To See a World in a Grain of Sand ── The Scientific Life of Shoucheng Zhang
Preface
Our friend and colleague, Prof. Shoucheng Zhang, passed away in 2018, which was a great loss for the entire physics community. For all of us who knew Shoucheng, it is difficult to overcome the sadness and shock of his early departure. However, we are very fortunate that Shoucheng has left us such a rich legacy and so many memories in his 55 years of life as a valuable friend, a world-leading physicist, a remarkable advisor, and a great thinker. We hope this small exhibition will provide some snapshots of Shoucheng’s wonderful scientific life.
---Biao Lian, Chao-Xing Liu, Xiaoqi Sun, Steven Kivelson, Eugene Demler and Xiao-Liang Qi Shoucheng Zhang A brief history 1963 - 2018
1963 • Shoucheng was born in Shanghai, China to Manfan Ding and Hongfan Zhang
1978 • Admitted to Fudan University 1980 • Began studying at Free University of Berlin 1983 • Graduated from Free University of Berlin • Started Ph.D. at Stony Brook University 1986 • Started working on condensed matter physics 1987 • Received Ph.D. and started postdoc at ITP, UC Santa Barbara • Married his childhood sweetheart, Barbara Yu • Chern-Simons theory of fractional quantum Hall states 1989 • Started at IBM Almaden Research Center, San Jose • Global phase diagram of fractional quantum Hall states 1993 • Joined the faculty of Stanford University • Shoucheng’s son Brian was born 1996 • Shoucheng’s daughter Stephanie was born 1997 • Proposed the SO(5) theory of high Tc superconductors
2001 • Generalized quantum Hall effect to 4 dimensions 2003 • Proposed the intrinsic spin Hall effect 2004 • Early models of quantum spin Hall effect (2004- 2005 2005) 2006 • Predicted the quantum spin Hall effect in HgTe 2007 • Quantum spin Hall effect realized in HgTe Honors and Awards 2008 • Topological magneto-electric effect • Fellow, American Physical Society (2005) 2009 • Prediction and realization of Bi2Te3 family of topological insulators • Guggenheim Fellow (2007) • Prediction of new topological superconductors • Alexander von Humboldt Research Prize (2009) 2010 • Prediction of quantum anomalous Hall effect • Europhysics Prize (2010) 2013 • Quantum anomalous Hall effect realized • Fellow, American Academy of Arts and • Founded venture capital firm DHVC Sciences (2011) • Oliver Buckley Prize (2012) • Celebrated 50th birthday with his friends, • Dirac Medal (2012) current and past group members. • Physics Frontier Prize (2013) 2017 • Experimental evidence of chiral topological superconductor reported • Foreign Member, Chinese Academy of Sciences (2013) 2018 • Founded the Stanford Center for Topological Quantum Physics • Benjamin Franklin Medal (2015) • Passed away on December 1, 2018 • Member, National Academy of Sciences (2015) Early Experience
Childhood and School Age Upon observing the equations on the tombstones of Max Planck, Otto Hahn, and Max Born, Shoucheng wrote that Shoucheng Zhang (张首晟) was born in he would “spend my energy on the pursuit of science, 1963 in Shanghai, China. In his hoping that I too would leave behind a life’s work that childhood, Shoucheng already showed could be summed up in a simple equation.” exceptional talent and a strong interest in various fields of knowledge. In the Ph.D. at Stony Brook University attic of Shoucheng’s home there were many books on art, history, philosophy, Shoucheng began his Ph.D. and science left by his grandfather and studies on supergravity at others of his parent’s generation. These Shoucheng at the State University of New included books on the philosophy of age 2 York at Stony Brook in 1983, Russell and Kant and the art of da Vinci and Rodin. advised by Peter van Shoucheng’s favorite activity after school was to read Nieuwenhuizen. In the final books in the attic. In an era when educational resources year of Shoucheng’s Ph.D. were scarce, these books opened a new world to him. In Shoucheng with his advisor (1986-1987), following Prof. Peter van Nieuwenhuizen 1976, Shoucheng’s father bought Shoucheng a set of high Chen-Ning Yang’s suggestion, (middle) and his future wife he started shifting his school textbooks on mathematics, physics, and chemistry. Barbara (left) at their He was immediately attracted by the amazing beauty of graduation ceremony, 1987 science. research direction to College times in Shanghai and Berlin condensed matter physics. He began a collaboration with Steven Kivelson, a In 1977, the National Higher faculty member at Stony Education Entrance Exam of Brook who later became China was restarted after the Barbara, Shoucheng, Chen- Shoucheng’s colleague and end of the Cultural Ning Yang, and Shoucheng’s lifelong friend at Stanford. Revolution. Without father Hongfan, 1987 attending high school, Santa Barbara, IBM and Stanford Shoucheng took the first exam and got admitted to After receiving his Ph.D. in 1987 the Physics Department of Shoucheng (third from the from Stony Brook, Shoucheng Fudan University in Shanghai right in the second row) and became a postdoctoral fellow at in 1978. At the age of 15, he his roommates at Fudan the Institute of Theoretical was the youngest student in University, 1978 Physics (ITP) in UC Santa his class. One year later, in Barbara. In 1987, he married his recognition of his excellent academic performance, childhood sweetheart Barbara Shoucheng and J. Robert Shoucheng was selected for an exchange program to study Yu. He then joined IBM Almaden Schrieffer at Santa Barbara, abroad at the Free University of Berlin, where he received Research Center as a Research 1989 his Diplom-Physiker (Bachelor of Science degree) in 1983. Staff Member from 1989 to 1993. Thereafter, he joined the During his college time in faculty of Stanford University, which remained his academic Germany, besides studying home for the rest of his life. physics, Shoucheng also had During the early years of his rich exposure to German career, Shoucheng focused culture. On a trip back from primarily on the theory of Bonn to Berlin in 1981, he and fractional quantum Hall some friends visited the states and high temperature Stadtfriedhof cemetery in superconductivity. His Shoucheng in front of the Göttingen that houses the scientific achievements in grave of nuclear physicist different areas of physics Otto Hahn in Göttingen, 1981 graves of many scientists. Even A group photo of Shoucheng and after many years, Shoucheng his students after a class at will be overviewed in the often remembered this visit as a source of inspiration. Stanford in 1994 following pages. Fractional Quantum Hall Effect
The Fractional Quantum Hall Effect The key idea of this theory is flux attachment, which When Shoucheng began to study condensed matter is achieved by introducing physics in the late 1980s, one of the first topics he became a gauge field with a interested in was the fractional quantum Hall effect Chern-Simons term. The (FQHE). dynamics of this gauge field attaches 2푘 − 1 magnetic fluxes to each The FQHE, experimentally discovered in 1982 by D. C. Tsui, Shoucheng with James H. electron, which H. L. Stormer, and A. C. Gossard, is a remarkable quantum Simons (middle), one of the phenomenon of two dimensional (2D) metals near the founders of Chern-Simons transmutes them into absolute zero of temperature in a strong magnetic field 퐵, theory, and Edward Witten bosons. The FQHE is then where the Hall resistance 푅 is quantized at values (right) at Stanford in 2010. interpreted as the 푥푦 condensate of this boson. ℎ In an interview with Stanford News after Shoucheng’s 푅 = 푥푦 휈푒2 death, Steven Kivelson described this discovery: “One day Shoucheng came to visit and he said, ‘Look what I figured Here 휈 is one of a particular set of rational fractions, ℎ is out.’ He then sketched Planck’s constant, and 푒 is the electron charge. out a basic idea of the theory and all of these mysterious features of the fractional quantum Hall effect just dropped in your lap incredibly simply,” Kivelson said. “That’s not how physics usually works. You usually Shoucheng with Per Bak, slave away at things. But The experiment of Hall Steven Kivelson, and Steven’s The FQHE with 푅 = ℎ/휈푒2, from resistance 푅 = 푉/퐼, from 푥푦 on that day Shoucheng’s wife Pamela Davis at Stony 푥푦 Eisenstein, Stormer, Science 1990 Kosmos 1986 idea was just so focused Brook in 1987, celebrating and so perfect.” Shoucheng’s thesis defense The understanding of the FQHE fundamentally challenged Global phase diagram conventional condensed matter theory. In 1983, Robert B. Laughlin proposed the Laughlin wavefunction which Along the same direction, in 1992 Shoucheng successfully described the ground state of fractional 1 collaborated with Steven Kivelson and Dung-Hai Lee to quantum Hall fluids with 휈 = (푘 = 1,2, ⋯); this laid 2푘−1 propose a global phase diagram of the FQHE, which the foundation for the theory of the FQHE. derived a set of interrelations among various FQHE states. The Chern-Simons Chern-Simons Ginzburg-Landau Theory effective theory and the global phase diagram also Shoucheng was interested in looking for a quantum field revealed the relation theory (QFT) for the FQHE. The presence of the between fractional background magnetic field 퐵 implies that this field theory quantum Hall physics must be quite different from those that were familiar in and various forms of high energy physics. The field theory needed to capture particle-vortex essential features of the FQH state such as the quantized Global phase diagram of FQHE, from S. duality, which still Hall conductance, fractional charged excitations, etc. Kivelson, D.-H. Lee, and S. C. Zhang remains an active Phys. Rev. B 46, 2223 (1992) research topic after In 1989, Shoucheng and his collaborators Thors Hans 30 years. Hansson and Steven Kivelson, proposed the Chern- Simons-Landau-Ginzburg field theory of the FQHE: References S. C. Zhang, T. H. Hansson, S. Kivelson, Phys. Rev. Lett. 62, 82 (1989) 푒휋 휇휈휆 ℒ = ℒ퐺퐿 휙, 𝑖휕휇 − 퐴휇 + 푎휇 휙 + 휖 푎휇휕휈푎휆 . S. C. Zhang, Int. J. Mod. Phys. 6, 25 (1992) 2휃Φ0 S. Kivelson, D.-H. Lee, S. C. Zhang, Phys. Rev. B 46, 2223 (1992) SO(5) Theory of High 푇푐 Superconductivity
both antiferromagnetism and superconductivity, the main The SO(5) theory of high T cuprates c two competing orders in the cuprates. This theory is The discovery of high temperature superconductivity in known as the SO(5) theory, in which spin rotation copper oxides (known as cuprates) in 1987 was one of the symmetry SO(3) and charge conservation symmetry most important breakthroughs in condensed matter U(1)≃O(2) are considered as subgroups of SO(5). This physics. Despite enormous theoretical and experimental theory offered a simple possible framework for efforts, many questions about cuprates remain open today. interpreting many of the rich phenomena in cuprate superconductors, and it clearly reflected Shoucheng’s unique style: relating simple and universal principles to experimental reality in condensed matter physics.
Further developments Although the SO(5) theory was proposed as a phenomenological effective field theory, Shoucheng and A levitated magnet above Schematic phase diagram of collaborators also identified and studied microscopic superconducting YBa2Cu3O7, Cuprates, from Damascelli, from Science photo library Hussain, Shen RMP 2003 models with an exact SO(5) symmetry (see e.g. S. Rabello et al., Phys. Rev. Lett. 80, 3586 (1998); C. Wu, J. P. Hu, S. C. Shoucheng started to work in this field in 1990, when he Zhang, Phys. Rev. Lett. 91, 186402 (2003)). The latter paper collaborated with C. N. Yang to point out that the Hubbard further generalized the model to even larger symmetry, model, a model that is widely believed to capture the which is realizable in large-spin ultra-cold fermion systems. essential physics of cuprate superconductors, has an (See e.g., S. Taie et al., Phys. Rev. Lett. 105, 190401 (2010)) enhanced SO(4) symmetry (C. N. Yang & S. C. Zhang, Mod. Phys. Lett. B, 04, 759 (1990)). Starting from this work, Shoucheng and collaborators explored the possibility of a (broken) enhanced
symmetry in high Tc superconductors and its physical C. N. Yang and Shoucheng’s paper on consequences. For SO(4) symmetry of Hubbard model. example, in 1995, Eugene Demler and Shoucheng studied the relation of a Shoucheng’s note about an SO(5) model in neutron scattering feature in cuprates with enhanced 1997, with Dirac Γ matrices. symmetry (E. Demler and S. C. Zhang, Phys. Rev. Lett. 75, 4126 (1995)). A single electron carries the spinor representation of SO(5), in a manner analogous to the case of the spatial Based on these works, in 1997 Shoucheng proposed a new rotation group. Interestingly, the four-component SO(5) effective field theory for high Tc superconductivity. As an spinors are intrinsically related to the Dirac equation. elegant application of symmetry principles, he proposed a Surprisingly, starting from these works, the SO(5) five-component vector order parameter that incorporates Clifford algebra and SO(5) spinors formed a theme that appeared again and again in Shoucheng’s significant contributions in seemingly unrelated topics, including the four-dimensional quantum Hall effect, the intrinsic spin Hall effect, the quantum spin Hall effect, and topological insulators.
References E. Demler, S. C. Zhang, Phys. Rev. Lett. 75, 4126 (1995) (Left) The SO(5) superspin, from E. Demler et al., Rev. Mod. Phys. S. C. Zhang, Science 275, 1089 (1997) 76 909 2004 S. Rabello et al., Phys. Rev. Lett. 80, 3586 (1998) (Right) The SO(5) phase diagram, from S. C. Zhang, Science 275, C. Wu, J. P. Hu, and S. C. Zhang, Phys. Rev. Lett. 91, 186402 (2003) 1090 (1997) E. Demler, W. Hanke, S. C. Zhang, Rev. Mod. Phys. 76, 909 (2004) The Four Dimensional Quantum Hall Effect
From quaternions to 4D quantum Hall effect Relation to more recent developments
Shoucheng was deeply committed to the pursuit of Noting that the 4D QH state has edge states consisting of mathematical beauty in physical theories, and his theory relativistic particles with different spins living in 3D space, of four dimensional (4D) quantum Hall effect (QHE) is one Zhang and Hu proposed it as a candidate “unified theory” of the most vivid examples of the power of this approach. of fundamental interactions and particles. This theory is also one of the earliest studies of topological states of A remarkable feature of the two-dimensional (2D) QHE is matter in high dimensions, and its idea has been that the QHE wave functions can be written as functions generalized in many directions, including in numerous of complex numbers 푧 = 푥 + 푖푦, where 푥, 푦 represents studies of other high dimensional topological states. the 2D coordinate of an electron. This motivated Shoucheng to ask: is there a QHE wave The 4D QHE theory partially motivated Shoucheng’s later function of quaternions instead of work on the quantum spin Hall (QSH) effect and complex numbers? topological insulators (TI). Around 2002, Shoucheng already realized that the SU(2) Hall current of the 4D QHE In mathematics, in addition to the real is nothing but a spin Hall current of electrons. In 2008, numbers 푅, and complex numbers 퐶, work from his group clarified that the 3D TI and 2D QSH there are also quaternions 퐻 invented can be obtained from a 4D QH state via dimensional by William Rowan Hamilton, which William Rowan reduction (Qi, Hughes, Zhang, Phys. Rev. B 78, 195424 have the form: Hamilton invented (2008)). 푢 = 푎 + 푏풊 + 푐풋 + 푑풌 , quaternions with 풊2 = 풋2 = 풌2 = −1. Since a quaternion consists of 4 The spin Hall current expression of the 4D QHE in Zhang, Hu (2001) components, Shoucheng was led to consider the 4D QHE with 4 coordinates. In theoretical physics, 풊, 풋, 풌 can be Experimental realization written as the 2 × 2 Pauli matrices 푖휎1, 푖휎2, 푖휎3, which are the generators of the spin-1/2 representation of the SU(2) Since we only have 3 spatial dimensions in reality, one group. Therefore, the 4D QHE quaternion wave function might not expect a 4D QHE to be experimentally couples to an SU(2) Yang-Mills gauge field, instead of the accessible. However, in 2018 experiments on photonic U(1) gauge field in 2D quantum Hall states. crystal (Zilberberg et al., Nature 2018) and cold atom systems (Lohse et al., Nature 2018) effectively “realized” the 4D QHE. The key idea is to use control parameters to create synthetic dimensions, much in the way Thouless pumping in 1D is related to the 2D QHE. These experiments effectively constructed a 4D system with a non-zero 2nd Chen number by introducing two pumping parameters tunable by laser beams, together with two spatial dimensions.
The 4D QHE paper of Shoucheng Zhang and Jiangping Hu published in Science (2001) The original paper of Shoucheng and his student Jiangping Hu on the 4D QHE (2001) is a beautiful synthesis of mathematics and physics. They employed the 2nd Hopf map, a topological map from the 7D sphere to the 4D sphere, to construct the 4D QHE quaternion wave function. This revealed that the 4D QH state is 4D QHE edge state pumping observed in the photonic crystal experiment (O. Zilberberg et al., Nature 2018) a topological state with a nonzero 2nd Chern number, which leads to References an SU(2) quantized Hall S. C. Zhang, J. Hu, Science 294, 823 (2001) effect. M. Lohse et al., Nature 553, 55 (2018) Shoucheng’s note in 2001 on 4D QHE O. Zilberberg et al., Nature 553, 59 (2018) Intrinsic Spin Hall Effect
Intrinsic spin Hall effect
Shoucheng’s work on the 4D QH effect implied the possibility of topological and dissipationless spin transport in 3D systems. Although the original 4D QH model is not directly related to any experimental system, (a) Experimental device and (b) observed edge spin accumulation (by measurement of light polarization) due to spin Hall effect in a this connection 2D hole gas system (J. Wunderlich et al., Phys. Rev. Lett. 94, motivated Shoucheng 047204 (2005)). to start thinking about spin-orbit coupling and A sentence near the end of the 4DQH paper (S. C. Zhang and J. Hu, Science Spin Hall insulators its role in dissipation 294, 823) -less transport. In 2004, Murakami, Nagaosa, and Shoucheng In 2003, Shuichi Murakami, Naoto Nagaosa and generalized their results Shoucheng predicted an intrinsic spin Hall effect in to a family of materials semiconductors (Science 301, 1348 (2001)), which referred that they named “spin to a transverse dissipationless spin current induced by an Hall insulators,” which are electric field in semiconductor compounds with strong systems with zero charge spin-orbit coupling. The spin Hall current is described by conductivity but finite the following formula: spin Hall conductivity. The Spin Hall conductivity and band 푗푖 = 휎 휖푖푗푘퐸 structure of the spin Hall insulator 푗 푠 푘 materials mentioned in As was stated at the beginning of their paper, this work “is (S. Murakami et al., Phys. Rev. this paper are zero gap Lett. 93, 156804 (2004) driven by the confluence of the important technological and narrow gap goals of quantum spintronics with the quest of semiconductors, including HgTe, HgSe, 훽-HgS, 훼-Sn, PbTe, generalizing the quantum Hall effect (QHE) to higher PbSe, PbS. Many of the materials mentioned here were dimensions.” later identified as topological insulators, although the Different from connection to topological physics had not yet been charge current, uncovered in this work. spin current is even under time Other works on spintronics reversal, which Besides the spin Hall effect, Shoucheng also worked more allows this effect broadly in the field of spintronics. For example, in his to occur in collaboration with B. A. Bernevig and J. Orenstein (Phys. Discussion in the first paragraph of the materials without Rev. Lett. 97, 236601 (2006)), they predicted, and then Murakami-Nagaosa-Zhang paper that a magnetic field. subsequently experimentally realized, a “persistent spin referred to the Zhang-Hu 4DQH work. Spin-orbit helix” which supported a long-lived helical spin coupling replaced the role of magnetic field in QHE, and configuration due to an emergent SU(2) symmetry. leads to the nontrivial Berry’s phase that is the essential Shoucheng’s works brought many fundamental physics reason of the spin Hall effect. The 4DQH and intrinsic spin ideas into the field of semiconductors and spintronics. Hall effect the first proposals of dissipationless transport in higher than two dimensions, which is the overture of the upcoming breakthrough of new topological insulators and topological superconductors. (Intrinsic spin Hall effect of a different type was also proposed in the concurrent work of J. Sinova et al., Phys. Rev. Lett. 92, 126603 (2004).) (Left) Theoretical prediction of the special Fermi surface that leads to persistent spin helix. (Right) Experimental realization of the spin Soon after its theoretical proposal, helix in J. D. Koralek et al., Nature 458, 610 (2009). the spin Hall effect was observed experimentally in hole doped semi- Schematic illustration of the References mechanism of intrinsic spin S. Murakami, N. Nagaosa, S. C. Zhang, Science 301, 1348 (2001) conductors (J. Wunderlich et al., Hall effect in the Murakami- S. Murakami, N. Nagaosa, S. C. Zhang, Phys. Rev. Lett. 93, 156804 (2004) Phys. Rev. Lett. 94, 047204 (2005)). Nagaosa-Zhang work. B. A. Bernevig, J. Orenstein, S. C. Zhang, Phys. Rev. Lett. 97, 236601 (2006) Quantum Spin Hall Effect
Early models of quantum spin Hall effect HgTe theory and experiments
The 4D QH and the intrinsic spin Hall effect are both The earliest models reviewed above set up the theoretical proposed as generalizations of the quantum Hall effect. In foundation of QSH, but they are difficult to realize 2004, Shoucheng started to consider a more direct experimentally for various reasons. In 2006, Shoucheng generalization of the QHE. B. A. Bernevig and Shoucheng and his students B. A. Bernevig and T. L. Hughes (BHZ) proposed that in a semiconductor with a spatially made the first realistic proposal of a QSH material: inhomogeneous strain, the spin-orbit coupling can realize HgTe/CdTe quantum wells (Science 314, 1757 (2006)). They a uniform magnetic field that is opposite for spin up and obtained the low energy effective theory of this quantum spin down electrons. This manifests as a quantized spin well (known as the BHZ model), and predicted that the Hall effect (QSH) preserving time reversal symmetry (Phys. QSH phase could be identified by a topological phase Rev. Lett. 96, 106802 (2006)). transition controlled by the thickness of the quantum well. The BHZ work not only proposed the first realistic QSH material, but also proposed a general mechanism to identify topological materials, which they named “band inversion.”
Soon after the proposal, the quantum spin Hall effect in HgTe quantum Shoucheng’s note about “the quantized spin wells was verified in (Top) The band edge energy of narrow Hall effect” on Jun. 12, 2004. Laurens Molenkamp’s and wide quantum wells. (Bottom) Illustration of the band inversion and Concurrently, a different quantum spin Hall model was group (Koenig et al., topological phase transition. From independently proposed by Charles L. Kane and J. Eugene Science 318, 766 (2007)). Bernevig, Hughes, Zhang, Science Mele, who studied spin-orbit coupling in graphene (Phys. They indeed observed a (2006). 2 Rev. Lett. 95, 226801 (2005)). This model is a time-reversal 2e /h conductance for d>dc while the resistance is much invariant generalization of a model proposed by F. D. M. higher when d
References B. A. Bernevig and S. C. Zhang, Phys. Rev. Lett. 96, 106802 (2006) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005) The two earliest models of the quantum spin Hall effect. (Left) The B. A. Bernevig, T. L. Hughes and S. C. Zhang, Science 314, 1757 illustration of the Bernevig-Zhang proposal. (Right) The energy (2006) spectrum of the Kane-Mele model. M. Koenig et al., Science 318, 766 (2007) From Quantum Spin Hall to Topological Insulator
Other quantum spin Hall materials In 2008, Shoucheng and collaborators proposed a new family of topological insulators, including three materials:
Bi2Te 3, Bi2Se3, and Sb2Te 3 (H. Zhang et al., Nature Physics 5, 438 (2009)). These materials are described by a simple effective model similar to the BHZ model. This family of TI
was realized experimentally in 2009. The Bi2Se3 family of Discovery of the quantum spin Hall TI is easy to grow and has a large gap and simple surface effect was listed by Science states, which immediately became the “gold standard” of magazine as one of the top 10 Breakthroughs of the Year in 2007 TI. They remain the most widely studied TI materials to A Perspective written by C. L. Kane this day. and E. J. Mele for the BHZ work. (Science 314 1692 (2006))
The discovery of HgTe QSH heralded the beginning of a new era with tremendous theoretical and experimental developments in topological states of matter. Since then Shoucheng and collaborators have also predicted several other QSH materials, such as InAs/GaSb quantum wells (C. X. Liu et al. Phys. Rev. Lett. 100, 236601 (2008)) and stanene, a single-layer thin film of Sn (Y. Xu et al., Phys. (Left) Crystal structure of the Bi2Se3 family of TI. (Right) Theoretical Rev. Lett. 111, 136804 (2013)). prediction of the surface state dispersion relation based on ab initio calculation. From H. Zhang et al., Nature Physics 5, 438 (2009).
Stanene and its non-trivial band structure, from Xu et al., Phys. Rev. Lett. 2013 Phase diagram of the InAs/GaSb quantum wells with QSH effect, Experimental observation of topological surface states in angle- from Liu et al., Phys. Rev. Lett. 2008. resolved photoemission. (Left) Bi2Se3 from Y. Xia et al., Nature Shoucheng’s group has also investigated many topological Physics 5, 398 (2009). (Right) Bi2Te 3 from Y. L. Chen et al., Science effects in QSH systems, such as electron interaction effects 325, 178 (2009). on the edge, and fractional charge on an edge magnetic domain wall, spin-charge separation induced by a 휋 flux, Other 3D TI materials etc. Since 3D TI’s are single crystals rather than quantum Generalization to three dimensions wells, it is easier to realize experimentally. Shoucheng The understanding of the QSH and 푍2 topological protection implied that the QSH could be generalized to and collaborators have three dimensional insulators, with robust two-dimensional predicted many of the early surface states. The surface electrons are “helical”, with 3D TI materials. This includes their spin direction locked with the momentum. This new strained HgTe (Xi Dai et al., state, named 3D topological insulators (TI), was proposed Phys. Rev. B 77, 125319 (2008), realized in L. in 2006 by three papers (J. E. Moore and L. Balents, Phys. Half-Heusler family of Rev. B 75, 121306(R) (2007), L. Fu, C. L. Kane, and E. J. Molenkamp’s group), half- topological insulators. From S. Mele., Phys Rev Lett. 98, 106803 (2007), R. Roy, Phys. Rev. Heusler compounds, TlBiSe2 Chadov et al., Nature B 79, 195322 (2009)). family, filled skutterudites, Materials 9, 541 (2010). Actinide compounds, etc. The first 3D TI was the alloy of Bi and Sb, proposed by L. References Fu and C. L. Kane, and C. X. Liu et al., Phys. Rev. Lett. 100, 236601 (2008) Y. Xu et al., Phys. Rev. Lett. 111, 136804 (2013) realized by M. Z. Hasan’s An illustration of the spin- H. Zhang et al., Nature Physics 5, 438 (2009) group in Princeton. momentum locking of topological Y. L. Chen et al., Science 325, 178 (2009) insulator surface states. B. Yan, S. C. Zhang, Rep. Prog. Phys. 75, 096501 (2012)
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identified as the “parent state” from which 3DTI and QSH QSH and 3DTI which from state” “parent the as identified with a particular coupling with the electromagnetic field field electromagnetic the with coupling particular a with
topological invariant) is invariant) topological The axion is a hypothetical particle in high energy physics physics energy high in particle hypothetical a is axion The
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related to electromagnetic duality, and has appeared appeared has and duality, electromagnetic to related Family tree of topological topological of tree Family
insulators, in which the 4D QH QH 4D the which in insulators, (2008). Surprisingly, the field theory describing TI is is TI describing theory field the Surprisingly, (2008).
a family tree of topological topological of tree family a 195424 195424 78 B Rev Phys Zhang, Hughes, (Qi, TI 3D of theory
Hughes Zhang paper proposed proposed paper Zhang - developed the topological field field topological the developed Shoucheng and Hughes
and symmetry classes. The Qi The classes. symmetry and - Liang Qi, Taylor L. L. Taylor Qi, Liang - Xiao 2008, early In features. universal
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unified view to topological topological to view unified states QSH and 3D TI should be described by some some by described be should TI 3D and QSH states
approach also provided a a provided also approach investigated in the fractional quantum Hall effect, the new new the effect, Hall quantum fractional the in investigated
The topological field theory theory field topological The Shoucheng that theory Simons - Chern the to Similar
Other quantum spin Hall materials Hall spin quantum Other electrodynamics axion and TI 3D topological insulators and beyond and insulators topological 3D Quantum Anomalous Hall Effect
Quantum anomalous Hall effect Based on further developments in TI, Shoucheng and collaborators made two more realistic proposals of QAH: The quantum anomalous Hall (QAH) effect refers to QHE in one in Mn doped HgTe/CdTe quantum wells, making use of a band insulator without an external magnetic field. QSH (C. X. Liu et al., Phys. Rev. Lett. 101, 146802 (2008));
Topologically it belongs to the same phase as integer QH, the other in magnetically doped Bi2Se3 (or Sb2Te 3, Bi2Te 3) but it has dispersing bands and no Landau levels. The first thin films, making use of the surface state of 3D TI (R. Yu et model for QAH was proposed by F. D. M. Haldane in 1988. al., Science 329, 61 (2010)). The Hg(Mn)Te material is paramagnetic and therefore requires an external field, while
the magnetically doped Bi2Se3 film was proposed to be ferromagnetic, which was experimentally realized in 2013.
The title of Haldane’s work on QAH (Phys. Rev. Lett. 61, 2015 (1988)) Although in principle QAH does not require quantum spin Hall and spin-orbit coupling, it is only after the proposal of QSH and TI that a path to realize the QAH effect was found. In a 2006 paper Shoucheng collaborated on with X. L. Qi Band inversion mechanism for QAH effect in Mn doped HgTe and Y. S. Wu about a quantum spin Hall model (Phys. Rev. B (left) and magnetically doped Bi2Se3 film. From C. X. Liu et al., Phys. Rev. Lett. (2008) and Yu et al., Science (2010). 74, 085308 (2006)), the authors also proposed a new model of QAH. (This paper is possibly the origin of the Experimental Realization name “quantum anomalous Hall effect”.) The QAH in magnetically
doped Bi2Se3 family TI was first realized in Cr-
doped (Bi,Sb)2Te 3 thin films in Qikun Xue’s group in Tsinghua University (C.- Z. Chang et al., Science 340, 167 (2013)). Since then, the QAH state has Qikun Xue and Shoucheng in a Shoucheng’s note on a list of possible research discussion at Stanford, 2014 projects related to QAHE in June 5, 2005. The been firmly verified in first project is on fractional QAHE, which became more and more studies. Many unique features of the QAH an active research field later in 2011. The second state, such as zero Hall plateau, universal scaling behavior project of QAHE at room temperature is still one at the plateau transition, QAH state with higher plateaus, major goal of the field. and anomalous edge transport due to coexisting chiral QAH can be realized by making use of TI because the latter and helical modes, were studied by Shoucheng’s group. already has nontrivial Berry curvature that contributes to These works guided the rapid development of this field. the Hall conductance. One just needs to break time- reversal in an appropriate way to avoid the cancellation.
(Left) Hall resistivity vs magnetic field. (Right) Gate voltage dependence of Hall resistivity and longitudinal resistivity. From C.- Z. Chang et al., Science (2013).
References C. X. Liu et al., Phys. Rev. Lett. 101, 146802 (2008) Table of three Hall effects and their quantum versions. From C.-X. R. Yu et al., Science 329, 61 (2010) Liu, S. C. Zhang, X. L. Qi, Annu. Rev. Cond. Matt. Phys. 7, 301 (2016). C. Z. Chang et al., Science 323, 1184 (2009) Topological Superconductors and More
Topological superconductors in a hybrid device of QAH and TSC (S. B. Chung et al., Phys. Rev. B 83, 100512(R) (2011), J. Wang, et al., Phys. Rev. B 92, The BCS mean-field theory of superconductivity describes 064520 (2015)). Experimental evidence of half quantum quasiparticles in a similar way as electrons in band Hall plateau was reported in 2017 (He et al., Science 357 insulators. When the superconductor has a full gap, the 294 (2017)), although the interpretation remains classification is very similar to that of insulators, which controversial (See M. suggested the concept of topological superconductors Kayyalha et al., arXiv (TSC). (This is analogous to how Ettore Majorana 1904.06463 (2019)). Based generalized the Dirac equation to his Majorana equation on this proposal, in 2018 in 1937.) Historically, the first topologically nontrivial Shoucheng’s group further superconductors discovered were 2D p+ip proposed the possibility of superconductors (Read and Green, Phys. Rev. B 61, 10267 realizing non-Abelian (2000)) and 1D p-wave superconductors (Kitaev (2001)). quantum gates with chiral A single qubit quantum gate Majorana fermions (Lian et Following the discovery of chiral Majorana fermions of time reversal (Lian et al. 2018) al., PNAS 115 10938 (2018)). invariant (TRI) topological insulators, Other areas of condensed matter physics new TSC with time reversal symmetry were Besides the directions discussed so far, Shoucheng also Comparison between 2D time- proposed in 2008-2009, conducted research in a variety of other topics in physics, reversal invariant TSC and QSH, from by independent works such as topological semimetals, interacting topological Qi et al Phys Rev Lett. (2009) from Shoucheng’s group states, Fe-pnictide superconductors, the role of symmetry and other groups (X. L. Qi et al., Phys. Rev. Lett. 102, in quantum Monte Carlo method, etc. In lieu of countless 187001 (2009), A. Schnyder et al., Phys. Rev. B 78, 195125 other posters, we end with a selection of figures from his (2008), A. Kitaev, AIP Conf. Proc. 1134, 22 (2009)). papers, which provide a glance into Shoucheng’s diverse Interestingly, the B phase of 3He superfluid, known from research interests. many years ago, was proposed as a 3D TRI topological superfluid.
2D chiral TSC from QAH
The 2D p+ip TSC (also known as chiral TSC) is an interesting system for quantum computation, because of non-Abelian statistics of vortices with Majorana zero
modes. A candidate material for chiral TSC is Sr2RuO4, but it has not been confirmed. In 2010, Shoucheng’s group proposed a new mechanism for realizing chiral TSC, by making use of QAH effect (X. L. Qi, T. L. Hughes, S. C. Zhang, Phys. Rev. B 82, 184516 (2010)). The proposal is that in general a quantum Hall phase transition will be broadened into a TSC phase when superconducting proximity effect is introduced. If realized, this proposal also enables new way to probe TSC through the QAH edge Snapshots of Shoucheng’s works on (starting from top left) giant- states. Shoucheng’s group proposed that the chiral magnetoresistive materials (Y. B. Bazaliy, B. A. Jones & S. C. Zhang 2010), fractional TI (J. Maciejko et al. 2010), Weyl semimetals and Majorana fermions yield a half quantum Hall plateau axion strings (Z. Wang & S. C. Zhang 2013), Fe-pnictide topological vortex (G. Xu et al. 2016), axion theory of TSC (X. L. Qi, E. Witten & S. C. Zhang 2013), nodal semimetals and Chern-Simons theory (B. Lian et al. 2017), Fe-pnictide effective model (S. Raghu et al. 2008), time reversal and quantum Monte Carlo (C. Wu and S. C. Zhang 2005).
References X. L. Qi, T. L. Hughes, S. C. Zhang, Phys. Rev. B 82,184516 (2010) Shoucheng’s proposals for 2D chiral TSC and device for detecting S. B. Chung, et al., Phys. Rev. B 83, 100512 (2011) chiral Majorana fermion (Qi, Hughes, Zhang 2010, Wang et al. B. Lian, et al., PNAS 115, 10938 (2018) 2015) Algorithms for the Future
Exploration beyond physics Another area of Shoucheng’s recent interest was blockchain technology. Blockchain (and its generalizations) Shoucheng always believed that science should have no is a mathematical mechanism to reach consensus on a disciplinary borders. He often told his students to spend record without requiring any centralized agency. In 80% of their time on their research focus, and 20% time analogy with the concept exploring broader directions and thinking about big of information entropy, questions. In recent years, Shoucheng investigated many Shoucheng emphasized directions beyond physics such as artificial intelligence (AI), that consensus has an blockchain, distributed computing, bio-informatics, and intrinsic value, and more. He had the vision of developing new technologies envisioned that from the perspectives of both fundamental principles and “humanity is now social good, which motivated him to found the investment reaching a new era firm Danhua Venture Capital (DHVC) in 2013. where trust and From Shoucheng’s notes in 2018, we can see how many A slide from Shoucheng’s talk “In consensus is built upon different directions he was exploring actively until the last Math We Trust – Foundations of math,” a viewpoint that Crypto-economic Science” in Sep. he summarized in the days of his life. The following is a small fraction of this 2018. precious record. motto “in math we trust.” Shoucheng believed that new consensus mechanisms would help to build a fairer, more efficient, and more diverse society.
Shoucheng was also interested in many other areas related to how mathematics and algorithms could improve human society. An example is the application of A few pages from secure multi-party computation in bioinformatics and Shoucheng’s 2018 notes medical sciences. Shoucheng’s son Brian Zhang suggested on various topics. From a common theme behind the different areas of top: CRISPR (clustered Shoucheng’s latest interests, which is “algorithms for the regularly interspaced future.” short palindromic repeats); vector clock and causality, DNA finger To see a world in a grain of sand printing; DAG (directed acyclic graph) and its There are many aspects of Shoucheng’s legendary application in block chain; scientific life that we are not able to cover in our brief zero knowledge proofs; overview. Despite his abrupt departure, his Maxwell’s demon. groundbreaking scientific works; his dreams and vision for science, life, and humanity; and his inspiration to all of us Algorithms for the future will forever be preserved as an eternal monument to him. We would like to end this exhibition by his favorite quote Shoucheng’s ideas often from William Blake’s poem: uncovered new connections between fundamental physics principles and various other fields. In 2018, Shoucheng and collaborators used AI to study chemical Illustration of the Atom2Vec compounds (Q. Zhou et algorithm. From Q. Zhou et al., PNAS (2018). al., PNAS 115, 6411 Acknowledgement (2018)). They developed an “Atom2Vec” algorithm and We would like to thank the Zhang family members Barbara, Stephanie, Brian, showed that the AI did not only rediscover the periodic and Ruth for sharing with us the precious photos and notes of Shoucheng, and for providing a lot of helpful suggestions for this exhibition. We would like to table, but was also able to predict new chemical thank Yaroslaw Bazaliy, B. Andrei Bernevig, Leonid Pryadko, and Congjun Wu compounds from the properties of elements it discovered. for helpful suggestions. We have used a poster template provided by MakeSigns.