Dome structure in pressure dependence of superconducting transition temperature for HgBa₂Ca₂Cu₃O₈ — Studies by ab initio low-energy effective Hamiltonian

At $P_{\rm amb}$, known values of $T_{c}^{\rm opt}$ range from $T_{c}^{\rm opt}\simeq 6$ K in Bi₂Sr₂CuO₆ (Bi2201) Torrance et al. (1988) to $T_{c}^{\rm opt}\simeq 138$ K in HgBa₂Ca₂Cu₃O₈ (Hg1223) Gao et al. (1994); Dai et al. (1995). $T_{c}^{\rm opt}$ further increases under pressure. In the case of Hg1223 and other Hg-based cuprates, $T_{c}^{\rm opt}$ has a dome-like structure as a function of $P$ Gao et al. (1994); Yamamoto et al. (2015). An example is shown in Fig. 1(b) for Hg1223: $T_{c}^{\rm opt}$ increases with pressure and shows the maximum 164 K at $P_{\rm opt}\simeq 30$ GPa Nuñez-Regueiro et al. (1993); Gao et al. (1994), which is the highest known value of $T_{c}^{\rm opt}$ in the cuprates.

For example, for Y-based Crommie et al. (1989); Meingast et al. (1991); Belenky et al. (1991); Welp et al. (1992); Meingast et al. (1993); Mito et al. (2012, 2014a, 2014b, 2016) and Hg-based Hardy et al. (2010); Mito et al. (2017) high-$T_{c}$ cuprates, the uniaxial pressures $P_{a}$ and $P_{c}$ were applied. (In this paper, $P_{a}$ refers to the simultaneous compression along axes ${\bf a}$ and ${\bf b}$ in Fig. 2, while keeping $|{\bf a}|=|{\bf b}|$, and $P_{c}$ refers to the compression along axis ${\bf c}$. The axes are represented in Fig. 2 for the tetragonal cell in Hg1223.) This decomposition of pressure revealed, in the case of HgBa₂CuO₄ (Hg1201, $T_{c}^{\rm opt}\simeq 94$ K Putilin et al. (1993)), that $T_{c}^{\rm opt}$ decreases with out-of-CuO₂ plane contraction caused by $P_{c}$ ($\partial T_{c}^{\rm opt}/\partial P_{c}\simeq-3$ K/GPa) but increases with in-plane contraction caused by $P_{a}$ ($\partial T_{c}^{\rm opt}/\partial P_{a}\simeq 5$ K/GPa) Hardy et al. (2010).

Since it is difficult to isolate these hidden mechanisms by experiments only, further theoretical studies of cuprates under $P$ are desirable.

For ab initio studies, the density functional theory (DFT) has been widely applied in the history Hohenberg and Kohn (1964); Kohn and Sham (1965). However, its insufficiency in strongly correlated electron systems is also well known. Instead, we apply the multiscale ab initio scheme for correlated electrons (MACE) Aryasetiawan et al. (2004, 2006); Imada and Miyake (2010); Hirayama et al. (2013, 2015, 2018, 2019), which has succeeded in correctly reproducing the SC properties of the cuprates Hirayama et al. (2019); Ohgoe et al. (2020); Morée et al. (2022); Schmid et al. (2023) at ambient pressure and has motivated further studies on hypothetical Ag-based compounds Hirayama et al. (2022).

MACE consists of a three-step procedure that determines the LEH parameters for the single-band AB Hamiltonian; this procedure has several different accuracy levels, which are defined below and whose details are given in Appendix A. At the earliest stage of the MACE, the simplest level denoted as LDA+cRPA Aryasetiawan et al. (2004, 2006) or GGA+cRPA was employed; at this level, we start from the electronic structure at the Local Density Approximation (LDA) or Generalized Gradient Approximation (GGA) level, and the effective interaction parameters are calculated on the level of the constrained random phase approximation (cRPA) Aryasetiawan et al. (2004). The next level is denoted as c$GW$-SIC Hirayama et al. (2018), in which the starting electronic structure is preprocessed from the LDA or GGA level to the one-shot $GW$ level, and the one-particle part is improved by using the constrained $GW$ (c$GW$) Hirayama et al. (2013) and the self-interaction correction (SIC) Hirayama et al. (2015). The most recent and accurate level is denoted as c$GW$-SIC+LRFB Hirayama et al. (2019), which is essentially the same as the c$GW$-SIC, except that the $GW$ electronic structure is further improved: The level renormalization feedback (LRFB) Hirayama et al. (2019) is used to correct the onsite Cu$3d_{x^{2}-y^{2}}$ and O$2p_{\sigma}$ energy levels.

Although the c$GW$-SIC+LRFB level is the most accurate and was used to reproduce the SC properties of the cuprates Hirayama et al. (2019); Ohgoe et al. (2020); Morée et al. (2022); Schmid et al. (2023), we mainly employ the simplest GGA+cRPA version for the purpose of the present paper, because the qualitative trend of the parameters can be captured by this simplest framework. (See Appendix A for a more detailed discussion.) We also reinforce the analysis by deducing more refined c$GW$-SIC+LRFB level in a limited case from the explicit cGW-SIC level calculations to remove the known drawback of GGA+cRPA as we detail later.

Other LEH parameters are given in the Supplemental Material (S1) hg (1). In the following, we mainly discuss $|t_{1}^{\rm avg}|$ and $u^{\rm avg}$, which are the ab initio values of $|t_{1}|$ and $u$ at GGA+cRPA level, averaged over the inner and outer CuO₂ planes. (See Fig. 2 for a representation of the CuO₂ planes.)

This paper is organized as follows. In Sec. II, the central results of the present paper are outlined to capture the essence of the results before detailed presentation. In Sec. III, we give the crystal structure of Hg1223, the hole concentration and a reminder of the GGA+cRPA scheme. In Sec. IV, we give the DFT electronic structure at the GGA level as a function of $P$. In Sec. V, we show the pressure dependence of AB LEH parameters at the GGA+cRPA level. In Sec. VI, we discuss the adequacy of the assumptions made in Sec. II. We also discuss the consistency of our results with the experimental $P$ dependence of $T_{c}^{\rm opt}$ in Fig. 1. Summary and Conclusion are given in Sec. VII. In Appendix A, methodological details of MACE scheme are summarized. In Appendix B, computational details used in this paper are described. In Appendices C and D, we detail the corrections used in Secs. II and VI. In Appendix E, we discuss in detail the $P$ dependence at the intermediate stage of the present procedure. In Appendix F, we detail the dependence of AB LEH parameters on crystal parameters (CP) around optimal pressure.

The main results obtained in this paper are summarized as (I) and (II) below.

The nontrivial pressure dependence of $T_{c}^{\rm opt}$ can be understood from (I,II), which is derived from our ab initio Hamiltonian even at the preliminary level GGA+cRPA, if we assume the following (A) and (B). [The reality of (A) and (B) will be discussed later in Sec. VI.]

(A)

The universal scaling for $T_{c}^{\rm opt}$ given theoretically as

$$T_{c}^{\rm est}\simeq 0.16|t_{1}|F_{\rm SC}$$

(1)

recently proposed for the cuprates with $N_{\ell}=$ 1, 2 and $\infty$ Morée et al. (2022); Schmid et al. (2023) is also valid for Hg1223 with $N_{\ell}=3$.

(B)

$F_{\rm SC}$ follows a universal $u$ dependence revealed in Ref. Schmid et al. (2023), where $F_{\rm SC}$ has a peak at $u=u_{\rm opt}\simeq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}8.0-}8.5$. In addition, at $P_{\rm amb}$, Hg1223 is located at slightly strong coupling side $u\gtrsim u_{\rm opt}$, while the highest pressure $P=60$ GPa applied so far is in the weak coupling side $u<u_{\rm opt}$. In fact, we justify later $u{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\ \simeq\ }u_{\rm opt}$ at optimal pressure $P_{\rm opt}\simeq 30$ GPa for Hg1223.

To understand the consequences of the assumptions (A) and (B) appropriately and to complement the consequences quantitatively, we correct the errors anticipated in our ab initio GGA+cRPA calculation by using the following (C) and (D). [Details of (C) and (D) are given in Appendices C and D, respectively.]

(C)

We correct the values of $u^{\rm avg}$ and $|t_{1}^{\rm avg}|$ obtained at the GGA+cRPA level by deducing the most sophisticated c$GW$-SIC+LRFB level. Since GGA+cRPA is known to underestimate $u$ in Bi2201 and Bi₂Sr₂CaCu₂O₈ (Bi2212), it is desirable to improve the AB LEH to the more accurate c$GW$-SIC+LRFB level. However, the explicit calculation at the c$GW$-SIC+LRFB level is computationally demanding, while the corrections from the explicitly calculated c$GW$-SIC to the c$GW$-SIC+LRFB levels are known to be small and are relatively materials insensitive. Thus we represent the correction by a universal constant with admitted uncertainty. The estimates of $u$ and $|t_{1}|$ improved in such ways are denoted as $u_{{\rm c}GW{\rm-SIC+``LRFB"}}$ and $|t_{1}|_{{\rm c}GW{\rm-SIC+``LRFB"}}$ The procedure consists in the two steps (C1) and (C2):

(C1)

c$GW$-SIC calculation: Starting from the whole and detailed pressure dependence of $u^{\rm avg}$ and $|t_{1}^{\rm avg}|$ for Hg1223 calculated at the GGA+cRPA level, we calculate explicitly the level of the c$GW$-SIC denoted as $u_{{\rm c}GW{\rm-SIC}}$ and $|t_{1}|_{{\rm c}GW{\rm-SIC}}$ in limited cases of pressure choices of Hg1223 to reduce the computational cost.

(C2)

Estimate at the c$GW$-SIC+LRFB level: We use

$$u_{{\rm c}GW{\rm-SIC+``LRFB"}}=x_{\rm LRFB}u_{{\rm c}GW{\rm-SIC}}$$

(2)

and

$$|t_{1}|_{{\rm c}GW{\rm-SIC+``LRFB"}}=y_{\rm LRFB}|t_{1}|_{{\rm c}GW{\rm-SIC}}$$

(3)

and estimate constants $x_{\rm LRFB}$ and $y_{\rm LRFB}$ in Hg1223 from the already explicitly calculated results  for other compounds Hg1201, CaCuO₂, Bi2201 and Bi2212. The estimated values are $x_{\rm LRFB}=0.95$ (with the range of uncertainty $0.91-0.97$) and $y_{\rm LRFB}=1.0$. See Appendix C for detailed procedure to estimate $x_{\rm LRFB}$ and $y_{\rm LRFB}$ for the case of Hg1223. The concrete effect of (C) for Hg1223 is to increase $u$ from the cRPA level by the ratio $u_{{\rm c}GW{\rm-SIC+``LRFB"}}/u^{\rm avg}\simeq 1.29$ at $P_{\rm amb}$, $\simeq 1.15$ at 30 GPa, and $\simeq 1.08$ at 60 GPa; also, the $\simeq 13-14\%$ increase in $|t_{1}^{\rm avg}|$ from $P_{\rm amb}$ to 30 GPa becomes $\simeq 17\%$ by this correction.

(D)

After applying (C), we further correct the value of $|t_{1}|_{{\rm c}GW{\rm-SIC+``LRFB"}}$ by considering the plausible error in crystal parameters at high pressure. Structural optimization by ab initio calculation is known to show quantitative error and it is preferable to correct it if experimental value is known. We compare our structural optimization and the experimental cell parameter $a$ if it is available (this is the case at $P<8.5$ GPa) and assume that this trend of the deviation continues for $P>8.5$ GPa, where experimental data are missing. Namely, at $P>8.5$ GPa, we assume that our calculation overestimates the experimental $a$ by $\simeq 0.05$ Å, and we correct $a$ by $\Delta a=-0.05$ Å  accordingly. The concrete effect of (D) is that the increase in $|t_{1}|_{{\rm c}GW{\rm-SIC+``LRFB"}}$ from $P_{\rm amb}$ to 30 GPa is now $\simeq 22\%$.

The final estimates of $u_{{\rm c}GW{\rm-SIC+``LRFB"}}$ and $|t_{1}|_{{\rm c}GW{\rm-SIC+``LRFB"}}$ are shown in Fig. 1(a). Since (C1) is computationally demanding, we perform (C) and (D) only at $P_{\rm amb}$, $30$ GPa and $60$ GPa, and infer the correction at other pressures by linear interpolation for the pressure dependence.

Even by considering only (A) and (B) above, the present mechanism qualitatively accounts for the microscopic trend of the dome structure: At $P<P_{\rm opt}$, (I), namely the increase in $|t_{1}|$, plays the role to increase $T_{c}$, whereas the decrease in $u$ does not appreciably affect $F_{\rm SC}$ and thus $T_{c}$, because $F_{\rm SC}$ passes through the broad peak region in the $u$ dependence. At $P>P_{\rm opt}$, (II), namely the decrease in $u$, drives the decrease in $F_{\rm SC}$ and thus $T_{c}$ surpassing the increase in $|t_{1}|$, which generates a dome structure. If we take into account (C) and (D) in addition to (A) and (B), the dome structure in the $P$ dependence of experimental $T_{c}^{\rm opt}$ is more quantitatively reproduced (see Fig. 1). In addition to the above results, we discuss the dependence of AB LEH parameters on each CP, which provides us with hints for future designing of even higher $T_{c}^{\rm opt}$ materials.

We abbreviate the inner and outer CuO₂ planes shown in Fig. 2 as IP and OP, respectively. The crystal structure is entirely determined by the seven CPs defined in Table 1, which consist of the two cell parameters $a$ and $c$ and the five characteristic distances $d^{z}_{l}$. The CP values considered in this paper are listed in Fig. 2, as a function of $P$. In the main analyses of this paper, we consider (i) CP values obtained by a structural optimization, which are denoted as optimized CP values. For comparison, we also consider (ii) the theoretical calculation of the CP values in Zhang et al. Zhang et al. (1997) for the region between $P_{\rm amb}$ and $20$ GPa, and extrapolate the pressure dependence up to $60$ GPa. Details about (i,ii) are given in Appendix B.1. We also consider (iii) the experimental CP values from Armstrong et al. Armstrong et al. (1995) between $P_{\rm amb}$ and $8.5$ GPa. (The values at $P_{\rm amb}$ correspond to the SC phase with the experimental SC transition temperature $T_{c}^{\rm exp}\simeq 135$ K close to $T_{c}^{\rm opt}\simeq 138$ K.) It is known that the optimized CP values slightly deviate from the experimental values and it is indeed seen in Fig. 2. From the comparison of the optimized and experimental CPs, we take into account the correction (D) addressed in Sec. I.

We use the same procedure as that in Ref. Morée et al., 2022 employed for Hg1201: We partially substitute Hg by Au. We consider the chemical formula Hg${}_{1-x_{\rm s}}$Au${}_{x_{\rm s}}$Ba₂Ca₂Cu₃O₈ with $x_{\rm s}=0.6$ in order to realize the average hole concentration per CuO₂ plane $p_{\rm av}=0.2$ Bordet et al. (1996); Kotegawa et al. (2001); Yamamoto et al. (2015). This choice is discussed and justified in Appendix B.2.

The nontrivial point is: Experimentally, what are the variations in CP values when the crystal structure is compressed along ${\bf a}$ (${\bf c}$) ? First, the compression along ${\bf a}$ obviously modifies the cell parameter $a$ as well as the amplitude $|d^{z}_{\rm buck}|$ of the Cu-O-Cu bond buckling in the OP, but it should not affect the other CPs $d^{z}_{\rm Cu}$, $d^{z}_{\rm Ca}$, $d^{z}_{\rm Ba}$, and $d^{z}_{\rm O(ap)}$. Thus, we define the uniaxial pressure $P_{a}^{\rm buck}$ along ${\bf a}$ as follows: The compression along ${\bf a}$ modifies the values of $a$ and $d^{z}_{\rm buck}$, and all other CP values are those at $P_{\rm amb}$. We also consider a simplified definition, denoted as $P_{a}$: The compression along ${\bf a}$ modifies only the value of $a$, and all other CP values are those at $P_{\rm amb}$. As we will see, $P_{a}$ is sufficient to describe the main effect of the compression along ${\bf a}$. Second, the compression along ${\bf c}$ modifies the values of $d^{z}_{l}$, that is, all CP values except that of $a$. This uniaxial pressure is denoted as $P_{c}$. For completeness, we also consider a second definition, denoted as $P_{c}^{\overline{\rm buck}}$: The compression along ${\bf c}$ modifies all CP values except those of $a$ and $d^{z}_{\rm buck}$. This allows to discuss the effect of the relatively large value of $|d^{z}_{\rm buck}|$ at $P>P_{\rm opt}$. In the main analyses of this paper, we consider $P_{a}$ ($P_{c}$) to simulate the compression along ${\bf a}$ (${\bf c}$). We also give complementary results by considering $P_{a}^{\rm buck}$ and $P_{c}^{\overline{\rm buck}}$.

The $P$ dependence of the GGA band structure is demonstrated in Fig. 3, from which we derive the LEH spanned by the Cu$3d_{x^{2}-y^{2}}$/O2$p_{\sigma}$ AB bands by employing the GGA+cRPA scheme sketched in Appendix A. Computational details of DFT and GGA+cRPA scheme are described in Appendix B.

In the AB LEH for multi-layer cuprates Morée et al. (2022), there is only one AB orbital centered on each Cu atom. Then the AB LEH reads

where $l,l^{\prime}=\{i,o,o^{\prime}\}$ with $i$ being an IP site, and $o,o^{\prime}$ belonging to the two equivalent OPs. in which we distinguish the hopping and interaction parts between planes $l$ and $l^{\prime}$, as, respectively,

where $\sigma,\sigma^{\prime}$ are the spin indices. By using these notations, ($l\sigma{\bf R}$) is the AB spin-orbital in the plane $l$ and in the unit cell at ${\bf R}$, with spin $\sigma$. $c^{\dagger}_{l\sigma{\bf R}}$, $c_{l\sigma{\bf R}}$ and $\hat{n}_{l\sigma{\bf R}}$ are respectively the creation, annihilation and number operators in ($l\sigma{\bf R}$), and $t^{l,l^{\prime}}({\bf R^{\prime}-R})$ and $U^{l,l^{\prime}}({\bf R^{\prime}-R})$ are respectively the hopping and direct interaction parameters between ($l\sigma{\bf R}$) and ($l^{\prime}\sigma^{\prime}{\bf R^{\prime}}$). The translational symmetry allows to restrict the calculation of LEH parameters to $t^{l,l^{\prime}}_{\sigma,\sigma^{\prime}}({\bf R})$ and $U^{l,l^{\prime}}_{\sigma,\sigma^{\prime}}({\bf R})$ between ($l\sigma{\bf 0}$) and ($l^{\prime}\sigma^{\prime}{\bf R}$).

(Other LEH parameters are given in the Supplemental Material (S1) hg (1).) Then within this restricted range, $\mathcal{H}^{l}$ is rewritten as

in which $\tilde{\mathcal{H}}_{\rm hop}^{l}=\mathcal{H}_{\rm hop}^{l}/|t_{1}^{l}|$ and $\tilde{\mathcal{H}}_{\rm int}^{l}=\mathcal{H}_{\rm int}^{l}/U^{l}$ are the dimensionless hopping and interaction parts, expressed in units of their respective characteristic energies $|t_{1}^{l}|$ and $U^{l}$. The full dimensionless intraplane LEH is $\tilde{\mathcal{H}}^{l}=\mathcal{H}^{l}/|t_{1}^{l}|$, and the dimensionless ratio $u^{l}=U^{l}/|t_{1}^{l}|$ encodes the correlation strength. As mentioned in Sec. I, we also discuss the values of $|t_{1}^{\rm avg}|=(|t_{1}^{i}|+|t_{1}^{o}|)/2$ and $u^{\rm avg}=(u^{i}+u^{o})/2$. Average values of other quantities with the superscript $l$ are defined similarly.

We use the RESPACK code Nakamura et al. (2020); Morée et al. (2022). The standard calculation procedure is presented in detail elsewhere Nakamura et al. (2020); Morée et al. (2022). First, we compute $t_{1}^{l}$ as

in which $w_{l{\bf R}}$ is the Wannier function of the AB orbital $(l{\bf R})$, $R_{1}=[100]$, $\Omega$ is the unit cell, and $h$ is the one-particle part at the GGA level. Then, we compute $U^{l}$ as follows. We compute the cRPA effective interaction $W_{\rm H}$, whose expression is found in Appendix B.5, Eq. (B15). We use a plane wave cutoff energy of $8$ Ry. We deduce the onsite effective Coulomb interaction as

We also deduce the onsite bare Coulomb interaction $v^{l}$ by replacing $W_{\rm H}$ by the bare Coulomb interaction $v$ in Eq. (9), and the cRPA screening ratio $R^{l}=U^{l}/v^{l}$. The obtained values of $|t_{1}^{l}|$, $U^{l}$, $v^{l}$ and $R^{l}$ are plotted in Fig. 4.

The band dispersion is shown in Fig. 3(a-l). We also show in Fig. 3(o-q) the onsite energy of the Cu$3d_{x^{2}-y^{2}}$ and in-plane O$2p_{\sigma}$ atomic-like Wannier orbital (ALWO). (As explained in Appendix B, we denote these ALWOs as M-ALWOs because they are in the M space.) We also show the Cu$3d_{x^{2}-y^{2}}$/O$2p_{\sigma}$ hopping amplitude $|t_{xp}^{l}|=|t^{{\rm Cu(}l{\rm),O(}l{\rm)}}_{x^{2}-y^{2},p_{\sigma}}|$ in the unit cell.

This causes two distinct effects: First, the electrons in the CuO₂ plane feel the stronger Madelung potential from ions in the crystal. Indeed, the amplitude of the Madelung potential scales as $1/d$, where $d$ is the interatomic distance between the ion and the Cu or O atom in the CuO₂ plane. The variation in Madelung potential modifies the M-ALWO onsite energies and causes [M$\epsilon$] (for details, see Appendix E.1). Second, the overlap and hybridization between M-ALWOs increases, which causes [M$t$]. Both [M$\epsilon$] and [M$t$] increase the splitting of the B/NB (bonding/nonbonding) and AB bands, which causes [M$W$]: The bandwidth $W$ of the M bands increases from $W\simeq 9$ eV at $P_{\rm amb}$ to $W\simeq 12$ eV at $P=60$ GPa [see Fig. 3(a-g)]. Simultaneously, the bandwidth $W_{\rm AB}$ of the AB band increases from $W_{\rm AB}\simeq 4$ eV at $P_{\rm amb}$ to $W_{\rm AB}\simeq 5.5$ eV at $P=60$ GPa, which is caused by [M$t$]. Indeed, the increase in $|t_{1}^{l}|$ and thus $W_{\rm AB}\simeq 8|t_{1}|$ originates from the increase in $|t_{xp}^{l}|$, as discussed later in Sec. V.1.

For instance, [M$W$] is caused by $P_{a}$ rather than $P_{c}$ [see Fig. 3(h-n)], because the the AB bandwidth $W_{\rm AB}$ and $W$ are mainly determined by the overlap between Cu$3d_{x^{2}-y^{2}}$ and O$2p_{\sigma}$ ALWOs in a CuO₂ plane. This increase in the bandwidth with $P_{a}$ was also mentioned in Ref. Sakakibara et al., 2012 in the case of Hg1201. On the other hand, the application of $P_{c}$ shifts a few specific bands: Hg$5d$-like bands are shifted from $-4/-5$ eV at $P_{\rm amb}$ to $-7$ eV at $P_{c}=30$ GPa. However, $P_{c}$ does not modify $W_{\rm AB}$. Effects of uniaxial pressure on [M$\epsilon$] and [Mt] are also obviously and intuitively understood in a similar fashion: We clearly see in Fig. 3(o-q) that [M$\epsilon$] and [Mt] are caused by $P_{a}$ rather than $P_{c}$. For more details of the pressure effects, see Appendix E.1.

In this section, we discuss the mechanisms of (I,II) that are summarized in Table 3, and demonstrate that (I,II) are indeed physical and robust. We discuss mainly $|t_{1}^{\rm avg}|$ and $u^{\rm avg}$, and discuss briefly the difference between values in the IP and OP. A comparison with experiments will be made separately in Sec. VI.

Here, we discuss in detail how the experimental $P$ dependence of $T_{c}^{\rm opt}$ is predicted by considering (I,II) together with the assumptions (A,B) and the corrections (C,D) in Sec. I. We also discuss that (A) through (D) are all physically sound.

First, we emphasize that only by considering (A) and (B), the dome structure in the $P$ dependence of $T_{\rm c}^{\rm opt}$ is qualitatively understood. Since (B) implies that $F_{\rm SC}$ stays at a plateau region around the peak of parabolic $P$ dependence between $P_{\rm amb}$ and $P_{\rm opt}$ as is seen in Fig. 1(a). Then the dominant $P$ dependence of $F_{\rm SC}$ arises from $t_{1}$, which causes increase in $T_{\rm c}^{\rm est}$ in Eq. (1). On the other hand, $F_{\rm SC}$ rather rapidly decreases with increasing $P$ above $P_{\rm opt}$, which dominates over the effect of increase in $|t_{1}|$.

The location of Hg1223 assumed in (B) is justified from (C). Without (C), we would have $u^{\rm avg}\simeq 7.2$ at $P_{\rm amb}$ and $\simeq 6.8$ at $P_{\rm opt}$: Both values are below $u_{\rm opt}\simeq 8.0-8.5$, so that $F_{\rm SC}$ would quickly decrease with $P$, and (B) would not be valid. On the other hand, if we apply (C), we have $u\simeq 9.3\gtrsim u_{\rm opt}$ at $P_{\rm amb}$ and $\simeq 7.8\simeq u_{\rm opt}$ at $P_{\rm opt}$ [see Fig. 6(a)], so that (B) becomes valid.

Let us discuss more quantitative aspects. Although $F_{\rm SC}$ does not vary substantially with increasing $P$ below $P_{\rm opt}$, there is a small ($\simeq 5\%$) decrease in $F_{\rm SC}$ from $P_{\rm amb}$ to $P_{\rm opt}$ even after applying (C) [see Fig. 1(a)]. If we apply (C) without (D), the $\simeq 13-14\%$ increase in $|t_{1}^{\rm avg}|$ from $P_{\rm amb}$ to $P_{\rm opt}$ becomes the $\simeq 17\%$ increase in $|t_{1}|$. However, the increase in $T_{\rm c}^{\rm est}$ estimated from Eq.(1) is only $\simeq 10\%$ due to the $\simeq 5\%$ decrease in $F_{\rm SC}$. If we apply (D) after (C), the increase in $|t_{1}|$ becomes $\simeq 22\%$, so that the increase in $T_{\rm c}^{\rm est}$ becomes $\simeq 17\%$ and reproduces that in $T_{\rm c}^{\rm opt}$. Note that the quantitative agreement between the increases in $T_{\rm c}^{\rm est}$ and $T_{\rm c}^{\rm opt}$ is very good at $x_{\rm LRFB}^{\rm est}=0.95$ at least for small $P$. [see Fig. 1(b)]. For completeness, note that (D) has a limitation: It relies on the $a$ dependence of $|t_{1}|$ at the GGA+cRPA level. [For more details, see the last paragraph of Appendix D.]

Now, we argue that (A,B,C,D) are adequate from the physical point of view. On (A), it was shown that the scaling Eq. (1) is equally satisfied for $N_{\ell}=1,2$ and $\infty$ Schmid et al. (2023). This is on the one hand due to the fact that the interlayer coupling is small for all the cases and within a CuO₂ layer on the other hand, the superconductivity is mainly dependent on $t_{1}$ and $U$ only and the dependence on other parameters is weak within the realistic range. In the present case of Hg1223 with $N_{\ell}=3$, the interlayer coupling is again small. For instance, the ratio between the interlayer offsite Coulomb repulsion $V^{i,o}$ and $U^{\rm avg}$ is $V^{i,o}/U^{\rm avg}=0.13$ at $P_{\rm amb}$ and the superconducting strength is expected to be governed by the single layer physics, which is the same as the cases of $N_{\ell}=1,2$ and $\infty$.

On (B), the statement that $u_{{\rm c}GW{\rm-SIC+``LRFB"}}$ at $P_{\rm amb}$ is above $u_{\rm opt}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\ \simeq\ }8.0$-8.5 is indeed satisfied in the ab initio estimate by considering the correction (C). As mentioned earlier, the GGA+cRPA estimate is $u^{\rm avg}\simeq 7.2<u_{\rm opt}$ at $P_{\rm amb}$. However, (C) yields $u_{{\rm c}GW{\rm-SIC+``LRFB"}}\simeq 9.3\gtrsim u_{\rm opt}$ at $P_{\rm amb}$, and $u_{{\rm c}GW{\rm-SIC+``LRFB"}}\simeq 7.8\simeq u_{\rm opt}$ at $P_{\rm opt}$. [See Fig. 6(a).]

On (C), the calculation of $u_{{\rm c}GW{\rm-SIC+``LRFB"}}$ and $|t_{1}|_{{\rm c}GW{\rm-SIC+``LRFB"}}$ is detailed in Appendix C.

On (D), it is plausible that our calculation overestimates $a$ by $\simeq 0.05$ Å at $P>8.5$ GPa, because the same overestimation is already observed at $P=8.5$ GPa in Fig. 2. The derivation of improved $|t_{1}|_{{\rm c}GW{\rm-SIC+``LRFB"}}$ is detailed in Appendix D.

In Fig. 1(b), although the pressure dependence of $T_{c}^{\rm opt}$ is nicely reproduced for $P<P_{\rm opt}$, the estimated $T_{c}^{\rm est}$ decreases more rapidly than the experimental $T_{c}^{\rm opt}$ at $P>P_{\rm opt}$. The origin of this discrepancy is not clear at the moment. One possible origin is of course the uncertainty of the crystal parameters at high pressure because there exist no experimental data. Another origin would be the limitation of the inference for the LRFB correction taken simply by the constants $x_{\rm LRFB}$ and $y_{\rm LRFB}$. The third possibility is the possible inhomogeneity of the pressure in the experiments. The complete understanding of the origin of the discrepancy is an intriguing future issue.

We have proposed the microscopic mechanism for the dome-like $P$ dependence of $T_{c}^{\rm opt}$ in Hg1223 as the consequence of (I) and (II) obtained in this paper together with the assumptions (A,B) and the corrections (C,D) mentioned in Sec. II and supported in Sec. VI. We have also elucidated the microscopic origins of (I,II), which are summarized below.

The elucidation of the above mechanisms offers a platform for future studies on cuprates under $P$ and design of new compounds with even higher $T_{c}^{\rm opt}$: For instance, $T_{c}$ may be controlled by controlling $|t_{1}|$ via the cell parameter $a$. However, the increase in $|t_{1}|$ is a double-edged sword for the increase in $T_{c}$: On one hand, it is the direct origin of the increase in $T_{c}^{\rm opt}\propto|t_{1}|$ at $P<P_{\rm opt}$ in Hg1223. On the other hand, it is a prominent cause of the decrease in $u$ and thus $F_{\rm SC}$ and $T_{c}^{\rm opt}$ at $P>P_{\rm opt}$. Conversely, in the OP, the buckling of Cu-O-Cu bonds reduces $|t_{1}|$: This reduces $T_{c}^{\rm opt}\propto|t_{1}|$, but this may also increase $F_{\rm SC}$ and thus $T_{c}^{\rm opt}$ if the value of $u$ is in the weak-coupling region [$u<{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}7.5}$ in Fig. 6(b)]. For instance, the buckling may be identified as the main origin of the higher $T_{c}^{\rm opt}$ in Bi2212 ($T_{c}^{\rm opt}\simeq 84$ K Torrance et al. (1988)) compared to Bi2201 ($T_{c}^{\rm opt}\simeq 6$ K Torrance et al. (1988)): The buckling reduces $|t_{1}|$ and thus increases $u$ in Bi2212 with respect to Bi2201 Morée et al. (2022), so that Bi2212 is near the optimal region whereas Bi2201 is in the weak-coupling region Schmid et al. (2023). This explains the larger $|t_{1}|F_{\rm SC}$ in Bi2212 Schmid et al. (2023) despite the smaller $|t_{1}|$.

We thank Michael Thobias Schmid for useful discussions. This work was supported by MEXT as Program for Promoting Researches on the Supercomputer Fugaku (Basic Science for Emergence and Functionality in Quantum Matter ­Innovative Strongly-Correlated Electron Science by Integration of Fugaku and Frontier Experiments­, JPMXP1020200104 and JPMXP1020230411) and used computational resources of supercomputer Fugaku provided by the RIKEN Center for Computational Science (Project ID: hp200132, hp210163, hp220166 and hp230169). We also acknowledge the financial support of JSPS Kakenhi Grant-in-Aid for Transformative Research Areas, Grants Nos. JP22H05111 and JP22H05114 (“Foundation of Machine Learning Physics”). Part of the results were obtained under the Special Postdoctoral Researcher Program at RIKEN. The left panel of Fig.  2 was drawn by using software VESTA Momma and Izumi (2011).