当然，我们这里不是诗词鉴赏栏目，我们只谈对于生命中美好事物的追求。且看其中第四首：

《鹧鸪天·元夕有所梦》

肥水东流无尽期，当初不合种相思。梦中未比丹青见，暗里忽惊山鸟啼。

春未绿，鬓先丝，人间别久不成悲。谁教岁岁红莲夜，两处沉吟各自知。

头两句就是我们这篇文字题目的来源。姜夔在睡梦中追忆自己青年时的恋人，在我们后人看来，除了个人生活中的伤情，这好似淡漠的句子中其实寄托着人生绵绵不绝的悲凉、久违和一再错过的沧桑，是普适的人类情感和人生体验。词人通过他的艺术，对于困厄现实进行了美的升华，也让自己在对于美好事物的追求中，超越个体的悲欢。这才是艺——也包括科学—— 让一代代人前赴后继的动力：让我们从个体的悲欢起，超脱到人类的悲欢止，这追求本身，就是寄托。

书归正题，最近有五个人完成了一项凝聚态物理学的研究工作，承蒙编辑部和 Referee 的支持，发表在最近一期的Chinese Physics Letter [1]。

**文章发展了动量空间量子蒙特卡洛算法，处理转角石墨烯等二维量子材料中涌现出的长程库伦相互作用问题**。长程的库伦相互作用，是关联电子系统中长期以来的普遍性难题之一（所以也就有意思，有它的美妙之处），比如量子霍尔效应或者分数量子霍尔效应这样的二维系统，电子之间存在的就是长程库伦相互作用。对于这样的系统，幸好有 Laughlin 猜出的波函数，作为很多情况下的理论极限，让人们有可以下手的地方，否则真是望费米海而兴叹了。而从量子多体问题严格数值计算的角度出发，发展出具有普适性、系统性的解决方案的尝试，一直没有成功。与具有短程相互作用（屏蔽之后）的 Hubbard 模型、海森堡模型等问题中量子蒙卡、张量网络大显身手不断取得进展相比，具有长程库伦相互作用的二维关联电子系统的数值求解，一直没有方法上的突破，显得寂寞和冷清。

近几年来，随着转角石墨烯，过渡金属二硫化物等更加优异的二维关联量子材料的出现，尤其是这些材料中也观察到了关联绝缘体、超导等短程相互作用材料（如高温超导）和模型（如 Hubbard 模型）中常见的典型量子多体物态（见文献 [2]），**如何求解长程库伦相互作用，如何发展出像严格求解实空间短程相互作用的 Hubbard 模型的行列式量子蒙特卡洛算法等问题，就又被大家提起和关注**。人们都知道，要想求解长程相互作用，需要走向动量空间。这是因为在实空间中，无论如何扩大截断范围，仍是短程相互作用，而与扩大截断范围所伴随的计算复杂度的增长，很快就会超出目前计算能力的极限。问题是，谁能成功地走出第一步，成功地在动量空间的相互作用模型中发展出量子蒙特卡洛算法，得到正确的结果，为后续的进展指出方向。

这篇文章就是回应这普适追求的一个尝试，只是起步，得到了一点有益的结果。我们发现采用动量空间的转角石墨烯有效模型，考虑长程库伦相互作用（投影到平带），在一些参数范围内可以用行列式量子蒙特卡洛方法严格求解，也从数学上证明了算法可以有效工作的原因（术语叫如何避免符号问题）。具体的推导和结果，感兴趣的同学和专家可以深入文章 [1]，与已知的严格结论的 benchmark 对比，在图1中展示。当然这只是开始，还有太多需要完善的地方，比如如何考虑材料中实际的能带效应，如何在动量空间中计算不同的物理可观测量等。这些具有普遍意义的问题就是吸引着从业人员不断追求下去的动力。

“库伦作用无尽期，动量蒙卡寄相思”，一如八百年前的姜白石，正是这对于具有普适意义的科学问题的追求，给了我们超越个体悲欢的寄托，推动着领域前进。

参考文献：

[1] Momentum space quantum Monte Carlo on twisted bilayer Graphene

Xu Zhang, Gaopei Pan, Yi Zhang, Jian Kang, Zi Yang Meng

Chin. Phys. Lett. 38, 077305 (2021) Express Letters

[2] 转角石墨烯的三昧，物理, 2021, 50(5): 348-352，https://mp.weixin.qq.com/s/bwy6beaE7gILl6RAVqqZFQ

]]>Our large-scale simulations reveal a rich phase diagram, establishing the existence of three distinct insulating phases whose appearance can be controlled by tuning different types of interactions between the electrons. The first phase, called a quantum valley Hall state, has an insulating bulk and a dissipationless electronic current on its edge. The second phase, called an intervalley coherent state, is an insulator in which electron pairs emerge that behave in the same way as in a ferromagnet. The third phase, called a valence bond solid, is a phase realized in single-layer graphene if there were strong Coulomb interactions and electrons with degrees of freedom beyond just charge and spin (known as multiflavor electrons).

These results offer an unbiased solution for pristine TBG at charge neutrality, that is, the system has no extra electronic charge, and serves as the foundation for explanation of the more exotic behaviors of this fascinating material.

Full Article: https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.011014

]]>Three passions, simple but overwhelmingly strong, have governed my life: the longing for love, the search of knowledge, and unbearable pity for the suffering of mankind.

—Bertrand Russell

As Russell nicely put it, scientists are often driven by strong passions for the search of knowledge, such search not only benefits the human society, but often times brings the ecstasy to themselves -- the ecstasy for instance of understanding the hearts of men, knowing why the stars shine, and apprehending the Pythagorean power by which number holds sway above the flux -- that is so great that scientists would often have endured the long hours of working and sleepless nights for the pursuit of such joy.

The Landau’s theory of Fermi liquid (FL) (Note 1), established in the first half of the 20th century, is the foundation of the scientific and industrial usage of the metallic materials in our society. It is also the basis of our current understanding of metals. However, in the second half of the 20th century, more and more metallic materials were discovered which behave very differently. The non-Fermi liquid (NFL) behaviour of these “strange metals” remains a puzzle to physicists, and there is no established theory to explain them.

Recently, a joint research team comprising members including Dr Zi Yang MENG, Associate Professor of Department of Physics at the University of Hong Kong (HKU), Dr Avraham KLEIN and Professor Andrey CHUBUKOV from the University of Minnesota, Dr Kai SUN, Associate Professor from the University of Michigan, and Dr Xiao Yan XU from the University of California at San Diego, has solved the puzzle of the NFL behaviour in interacting electrons systems, and provided a protocol for the establishment of new paradigms in quantum metals, through quantum many-body computation and analytical calculations. The findings have recently been published in Npj Quantum Materials. The work was supported by the Research Grants Council of HKSAR, and the Ministry of Science and Technology of China.

The Landau’s theory of Fermi liquid (FL) successfully explained many features of simple metals like Copper, Silver, Gold and Iron, such as when temperature changes, their resistivity, heat capacity and other properties follow simple function form with respect to temperature T (for example, resistivity follows ρ~T^{2} and heat capacity follows C~T, independent of material details). The success of the Fermi liquid theory lies in the central assumption that the electrons, the droplets in the Fermi liquid are not interacting with each other, but behave identically in the material.

However, many metallic materials which were discovered after FL was established, do NOT behave as FL. For example, in the so-called high-temperature superconductor compounds - copper oxides and iron pnictides - their resistivities are linear in temperature ρ~T before the system becomes superconducting (resistivity is then zero), and such systems are in general dubbed Non-Fermi-Liquids (NFL). Different from the simple FL, the electrons of NFL, the droplets, strongly interact with each other.

The physicists still do not have much clue about NFL, which makes it very difficult to make concrete predictions. Still, these systems are essential for the continued prosperity of human society, as NFLs hold the key in making use of high-temperature superconducting material that will solve the energy crisis. Currently, the so-called high-temperature superconducting materials still only work at temperature scale of-100 Celsius - they are called high-temperature in comparison with the FL superconductors, which work at temperature scale of -200 Celsius - so it is still hard to put high-temperature superconductors into daily usage at room temperature, but only then can we enjoy the nice properties of such material that the electronic power will not be loss in heat due to resistivity. Only when we understand how the NFL in high-temperature superconductor works at -100 Celsius, can we then design the ultimate material to work at room temperature. Therefore, the complete understanding of NFL is of vital importance.

Physicists from analytical background have been trying to understand NFL for about half a century. The problem of analytical calculation is that, due to the quantum many-body nature of the NFL, the convergence and accuracy of many theoretical predictions cannot be controlled or guaranteed; one would need unbiased quantum computation to verify these prepositions.

At the numerical front, there have been many previous attempts, but the problem is that the results obtained are always different from the analytical prediction. For example, the most important quantity of the NFL, the self-energy Σ , which describes the level of the electron interactions in the material, is expected to have a power-law frequency dependence such as Σ~ω2/3in the model exhibited in Fig.1. However, the computed self-energy doesn’t follow such as power-law, it shows a slow diverging behaviour, that is the self-energy computed doesn’t go to zero as frequency is reduced, but instead gets larger and large, as the data in Fig.2 (b) indicated. Such difference makes the situation even more perplexing.

After a very inspirational discussion between Dr Meng, Professor Chubukov, and Dr Klein, they realized that the setting of the numerical simulation is actually different from that of the analytical calculation. Such subtlety comes from the fact that the model simulations are performed on the finite system at finite temperature, that is T≠0, whereas the analytical expectations are strictly at the zero temperature T=0. In other words, the numerical data actually contain both the zero temperature NFL contribution and the contribution from the fluctuations at finite temperature. To be able to reveal the NFL behaviour from the lattice model simulation such as the setting in Fig.1, one would need to deduce the finite temperature contribution.

This turns out to be the key revelation to the puzzle of NFL. Dr Klein, Dr Sun and Prof Chubukov derived the analytical form of the finite temperature contribution (with the input from the lattice model in Fig.1 designed by Dr Meng and Dr Xu) for Dr Meng and Dr Xu to employ and deduce from the numerical data, the results are shown as the black dashed line and the data round it in Fig. 2(c). To everyone’s surprise and ecstasy, the results after the deduction perfectly exhibit the expected NFL behaviour, from finite temperature all the way to zero temperature, the power-law is revealed. It is the first time that such clear NFL behaviour has been obtained from unbiased numerical simulation.

Dr Meng said it is expected that this work will inspire many follow-up theoretical and experimental researches, and in fact, promising results for further identification of NFL behaviour in another model system have been obtained by the further investigations, he said: “This research work reveals the puzzle of Non-Fermi-liquid for several decades and paves the avenue for the establishment of new paradigm of quantum metals beyond those more than half-a-century ago. Eventually, we will be able to understand the NFL materials such as high-temperature superconductors as we understand simple metals such as Cooper and Sliver now, and such new understanding will solve the energy crisis and bring better industrial and personal applications to the society.”

Link of the journal paper: https://www.nature.com/articles/s41535-020-00266-6

Source: https://www.hku.hk/press/news_detail_21752.html

Note 1: Lev Landau is a Soviet physicist and among the founders of condensed matter physics. Enrico Fermi is an American-Italian physicist and one of the creators of the quantum mechanics.

]]>DQMC Note 2：从Hubbard Model开始的DQMC

DQMC Note 5：Checkerboard分解、Self-learning

夫人之相与，俯仰一世，或取诸怀抱，

悟言一室之内；或因寄所托，放浪形骸之外。

虽趣舍万殊，静躁不同，当其欣于所遇，

暂得于己，快然自足，不知老之将至。

——王羲之《兰亭集序》

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Quantum materials are becoming the cornerstone for the continuous prosperity of human society. From the next-generation AI computing chips that go beyond Moore’s law (the law is the observation that the number of transistors in a dense integrated circuit doubles about every two years, our PC and smartphone are all based on the success of it. Nevertheless, as the size of the transistors are becoming smaller to the scale of nanometer, the behaviour of electrons are subject to quantum mechanics, the Moore’s law is expecting to breakdown very soon), to the high speed Maglev train and the topological unit for quantum computers, investigations along these directions all belong to the arena of quantum material research.

However, such research is by no means easy, the difficulty lies in the fact that scientists have to solve the millions of thousands of the electrons in the material in a quantum mechanical way (hence quantum materials are also called quantum many-body systems), this is far beyond the time of paper and pencil, and requires instead modern quantum many-body computational techniques and advanced analysis. Thanks to the fast development of the supercomputing platforms all over the world, scientists and engineers are now making great use of these computation facilities and advanced mathematical tools to discover better materials to benefit our society.

The research is inspired by the KT phase theory avocated by J Michael Kosterlitz, David J Thouless and F Duncan M Haldane, laureates of the Nobel Prize in Phyiscs 2016. They were awarded for their theoretical discoveries of topological phase and phase transitions of matter. Topology is a new way of classifying and predicting the properties of materials in condensed matter physics, and is now becoming the main stream of quantum material research and industry, with broad potential applications in quantum computer, lossless transmission of signals for information technology, etc. Back to 1970s, Kosterlitz and Thouless had predicted the existence of topological phase, hence named after them as the KT phase, in quantum magnetic materials. However, although such phenomena have been found in superfluids and superconductors, KT phase has yet been realized in bulk magnetic material.

The joint team is led by Dr Zi Yang Meng from HKU, Dr Wei Li from Beihang Univeristy and Professor Yang Qi from Fudan University. Their joint effort has revealed the comprehensive properties of the material TMGO. For example, in Figure 2 (see below), by self-adjustable tensor network calculation, they computed the properties of the model system at different temperature, magnetic field, and by comparing with the corresponding experimental results of the material, they identify the correct microscopic model parameters. With the correct microscopic model at hand, they then performed quantum Monte Carlo simulation and obtained the neutron scattering magnetic spectra at different temperatures (neutron scattering is the established detection method for material structure and their magnetic properties, the closest such facility to Hong Kong is the China Spallation Neutron Source in Dongguan, Guangdong). As shown in Figure 3 (see below), the magnetic spectra with its unique signature at the M point is the dynamical fingerprint of the topological KT phase that has been proposed more than half-a-century ago.

“This research work provides the missing piece of topological KT phenomena in the bulk magnetic materials, and has completed the half-a-century pursuit which eventually leads to the Nobel Physics Prize of 2016. Since the topological phase of matter is the main theme of condensed matter and quantum material research nowadays, it is expected that this work will inspire many follow-up theoretical and experimental researches, and in fact, promising results for further identification of the topological properties in quantum magnet have been obtained among the joint team and our collaborators,” said Dr Meng.

Dr Meng added: “The joint team research across Hong Kong, Beijing and Shanghai, also sets up the protocol of modern quantum material research, such protocol will certainly lead to more profound and impactful discoveries in quantum materials. The computation power of our smartphone nowadays is more powerful than the supercomputers 20 years ago, one can optimistically foresee that with the correct quantum material as the building block, the personal devices in 20 years can certainly be more powerful than the fastest supercomputers right now, with minimal energy cost of everyday battery.”

Link of journal paper: https://www.nature.com/articles/s41467-020-14907-8

HKU Press Releases: https://www.hku.hk/press/press-releases/detail/21179.html

]]>Superconductivity, in which electrons pass through a material without resistance, was an important experimental discovery in the early 1900s because it holds great potential for better energy transportation and materials science. However, the ultra-cold temperature required – initially about -200 degrees Celsius – was for many years an obstacle to these developments. In the 1980s, scientists discovered new materials that exhibit superconductivity at temperatures around -100 degrees Celsius. This “high-temperature” superconductivity is considered one of the greatest success in modern quantum materials, but until recently it was not really understood why this “high-temperature” superconductivity occurred. This lack of understanding has hindered further advances both in the quantum many-body theory and materials realisations. The key of understanding the high-temperature superconductors, lies in the modeling and understanding the metallic states in which electrons are interacted in a quantum many-body manner, from which the superconductivity emerges.

Now, new work by Dr Meng Zi Yang and his collaborators has helped to fill in some of the blanks about the quantum critical metals from which the high-temperature superconductors emerge. The key lies in a proper modelling and an unbiased solution of the model that gives rise to the understanding the metallic states in which electrons interact in a quantum many-body manner.

The scholars designed a lattice model of interacting electrons and solved it with state-of-art quantum computation platforms – the Tianhe supercomputer in China’s national supercomputer centre.

Their goal was to understand the behaviour of interacting electrons in metal, especially close to a critical point in the transition between magnetic metal and non-magnetic metal. This was not possible using traditional simple analytical methods (such as mean-field treatments) because of the scale involved – quantum fluctuations have to be observed among a huge number of electrons (at 10²³).

Dr Meng brought in computational expertise to the problem, which involved applying a quantum Monte Carlo method – essentially, gambling where an electron sits in the system according to the fundamental principles of quantum mechanics – to engineer the transition of a magnetic metal to a non-magnetic metal. The team then used the Tianhe supercomputer to solve the problem.

“The magic happens when there is a transition from magnetic metal to non-magnetic metal because the interaction between electrons is strongest at that point and it is here where interesting phenomena such as critical metal and eventually superconductivity is generated,” Dr Meng said.

This kind of quantum critical metallic state has not been solved in an unbiased manner in research before and it gives scientists a new way of understanding the parent state of high-temperature superconductivity, with major implications for both quantum many-body theories and future efficient energy transportation.

Liu Z.H., Pan G., Xu X.Y., Sun K. and Meng Z.Y., "Itinerant quantum critical point with fermion pockets and hotspots", Proceedings of the National Academy of Sciences of the USA (PNAS), 2019, 116 (34), 16760-16767.

HKU Research Stories: https://www.hku.hk/research/stories/20645/

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