Arkya Chatterjee

About Me

I am a graduate student in the Department of Physics at the Massachusetts Institute of Technology (MIT), working in the area of condensed matter theory supervised by Professor Xiao-Gang Wen. Before this, I was an undergraduate at the Indian Institute of Technology (IIT) Bombay where I earned a major in Physics and a minor in Mathematics. My current research is aimed at developing theoretical tools that provide insights into the organizing principles behind the rich emergent phenomena realized in quantum many-body systems. In the process, I have used concepts from topology and category theory for various physics questions. I also maintain a general interest in applications of statistical physics in studying emergent phenomena in complex systems more broadly. In the past, I have explored this interest in topics such as cellular growth, self-organized collective motion, dynamics of internet memes, etc. Outside of work, I enjoy cooking, reading, listening to (and playing) music, and learning languages.

Non-technical summary of research interests:

My research is motivated by the age-old question of "What is stuff made out of?" It is a question of classification of the dazzling array of beautiful materials we see around us every day, both natural and human-made. Ancient civilizations classified materials into four classes: Earth, Fire, Air, Water, based on the experience that all substances seen or used in daily life are hot, cold, dry, or wet. The development of new classes of materials, over the course of human history, ushered in new eras of technological advancement, such as Stone age, Bronze age, and Iron age. In the era of scientific understanding, humans obtained an increasingly sophisticated understanding of the microscopic building blocks of matter. Hand-in-hand with this progress, our understanding of matter on the macro-scale has also undergone many revolutions. One major milestone in this process was Landau's theory of phases of matter based on symmetries. Using this framework, we were able to understand exactly why ice and liquid water are so different even though they are made of the same molecules, and why mercury when cooled to near-absolute-zero temperatures suddenly allows current to flow through it without any resistance. Despite this success, towards the end of the last century it became clear that this so-called "Landau paradigm" might not be enough. Revolutionary experiments in the 1980s discovered a special class of quantum materials which one could not tell apart simply by their symmetries. This motivated the definition of topological order as a new classifying principle. This serves as another important cornerstone of the modern understanding of quantum phases of matter. Despite these successes, there are a class of materials that still lie beyond the reach of existing classifying principles, called "gapless states". These are particularly interesting because they have low-energy modes which we humans may be able to coax to do our bidding. Metals are one class of gapless matter, which have a large number of electrons that are almost free and can be forced to march in unison down an externally applied potential drop: this is how we have electricity! There is no known systematic understanding of gapless states at present. One of the goals of my PhD research is aimed at developing tools to identify the organizing principles of such states of matter to eventually reach a complete classification.

Technical summary of research interests:

My PhD research is largely aimed at understanding symmetries in quantum many body systems from a modern point of view. We know from experience that symmetry is a very useful non-perturbative tool in condensed matter. Most famously, Landau's theory of phase transitions is based on spontaneous breaking of a system's underlying symmetries. For the past few decades however, many quantum phases and phase transitions have been discovered that are beyond this framework. A general classification of gapped phases is now understood reasonably well. However continuous phase transitions as well as gapless phases are still quite mysterious, due to the lack of a general theory. In recent years, there has been a revolution in our understanding of symmetries in quantum field theories. Various generalized notions of symmetry have been fruitfully introduced. In the infrared limit, phases of many body quantum systems are often described by QFTs, therefore these advances have also taught us various lessons about emergent symmetries in phases of quantum many body systems. With the many different kinds of symmetries that have been discovered, they seemingly require different mathematical descriptions. It is conceptually desirable to develop a unifying framework that puts the physical content of the symmetries on the foreground. In collaboration with my advisor, I have been exploring the so called Symmetry/Topological Order (Sym/TO) correspondence, which proposes a correspondence between symmetry charges and symmetry defects in a d-dimensional quantum system and the fusion and braiding data of a d+1-dimensional topological order. We find that the low energy consequences of apparently different symmetries can be equivalent, and this equivalence can be uncovered by said correspondence. We also find that the SymTO can concisely capture the universal data associated with emergent symmetries in conformal field theories describing continuous phase transitions in 1+1D. A long-term goal is to leverage this framework to obtain a classification of gapless phases more broadly.

Research

My research is listed in various places on the web:
Google scholar, INSPIRE, arXiv, ResearchGate, ORCID

Emergent generalized symmetry and maximal symmetry-topological-order

In collaboration with: Wenjie Ji and Xiao-Gang Wen

A characteristic property of a gapless liquid state is its emergent symmetry and dual symmetry, associated with the conservation laws of symmetry charges and symmetry defects respectively. These conservation laws, considered on an equal footing, can't be described simply by the representation theory of a group (or a higher group). They are best described in terms of a topological order (TO) with gappable boundary in one higher dimension; we call this the symTO of the gapless state. The symTO can thus be considered a fingerprint of the gapless state. We propose that a largely complete characterization of a gapless state, up to local-low-energy equivalence, can be obtained in terms of its maximal emergent symTO. In this paper, we review the symmetry/topological-order (Sym/TO) correspondence and propose a precise definition of maximal symTO. We discuss various examples to illustrate these ideas. We find that the 1+1D Ising critical point has a maximal symTO described by the 2+1D double-Ising topological order. We provide a derivation of this result using symmetry twists in an exactly solvable model of the Ising critical point. The critical point in the 3-state Potts model has a maximal symTO of double (6,5)-minimal-model topological order. As an example of a noninvertible symmetry in 1+1D, we study the possible gapless states of a Fibonacci anyon chain with emergent double-Fibonacci symTO. We find the Fibonacci-anyon chain without translation symmetry has a critical point with unbroken double-Fibonacci symTO. In fact, such a critical theory has a maximal symTO of double (5,4)-minimal-model topological order. We argue that, in the presence of translation symmetry, the above critical point becomes a stable gapless phase with no symmetric relevant operator. Our results are available as a preprint on arXiv.[EM1]

[EM1] Chatterjee, Arkya, Wenjie Ji, and Xiao-Gang Wen. Emergent generalized symmetry and maximal symmetry-topological-order. arXiv preprint arXiv:2212.14432. [Preprint]

maxSymTO — Resolving the symmetries of a massless QFT (gapless state) in terms of two different SymTO's

Holographic theory for continuous phase transitions: the emergence and symmetry protection of gaplessness

In collaboration with: Xiao-Gang Wen

Two global symmetries are holo-equivalent if their algebras of local symmetric operators are isomorphic. A holo-equivalent class of global symmetries is described by a gappable-boundary topological orders (TO) in one higher dimension (called symmetry TO), which leads to a symmetry/topological-order (Symm/TO) correspondence. We establish that:

For systems with a symmetry described by symmetry TO M, their gapped and gapless states are classified by condensable algebras A, formed by elementary excitations in M with trivial self-/mutual statistics. These so-called A-states can describe symmetry breaking orders, symmetry protected topological orders, symmetry enriched topological orders, gapless critical points, etc., in a unified way.
The local low-energy properties of an A-state can be calculated from its reduced symmetry TO M_/A, using holographic modular bootstrap (holoMB) which takes M_/A as an input. Here M_/A is obtained from M by condensing excitations in A. Notably, an A-state must be gapless if M_/A is nontrivial. This provides a unified understanding of the emergence and symmetry protection of gaplessness that applies to symmetries that are anomalous, higher-form, and/or non-invertible.
The relations between condensable algebras constrain the structure of the global phase diagram. We find that, for 1 + 1D Z₂×Z₂' symmetry with mixed anomaly, there is a stable continuous transition (deconfined quantum critical point) between the Z₂-breaking-Z₂'-symmetric phase and the Z₂-symmetric-Z₂'-breaking phase. The critical point is the same as a Z₄ symmetry breaking critical point.
1+1D bosonic systems with S3 symmetry have four gapped phases with unbroken symmetries S₃, Z₃, Z₂, and Z₁. We find a duality between two transitions S₃ ↔ Z₁ and Z₃ ↔ Z₂: they are either both first order or both (stably) continuous, and in the latter case, they are described by the same conformal field theory (CFT).
The gapped and gapless states for 1 + 1D bosonic systems with anomalous S₃ symmetries are obtained as well. For example, anomalous S₃⁽¹⁾ and S₃⁽²⁾ symmetries can have symmetry protected chiral gapless states with only symmetric irrelevant and marginal operators.

Our results have been published in Physical Review B.[HT1]

[HT1] Chatterjee, Arkya, and Xiao-Gang Wen. "Holographic theory for continuous phase transitions: Emergence and symmetry protection of gaplessness." Physical Review B 108, no. 7 (2023): 075105. [Journal, Preprint]

Gapless-SymTO-boundary — Gapless boundaries of the 2+1D SymTO, associated with non-Lagrangian condensable algebras, correspond to gapless states of the 1+1D theory

Symmetry as a shadow of topological order and a derivation of topological holographic principle

In collaboration with: Xiao-Gang Wen

Symmetry is usually defined via transformations described by a (higher) group. But a symmetry really corresponds to an algebra of local symmetric operators, which directly constrains the properties of the system. In this paper, we point out that the algebra of local symmetric operators contains a special class of extended operators -- transparent patch operators, which reveal the selection sectors and hence the corresponding symmetry. The algebra of those transparent patch operators in n-dimensional space gives rise to a non-degenerate braided fusion n-category, which happens to describe a topological order in one higher dimension (for finite symmetry). Such a holographic theory not only describes (higher) symmetries, it also describes anomalous (higher) symmetries, non-invertible (higher) symmetries (also known as algebraic higher symmetries), and non-invertible gravitational anomalies. Thus, topological order in one higher dimension, replacing group, provides a unified and systematic description of the above generalized symmetries. This is referred to symmetry/topological-order (Symm/TO) correspondence. Our approach also leads to a derivation of topological holographic principle: \emph{boundary uniquely determines the bulk}, or more precisely, the algebra of local boundary operators uniquely determines the bulk topological order. As an application of the Symm/TO correspondence, we show the equivalence between Z₂×Z₂ symmetry with mixed anomaly and Z₄ symmetry, as well as between many other symmetries, in 1-dimensional space. Our results have been published in Physical Review B.[LO1]

[LO1] Chatterjee, Arkya, and Xiao-Gang Wen. "Symmetry as a shadow of topological order and a derivation of topological holographic principle." Physical Review B 107.15 (2023): 155136. [Journal, Preprint]

Dual view — Symmetry as a shadow of topological order

Active Gel Physics of Actomyosin Cortex

In collaboration with: Mainak Chatterjee, Amitabha Nandi, and Anirban Sain

Liquid crystals, which are ubiquitous today in the form of Liquid Crystal Displays (LCDs), have the fascinating property of simultaneously showing fluidity and long-range order. One of the classes of liquid crystals is the class of nematics. The dynamics of nematics can be described by a theory due to Pierre-Giles de Gennes (Nobel Prize in Physics, 1991). A nonequilibrium extension to this liquid crystal framework, namely, active gel theory, has become quite popular in the biophysics community over the past decade. For this project, I studied the dynamics of cytokinetic ring closure, a phenomenon that is responsible for dividing the cytoplasm of a parent cell into two daughter cells. Using an active gel theory framework, we worked out the quasi-static dynamics of the ring closure phenomenon, and quantified the stability of the process by analyzing angular perturbation modes. We were able to identify specific mechanisms for the well-documented [AG1, AG2] slow-down of the ring contraction at late times. In experiments, such as Silva et. al.[AG3], it was also found that the closure of the cytokinetic ring often starts out in a non-circular/asymmetric manner but becomes more circular as it contracts. Consistent with this observation, we found that some of the angular deformation modes are unstable at a large ring radius but all of them become stable at small values of the ring radius. The results of this work (which formed a major portion of my senior undergraduate thesis) have been published in Physical Review Letters.[AG4]

[AG1] A. Zumdieck, K. Kruse, H. Bringmann, A. A. Hyman, and F. Julicher, PloS one 2 (2007).
[AG2] A. S. Maddox, L. Lewellyn, A. Desai, and K. Oegema, Developmental cell 12, 827 (2007).
[AG3] A. M. Silva et. al., “Robust gap repair in the contractile ring ensures timely completion of cytokinesis.” J Cell Biol (2016): jcb-201605080.
[AG4] M. Chatterjee, A. Chatterjee, A. Nandi, A. Sain, “Dynamics and Stability of the Contractile Actomyosin Ring in the Cell” Physical Review Letters, 128(6), 068102. [Journal, Preprint]

Contractile Ring — Contractile ring (*Source: Tutorialspoint*)

Mean-Field Theory of Motility-Induced Phase Separation (MIPS)

In collaboration with: Andreas Fischer and Thomas Speck

In Summer 2018, supported by the DAAD-WISE fellowship, I did a research internship in the condensed matter theory group at JGU Mainz. My project involved developing a theoretical model to improve on existing Active Brownian Particle (ABP)-based models for the phenomenon of motility-induced phase separation (MIPS). We were able to qualitatively explain the deviation of experiments from the previous ABP models, using a constant-affiinity ensemble approach. I also worked on the problem of sedimentation in active ideal gases and was able to predict a first-order correction to the previous theoretical estimates of the sedimentation length in such systems. A paper based on our work has been published in the Journal of Chemical Physics.[MF1]

[MF1] Fischer, A., Chatterjee, A., and Speck, T., 2019. Aggregation and sedimentation of active Brownian particles at constant affinity. The Journal of chemical physics, 150(6), p.064910. [Journal, Preprint]

Motility Induced Phase Separation (*Source: Redner, G.S. et al., 2013. Structure and dynamics of a phase-separating active colloidal fluid.*)

Active Brownian Particles (ABPs) in a Magnetic Field

In collaboration with: Lokrshi P. Dadhichi and Sriram Ramaswamy

In Summer 2017, supported by a scholarship from the Indian Academy of Sciences, I did a research internship at IISc. I investigated charged ABPs placed in a confining potential in 2 dimensions, in the presence of an external magnetic field. To that end, I studied the theory of stochastic processes, and its applications to Brownian motion. I also delved into linear response theory, and some introductory stochastic thermodynamics. Furthermore, I explored the possibility of applying a generalized fluctuation-dissipation theorem on this particular system. This was essentially an attempt to quantify the extent of the non-equilibrium behaviour of the system.