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<p class="MsoNormal" align="center" style="text-align:center"><a name="_Hlk536455403"><img width="624" height="198" style="width:6.5in;height:2.0625in" id="Picture_x0020_2" src="cid:image001.png@01D95591.5CFBE050"><o:p></o:p></a></p>

<p class="Default"><span style="mso-bookmark:_Hlk536455403"><o:p> </o:p></span></p>

<p class="MsoNormal" align="center" style="text-align:center"><span style="mso-bookmark:_Hlk536455403"><b><span style="font-size:26.0pt;font-family:"Times New Roman",serif;color:#006FC0">Special Condensed Matter Seminar<br>

</span></b></span><span style="mso-bookmark:_Hlk536455403"><i><span style="font-size:26.0pt;font-family:"Times New Roman",serif">Unconventional Flux Quantization of Topological Superconductors<o:p></o:p></span></i></span></p>

<p class="MsoNormal" align="center" style="text-align:center"><span style="mso-bookmark:_Hlk536455403"><b><i><span style="font-size:22.0pt;font-family:"Times New Roman",serif">Yufan Li</span></i></b></span><span style="mso-bookmark:_Hlk536455403"><b><i><span style="font-size:18.0pt;font-family:"Times New Roman",serif"><br>

</span></i></b></span><span style="mso-bookmark:_Hlk536455403"><i><span style="font-size:18.0pt;font-family:"Times New Roman",serif">Department of Physics, the Chinese University of Hong Kong<o:p></o:p></span></i></span></p>

<p class="MsoNormal" style="line-height:115%"><span style="mso-bookmark:_Hlk536455403"><b><span style="font-size:12.0pt;line-height:115%;font-family:"Times New Roman",serif;color:black;background:white">Abstract:</span></b></span><span style="mso-bookmark:_Hlk536455403"><span style="font-size:12.0pt;line-height:115%;font-family:"Times New Roman",serif;color:black;background:white">

</span></span><span style="mso-bookmark:_Hlk536455403"><span style="font-size:10.0pt;line-height:115%;font-family:"Times New Roman",serif">The most distinctive feature of superconductivity is the presence of the complex-valued many-particle wave function that

 sustains phase coherence over macroscopic distances. One of its experimental manifestations is the flux quantization in a superconducting ring. The single-value nature of the wave function dictates a universal phase change of 2</span></span><span style="mso-bookmark:_Hlk536455403"><m:r></m:r></span><!--[if gte msEquation 12]><m:oMath><span style='mso-bookmark:_Hlk536455403'><m:r><i><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>π</span></i></m:r></span></m:oMath><![endif]--><![if !msEquation]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-ligatures:standardcontextual;mso-fareast-language:EN-US"><img width="8" height="16" style="width:.0833in;height:.1666in" id="_x0000_i1025" src="cid:image002.png@01D95591.5CFBE050"></span><![endif]><span style="mso-bookmark:_Hlk536455403"><span style="font-size:10.0pt;line-height:115%;font-family:"Times New Roman",serif">

 for any closed path around the ring. As a result, the fluxoid can only take on quantized values in integer steps of

</span></span><span style="mso-bookmark:_Hlk536455403"></span><!--[if gte msEquation 12]><m:oMath><span style='mso-bookmark:_Hlk536455403'></span><m:sSub><m:sSubPr><span style='font-family:"Cambria Math",serif;mso-fareast-language:ZH-CN;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><span style='mso-bookmark:_Hlk536455403'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>Φ</span></m:r></span><span style='mso-bookmark:_Hlk536455403'></span></m:e><m:sub><span style='mso-bookmark:_Hlk536455403'><m:r><i><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>0</span></i></m:r></span><span style='mso-bookmark:_Hlk536455403'></span></m:sub></m:sSub><span style='mso-bookmark:_Hlk536455403'><m:r><i><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>=h/2</span></i></m:r><m:r><i><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>e</span></i></m:r></span></m:oMath><![endif]--><![if !msEquation]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-ligatures:standardcontextual;mso-fareast-language:EN-US"><img width="62" height="17" style="width:.6458in;height:.177in" id="_x0000_i1025" src="cid:image003.png@01D95591.5CFBE050"></span><![endif]><span style="mso-bookmark:_Hlk536455403"><span style="font-size:10.0pt;line-height:115%;font-family:"Times New Roman",serif">,

 where h is the Planck constant and the denominator of </span></span><span style="mso-bookmark:_Hlk536455403"><m:r></m:r></span><!--[if gte msEquation 12]><m:oMath><span style='mso-bookmark:_Hlk536455403'><m:r><i><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>2</span></i></m:r><m:r><i><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>e</span></i></m:r></span></m:oMath><![endif]--><![if !msEquation]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-ligatures:standardcontextual;mso-fareast-language:EN-US"><img width="14" height="16" style="width:.1458in;height:.1666in" id="_x0000_i1025" src="cid:image004.png@01D95591.5CFBE050"></span><![endif]><span style="mso-bookmark:_Hlk536455403"><span style="font-size:10.0pt;line-height:115%;font-family:"Times New Roman",serif">

 is the signature of electron pairing. There are two hallmarks of the well established integer flux quantization: the quantization is

</span></span><span style="mso-bookmark:_Hlk536455403"></span><!--[if gte msEquation 12]><m:oMath><span style='mso-bookmark:_Hlk536455403'></span><m:sSub><m:sSubPr><span style='font-family:"Cambria Math",serif;mso-fareast-language:ZH-CN;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><span style='mso-bookmark:_Hlk536455403'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>Φ</span></m:r></span><span style='mso-bookmark:_Hlk536455403'></span></m:e><m:sub><span style='mso-bookmark:_Hlk536455403'><m:r><i><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>0</span></i></m:r></span><span style='mso-bookmark:_Hlk536455403'></span></m:sub></m:sSub></m:oMath><![endif]--><![if !msEquation]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-ligatures:standardcontextual;mso-fareast-language:EN-US"><img width="16" height="17" style="width:.1666in;height:.177in" id="_x0000_i1025" src="cid:image005.png@01D95591.5CFBE050"></span><![endif]><span style="mso-bookmark:_Hlk536455403"><span style="font-size:10.0pt;line-height:115%;font-family:"Times New Roman",serif">

 periodic, and energy minima are achieved at </span></span><span style="mso-bookmark:_Hlk536455403"><m:r><m:rPr><m:scr m:val="roman" /><m:sty m:val="p" /></m:rPr></m:r></span><!--[if gte msEquation 12]><m:oMath><span style='mso-bookmark:_Hlk536455403'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>Φ</span></m:r></span></m:oMath><![endif]--><![if !msEquation]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-ligatures:standardcontextual;mso-fareast-language:EN-US"><img width="10" height="16" style="width:.1041in;height:.1666in" id="_x0000_i1025" src="cid:image006.png@01D95591.5CFBE050"></span><![endif]><span style="mso-bookmark:_Hlk536455403"><span style="font-size:10.0pt;line-height:115%;font-family:"Times New Roman",serif">

= n <span style="mso-bookmark:_Hlk536455403"><![if !msEquation]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-ligatures:standardcontextual;mso-fareast-language:EN-US"><img width="16" height="17" style="width:.1666in;height:.177in" id="_x0000_i1025" src="cid:image005.png@01D95591.5CFBE050"><![endif]><span style="mso-bookmark:_Hlk536455403"><span style="font-size:10.0pt;line-height:115%;font-family:"Times New Roman",serif">.

 Our experiments in topological superconductors found that both hallmarks of flux quantization may be violated. For topological superconductors, the spin-triplet pairing state may give rise to half-integer flux quantization

</span></span><span style="mso-bookmark:_Hlk536455403"><m:r><m:rPr><m:scr m:val="roman" /><m:sty m:val="p" /></m:rPr></m:r></span><!--[if gte msEquation 12]><m:oMath><span style='mso-bookmark:_Hlk536455403'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>Φ'</span></m:r></span></m:oMath><![endif]--><![if !msEquation]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-ligatures:standardcontextual;mso-fareast-language:EN-US"><img width="14" height="16" style="width:.1458in;height:.1666in" id="_x0000_i1025" src="cid:image007.png@01D95591.5CFBE050"></span><![endif]><span style="mso-bookmark:_Hlk536455403"><span style="font-size:10.0pt;line-height:115%;font-family:"Times New Roman",serif">

 = (<i>n+</i>�) </span></span><span style="mso-bookmark:_Hlk536455403"></span><!--[if gte msEquation 12]><m:oMath><span style='mso-bookmark:_Hlk536455403'></span><m:sSub><m:sSubPr><span style='font-family:"Cambria Math",serif;mso-fareast-language:ZH-CN;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><span style='mso-bookmark:_Hlk536455403'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>Φ</span></m:r></span><span style='mso-bookmark:_Hlk536455403'></span></m:e><m:sub><span style='mso-bookmark:_Hlk536455403'><m:r><i><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>0</span></i></m:r></span><span style='mso-bookmark:_Hlk536455403'></span></m:sub></m:sSub></m:oMath><![endif]--><![if !msEquation]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-ligatures:standardcontextual;mso-fareast-language:EN-US"><img width="16" height="17" style="width:.1666in;height:.177in" id="_x0000_i1025" src="cid:image005.png@01D95591.5CFBE050"></span><![endif]><span style="mso-bookmark:_Hlk536455403"><span style="font-size:10.0pt;line-height:115%;font-family:"Times New Roman",serif">,

due to the odd parity of the superconducting gap function, or <span style="mso-bookmark:_Hlk536455403"><![if !msEquation]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-ligatures:standardcontextual;mso-fareast-language:EN-US"><img width="58" height="17" style="width:.6041in;height:.177in" id="_x0000_i1025" src="cid:image008.png@01D95591.5CFBE050"><![endif]><span style="mso-bookmark:_Hlk536455403"><span style="font-size:10.0pt;line-height:115%;font-family:"Times New Roman",serif">.

 The half-quantum flux has been observed in polycrystalline rings of superconducting

</span></span><span style="mso-bookmark:_Hlk536455403"><m:r></m:r></span><!--[if gte msEquation 12]><m:oMath><span style='mso-bookmark:_Hlk536455403'><m:r><i><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>β</span></i></m:r></span></m:oMath><![endif]--><![if !msEquation]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-ligatures:standardcontextual;mso-fareast-language:EN-US"><img width="8" height="16" style="width:.0833in;height:.1666in" id="_x0000_i1025" src="cid:image009.png@01D95591.5CFBE050"></span><![endif]><span style="mso-bookmark:_Hlk536455403"><span style="font-size:10.0pt;line-height:115%;font-family:"Times New Roman",serif">-Bi<sub>2</sub>Pd</span></span><span style="mso-bookmark:_Hlk536455403"></span><span style="mso-bookmark:_Hlk536455403"><span style="font-size:10.0pt;line-height:115%;font-family:"Times New Roman",serif"> [1]</span></span><span style="mso-bookmark:_Hlk536455403"></span><span style="mso-bookmark:_Hlk536455403"><span style="font-size:10.0pt;line-height:115%;font-family:"Times New Roman",serif">

and �-BiPd<span style="mso-bookmark:_Hlk536455403"><span style="mso-bookmark:_Hlk536455403"><span style="font-size:10.0pt;line-height:115%;font-family:"Times New Roman",serif"> [2]<span style="mso-bookmark:_Hlk536455403"><span style="mso-bookmark:_Hlk536455403"><span style="font-size:10.0pt;line-height:115%;font-family:"Times New Roman",serif">.

 Moreover, we have observed multiple quantization periodicities of of </span></span><span style="mso-bookmark:_Hlk536455403"><m:r></m:r></span><!--[if gte msEquation 12]><m:oMath><span style='mso-bookmark:_Hlk536455403'><m:r><i><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>2</span></i></m:r></span><span style='mso-bookmark:_Hlk536455403'></span><m:sSub><m:sSubPr><span style='font-family:"Cambria Math",serif;mso-fareast-language:ZH-CN;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><span style='mso-bookmark:_Hlk536455403'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>Φ</span></m:r></span><span style='mso-bookmark:_Hlk536455403'></span></m:e><m:sub><span style='mso-bookmark:_Hlk536455403'><m:r><i><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>0</span></i></m:r></span><span style='mso-bookmark:_Hlk536455403'></span></m:sub></m:sSub></m:oMath><![endif]--><![if !msEquation]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-ligatures:standardcontextual;mso-fareast-language:EN-US"><img width="24" height="16" style="width:.25in;height:.1666in" id="_x0000_i1025" src="cid:image010.png@01D95591.5CFBE050"></span><![endif]><span style="mso-bookmark:_Hlk536455403"><span style="font-size:10.0pt;line-height:115%;font-family:"Times New Roman",serif">,

</span></span><span style="mso-bookmark:_Hlk536455403"><m:r></m:r></span><!--[if gte msEquation 12]><m:oMath><span style='mso-bookmark:_Hlk536455403'><m:r><i><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>3</span></i></m:r></span><span style='mso-bookmark:_Hlk536455403'></span><m:sSub><m:sSubPr><span style='font-family:"Cambria Math",serif;mso-fareast-language:ZH-CN;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><span style='mso-bookmark:_Hlk536455403'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>Φ</span></m:r></span><span style='mso-bookmark:_Hlk536455403'></span></m:e><m:sub><span style='mso-bookmark:_Hlk536455403'><m:r><i><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>0</span></i></m:r></span><span style='mso-bookmark:_Hlk536455403'></span></m:sub></m:sSub></m:oMath><![endif]--><![if !msEquation]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-ligatures:standardcontextual;mso-fareast-language:EN-US"><img width="24" height="16" style="width:.25in;height:.1666in" id="_x0000_i1025" src="cid:image011.png@01D95591.5CFBE050"></span><![endif]><span style="mso-bookmark:_Hlk536455403"><span style="font-size:10.0pt;line-height:115%;font-family:"Times New Roman",serif">

 and </span></span><span style="mso-bookmark:_Hlk536455403"><m:r></m:r></span><!--[if gte msEquation 12]><m:oMath><span style='mso-bookmark:_Hlk536455403'><m:r><i><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>4</span></i></m:r></span><span style='mso-bookmark:_Hlk536455403'></span><m:sSub><m:sSubPr><span style='font-family:"Cambria Math",serif;mso-fareast-language:ZH-CN;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><span style='mso-bookmark:_Hlk536455403'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>Φ</span></m:r></span><span style='mso-bookmark:_Hlk536455403'></span></m:e><m:sub><span style='mso-bookmark:_Hlk536455403'><m:r><i><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>0</span></i></m:r></span><span style='mso-bookmark:_Hlk536455403'></span></m:sub></m:sSub></m:oMath><![endif]--><![if !msEquation]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-ligatures:standardcontextual;mso-fareast-language:EN-US"><img width="24" height="16" style="width:.25in;height:.1666in" id="_x0000_i1025" src="cid:image012.png@01D95591.5CFBE050"></span><![endif]><span style="mso-bookmark:_Hlk536455403"><span style="font-size:10.0pt;line-height:115%;font-family:"Times New Roman",serif">,

<span style="color:black">in addition to </span>the conventional </span></span><span style="mso-bookmark:_Hlk536455403"></span><!--[if gte msEquation 12]><m:oMath><span style='mso-bookmark:_Hlk536455403'></span><m:sSub><m:sSubPr><span style='font-family:"Cambria Math",serif;mso-fareast-language:ZH-CN;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><span style='mso-bookmark:_Hlk536455403'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>Φ</span></m:r></span><span style='mso-bookmark:_Hlk536455403'></span></m:e><m:sub><span style='mso-bookmark:_Hlk536455403'><m:r><i><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>0</span></i></m:r></span><span style='mso-bookmark:_Hlk536455403'></span></m:sub></m:sSub></m:oMath><![endif]--><![if !msEquation]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-ligatures:standardcontextual;mso-fareast-language:EN-US"><img width="16" height="17" style="width:.1666in;height:.177in" id="_x0000_i1025" src="cid:image005.png@01D95591.5CFBE050"></span><![endif]><span style="mso-bookmark:_Hlk536455403"><span style="font-size:10.0pt;line-height:115%;font-family:"Times New Roman",serif">,

 in mesoscopic rings of epitaxial </span></span><span style="mso-bookmark:_Hlk536455403"><m:r></m:r></span><!--[if gte msEquation 12]><m:oMath><span style='mso-bookmark:_Hlk536455403'><m:r><i><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math",serif'>β</span></i></m:r></span></m:oMath><![endif]--><![if !msEquation]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;position:relative;top:2.5pt;mso-text-raise:-2.5pt;mso-ligatures:standardcontextual;mso-fareast-language:EN-US"><img width="8" height="16" style="width:.0833in;height:.1666in" id="_x0000_i1025" src="cid:image009.png@01D95591.5CFBE050"></span><![endif]><span style="mso-bookmark:_Hlk536455403"><span style="font-size:10.0pt;line-height:115%;font-family:"Times New Roman",serif">-Bi<sub>2</sub>Pd</span></span><span style="mso-bookmark:_Hlk536455403"></span><span style="mso-bookmark:_Hlk536455403"><span style="font-size:10.0pt;line-height:115%;font-family:"Times New Roman",serif"> [3]</span></span><span style="mso-bookmark:_Hlk536455403"></span><span style="mso-bookmark:_Hlk536455403"><span style="font-size:10.0pt;line-height:115%;font-family:"Times New Roman",serif">,

 which may reflect its topological properties. The highly unconventional flux quantization are distinctive experimental features for topological superconductors.

<o:p></o:p></span></span></p>

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