From Purdue University: “A new duality discovered at Purdue University solves a physics mystery”

From Purdue University
5.23.22
Cheryl Pierce
Communications
pierce81@purdue.edu

A Poincaré half-plane can be viewed in the background which demonstrates a curved surface. The white geodesics of the curved surface are shown as an analog of straight lines on a flat space. White balls moving in the right direction demonstrate the geometric origin of an extraordinary skin effect in non-Hermitian physics. Graphic provided by Chenwei Lv and Ren Zhang.
In conventional wisdom, producing a curved space requires distortions, such as bending or stretching a flat space. A team of researchers at Purdue University have discovered a new method to create curved spaces that also solves a mystery in physics. Without any physical distortions of physical systems, the team has designed a scheme using non-Hermiticity**, which exists in any systems coupled to environments, to create a hyperbolic surface and a variety of other prototypical curved spaces.
“Our work may revolutionize the general public’s understanding of curvatures and distance,” says Qi Zhou, Professor of Physics and Astronomy. “It has also answered long-standing questions in non-Hermitian quantum mechanics by bridging non-Hermitian physics and curved spaces. These two subjects were assumed to be completely disconnected. The extraordinary behaviors of non-Hermitian systems, which have puzzled physicists for decades, become no longer mysterious if we recognize that the space has been curved. In other words, non-Hermiticity and curved spaces are dual to each other, being the two sides of the same coin.”
The Team recently published their findings in Nature Communications. Of the members of the team, most work at Purdue University’s West Lafayette campus. Chenwei Lv, graduate student, is the lead author, and other members of the Purdue team include Prof. Qi Zhou, and Zhengzheng Zhai, postdoctoral fellow. The co-first author, Prof. Ren Zhang from Xi’an Jiaotong University, was a visiting scholar at Purdue when the project was initiated.
In order to understand how this discovery works, first one must understand the difference between Hermitian and non-Hermitian systems in physics. Zhou explains it using an example in which a quantum particle can “hop” between different sites on a lattice. If the probability for a quantum particle to hop in the right direction is the same as the probability to hop in the left direction, then the Hamiltonian is Hermitian. If these two probabilities are different, the Hamiltonian is non-Hermitian. This is the reason that Chenwei and Ren Zhang have used arrows with different sizes and thicknesses to denote the hopping probabilities in opposite directions in their plot.
“Typical textbooks of quantum mechanics mainly focus on systems governed by Hamiltonians* that are Hermitian,” says Lv. “A quantum particle moving in a lattice needs to have an equal probability to tunnel along the left and right directions. Whereas Hermitian Hamiltonians are well-established frameworks for studying isolated systems, the couplings with the environment inevitably lead to dissipations in open systems, which may give rise to Hamiltonians that are no longer Hermitian. For instance, the tunneling amplitudes in a lattice are no longer equal in opposite directions, a phenomenon called nonreciprocal tunneling. In such non-Hermitian systems, familiar textbook results no longer apply and some may even look completely opposite to that of Hermitian systems. For instance, eigenstates of non-Hermitian systems are no longer orthogonal, in sharp contrast to what we learned in the first class of an undergraduate quantum mechanics course. These extraordinary behaviors of non-Hermitian systems have been intriguing physicists for decades, but many outstanding questions remain open.”
He further explains that their work provides an unprecedented explanation of fundamental non-Hermitian quantum phenomena. They found that a non-Hermitian Hamiltonian has curved the space where a quantum particle resides. For instance, a quantum particle in a lattice with nonreciprocal tunneling is in fact moving on a curved surface. The ratio of the tunneling amplitudes along one direction to that in the opposite direction controls how large the surface is curved. In such curved spaces, all the strange non-Hermitian phenomena, some of which may even appear unphysical, immediately become natural. It is the finite curvature that requires orthonormal conditions distinct from their counterparts in flat spaces. As such, eigenstates would not appear orthogonal if we used the theoretical formula derived for flat spaces. It is also the finite curvature that gives rise to the extraordinary non-Hermitian skin effect that all eigenstates concentrate near one edge of the system.
“This research is of fundamental importance and its implications are two-fold”, says Zhang. “On the one hand, it establishes non-Hermiticity as a unique tool to simulate intriguing quantum systems in curved spaces,” he explains. “Most quantum systems available in laboratories are flat and it often requires significant efforts to access quantum systems in curved spaces. Our results show that non-Hermiticity offers experimentalists an extra knob to access and manipulate curved spaces. An example is that a hyperbolic surface could be created and further be threaded by a magnetic field. This could allow experimentalists to explore the responses of quantum Hall states to finite curvatures, an outstanding question in condensed matter physics. On the other hand, the duality allows experimentalists to use curved spaces to explore non-Hermitian physics. For instance, our results provide experimentalists a new approach to access exceptional points using curved spaces and improve the precision of quantum sensors without resorting to dissipations.”
Now that the team has published their findings, they anticipate it spinning off into multiple directions for further study. Physicists studying curved spaces could implement their apparatuses to address challenging questions in non-Hermitian physics. Also, physicists working on non-Hermitian systems could tailor dissipations to access non-trivial curved spaces that cannot be easily obtained by conventional means. The Zhou research group will continue to theoretically explore more connections between non-Hermitian physics and curved spaces. They also hope to help bridge the gap between these two physics subjects and bring these two different communities together with future research.
According to the team, Purdue University is uniquely qualified to foster this type of quantum research. Purdue has been growing strong in quantum information science at a fast pace over the past few years. The Purdue Quantum Science and Engineering Institute paired with the Department of Physics and Astronomy, allow the team to collaborate with many colleagues with diverse expertise and foster interdepartmental and collegiate growth on a variety of platforms that exhibit dissipations and nonreciprocal tunneling.
*”Hamiltonian simulation”: (also referred to as quantum simulation) is a problem in quantum information science that attempts to find the computational complexity and quantum algorithms needed for simulating quantum systems.
**”Non-Hermiticity”: a subject in Quantum Mechanics which is too complex for consideration here. Not reducible to a sentence and only a subject for a professional quantum physicist.
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Purdue University is a public land-grant research university in West Lafayette, Indiana, and the flagship campus of the Purdue University system. The university was founded in 1869 after Lafayette businessman John Purdue donated land and money to establish a college of science, technology, and agriculture in his name. The first classes were held on September 16, 1874, with six instructors and 39 students.
The main campus in West Lafayette offers more than 200 majors for undergraduates, over 69 masters and doctoral programs, and professional degrees in pharmacy and veterinary medicine. In addition, Purdue has 18 intercollegiate sports teams and more than 900 student organizations. Purdue is a member of the Big Ten Conference and enrolls the second largest student body of any university in Indiana, as well as the fourth largest foreign student population of any university in the United States.
Purdue University is a member of the Association of American Universitiesand is classified among “R1: Doctoral Universities – Very high research activity”. Purdue has 25 American astronauts as alumni and as of April 2019, the university has been associated with 13 Nobel Prizes.
In 1865, the Indiana General Assembly voted to take advantage of the Morrill Land-Grant Colleges Act of 1862 and began plans to establish an institution with a focus on agriculture and engineering. Communities throughout the state offered facilities and funding in bids for the location of the new college. Popular proposals included the addition of an agriculture department at Indiana State University, at what is now Butler University. By 1869, Tippecanoe County’s offer included $150,000 (equivalent to $2.9 million in 2019) from Lafayette business leader and philanthropist John Purdue; $50,000 from the county; and 100 acres (0.4 km^2) of land from local residents.
On May 6, 1869, the General Assembly established the institution in Tippecanoe County as Purdue University, in the name of the principal benefactor. Classes began at Purdue on September 16, 1874, with six instructors and 39 students. Professor John S. Hougham was Purdue’s first faculty member and served as acting president between the administrations of presidents Shortridge and White. A campus of five buildings was completed by the end of 1874. In 1875, Sarah A. Oren, the State Librarian of Indiana, was appointed Professor of Botany.
Purdue issued its first degree, a Bachelor of Science in chemistry, in 1875, and admitted its first female students that autumn.
Emerson E. White, the university’s president, from 1876 to 1883, followed a strict interpretation of the Morrill Act. Rather than emulate the classical universities, White believed Purdue should be an “industrial college” and devote its resources toward providing a broad, liberal education with an emphasis on science, technology, and agriculture. He intended not only to prepare students for industrial work, but also to prepare them to be good citizens and family members.
Part of White’s plan to distinguish Purdue from classical universities included a controversial attempt to ban fraternities, which was ultimately overturned by the Indiana Supreme Court, leading to White’s resignation. The next president, James H. Smart, is remembered for his call in 1894 to rebuild the original Heavilon Hall “one brick higher” after it had been destroyed by a fire.
By the end of the nineteenth century, the university was organized into schools of agriculture, engineering (mechanical, civil, and electrical), and pharmacy; former U.S. President Benjamin Harrison served on the board of trustees. Purdue’s engineering laboratories included testing facilities for a locomotive, and for a Corliss steam engine—one of the most efficient engines of the time. The School of Agriculture shared its research with farmers throughout the state, with its cooperative extension services, and would undergo a period of growth over the following two decades. Programs in education and home economics were soon established, as well as a short-lived school of medicine. By 1925, Purdue had the largest undergraduate engineering enrollment in the country, a status it would keep for half a century.
President Edward C. Elliott oversaw a campus building program between the world wars. Inventor, alumnus, and trustee David E. Ross coordinated several fundraisers, donated lands to the university, and was instrumental in establishing the Purdue Research Foundation. Ross’s gifts and fundraisers supported such projects as Ross–Ade Stadium, the Memorial Union, a civil engineering surveying camp, and Purdue University Airport. Purdue Airport was the country’s first university-owned airport and the site of the country’s first college-credit flight training courses.
Amelia Earhart joined the Purdue faculty in 1935 as a consultant for these flight courses and as a counselor on women’s careers. In 1937, the Purdue Research Foundation provided the funds for the Lockheed Electra 10-E Earhart flew on her attempted round-the-world flight.
Every school and department at the university was involved in some type of military research or training during World War II. During a project on radar receivers, Purdue physicists discovered properties of germanium that led to the making of the first transistor. The Army and the Navy conducted training programs at Purdue and more than 17,500 students, staff, and alumni served in the armed forces. Purdue set up about a hundred centers throughout Indiana to train skilled workers for defense industries. As veterans returned to the university under the G.I. Bill, first-year classes were taught at some of these sites to alleviate the demand for campus space. Four of these sites are now degree-granting regional campuses of the Purdue University system. On-campus housing became racially desegregated in 1947, following pressure from Purdue President Frederick L. Hovde and Indiana Governor Ralph F. Gates.
After the war, Hovde worked to expand the academic opportunities at the university. A decade-long construction program emphasized science and research. In the late 1950s and early 1960s the university established programs in veterinary medicine, industrial management, and nursing, as well as the first computer science department in the United States. Undergraduate humanities courses were strengthened, although Hovde only reluctantly approved of graduate-level study in these areas. Purdue awarded its first Bachelor of Arts degrees in 1960. The programs in liberal arts and education, formerly administered by the School of Science, were soon split into an independent school.
The official seal of Purdue was officially inaugurated during the university’s centennial in 1969.

Consisting of elements from emblems that had been used unofficially for 73 years, the current seal depicts a griffin, symbolizing strength, and a three-part shield, representing education, research, and service.
In recent years, Purdue’s leaders have continued to support high-tech research and international programs. In 1987, U.S. President Ronald Reagan visited the West Lafayette campus to give a speech about the influence of technological progress on job creation.
In the 1990s, the university added more opportunities to study abroad and expanded its course offerings in world languages and cultures. The first buildings of the Discovery Park interdisciplinary research center were dedicated in 2004.
Purdue launched a Global Policy Research Institute in 2010 to explore the potential impact of technical knowledge on public policy decisions.
On April 27, 2017, Purdue University announced plans to acquire for-profit college Kaplan University and convert it to a public university in the state of Indiana, subject to multiple levels of approval. That school now operates as Purdue University Global, and aims to serve adult learners.
Campuses
Purdue’s campus is situated in the small city of West Lafayette, near the western bank of the Wabash River, across which sits the larger city of Lafayette. State Street, which is concurrent with State Road 26, divides the northern and southern portions of campus. Academic buildings are mostly concentrated on the eastern and southern parts of campus, with residence halls and intramural fields to the west, and athletic facilities to the north. The Greater Lafayette Public Transportation Corporation (CityBus) operates eight campus loop bus routes on which students, faculty, and staff can ride free of charge with Purdue Identification.
Organization and administration
The university president, appointed by the board of trustees, is the chief administrative officer of the university. The office of the president oversees admission and registration, student conduct and counseling, the administration and scheduling of classes and space, the administration of student athletics and organized extracurricular activities, the libraries, the appointment of the faculty and conditions of their employment, the appointment of all non-faculty employees and the conditions of employment, the general organization of the university, and the planning and administration of the university budget.
The Board of Trustees directly appoints other major officers of the university including a provost who serves as the chief academic officer for the university, several vice presidents with oversight over specific university operations, and the regional campus chancellors.
Academic divisions
Purdue is organized into thirteen major academic divisions.
College of Agriculture
The university’s College of Agriculture supports the university’s agricultural, food, life, and natural resource science programs. The college also supports the university’s charge as a land-grant university to support agriculture throughout the state; its agricultural extension program plays a key role in this.
College of Education
The College of Education offers undergraduate degrees in elementary education, social studies education, and special education, and graduate degrees in these and many other specialty areas of education. It has two departments: (a) Curriculum and Instruction and (b) Educational Studies.
College of Engineering
The Purdue University College of Engineering was established in 1874 with programs in Civil and Mechanical Engineering. The college now offers B.S., M.S., and Ph.D. degrees in more than a dozen disciplines. Purdue’s engineering program has also educated 24 of America’s astronauts, including Neil Armstrong and Eugene Cernan who were the first and last astronauts to have walked on the Moon, respectively. Many of Purdue’s engineering disciplines are recognized as top-ten programs in the U.S. The college as a whole is currently ranked 7th in the U.S. of all doctorate-granting engineering schools by U.S. News & World Report.
Exploratory Studies
The university’s Exploratory Studies program supports undergraduate students who enter the university without having a declared major. It was founded as a pilot program in 1995 and made a permanent program in 1999.
College of Health and Human Sciences
The College of Health and Human Sciences was established in 2010 and is the newest college. It offers B.S., M.S. and Ph.D. degrees in all 10 of its academic units.
College of Liberal Arts
Purdue’s College of Liberal Arts contains the arts, social sciences and humanities programs at the university. Liberal arts courses have been taught at Purdue since its founding in 1874. The School of Science, Education, and Humanities was formed in 1953. In 1963, the School of Humanities, Social Sciences, and Education was established, although Bachelor of Arts degrees had begun to be conferred as early as 1959. In 1989, the School of Liberal Arts was created to encompass Purdue’s arts, humanities, and social sciences programs, while education programs were split off into the newly formed School of Education. The School of Liberal Arts was renamed the College of Liberal Arts in 2005.
Krannert School of Management
The Krannert School of Management offers management courses and programs at the undergraduate, master’s, and doctoral levels.
College of Pharmacy
The university’s College of Pharmacy was established in 1884 and is the 3rd oldest state-funded school of pharmacy in the United States. The school offers two undergraduate programs leading to the B.S. in Pharmaceutical Sciences (BSPS) and the Doctor of Pharmacy (Pharm.D.) professional degree. Graduate programs leading to M.S. and Ph.D. degrees are offered in three departments (Industrial and Physical Pharmacy, Medicinal Chemistry and Molecular Pharmacology, and Pharmacy Practice). Additionally, the school offers several non-degree certificate programs and post-graduate continuing education activities.
Purdue Polytechnic Institute
The Purdue Polytechnic Institute offers bachelor’s, master’s and Ph.D. degrees in a wide range of technology-related disciplines. With over 30,000 living alumni, it is one of the largest technology schools in the United States.
College of Science
The university’s College of Science houses the university’s science departments: Biological Sciences; Chemistry; Computer Science; Earth, Atmospheric, & Planetary Sciences; Mathematics; Physics & Astronomy; and Statistics. The science courses offered by the college account for about one-fourth of Purdue’s one million student credit hours.
College of Veterinary Medicine
The College of Veterinary Medicine is accredited by the AVMA to offer the Doctor of Veterinary Medicine degree, associate’s and bachelor’s degrees in veterinary technology, master’s and Ph.D. degrees, and residency programs leading to specialty board certification. Within the state of Indiana, the Purdue University College of Veterinary Medicine is the only veterinary school, while the Indiana University School of Medicine is one of only two medical schools (the other being Marian University College of Osteopathic Medicine). The two schools frequently collaborate on medical research projects.
Honors College
Purdue’s Honors College supports an honors program for undergraduate students at the university.
The Graduate School
The university’s Graduate School supports graduate students at the university.
Research
The university expended $622.814 million in support of research system-wide in 2017, using funds received from the state and federal governments, industry, foundations, and individual donors. The faculty and more than 400 research laboratories put Purdue University among the leading research institutions. Purdue University is considered by the Carnegie Classification of Institutions of Higher Education to have “very high research activity”. Purdue also was rated the nation’s fourth best place to work in academia, according to rankings released in November 2007 by The Scientist magazine. Purdue’s researchers provide insight, knowledge, assistance, and solutions in many crucial areas. These include, but are not limited to Agriculture; Business and Economy; Education; Engineering; Environment; Healthcare; Individuals, Society, Culture; Manufacturing; Science; Technology; Veterinary Medicine. The Global Trade Analysis Project (GTAP), a global research consortium focused on global economic governance challenges (trade, climate, resource use) is also coordinated by the University. Purdue University generated a record $438 million in sponsored research funding during the 2009–10 fiscal year with participation from National Science Foundation, National Aeronautics and Space Administration, and the Department of Agriculture, Department of Defense, Department of Energy, and Department of Health and Human Services. Purdue University was ranked fourth in Engineering research expenditures amongst all the colleges in the United States in 2017, with a research expenditure budget of 244.8 million. Purdue University established the Discovery Park to bring innovation through multidisciplinary action. In all of the eleven centers of Discovery Park, ranging from entrepreneurship to energy and advanced manufacturing, research projects reflect a large economic impact and address global challenges. Purdue University’s nanotechnology research program, built around the new Birck Nanotechnology Center in Discovery Park, ranks among the best in the nation.
The Purdue Research Park which opened in 1961 was developed by Purdue Research Foundation which is a private, nonprofit foundation created to assist Purdue. The park is focused on companies operating in the arenas of life sciences, homeland security, engineering, advanced manufacturing and information technology. It provides an interactive environment for experienced Purdue researchers and for private business and high-tech industry. It currently employs more than 3,000 people in 155 companies, including 90 technology-based firms. The Purdue Research Park was ranked first by the Association of University Research Parks in 2004.
Purdue’s library system consists of fifteen locations throughout the campus, including an archives and special collections research center, an undergraduate library, and several subject-specific libraries. More than three million volumes, including one million electronic books, are held at these locations. The Library houses the Amelia Earhart Collection, a collection of notes and letters belonging to Earhart and her husband George Putnam along with records related to her disappearance and subsequent search efforts. An administrative unit of Purdue University Libraries, Purdue University Press has its roots in the 1960 founding of Purdue University Studies by President Frederick Hovde on a $12,000 grant from the Purdue Research Foundation. This was the result of a committee appointed by President Hovde after the Department of English lamented the lack of publishing venues in the humanities. Since the 1990s, the range of books published by the Press has grown to reflect the work from other colleges at Purdue University especially in the areas of agriculture, health, and engineering. Purdue University Press publishes print and ebook monograph series in a range of subject areas from literary and cultural studies to the study of the human-animal bond. In 1993 Purdue University Press was admitted to membership of the Association of American University Presses. Purdue University Press publishes around 25 books a year and 20 learned journals in print, in print & online, and online-only formats in collaboration with Purdue University Libraries.
Sustainability
Purdue’s Sustainability Council, composed of University administrators and professors, meets monthly to discuss environmental issues and sustainability initiatives at Purdue. The University’s first LEED Certified building was an addition to the Mechanical Engineering Building, which was completed in Fall 2011. The school is also in the process of developing an arboretum on campus. In addition, a system has been set up to display live data detailing current energy production at the campus utility plant. The school holds an annual “Green Week” each fall, an effort to engage the Purdue community with issues relating to environmental sustainability.
Rankings
In its 2021 edition, U.S. News & World Report ranked Purdue University the 5th most innovative national university, tied for the 17th best public university in the United States, tied for 53rd overall, and 114th best globally. U.S. News & World Report also rated Purdue tied for 36th in “Best Undergraduate Teaching, 83rd in “Best Value Schools”, tied for 284th in “Top Performers on Social Mobility”, and the undergraduate engineering program tied for 9th at schools whose highest degree is a doctorate.
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