Strong particle-hole asymmetry of charge instabilities in doped Mott insulators

We study possible charge instabilities in doped Mott insulators by employing the two-dimensional t-J model with a positive value of the next nearest-neighbor hopping integral t' on a square lattice, which is applicable to electron-doped cuprates. Although the d-wave charge density wave (flux phase) and d-wave Pomeranchuk instability (nematic order) are dominant instabilities for a negative t' that corresponds to hole-doped cuprates, we find that those instabilities are strongly suppressed and become relevant only rather close to half filling. Instead, various types of bond orders with modulation vectors close to (pi,pi) are dominant in a moderate doping region. Phase separation is also enhanced, but it can be suppressed substantially by the nearest-neighbor Coulomb repulsion without affecting the aforementioned charge instabilities.


I. INTRODUCTION
of collective excitations in the optimal e-cuprates 39 suggests that a charge-order tendency can be present even in e-cuprates.
The e-cuprates were often discussed via a comparison with h-cuprates, focusing on specific aspects, e.g., pairing properties, 40,41 magnetic properties, 42,43 stability of charge stripes but with different conclusions, 40,44 and optical conductivity. 45 A recent comprehensive study by using variational Monte Carlo 46 showed that superconductivity is enhanced, but antiferromagnetism is suppressed in h-cuprates, whereas the opposite occurs in e-cuprates, nicely demonstrating the experimental fact. In spite of these works, charge-order tendencies in e-cuprates have not been clarified.
In the present paper, we study possible charge instabilities in e-cuprates. We extend our previous work for h-cuprates 30 to e-cuprates and take all possible charge instabilities into account on equal footing in the two-dimensional t-t ′ -J model. We find that charge-order tendencies exhibit a very strong particle-hole asymmetry. While h-cuprates have strong tendencies toward the dCDW and dPI, 30 these orders are substantially suppressed in ecuprates. Instead various bond orders with large momenta near (π, π) are favored in a moderated doping region. In Sec. II we define our model and explain our methods briefly.
Numerical results are presented in Sec. III. We discuss possible charge instabilities and the PG in e-cuprates in Sec. IV, followed by conclusions in Sec. V.

II. MODEL AND METHODS
We study charge instabilities in the two-dimensional t-t ′ -J model by including the nearestneighbor Coulomb interaction V , is the hopping integral between the first (second) nearest-neighbor sites on a square lattice; J and V are the exchange interaction and the Coulomb repulsion, respectively, between the nearest-neighbor sites.c † iσ andc iσ are the creation and annihilation operators of electrons with spin σ (σ =↓,↑), respectively, in Fock space without any double occupancy. n i is the electron density operator and S i is the spin operator in that space.
A role of the V term in the present study is to suppress the strong tendency toward PS and does not affect major charge instabilities originating from the J term. Because of the high complexity of the t-t ′ -J model, which comes mainly from the local constraint of the model, it is still unrealistic to perform a full analysis of the model. Since we are interested in possible charge instabilities, we employ a large-N expansion. In this scheme the two spin components are extended to N and an expansion in powers of the small parameter 1/N is performed, providing a controllable scheme without a perturbative expansion in any model parameters. In addition, different kinds of charge instabilities can be studied on equal footing. The greatest advantage of this technique is that charge susceptibilities contain collective effects already at leading order, whereas magnetic and paring channels do not. This property allows us to perform a comprehensive analysis of all possible charge instabilities in the t-t ′ -J model in leading order theory.
The large-N technique was already applied to the t-t ′ -J model in various formalisms. 9,47,48 Here we follow the path integral formalism developed in Refs. 30 and 48. In particular, we use the same notations as Ref. 30, unless explicitly stated otherwise.
The possible charge instabilities are formulated in a six-dimensional space spanned by the bosonic field δR i indicates charge fluctuations at site i around the average carrier density and δλ i describes fluctuations associated with a Lagrange multiplier around its mean value; the Lagrange multiplier comes from the local constraint that excludes the double occupancy of electrons at any site i. r η i and A η i are the real and imaginary parts of fluctuations of the bond variable between the nearest-neighbor sites and η denotes the bond direction, namely η 1 = (1, 0) and η 2 = (0, 1) on a square lattice. The general charge susceptibility is thus defined as a 6 × 6 matrix. This matrix depends on both momentum q and energy ω. We focus on the static limit in the present study. When an eigenvalue of the inverse of the matrix crosses zero at a given doping rate δ, temperature T , and q, a charge instability with a modulation vector q occurs and the ordering pattern is determined by the corresponding eigenvector V a .
iii) V a ∝ (0, 0, 1, 0, 0, 0), (0, 0, 0, 1, 0, 0), (0, 0, 1, 1, 0, 0), and (0, 0, 1, −1, 0, 0), which correspond to bond-order phase (BOP) 9,47,48 with q = (π, π) or close to it, with four different patterns: BOP x , BOP y , BOP xy , and BOP xȳ , respectively. BOP x(y) is a phase which has a bond amplitude along the x(y) direction and zero amplitude along the other direction, whereas BOP xy(xȳ) with (π, π) has a bond amplitude along both x and y directions and its relative phase is inphase (antiphase). Since the dPI and BOP xȳ belong to the same eigenvector (0, 0, 1, −1, 0, 0), the dPI with q ≈ (π, π) is equivalent to the BOP xȳ . However, the term of the dPI makes sense only for a small q and thus we use the term BOP xȳ when q is no longer close to (0, 0). iv) V a ∝ (1, 0, 0, 0, 0, 0), which corresponds to PS with q = (0, 0). Conventional CDW including charge stripes also belongs to the same eigenvector, but with a finite q. Such an instability was not found in the present study. The present work is in favor of Ref. The t-t ′ -J model with t ′ < 0 has been extensively studied in the context of h-cuprates. To apply the present methods to e-cuprates, we perform a particle-hole transformation. This is implemented by taking a positive value of t ′ .

III. RESULTS
Because we determine critical lines of charge instabilities by studying the susceptibility, the transition is always continuous and a possible first-order transition is not considered in the present study. We set t = 1 and all quantities with dimension of energy are in units of t.

A. phase separation
We first discuss PS. As seen in literature, 41,43,53 PS is strongly enhanced for a positive t ′ . Figure 1 Fig. 1(b) was also obtained in the Hubbard model in strong coupling. 54 We, however, note that our PS is not pure PS especially for high T . While the eigenvector of PS contains the component of (1, 0, 0, 0, 0, 0) almost 100 % close to zero temperature, the weight from other components, especially from (0, 0, 1, 1, 0, 0), increases with increasing T . For example, the weight of the (1, 0, 0, 0, 0, 0) component is reduced  200-100 K. Figure 2 shows phase diagrams in the plane of δ and T . As already seen in Fig. 1(b), PS occurs on the side of half-filling and is enhanced at high T . In contrast to PS, other charge instabilities are driven by the J term and occur at lower temperatures (0 < T < 0.04) below δ ≈ 0.14. This region is actually what we are interested in, in the context of e-cuprates.
We obtain three different types of charge instabilities: dCDW with q = (π, π), dPI with q = (0, 0), and various BOPs such as BOP x(y) , BOP xy , and BOP xȳ with q ≈ (π, π). It is the dPI which is suppressed most strongly and is stabilized only rather close to half-filling.  Fig. 2 should be interpreted as a hierarchy of different charge instabilities, that is, the outer the critical line is, the stronger the tendency toward the corresponding instability is.
While the dPI and dCDW instabilities always occur at q = (0, 0) and (π, π), respectively, the modulation vectors of BOP xy and BOP x(y) show the instabilities at q = (π, π) and shift at very low T ( 0.005) slightly toward the direction (π, π)-(π, 0) for BOP xy and BOP x and the direction (π, π)-(0, π) for BOP y . The modulation vector of BOP xȳ is shown in Fig. 3 along its critical line in Figs. 2(a)-(c); hence the doping rate also changes with changing T c .
Since the charge susceptibility corresponding to BOP xȳ is rather flat in momentum space, we plot the modulation vectors where the inverse of the charge susceptibility is less than 10 −4 .
The width of such a q region at a fixed temperature implies how sharp the susceptibility is in momentum space. The modulation vector does not extend to the side of (π, π)-(π, 0) direction because the eigenvector there changes to a different one. Because of the flat feature of the susceptibility at high T , the modulation vector cannot be specified uniquely in the susceptibility analysis. With decreasing T , the susceptibility becomes sharper. The modulation vector is then defined as q = (π, π) for t ′ = 0.2 and shifts toward the diagonal direction along (π, π)-(0, 0) for a larger t ′ .
Compared with the results for t ′ < 0 obtained in Ref. 30, charge instabilities, except for PS, show a much weaker dependence of t ′ for t ′ > 0. While BOPs are even weaker instabilities than the dPI and dCDW for t ′ < 0, various BOPs extend to a moderate doping for t ′ > 0 and become dominant there. The most striking difference lies in the dPI. Whereas the dPI is strongly enhanced with increasing |t ′ |(t ′ < 0), 30 it is substantially suppressed once the sign of t ′ is reversed. To understand such a drastic change, we closely study how the modulation vector of the dPI evolves by changing t ′ . In the left-hand panels in Fig. 4, we show the eigenvalue of the inverse of the susceptibility for the eigenvector (0, 0, 1, −1, 0, 0) at T c = 0.008 and 0.0001 for a sequence of t ′ . In the right-hand panels the corresponding modulation vector q is summarized by determining the momentum q at which the eigenvalue becomes less than 10 −4 at each temperature. For t ′ = −0.2 [ Figs. 4(a) and (f)] the dPI occurs at q = (0, 0) and slightly away from it at very low T . For t ′ = 0 [ Figs. 4(b) and (g)] the susceptibility of the dPI becomes flat along the direction (0, 0)-(π, π), but eventually an incommensurate q is favored along the direction (0, 0)-(π, 0) at low T . The flat feature is a special aspect of the dPI susceptibility, which becomes exactly flat along (0, 0)-(π, π) direction for any T and δ for t ′ = 0. 30 With the inclusion of a tiny t ′ (= 0.01) [Fig. 4(c)] the flat structure is slightly slanted and the eigenvalue at q = (π, π) becomes smaller than that at (0, 0). As a result, the instability occurs at q = (π, π) at high T , which is equivalent to BOP xȳ . Although the flat feature still remains at low T [ Fig. 4(c)], an incommensurate dPI develops along the direction of (0, 0)-(π, 0), similar to the results for t ′ = 0. A value of t ′ ≥ 0.10 is sufficient to completely destroy the dPI with a small q and stabilizes BOP xȳ with q ≈ (π, π) in the whole temperature region imply that the reason why the stabilization of the dPI changes rapidly by changing the sign of t ′ lies in the special feature of the dPI susceptibility, which exhibits a exactly flat structure along (0, 0)-(π, π) direction at t ′ = 0.

IV. DISCUSSIONS
The e-cuprates are characterized by a positive t ′ in the t-t ′ -J model. We first consider possible effects of superconductivity and antiferromagnetism on our phase diagram, which are not taken into account in our leading order theory. Typically, superconductivity in ecuprates occurs below 25 K, 38 which is around 0.004t in the present theory for t ∼ 500 meV.
Since our charge-order instabilities occur higher than this temperature, a major part of our results could not be affected by superconductivity although charge orders or their tendencies would be suppressed inside the superconducting state. Antiferromagnetism extends to the region of 11-14% doping in e-cuprates. 38 Since charge orders occur below 13-14% in our phase diagram, most of charge orders that we have found could be masked by antiferromagnetism, yet may be observed for, e.g., Pr 1−x LaCe x CuO, which has a lower critical doping rate of antiferromagnetism. The most relevant charge orders in a moderate doping region are various BOPs with q close to (π, π), which become dominant below T ∼ 0.01t ∼ 50K as seen in Fig. 2. Figure 2 also implies that the dCDW can become relevant to e-cuprates if the critical temperature of antiferromagnetism becomes lower than T c of the dCDW in a certain doping region. At present neither experimental evidence of BOP nor dCDW is obtained in ecuprates. However, given that evidence of some order competing with superconductivity was obtained quite recently in e-cuprates 56 and that a new type of charge order was also found quite recently in h-cuprates, [25][26][27][28] it may be too early to reach a conclusion about a possible charge instability in e-cuprates. In particular, BOPs with q close to (π, π) are not reported in h-cuprates and thus in this sense e-cuprates are attractive to explore a new type of charge order in cuprates. Even if charge-order instability does not occur, its fluctuation effect can be observed as collective excitations. It is interesting to explore a possible connection to a new collective mode recently found in optimal e-cuprates by resonant inelastic x-ray scattering. 39 There is growing evidence that the PG is related to some charge order or its strong fluctuations in h-cuprates. [5][6][7]57,58 In particular, the dCDW 8,10-12 and dPI [13][14][15] are some strong candidates. If the charge order is indeed responsible for the PG, the present theory suggests that the property of the PG should be different between hole-doping and electron-doping because of a strong particle-hole asymmetry of charge-order instabilities. While a PG was reported in the optical conductivity spectra in the non-superconducting crystals of e-cuprates, 59 the PG corresponding to the one observed in h-cuprates, namely in a doping region where the superconducting phase occurs with decreasing T , seems missing or at least much weaker.
It is quite interesting whether other scenarios of the PG such as fluctuations associated with Cooper pairing and antiferromagnetism can provide a natural explanation about the asymmetry of the PG between hole-doped and electron-doped cuprate superconductors.

V. CONCLUSIONS
We have performed a stability analysis of the paramagnetic phase in the two-dimensional t-t ′ -J model by employing a positive t ′ in a large-N expansion formalism. Our theoretical framework has the advantage to take all possible charge instabilities on equal footing into account and to allow us to perform a comprehensive study of charge instabilities in a controllable scheme. We have found that the dCDW and dPI become relevant rather close to half-filling and various types of BOPs with q close to (π, π) are dominant in a moderate doping region. PS is also enhanced but can be suppressed substantially by the nearest-neighbor Coulomb repulsion V although the instabilities associated with BOPs, dCDW, and dPI are almost intact even in the presence of large V .
The charge order tendencies we have found for t ′ > 0 are very different from those for t ′ < 0. 30 This strong particle-hole asymmetry implies that charge orders are less favorable in e-cuprates, although they can still occur. Furthermore, if charge orders are responsible for the PG, the present theory may naturally explain the reason why the PG phenomenon is very different between e-cuprates and h-cuprates.