Hiding a Devil's staircase in spin-crossover materials

 Collaborators and I just posted a preprint.

Hidden Devil's staircase in a two-dimensional elastic model of spin crossover materials  Gian Ruzzi, Jace Cruddas, Ross H. McKenzie, Ben J. Powell

Condensed matter physicists study diverse states of matter (crystals, antiferromagnets, liquid crystals, superfluids, …), the types of ordering and symmetry associated with each state, and the transitions that can occur between different states when external parameters such as temperature and magnetic field are varied. A challenge for theorists is to create simple models that can describe the orderings in broad classes of materials, taking into account properties at the microscopic (atomic) level. Spin-crossover materials are a broad class of materials that are chemically and structurally complex: composed of transition metal ions surrounded by large molecules. The quantum spin (magnetic) state of the metal ion can change as the temperature is varied, leading to changes in magnetism, colour, and structure. A diversity of spatially periodic orderings of the spin states has been observed and transitions between these ordered states can be smooth, or discontinuous with hysteresis. Chemists love these materials because of their chemical and structural complexity and their potential technological applications, including as switchable magnetic memories. 

We present a general procedure to derive a simple lattice Hamiltonian that can describe the diversity of spin-state orderings and transitions, based on physically realistic microscopic interactions. We show that two classes of models that have been considered previously, Ising models and ball-and-spring elastic models are equivalent. Furthermore, our work shows that the Ising interactions arise from elastic interactions, whereas previously the Ising interactions were just treated as empirical parameters with no clear physical origin. These interactions compete with one another (they are frustrated). At zero temperature there appears to be an infinite number of possible spin-state orderings and transitions between them are described by a Devil’s staircase. At finite temperature, some of this structure is washed out, but there are first-order transitions between some of these ordered states. This complexity is similar to that found in spin-crossover materials, and here is produced by a theoretical model with just a single elastic parameter, the ratio of the bulk and shear moduli. 

The physical insight and methodology advances in our work provide a foundation for more detailed microscopic descriptions (such as using electronic structure methods) and the establishment of structure-property relations that may guide the development of new materials tailored for specific technological applications.

 

This supersedes an earlier preprint.  (Aside: given the new paper is so different we thought this should be a new submission to the arxiv, but they did not agree).

We got so much helpful feedback on that work that we did more calculations and discovered the hidden Devil's staircase.

We welcome any comments.

Superconductivity in kagome metals

Condensed matter physics is driven by fashion (too much). Is it fair to say that the latest fashion is the vanadium-based kagome metals  AV3Sb3 (A=K,Rb,Cs)?

[The PRL reporting superconductivity was published less than six months ago and has already been cited 44 times.]

These are certainly fascinating materials and have probably attracted attention for the following reasons.

-Kagome lattices support rich physics such as flat bands, Dirac metals, massively degenerate ground states, and (possibly) spin liquids.

-unlike other Kagome metals these compounds have both inversion and time-reversal symmetries, there is a Z2 topological invariant associated with bands near the Fermi surface, and topologically non-trivial surface states

-they are superconducting; furthermore, there are two superconducting domes as a function of pressure

-an anomalous Hall effect has been observed, which may result from topological physics

-there may be several types of charge order, including chiral charge density wave order

-the materials may be a topological superconductor [which MAY mean that it can be used to construct qubits that are "topologically protected].

Here are a few papers that I have looked at to get a better feel for this topic. I add a few things I gleaned from the papers and some basic questions I have. I welcome suggestions of other papers, that may be more helpful introductions. 

CsV3Sb5: A Z2 Topological Kagome Metal with a Superconducting Ground State 

Brenden R. Ortiz, Samuel M. L. Teicher, Yong Hu, Julia L. Zuo, Paul M. Sarte, Emily C. Schueller, A. M. Milinda Abeykoon, Matthew J. Krogstad, Stephan Rosenkranz, Raymond Osborn, Ram Seshadri, Leon Balents, Junfeng He, and Stephen D. Wilson

The figure below shows a top-down view of a single layer. The V atoms (red) form a Kagome lattice. There are three V atoms per unit cell.

The authors present DFT-based band structure calculations, which are compared to ARPES data. The good agreement suggests to me that strong correlations are not important. 

The authors use their band structures to construct Wannier orbitals and a tight-binding model for the band structure. However, even in the Supplementary information, they provide no details of this. I would like to know answers to the following.

For bands near the Fermi energy what is the composition of the underlying atomic orbitals (especially, how much d on V and p on Sb)?

How much of the band structure is described by a simple tight-binding model on a Kagome lattice with only next-nearest neighbour hopping?

Is the hopping between V sites via the p orbitals on the intermediate Sb atoms (superexchange in chemistry language)?

What is the band filling? 

Simple charge counting suggests there is one electron per triangle (1/6 band filling).

At a temperature of 100 K the intralayer resistivity is about 10 microohm-cm, well below the Mott-Ioffe-Regel limit (where the mean-free path is comparable to the lattice spacing), also suggesting that strong correlations are not significant.

Electronic instabilities of kagome metals: saddle points and Landau theory Takamori Park, Mengxing Ye, Leon Balents

Section V. A. discusses a tight-binding model. I think it is for the Kagome lattice with only nearest-neighbour hopping.

Double-dome superconductivity under pressure in the V-based Kagome metals AV3Sb5 (A = Rb and K)

C. C. Zhu, X. F. Yang, W. Xia, Q. W. Yin, L. S. Wang, C. C. Zhao, D. Z. Dai, C. P. Tu, B. Q. Song, Z. C. Tao, Z. J. Tu, C. S. Gong, H. C. Lei, Y. F. Guo, S. Y. Li

Answers to the following questions may determine whether interest in these materials is sustained.

Is the superconductivity topological?

Is the superconductivity unconventional? There are two independent parts to this question: does the superconductivity result from electron-phonon coupling or purely electronic interactions? Is the order parameter s-wave?

[On the related question of whether there are nodes in the superconducting energy gap there are already preprints with contradictory conclusions].

Is there any significant connection between any of the following: the topological character of the metal, the superconductivity, charge density orderings, and electron correlations?

The emergence of wisdom

From a scientific point of view, the following questions are very important, particularly as "big data" and "machine learning" enter science.

What is the difference between data, information, knowledge, and wisdom? 

What is the relationship between them?

In chapter 1 of The Model Thinker, Scott Page has a helpful discussion on these questions, built around the diagram below.

I like the diagram because the hierarchy shows how the different entities emerge from one another. Here is Page's explanation.

At the bottom of the hierarchy lie data: raw, uncoded events, experiences, and phenomena. Births, deaths, market transactions, votes, music downloads, rainfall, soccer matches, and speciation events. Data can be long strings of zeros and ones, time stamps, and linkages between pages. Data lack meaning, organization, or structure.  

Information names and partitions data into categories. [It suggests something about how the world may be structured (or not). It identifies patterns.]

Plato defined knowledge as justified true belief. More modern definitions refer to it as understandings of correlative, causal, and logical relationships. Knowledge organizes information. Knowledge often takes model form. ...models explain and predict.

Atop the hierarchy lies wisdom, the ability to identify and apply relevant knowledge [to a particular problem]. Wisdom requires many-model thinking. Sometimes, wisdom consists of selecting the best model, as if drawing from a quiver of arrows. Other times, wisdom can be achieved by averaging models; this is common when making predictions... When taking actions, wise people apply multiple models like a doctor’s set of diagnostic tests. They use models to rule out some actions and privilege others. Wise people and teams construct a dialogue across models, exploring their overlaps and differences.

Aside 1. I note that this many-model approach is similar to the method of multiple hypotheses advocated by John Platt and that I have blogged about previously.

Aside 2. Page quotes these great lines 

“Where is the wisdom we have lost in knowledge? 

Where is the knowledge we have lost in information?”

taken from the opening stanza Choruses from The Rock, by T.S. Eliot. However, reading the whole stanza I feel that a lot is being missed as the poem is really about secularisation.

Increased competition for admission to USA PhD programs?

 We live in different times. There is some anecdotal evidence that this year admissions to leading graduate schools in the USA have become a lot more competitive, particularly for international applicants. Doug Natelson has discussed the issue, highlighting that it is important for unsuccessful applicants to know that these are exceptional times and their lack of success does not reflect on their ability and potential, but rather on structural issues.

I have a few questions for readers.

A. Is it your experience (whether as an applicant, recommender, or decider) that it is more competitive this year? Have you seen any articles about this?

B. If so, which of the following factors are particularly causing this crunch? (Doug mentions some of these factors.)

1. Fewer current Ph.D. students are graduating because of delays or lack of job opportunities due to the pandemic. This leaves less money for new students.

2. Universities are nervous about making offers to international students because of pandemic-related travel restrictions and uncertainty. There is a preference for domestic students.

3. Some universities are undergoing budget cuts or are very uncertain about their financial stability. This has flowed on to reduced admissions.

4. There are more applicants because of limited alternative job opportunities.

5. Other factors?

It will help all concerned if we can have a more accurate picture of what is going on. I raise the issue because I was surprised and disappointed that I encouraged a student to apply and wrote a glowing reference (neither of which I do very often) but he did not succeed. 

Please do share what you do know.

What to read after A Very Short Introduction?

On friday I finally sent my draft manuscript for Condensed Matter Physics: A Very Short Introduction off to the publisher. Yay!

At the end of the book there is the opportunity to make suggestions for what people might read next (assuming readers have been so inspired!).

This was not easy to write for condensed matter as there is a dearth of popular and accessible books. Here is my draft of Further Reading. What do you think?

Three years ago I asked about basic introductions to condensed matter for motivated and intelligent beginning undergrads and received some helpful suggestions. From this I will add to my list 

Fundamentals of Condensed Matter and Crystalline Physics: An Introduction for Students of Physics and Materials Science by David Sidebottom.

I welcome further suggestions.

A rich phase diagram for a Hubbard model on the decorated honeycomb lattice

 An important scientific idea is that simple rules can produce complex behaviour. In condensed matter theory, model Hamiltonians with just a few parameters can have rich phase diagrams with many competing ground states. My colleagues and I just completed a paper that is one more example of this.

Spin-0 Mott insulator to metal to spin-1 Mott insulator transition in the single-orbital Hubbard model on the decorated honeycomb lattice  H. L. Nourse, Ross H. McKenzie, B. J. Powell 

We study the interplay of strong electron correlations and intra-triangle spin exchange at two-thirds filling of the single-orbital Hubbard model on the decorated honeycomb lattice using rotationally invariant slave bosons (RISB). We find that the spin exchange tunes between a spin-1 Mott insulator, a metal, and a spin-0 Mott insulator when the exchange is antiferromagnetic. The Mott insulators occur from effective intra-triangle multi-orbital interactions and are adiabatically connected to the ground state of an isolated triangle. An antiferromagnetic spin exchange, as determined by the Goodenough-Kanamori rules, may occur in coordination polymers from kinetic exchange via the ligands. We characterize the magnetism in the regime where spin-triplets dominate. For small U a spin-1 Slater insulator occurs with antiferromagnetic order between triangles. Magnetism in the spin-1 Mott insulator is described by a spin-1 Heisenberg model on a honeycomb lattice, whose ground state is Néel ordered.

Comments are welcome.

Is this a good book preface?

 I have never written a book before. Hence, I have never written a preface. For what it is worth, according an article on Masterclass.com, Writing 101: How to Write a Preface for a Book, 

In one or two pages, an author’s preface is meant to:

• Explain why the author chose to write about this topic
• Reveal their motivation and inspiration for writing the book
• Describe the process of researching the topic of the book
• Outline the process of writing the book, including any challenges and how long it took.

Based on this I wrote the following draft of a Preface to Condensed Matter Physics: A Very Short Introduction. 

Preface

I love learning new things. I find joy in understanding, particularly if it is profound and important.  I love trying to communicate that understanding, particularly in ways that may appeal and inspire. I find beauty in ideas, images, and stories. I am a “big picture” person. I love seeing connections, whether deep or anecdotal, between different pieces of knowledge. Hence, it is not surprising that I was excited when I discovered the Very Short Introductions series. My eclectic interests are reflected in some of the VSI titles on my bookshelf including Marx, Depression, Complexity, Adam Smith, Water, Corruption, Social and Cultural Anthropology, and Epidemology.

Condensed matter physics is one of the largest and most vibrant sub-fields of physics. It is a rich source of ideas, concepts, and techniques that have cross-fertilised with not just other sub-fields of physics, such as elementary particles, but also other scientific disciplines (chemistry, biology, computer science, sociology, economics) and engineering (electrical, materials, chemical). In the past thirty years, the Nobel Prize in Physics has been awarded thirteen times for work on condensed matter. In the past twenty years, eight condensed matter physicists have received the Nobel Prize in Chemistry. 

The last thirty-five years I have worked in condensed matter physics: learning, teaching, and researching. I have come to love the discipline and to appreciate its’ importance, intellectual richness, beauty, and coherence. Given there are now more than 500 titles in the VSI series I was surprised and disappointed that there was no title on condensed matter. There were titles on subjects such as Magnetism, Superconductivity, Crystallography, and now Soft Matter, that are related to condensed matter, but they miss the unity of the whole discipline. Hence, I am glad to have the opportunity to contribute to the series.

Condensed matter physics is all about emergence; the whole is greater than the sum of the parts. A system composed of many interacting parts (atoms) has properties (a state of matter) that are qualitatively different from the properties of the individual parts. This commonality is why condensed matter physics has had such a fruitful cross-fertilisation with other sciences. It is arguably the field with the greatest success at understanding emergent phenomena in complex systems, particularly at the quantitative level. This is not because condensed matter physicists are smarter than sociologists, economists, or neuroscientists. It is because the materials we study are much “simpler” than societies, economies, and brains. 

Even after I finished a undergraduate degree in physics in Australia in 1982, I would not have been able to answer the question, “what is condensed matter physics?” There was no such course at my university. The only related one was in “solid state physics” and I did not take that course because the lecturer was boring and incompetent. I then went to Princeton University in the USA to pursue a Ph.D. in physics and ended up working in theoretical condensed matter physics. But, even then I don’t think I was clear what the field is really about. In hindsight, I think this failing arose due to my own lack of attention to broader intellectual issues.

In the past two decades several circumstances have shaped the emergence of the shape and content of this book. One of the best ways to learn and understand a subject is to try and teach it, particularly to novices. In 2000, I started working at the University of Queensland and I was assigned to teach a course for second year science undergraduates entitled, “Thermodynamics and Condensed Matter Physics.” As is too often the case in universities the curriculum for this course did not arise from of a grand and coherent educational vision but rather from historical circumstances involving politics and personnel. Nevertheless, serendipity is a powerful means of discovery. Order can emerge from chaos, and beauty can rise from ashes. In most undergraduate programs, condensed matter physics is only taught in the third or fourth year, if at all, and usually as an elective for physics majors. This is because courses build on advanced subjects such as quantum theory and statistical mechanics, and the associated mathematical machinery. However, many of the most important ideas in condensed matter can be understood without reference to these advanced subjects. This is the approach I take here, with some of the material drawing from the experience of teaching that course. 

In 2009, I started writing a blog, Condensed Concepts: Ruminations on Emergent Phenomena in Condensed Phases of Matter. A great beauty of blogging is that you can write about whatever you like and in whatever format you choose. I have now written more than two thousand posts on diverse topics, including technical commentary on specific scientific papers, presentations to school students, history, university politics, job advice, philosophy of science, and the relation of condensed matter to other sciences, from biology to economics. Blogging improved my ability to write for a broad audience and clarified my thinking about condensed matter physics in a broad intellectual context. Some of the material presented in this book is loosely based on my blog posts.

Over the past decade I have had significant and fruitful intellectual interactions with colleagues in a range of disciplines including law, economics, sociology, history, engineering, anthropology and biology. This has been facilitated and stimulated by being part of several informal networks (local, national, and international) of academics, with a particular interest in the relationship between our academic disciplines, Christian theology, and human flourishing. This has clarified my understanding of my own discipline and how to discuss it with those outside.

Condensed matter physics as a research field is not just defined by the objects that it studies (states of matter in materials), but rather by a particular approach to the study of these objects. The aim is to address fundamental questions and to find unifying concepts that can be used to describe and understand a wide range of phenomena in materials that are chemically and structurally diverse. My aim has not been to write a popular summary of the main topics covered in a textbook. Rather it is an essay that I hope brings joy and appreciation and a desire to learn more. I have chosen to emphasise concepts rather than trying to explain technical details. 

To make the book more interesting and engaging in most of the chapters I have consciously included several elements. A key concept, some connection to everyday experience, an interesting anecdote, a technological application, and a connection to another discipline. There are no mathematical equations and many figures.

This VSI is shaped by an important intellectual legacy that should be acknowledged and only became apparent to me when much of it was written. In his wonderful recent biography, Mind over Matter: Philip Anderson and the Physics of the Very Many, Andrew Zangwill states “more than any other twentieth-century physicist, he [Anderson] transformed the patchwork of ideas and techniques formerly called solid-state physics into the deep, subtle, and intellectually coherent discipline known today as condensed matter physics.” I hope that what I have written does honour this rich legacy of Phil Anderson (1923-2020).

I have really enjoyed writing this book. I hope I can convince you that condensed matter physics is fascinating, significant, exciting, beautiful, and profound.  

What do you think? Is this a good Preface? Would this motivate your aunt or your colleague in law to start reading the book?