Great Country Academician

After hearing his friend's inquiry, Witten took a deep breath and calmed down slowly.

Looking at the silver-white curtain on the lecture table, he said: "As a pure mathematician, it may be difficult for you to understand the influence of the mathematical basic theory of non-equilibrium strongly correlated electron systems on condensed matter physics."

"If I were to evaluate, the status of difficult problems in strongly correlated electron systems in condensed matter physics is like the Riemann Hypothesis in number theory."

"In two different systems, it may be difficult to compare the difficulty of solving them. But the influence is not weak at all."

"Strongly correlated electron systems in non-equilibrium states are the most classic problem in strongly charged electronic systems. It studies the dynamic behavior of strongly correlated systems in non-equilibrium states to reveal new physical phenomena and application potential."

"But so far, no one in the physics and mathematics circles has been able to provide a complete mathematical foundation, or even a complete mathematical tool."

Witten explained briefly, but never moved his eyes away. He kept staring closely at the reporting table. The unrest in his heart appeared on his face, which surprised Deligne.

Having worked with this good friend at the Institute for Advanced Study in Princeton for so many years, he rarely saw Witten lose his temper like this, especially as he grew older over the years.

But after listening to the explanation, he somewhat understood.

If the influence of a difficult problem can be compared with the Riemann Hypothesis in mathematics, then this difficult problem will inevitably have extremely high visibility and influence in the corresponding field.

Just like the Riemann Hypothesis, with the development of mathematics in recent years, there are thousands of mathematical formulas based on the establishment of this conjecture.

If the Riemann Hypothesis is proven to be true, then these thousands of formulas will be promoted into theorems together with them.

If it is disproven, the field of number theory will be followed by the biggest earthquake in history.

If the impact of the strong correlation field on condensed matter physics can reach this level, it is no wonder that Witten is so surprised.

Even just a partial result can affect the development of condensed matter physics.

In fact, Deligne's thinking was still too simple.

Compared with Witten, he is really a pure mathematician, mainly engaged in research work in algebraic geometry and number theory, and has never been away from mathematics in his life.

He really doesn't know much about physics. Although he knows about condensed matter physics and the strongly correlated electron system, it is not clear how much influence these two have in condensed matter physics.

Even Edward Witten was not entirely sure of the extent of the influence of strongly correlated electronic systems.

After all, his main research scope does not include condensed matter physics, and he only knows about mathematical physics and quantum theory.

In fact, the influence of strongly correlated electronic systems in the field of condensed matter physics, and even in the entire field of physics, is one of the largest branches.

The correlation of electrons will lead to a large number of rich quantum effects and phenomena such as high temperature, unconventional superconductivity, anomalous magnetism, metal-insulator phase transition, semimetals, giant thermoelectricity, multiferroics, heavy fermions, etc.

Exploring the microscopic mechanisms of these effects and phenomena and establishing a multi-body quantum theoretical system are one of the most active and challenging frontier research fields in condensed matter physics, quantum physics, chemical physics and other fields.

Perhaps the strongly correlated electronic system described by the Riemann Hypothesis is not a very appropriate explanation.

If you really want to use mathematics to find an approximate problem, then the NS equation should be the most similar.

The advancement and solution of the NS equation will elevate humankind's understanding of fluids to a great level, thereby bringing about tremendous development in all theories and technologies related to fluids.

From simulating cloud flow, ocean flow, to turbulence after airplane takeoff, blockage after rocket launch, to blood flow through the heart, etc.

All will be greatly improved.

For strongly correlated electronic systems, the solution of this entire set of systemic problems will make a qualitative leap in human understanding of condensed matter physics and microscopic particles.

In this field, what is affected is the development of materials.

For example, the most popular new materials in recent years are copper-based/iron-based superconductors, FeSe/STO interface superconductors, iridium oxides, Mott insulators, quantum antiferromagnets and other low-dimensional quantum materials, all of which are based on strongly correlated electrons. born under the system.

The emergence of each of these materials has made human science and technology a big step forward, and its significance is naturally self-evident.

On the lecture stage, Xu Chuan opened the PPT and turned to a new page.

“For us, mathematics is the study of concepts such as quantity, structure, change, and spatial models.”

"Through the use of abstraction and logical reasoning, arising from counting, computation, measurement, and observation of the shape and motion of objects. We extend these concepts in order to formulate new conjectures and build from appropriately chosen axioms and definitions Rigorously derived truth.”

"And these truths are applied in other fields, bringing technology and progress to mankind."

"What I want to talk about today is to use mathematical tools to bring a set of mathematical theories and calculation methods to the strongly correlated electron system in condensed matter physics, which can greatly promote the development of condensed matter physics and particle physics."

"Of course, in turn, the development of physics will inevitably lead to the advancement of mathematics."

"Just like Newton invented calculus to solve physical problems. Faraday studied electricity and magnetism, but due to his limited mathematical level, he was unable to further provide a profound connection between electricity and magnetism, while Maxwell used his superb mathematical talents to Just like the perfect unification of electricity and magnetism.”

"After all, we always need mathematics to explain these new phenomena and theories."

While talking, Xu Chuan flipped through the PPT.

"Okay, now I'm going to give a report on my thesis from the shallower to the deeper level."

"First principles calculation is an algorithm that directly solves the Schrödinger equation after some approximation processing based on the principles of the interaction between atomic nuclei and electrons and their basic motion laws, using the principles of quantum mechanics and starting from specific requirements."

"Whether it is ab initio calculations based on Hartree-Fock self-consistent field calculations or density functional theory (DFT) calculations, they all fall into it."

"Such as φM=-(Ve+εFe)."

"I believe that everyone here, even if they have not studied physics, can see that this is the minimum work required to calculate the Fermi level of metal M to cross the surface with no net charge to extract electrons."

"."

On the lecture stage, as Xu Chuan explained, lines of calculations were presented in front of everyone.

For the mathematicians sitting here today, as time goes by, not everyone can smoothly keep up with the rhythm and understand these things.

But in the lecture hall with hundreds of people, there are many top mathematicians and scholars in the field of mathematical physics, such as Edward Witten, Mr. Qiu, Deligne and so on.

These people were listening attentively.

The calculation of first principles is not difficult for the mathematicians here to understand. After all, it uses mathematics to calculate from scratch and does not require any experimental parameters. It only requires some basic physical constants to obtain the principle of the basic properties of the system's ground state. .

But as time goes by, not many can keep up.

For Xu Chuan, he did not expect to be able to explain things related to physics clearly to everyone in today's mathematics seminar.

What he did was to tell many mathematicians about the applications of mathematics and the relationship between mathematics and physics in the process.

Compared to mathematicians like his mentor Professor Deligne, he is actually far less pure.

If possible, he hopes that more mathematicians can enter the field of physics. Of course, I also hope that more physicists can accept more new mathematical knowledge.

Physics is a science that understands nature and provides abstract descriptions of the real world, while mathematics is the language of science. Neither can do without the other.

In the front row of the auditorium, watching Xu Chuan standing on the podium and reporting, Professor Deligne couldn't help but poke Witten next to him with his hand: "What is he doing?"

In the second half of the report meeting, he could no longer understand what his student was saying.

For a top mathematician like him, this feeling is too unacceptable.

Professor Witten stared at the lecture table without looking back and said: "To put it simply, he is using mathematics to explain the non-equilibrium strongly correlated electron system. Just like Einstein used Riemannian geometry to describe gravity. .”

"What he is doing now is using mathematics to describe non-equilibrium strongly correlated electronic systems."

Deligne was silent for a moment and asked: "How can I understand this?"

Hearing this, Edward Witten came back to his senses, thought for a moment, and said, "Perhaps this requires you to learn some knowledge of condensed matter physics?"

After a slight pause, he added: "You may also need some knowledge of quantum chemistry, quantum many-body physics, atomic and molecular physics, etc."

"But there is a high probability that it is already too late. What he reported today already involves the most cutting-edge condensed matter physics. Even for me, it is not that easy to understand."

Deligne: "."

PS: There will be another chapter tonight, please vote for me.