Spectral properties near the Mott transition in the two-dimensional t-J model

The single-particle spectral properties of the two-dimensional t-J model in the parameter regime relevant to cuprate high-temperature superconductors are investigated using cluster perturbation theory. Various anomalous features observed in cuprate high-temperature superconductors are collectively explained in terms of the dominant modes near the Mott transition in this model. Although the behavior of the dominant modes in the low-energy regime is similar to that in the two-dimensional Hubbard model, significant differences appear near the Mott transition for the high-energy electron removal excitations which can be considered to primarily originate from holon modes in one dimension. The overall spectral features are confirmed to remain almost unchanged as the cluster size is increased from 4x4 to 6x6 sites by using a combined method of the non-abelian dynamical density-matrix renormalization group method and cluster perturbation theory.


I. INTRODUCTION
Cuprate high-temperature (high-T c ) superconductors, which are obtained by doping Mott insulators containing CuO 2 planes, 1 exhibit various features that appear anomalous from conventional viewpoints. 2- 13 Because the anomalous features are considered to be related to high-T c superconductivity, the effects of electronic correlations near the Mott transition in two-dimensional (2D) systems have attracted considerable attention. [12][13][14][15][16][17] In particular, through an analysis of electronic correlations among relevant Cu and O orbitals, the 2D t-J model has been derived as a minimal model of high-T c cuprates. 18 However, its properties and their relationship to the anomalous features are not well understood, primarily because of difficulties in dealing with the nodouble-occupancy constraint.
Although this model can also be derived effectively from the 2D Hubbard model in the large-repulsion limit, 19 it is not clear how similar the two models' properties are in the parameter regime relevant to high-T c cuprates. In fact, the 2D t-J and Hubbard models have frequently been studied from different viewpoints: the former has been considered a doped Mott insulator accessible from slave-particle theories, 15,16 while the latter has been considered a strongly interacting electron system accessible from Fermi liquid theory. 17 In some studies, double occupancy, which is excluded in the t-J model, has been considered important to the anomalous features. [20][21][22][23][24] In this paper, by using cluster perturbation theory (CPT), 25,26 similarities and dissimilarities in the spectral features of these models are clarified. In addition, various anomalous features observed in high-T c cuprates are collectively explained in the 2D t-J model, which is an effective model of the CuO 2 plane and has no double occupancy. A method to reduce cluster-size effects is also introduced.

A. Models
We consider the 2D t-J model defined by the following Hamiltonian for t > 0 and J > 0: wherec i,σ = (1 − n i,−σ )c i,σ and n i = σ n i,σ for the annihilation operator c i,σ and number operator n i,σ of an electron with spin σ at site i. Here, S i denotes the spin operator at site i, and i, j means that sites i and j are nearest neighbors on a square lattice. At half-filling (doping concentration δ = 0), the model reduces to the Heisenberg model. The t-J model can also be effectively obtained for J = 4t 2 /U by neglecting the three-site term 16,19 in the large-U/t limit of the Hubbard model defined by the following Hamiltonian: In the ground state near the Mott transition, ferromagnetic fluctuations might be dominant in the very small-J/t regime [27][28][29] and phase separation occurs in the large-J/t regime. 13,16,[30][31][32][33][34][35][36][37][38] Here, focusing attention on the parameter regime relevant to high-T c cuprates (J/t ≈ 0.5), 13,16 where the CPT results exhibit no indication of phase separation [ Fig. 1(m)], we study the spectral function defined as A(k, ω) = −ImG(k, ω)/π. Here, G(k, ω) denotes the retarded single-particle Green function for momentum k and frequency ω at zero temperature. 12,13 B. Methods In this paper, CPT is employed: G(k, ω) is calculated by connecting cluster Green functions (j)-(l) The same as in (b)-(d) but for ω ≈ 0. (m) Chemical potential µ for J/t = 0.5, 0.4, and 0.3 (from above). (n) Energy of mode i ′ at (π, π) [ε(π, π)] (blue squares) for J/t = 0.5. (o) ε (0,0) (purple squares) and ε (π,π) (yellow triangles). The solid green line indicates √ 2vs/t (vs ≈ 1.18 √ 2J [39]). (p) εp (purple squares) and ε (π,0) (yellow triangles). The solid green line is proportional to J. The red circles in (n) and blue diamonds and red circles in (o) and (p) indicate corresponding energies in the 2D Hubbard model (U/t = 4t/J), taken from Ref. 40. Gaussian broadening with a standard deviation of 0.1t is used.
The DDMRG method and CPT work well together because of the following reasons. (1) It is not necessary to repeat DDMRG calculations because CPT does not impose self-consistency. (2) Data under open boundary conditions, which are obtained accurately in the DDMRG method, are used in CPT. The combined method can be regarded either as CPT where the DDMRG method is utilized as a cluster Green function solver or as the DDMRG method where momenta are interpolated based on CPT. Note that real-space cluster Green functions are used in the combined method [ Fig. 1(e)] in contrast with the random-phase approximation (RPA) from the decoupled-chain limit 48 using DDMRG results [Figs. 2(c) and 2(f)]. 40 In this paper, the RPA from the decoupledchain limit, which corresponds to the perturbation theory up to the first order with respect to interchain hopping, is only used to explain how the spectral weights are shifted by interchain hopping from the decoupled-chain limit and to trace the origins of the dominant modes of 2D systems back to those of one-dimensional (1D) systems (Secs. III A and III D).

C. Cluster-size effects
As shown in Figs. 1(c), 1(e), 1(i), 1(m), and 1(n), the results obtained using (6 × 6)-site CPT are almost the same as those using (4 × 4)-site CPT. This implies that the overall spectral features will not change significantly if the cluster size is increased.

B. Positive ω
Mode i ′ corresponds to the doping-induced states observed in high-T c cuprates and in theoretical calculations 3,4,12,13,20-24,32,40,42,60-76 with controversial interpretations. The CPT results indicate that the energy of this mode at (π, π) [ε(π, π)] does not reach zero even in the small-doping limit [ Fig. 1(n)]. The extrapolated value of ε(π, π) to δ → 0 [ε (π,π) ] behaves essentially as √ 2v s in the small-J/t regime, where v s  Fig. 1(o)], as in the 2D Hubbard model. 40 In addition, the spectral weight for ω > 0 (W + ) behaves exactly as δ [ Fig. 1(i)]. 32,75 These results imply that mode i ′ continuously leads to the magnetic excitation of the Mott insulator, while its spectral weight gradually disappears toward the Mott transition. This feature is essentially the same as that in the 1D and 2D Hubbard models 40,49,62 and is consistent with a general argument in the small-doping limit. 47 Thus, this feature will be related to transformation to the Mott insulator, which has low-energy spin excitation but no low-energy charge excitation (spin-charge separation), rather than being related to double occupancy.
For the electron operators with the no-doubleoccupancy constraint,c Because the spectral weight for ω < 0 (W − ) is equal to (1 − δ)/2 in the t-J and Hubbard models, n − (k) around k = π in the t-J model becomes larger to compensate for the reduction around k = 0. Here, 0 and π respectively indicate 0 and π for 1D systems and (0, 0) and (π, π) for 2D systems.
In the RPA from the decoupled-chain limit 48 the spectral weights shift upward and downward in the momentum regime for t ⊥ (k) > 0 and t ⊥ (k) < 0, respectively, 40,42,58,59 where G 1D (k x , ω) and t ⊥ (k)(= −2t ⊥ cos k y ) denote the Green function of a chain and Fourier transform of the interchain hopping integral, respectively. In addition, the mode shift becomes large if the mode carries large spectral weights in the large-|t ⊥ (k)| regime. 40,42,58,59 Because the spectral weight around k = 0 for ω < 0 in the 1D t-J model is significantly smaller, the downward spectral-weight shift around (0, 0) is smaller [ Fig. 2(f)] than that of the Hubbard model [ Fig. 2(c)].
This argument explains why the ω value around (0, 0) of mode iv ′ is higher than that of mode iv near the Mott transition [Figs. 1(c) and 2(g)]. For (π, π), a similar argument can also explain that the ω value around (π, π) of mode iv ′ is higher than that of mode iv. Near the Mott transition, however, the ω values of modes 4, 4 ′ , iv, and iv ′ around k = π appear to be better explained by considering that they are almost the same as those around k = 0 [Figs. 1(a)-1(c), 1(e), 2(a), 2(d), and 2(g)].
At half-filling, the ω value of mode 4 at k = π is exactly the same as that at k = 0 because the dispersion relation of the holon mode becomes symmetric with respect to k = π/2. 49,51,105 For the differences in the large negative ω regime, the neglect of the three-site term is relevant, 43,104 because this term contributes to hopping.

IV. DISCUSSION AND SUMMARY
In this study, the single-particle spectral properties of the 2D t-J model for J/t ≈ 0.5 near the Mott transition were investigated. In contrast with conventional exact diagonalization studies, where the spectral weights were calculated only at available k and δ points depending on the cluster size and boundary conditions, the spectralweight distributions for continuous k and ω near the Mott transition with small intervals of δ were calculated using CPT, and how the peaks of the spectral weights form dominant modes and how the modes transform to those of the Mott insulator as δ gradually decreases were clarified in the 2D t-J model. In addition, through comparisons of the (4 × 4)-site CPT results with the (6 × 6)-site CPT results obtained by combining the non-Abelian DDMRG method, the cluster-size effects were confirmed to be small enough to allow discussion on the overall spectral features.
Furthermore, the natures of the dominant modes in the 2D t-J model were clarified by investigating the δ and J/t dependences of the characteristic energies and spectral weights and by tracing the origins of the modes back to those of the 1D models. In terms of the dominant modes, various anomalous spectral features observed in high-T c cuprates, such as the doping-induced states, flat band, pseudogap, Fermi arc, spinon-like and holon-like branches, giant kink, and waterfall behavior, 2-13 were collectively explained in the 2D t-J model, as in the 2D Hubbard model. 40 The results for the natures of the dominant modes imply that these spectral features are primarily related to the proximity of the antiferromagnetic Mott insulator, which has a low-energy spin-wave mode 106 but no lowenergy charge excitation, and to the existence of different energy scales that characterize the bandwidths of the dominant modes rather than to double occupancy, which is completely removed in the 2D t-J model.
Although the spectral features of the 2D t-J model in the small-|ω| regime are similar to those of the 2D Hubbard model, significant differences appear in the large negative ω regime around (0, 0) and (π, π) near the Mott transition for the modes which can be considered to primarily originate from the 1D holon modes. In this study, the differences were interpreted as a consequence of the restriction on spectral weights.
Because of the limited resolution, this study could not clarify the properties in the very small-|ω| regime, such as the nature of the excitation in the small-|ω| limit, the accurate gapless points of the single-particle spectrum, and the presence or absence of a long-range order in the ground state, although a superconducting ground state has been suggested in a considerable number of studies for the 2D t-J model. [13][14][15][16][107][108][109][110][111] Further studies are needed to clarify more details and how the anomalous features are related to high-T c superconductivity.