numerical simulations [12,15,19] and in high-T c cuprates [2]. Below, we discuss its relationship to the pseudogap and Fermi arc.
Near the Mott transition, the main peak of A(ω)[≡ dkA(k, ω)/(2π) 2 ] is located below ω = 0 [ Fig. 1(k), rightmost panel]. Namely, the spectral weight decreases from the peak value as ω → 0. This feature is called a pseudogap. To discuss the pseudogap quantitatively, we define two energies: ǫ p as the energy difference between the peak in A(ω) and the top of the LHB at δ = 0 [ Fig.  1(l)], and ǫ (π,0) as the energy difference between the flat mode at (π,0) and the top of the LHB at δ = 0 [ Fig.  1(l)]. Figure 2(g) shows that the pseudogap defined in A(ω) is primarily due to the flat mode at (π,0) and that the pseudogap is induced by interactions even without further-neighbor hopping. In addition, this figure indicates that the pseudogap is proportional to J (= 4t 2 /U ) [ǫ p ≈ ǫ (π,0) ≈ 0.8J] in the large U/t regime, implying that it will be related to the antiferromagnetic fluctuation [12,15]. The pseudogaps defined by A(ω) and by the flat mode decrease as δ increases [ Figs. 1(j,k)], because the chemical potential is lowered with the flat mode almost unchanged. The pseudogap closes at a δ value where the peak in A(ω) or the flat mode crosses ω = 0.
Namely, the mode crossing ω = 0 in the (π,0)-(π,π) direction can be identified as that originating from the upper edge of the spinon-antiholon continuum, whose spectral weight fades away as δ → 0. As a result, the spectral weight almost disappears along this direction around ω = 0. This leads to the behavior that can be regarded as a Fermi arc (disconnected portion of the large Fermi surface) [2] [ Fig. 1(o)]. This behavior contrasts with that of the large Fermi surface in the large doping regime [ Fig.  1(m)]. Fully quantitative explanation for the Fermi arc observed in high-T c cuprates requires further study. Also, properties in a very small energy scale inaccessible with (4 × 4)-cluster CPT are beyond the scope of this Letter.
Mott gap.− In a (doped) Mott insulator, the Mott gap is due to Coulomb interactions, whereas the band gap in a (doped) band insulator is due to a chemical potential difference between sites or orbitals within a unit cell. A deeper insight into Mott physics can be obtained by tracing the origin back to 1D. In 1D, the quasiparticle responsible for the Mott gap has been identified as that defined by the k-Λ string in the Bethe ansatz and called the doublon [21], which can be regarded as a pair of electrons [30]. By the presence of the doublon, the UHB can be distinguished from the LHB [21]. The doublon is not a double occupancy because the latter exist in the LHB as well as the UHB for U/t < ∞. Because the Mott gap in 2D can change into that in 1D by reducing the interchain hopping, the quasiparticle that determines the energy scale of the UHB in 2D (doublon in 2D) should be a descendant of the 1D doublon, i.e., a pair of electrons rather than a double occupancy. The similarities in the spectral weight transferred by doping and those in the Mott gap at δ = 0 between 1D and 2D [ Fig. 2(h)] seem to support the traceability. Apart from the presence of the doublon, we can identify the dominant modes in the UHB [ Figs. 1(a-d)] as in the LHB for ω < 0, noting the particle-hole symmetry at δ = 0, as in 1D [21].
Noting that the high-energy magnetic excitations in a magnetic field in quasi-1D Heisenberg antiferromagnets, which can be regarded as repulsively interacting hardcore boson systems, have been explained using two-string solutions [24,31,32], we can generalize the concept of Mott physics: the generalized doublon (a pair of single particles definable by a string with a length of two in 1D [21,31,32] or its descendant in higher dimensions [24]) induced by repulsive interactions will be responsible for the high-energy states without multi-site (multi-orbital) unit cells, in contrast to band theory. The Mott physics could also be explored in cold atomic systems.
Summary.− Anomalous spectral features observed in high-T c cuprates, such as the pseudogap, Fermi arc, flat band, doping-induced states, spinon-like and holon-like branches, as well as kink and waterfall in the dispersion relation, were explained in a unified manner as properties of the 2D Hubbard model near the Mott transition. The properties near (π,0) are characterized by the flat mode unlike 1D features. The physics of the Mott gap was examined by tracing the origin back to 1D. The Mott transition is characterized by a dispersing mode that leads continuously to the spin-wave mode of the Mott insulator with the spectral weight fading away toward the Mott transition due to charge freezing. The loss of charge character from the dispersing mode will be a general feature of Mott transitions, which contrasts with the transition to a band insulator without spin-charge separation.
This work was supported by KAKENHI 22014015 and 23540428, and WPI for Materials Nanoarchitectonics.