The superconducting (SC) cuprate <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:msub><a:mi>HgBa</a:mi><a:mn>2</a:mn></a:msub><a:msub><a:mi>Ca</a:mi><a:mn>2</a:mn></a:msub><a:msub><a:mi>Cu</a:mi><a:mn>3</a:mn></a:msub><a:msub><a:mi mathvariant="normal">O</a:mi><a:mn>8</a:mn></a:msub></a:mrow></a:math> (Hg1223) has the highest SC transition temperature <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msub><c:mi>T</c:mi><c:mi>c</c:mi></c:msub></c:math> among cuprates at ambient pressure <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:msub><d:mi>P</d:mi><d:mi>amb</d:mi></d:msub></d:math>, namely, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:msubsup><e:mi>T</e:mi><e:mrow><e:mi>c</e:mi></e:mrow><e:mi>opt</e:mi></e:msubsup><e:mo>≃</e:mo><e:mn>138</e:mn></e:mrow></e:math> K experimentally at the optimal hole doping concentration. <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:msubsup><f:mi>T</f:mi><f:mrow><f:mi>c</f:mi></f:mrow><f:mi>opt</f:mi></f:msubsup></f:math> further increases under pressure <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mi>P</g:mi></g:math> and reaches 164 K at optimal pressure <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:msub><h:mi>P</h:mi><h:mi>opt</h:mi></h:msub><h:mo>≃</h:mo><h:mn>30</h:mn></h:mrow></h:math> GPa, then <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:msubsup><i:mi>T</i:mi><i:mrow><i:mi>c</i:mi></i:mrow><i:mi>opt</i:mi></i:msubsup></i:math> decreases with increasing <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mrow><j:mi>P</j:mi><j:mo>></j:mo><j:msub><j:mi>P</j:mi><j:mi>opt</j:mi></j:msub></j:mrow></j:math> generating a dome structure [Gao , ]. This nontrivial and nonmonotonic <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mi>P</k:mi></k:math> dependence of <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:msubsup><l:mi>T</l:mi><l:mrow><l:mi>c</l:mi></l:mrow><l:mi>opt</l:mi></l:msubsup></l:math> calls for a theoretical understanding and mechanism. To answer this open question, we consider the low-energy effective Hamiltonian (LEH) for the antibonding (AB) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>Cu</m:mi><m:mn>3</m:mn><m:msub><m:mi>d</m:mi><m:mrow><m:msup><m:mi>x</m:mi><m:mn>2</m:mn></m:msup><m:mo>−</m:mo><m:msup><m:mi>y</m:mi><m:mn>2</m:mn></m:msup></m:mrow></m:msub><m:mo>/</m:mo><m:mi mathvariant="normal">O</m:mi><m:mn>2</m:mn><m:msub><m:mi>p</m:mi><m:mi>σ</m:mi></m:msub></m:mrow></m:math> band derived generally for the cuprates. In the AB LEH for cuprates with <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"><o:mrow><o:msub><o:mi>N</o:mi><o:mi>ℓ</o:mi></o:msub><o:mo>≤</o:mo><o:mn>2</o:mn></o:mrow></o:math> laminated <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"><p:msub><p:mi>CuO</p:mi><p:mn>2</p:mn></p:msub></p:math> planes between block layers, it was proposed that <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"><q:msubsup><q:mi>T</q:mi><q:mi>c</q:mi><q:mi>opt</q:mi></q:msubsup></q:math> is determined by a universal scaling <r:math xmlns:r="http://www.w3.org/1998/Math/MathML"><r:mrow><r:msubsup><r:mi>T</r:mi><r:mrow><r:mi>c</r:mi></r:mrow><r:mi>opt</r:mi></r:msubsup><r:mo>≃</r:mo><r:mn>0.16</r:mn><r:mrow><r:mo>|</r:mo><r:msub><r:mi>t</r:mi><r:mn>1</r:mn></r:msub><r:mo>|</r:mo></r:mrow><r:msub><r:mi>F</r:mi><r:mi>SC</r:mi></r:msub></r:mrow></r:math> [Schmid , ], where <s:math xmlns:s="http://www.w3.org/1998/Math/MathML"><s:msub><s:mi>t</s:mi><s:mn>1</s:mn></s:msub></s:math> is the nearest-neighbor hopping, and the SC order parameter at optimal hole doping <t:math xmlns:t="http://www.w3.org/1998/Math/MathML"><t:msub><t:mi>F</t:mi><t:mi>SC</t:mi></t:msub></t:math> mainly depends on the ratio <u:math xmlns:u="http://www.w3.org/1998/Math/MathML"><u:mrow><u:mrow><u:mi>u</u:mi><u:mo>=</u:mo><u:mi>U</u:mi><u:mo>/</u:mo><u:mo>|</u:mo></u:mrow><u:msub><u:mi>t</u:mi><u:mn>1</u:mn></u:msub><u:mrow><u:mo>|</u:mo></u:mrow></u:mrow></u:math> where <v:math xmlns:v="http://www.w3.org/1998/Math/MathML"><v:mi>U</v:mi></v:math> is the onsite effective Coulomb repulsion: The <w:math xmlns:w="http://www.w3.org/1998/Math/MathML"><w:mi>u</w:mi></w:math> dependence of <x:math xmlns:x="http://www.w3.org/1998/Math/MathML"><x:msub><x:mi>F</x:mi><x:mi>SC</x:mi></x:msub></x:math> has a peak at <y:math xmlns:y="http://www.w3.org/1998/Math/MathML"><y:mrow><y:msub><y:mi>u</y:mi><y:mi>opt</y:mi></y:msub><y:mo>≃</y:mo><y:mn>8.5</y:mn></y:mrow></y:math> and a steep decrease with decreasing <z:math xmlns:z="http://www.w3.org/1998/Math/MathML"><z:mi>u</z:mi></z:math> in the region <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML"><ab:mrow><ab:mi>u</ab:mi><ab:mo><</ab:mo><ab:msub><ab:mi>u</ab:mi><ab:mi>opt</ab:mi></ab:msub></ab:mrow></ab:math> irrespective of materials dependent on other parameters. In this paper, we show that <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML"><bb:mrow><bb:mrow><bb:mo>|</bb:mo></bb:mrow><bb:msub><bb:mi>t</bb:mi><bb:mn>1</bb:mn></bb:msub><bb:mrow><bb:mo>|</bb:mo></bb:mrow></bb:mrow></bb:math> increases with <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML"><cb:mi>P</cb:mi></cb:math>, whereas <db:math xmlns:db="http://www.w3.org/1998/Math/MathML"><db:mi>u</db:mi></db:math> decreases with <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML"><eb:mi>P</eb:mi></eb:math> in the Hamiltonian of Hg1223. Based on these facts, we show that the domelike <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML"><fb:mi>P</fb:mi></fb:math> dependence of <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML"><gb:msubsup><gb:mi>T</gb:mi><gb:mrow><gb:mi>c</gb:mi></gb:mrow><gb:mi>opt</gb:mi></gb:msubsup></gb:math> can emerge at least qualitatively if we assume Hg1223 with <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML"><hb:mrow><hb:msub><hb:mi>N</hb:mi><hb:mi>ℓ</hb:mi></hb:msub><hb:mo>=</hb:mo><hb:mn>3</hb:mn></hb:mrow></hb:math> follows the same universal scaling for <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML"><ib:msubsup><ib:mi>T</ib:mi><ib:mrow><ib:mi>c</ib:mi></ib:mrow><ib:mi>opt</ib:mi></ib:msubsup></ib:math>, and Hg1223 is located at the slightly strong coupling region <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML"><jb:mrow><jb:mi>u</jb:mi><jb:mo>≳</jb:mo><jb:msub><jb:mi>u</jb:mi><jb:mi>opt</jb:mi></jb:msub></jb:mrow></jb:math> at <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML"><kb:msub><kb:mi>P</kb:mi><kb:mi>amb</kb:mi></kb:msub></kb:math> and <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML"><lb:mrow><lb:mi>u</lb:mi><lb:mo>≃</lb:mo><lb:msub><lb:mi>u</lb:mi><lb:mi>opt</lb:mi></lb:msub></lb:mrow></lb:math> at <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML"><mb:msub><mb:mi>P</mb:mi><mb:mi>opt</mb:mi></mb:msub></mb:math> by taking account of expected corrections to our calculation. The consequence of these assumptions is the following: With increasing <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML"><nb:mi>P</nb:mi></nb:math> within the range <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML"><ob:mrow><ob:mi>P</ob:mi><ob:mo><</ob:mo><ob:msub><ob:mi>P</ob:mi><ob:mi>opt</ob:mi></ob:msub></ob:mrow></ob:math>, the increase in <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML"><pb:msubsup><pb:mi>T</pb:mi><pb:mrow><pb:mi>c</pb:mi></pb:mrow><pb:mi>opt</pb:mi></pb:msubsup></pb:math> is accounted for by the increase in <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML"><qb:mrow><qb:mrow><qb:mo>|</qb:mo></qb:mrow><qb:msub><qb:mi>t</qb:mi><qb:mn>1</qb:mn></qb:msub><qb:mrow><qb:mo>|</qb:mo></qb:mrow></qb:mrow></qb:math>, whereas <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML"><rb:msub><rb:mi>F</rb:mi><rb:mi>SC</rb:mi></rb:msub></rb:math> is insensitive to the decrease in <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML"><sb:mi>u</sb:mi></sb:math> around <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML"><tb:mrow><tb:mo>≃</tb:mo><tb:msub><tb:mi>u</tb:mi><tb:mi>opt</tb:mi></tb:msub></tb:mrow></tb:math> and hence to <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML"><ub:mi>P</ub:mi></ub:math> as well. At <vb:math xmlns:vb="http://www.w3.org/1998/Math/MathML"><vb:mrow><vb:mi>P</vb:mi><vb:mo>></vb:mo><vb:msub><vb:mi>P</vb:mi><vb:mi>opt</vb:mi></vb:msub></vb:mrow></vb:math>, the decrease in <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML"><wb:msubsup><wb:mi>T</wb:mi><wb:mrow><wb:mi>c</wb:mi></wb:mrow><wb:mi>opt</wb:mi></wb:msubsup></wb:math> is accounted for by the decrease in <xb:math xmlns:xb="http://www.w3.org/1998/Math/MathML"><xb:mi>u</xb:mi></xb:math> below <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML"><yb:msub><yb:mi>u</yb:mi><yb:mi>opt</yb:mi></yb:msub></yb:math>, which causes a rapid decrease in <zb:math xmlns:zb="http://www.w3.org/1998/Math/MathML"><zb:msub><zb:mi>F</zb:mi><zb:mi>SC</zb:mi></zb:msub></zb:math> dominating over the increase in <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML"><ac:mrow><ac:mrow><ac:mo>|</ac:mo></ac:mrow><ac:msub><ac:mi>t</ac:mi><ac:mn>1</ac:mn></ac:msub><ac:mrow><ac:mo>|</ac:mo></ac:mrow></ac:mrow></ac:math>. We further argue the appropriateness of these assumptions based on the insight from studies on other cuprate compounds in the literature. In addition, we discuss the dependencies of <bc:math xmlns:bc="http://www.w3.org/1998/Math/MathML"><bc:mi>u</bc:mi></bc:math> and <cc:math xmlns:cc="http://www.w3.org/1998/Math/MathML"><cc:mrow><cc:mrow><cc:mo>|</cc:mo></cc:mrow><cc:msub><cc:mi>t</cc:mi><cc:mn>1</cc:mn></cc:msub><cc:mrow><cc:mo>|</cc:mo></cc:mrow></cc:mrow></cc:math> on each crystal parameter (CP), which provides hints for design of even higher <dc:math xmlns:dc="http://www.w3.org/1998/Math/MathML"><dc:msubsup><dc:mi>T</dc:mi><dc:mrow><dc:mi>c</dc:mi></dc:mrow><dc:mi>opt</dc:mi></dc:msubsup></dc:math> materials. Published by the American Physical Society 2024
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