# Publications by Siddharth Parameswaran

## Dynamics and transport at the threshold of many-body localization

Physics Reports Elsevier **862** (2020) 1-62

Many-body localization (MBL) describes a class of systems that do not approach thermal equilibrium under their intrinsic dynamics; MBL and conventional thermalizing systems form distinct dynamical phases of matter, separated by a phase transition at which equilibrium statistical mechanics breaks down. True many-body localization is known to occur only under certain stringent conditions for perfectly isolated one-dimensional systems, with Hamiltonians that have strictly short-range interactions and lack any continuous non-Abelian symmetries. However, in practice, even systems that are not strictly MBL can be nearly MBL, with equilibration rates that are far slower than their other intrinsic timescales; thus, anomalously slow relaxation occurs in a much broader class of systems than strict localization. In this review we address transport and dynamics in such nearly-MBL systems from a unified perspective. Our discussion covers various classes of such systems: (i) disordered and quasiperiodic systems on the thermal side of the MBL-thermal transition; (ii) systems that are strongly disordered, but obstructed from localizing because of symmetry, interaction range, or dimensionality; (iii) multiple-component systems, in which some components would in isolation be MBL but others are not; and finally (iv) driven systems whose dynamics lead to exponentially slow rates of heating to infinite temperature. A theme common to many of these problems is that they can be understood in terms of approximately localized degrees of freedom coupled to a heat bath (or baths) consisting of thermal degrees of freedom; however, this putative bath is itself nontrivial, being either small or very slowly relaxing. We discuss anomalous transport, diverging relaxation times, and other signatures of the proximity to MBL in these systems. We also survey recent theoretical and numerical methods that have been applied to study dynamics on either side of the MBL transition.

## Twisted bilayer graphene in a parallel magnetic field

Physical review B: Condensed matter and materials physics American Physical Society **101** (2020) 205116

We study the effect of an in-plane magnetic field on the non-interacting dispersion of twisted bilayer graphene. Our analysis is rooted in the chirally symmetric continuum model, whose zero-field band structure hosts exactly flat bands and large energy gaps at the magic angles. At the first magic angle, the central bands respond to a parallel field by forming a quadratic band crossing point (QBCP) at the moire Brillouin zone center. Over a large ´ range of fields, the dispersion is invariant with an overall scale set by the magnetic field strength. For deviations from the magic angle and for realistic interlayer couplings, the motion and merging of the Dirac points lying near charge neutrality are discussed in the context of the symmetries, and we show that small magnetic fields are able to induce a qualitative change in the energy spectrum. We conclude with a discussion on the possible ramifications of our study to the interacting ground states of twisted bilayer graphene systems.

## 'Unhinging' the surfaces of higher-order topological insulators and superconductors

Physical Review Letters American Physical Society **124** (2020) 046801

## Quantum oscillations probe the Fermi surface topology of the nodal-line semimetal CaAgAs

Physical Review Research American Physical Society **2** (2020) 012055(R)

Nodal semimetals are a unique platform to explore topological signatures of the unusual band structure that can manifest by accumulating a nontrivial phase in quantum oscillations. Here we report a study of the de Haas–van Alphen oscillations of the candidate topological nodal line semimetal CaAgAs using torque measurements in magnetic fields up to 45 T. Our results are compared with calculations for a toroidal Fermi surface originating from the nodal ring. We find evidence of a nontrivial π phase shift only in one of the oscillatory frequencies. We interpret this as a Berry phase arising from the semiclassical electronic Landau orbit which links with the nodal ring when the magnetic field lies in the mirror (ab) plane. Furthermore, additional Berry phase accumulates while rotating the magnetic field for the second orbit in the same orientation which does not link with the nodal ring. These effects are expected in CaAgAs due to the lack of inversion symmetry. Our study experimentally demonstrates that CaAgAs is an ideal platform for exploring the physics of nodal line semimetals and our approach can be extended to other materials in which trivial and nontrivial oscillations are present.

## Classical dimers on penrose tilings

Physical Review X American Physical Society **10** (2020) 011005

## Glide symmetry breaking and Ising criticality in the quasi-1D magnet CoNb2O6

Proceedings of the National Academy of Sciences National Academy of Sciences **117** (2020) 25219-25224

We construct a microscopic spin-exchange Hamiltonian for the quasi–one-dimensional (1D) Ising magnet CoNb2O6 that captures detailed and hitherto-unexplained aspects of its dynamic spin structure factor. We perform a symmetry analysis that recalls that an individual Ising chain in this material is buckled, with two sites in each unit cell related by a glide symmetry. Combining this with numerical simulations benchmarked against neutron scattering experiments, we argue that the single-chain Hamiltonian contains a staggered spin-exchange term. We further argue that the transverse-field–tuned quantum critical point in CoNb2O6 corresponds to breaking this glide symmetry, rather than an on-site Ising symmetry as previously believed. This gives a unified microscopic explanation of the dispersion of confined states in the ordered phase and quasiparticle breakdown in the polarized phase at high transverse field.

## Erratum: Charge Transport in Weyl Semimetals (Physical Review Letters (2012) 108 (046602) DOI: 10.1103/PhysRevLett.108.046602)

Physical Review Letters **123** (2019)

© 2019 American Physical Society. This erratum corrects errors in numerical factors in Eqs. (1), (7), and (8), and the overall scale of the dc resistivity plotted in Fig. 2. We recently discovered an algebraic error in Eq. (7), which led to incorrect numerical factors in Eqs. (1) and (8). The correct Eqs. (1), (7) and (8), respectively, are (Formula Presented). An error was also found in the overall scale of pdc = 1/σdc calculated from (1) and plotted in Fig. 2 of the Letter. With these corrections our theory underestimates ?dc of the samples in Ref. [12] of the Letter, which is understandable since the samples are polycrystalline while our theory specializes to single crystals. However, correcting both errors gives excellent agreement with recent experiments on Eu 0.96 Bi 0.04 Ir 2 O 7 [1] for reasonable values of parameters, as shown in Fig. 1. Moreover, Ref. [1] finds pdc ( T ) ∼ 1 / T , as predicted by our theory, only at low temperatures, which is where our theory is best applicable since it contains only Coulomb scattering but ignores phonon scattering. Thus, it is likely that the low-temperature transport in Eu 0.96 Bi 0.04 Ir 2 O 7 is dominated by Coulomb scattering. We thank Surjeet Singh and Prachi Telang for bringing the error in the computation of pdc to our attention (Figure Presented).

## Topology and symmetry-protected domain wall conduction in quantum Hall nematics

Physical review B: Condensed matter and materials physics American Physical Society **100** (2019) 165103

## Topological 'Luttinger' invariants for filling-enforced non-symmorphic semimetals

Journal of Physics: Condensed Matter IOP Publishing **31** (2019) 104001-

Luttinger’s theorem is a fundamental result in the theory of interacting Fermi systems: it states that the volume inside the Fermi surface is left invariant by interactions, if the number of particles is held fixed. Although this is traditionally justified in terms of analytic properties of Green’s functions, it can be viewed as arising from a momentum balance argument that examines the response of the ground state to the insertion of a single flux quantum [M. Oshikawa, Phys. Rev. Lett. 84, 3370 (2000)]. This reveals that the Fermi volume is a topologically protected quantity, whose change requires a phase transition. However, this sheds no light on the stability or lack thereof of interacting semimetals, that either lack a Fermi surface, or have perfectly compensated electron and hole pockets and hence vanishing net Fermi volume. Here, I show that semimetallic phases in non-symmorphic crystals possess additional topological ‘Luttinger invariants’ that can be nonzero even though the Fermi volume vanishes. The existence of these invariants is linked to the inability of non-symmorphic crystals to host band insulating ground states except at special fillings. I exemplify the use of these new invariants by showing that they distinguish various classes of twoand three-dimensional semimetals.

## Quantum Brownian motion in a quasiperiodic potential

Physical review B: Condensed matter and materials physics American Physical Society **100** (2019) 060301

We consider a quantum particle subject to Ohmic dissipation, moving in a bichromatic quasiperiodic potential. In a periodic potential the particle undergoes a zero-temperature localization-delocalization transition as dissipation strength is decreased. We show that the delocalized phase is absent in the quasiperiodic case, even when the deviation from periodicity is infinitesimal. Using the renormalization group, we determine how the effective localization length depends on the dissipation. We show that a similar problem can emerge in the strong-coupling limit of a mobile impurity moving in a periodic lattice and immersed in a one-dimensional quantum gas.

## Interacting multi-channel topological boundary modes in a quantum Hall valley system

Nature Springer Nature **566** (2019) 363-367

Symmetry and topology are central to understanding quantum Hall ferromagnets (QHFMs), two-dimensional electronic phases with spontaneously broken spin or pseudospin symmetry whose wavefunctions also have topological properties1,2. Domain walls between distinct broken-symmetry QHFM phases are predicted to host gapless one-dimensional modes—that is, quantum channels that emerge because of a topological change in the underlying electronic wavefunctions at such interfaces. Although various QHFMs have been identified in different materials3,4,5,6,7,8, interacting electronic modes at these domain walls have not been probed. Here we use a scanning tunnelling microscope to directly visualize the spontaneous formation of boundary modes at domain walls between QHFM phases with different valley polarization (that is, the occupation of equal-energy but quantum mechanically distinct valleys in the electronic structure) on the surface of bismuth. Spectroscopy shows that these modes occur within a topological energy gap, which closes and reopens as the valley polarization switches across the domain wall. By changing the valley flavour and the number of modes at the domain wall, we can realize different regimes in which the valley-polarized channels are either metallic or develop a spectroscopic gap. This behaviour is a consequence of Coulomb interactions constrained by the valley flavour, which determines whether electrons in the topological modes can backscatter, making these channels a unique class of interacting one-dimensional quantum wires. QHFM domain walls can be realized in different classes of two-dimensional materials, providing the opportunity to explore a rich phase space of interactions in these quantum wires.

## Kosterlitz-Thouless scaling at many-body localization phase transitions

Physical Review B: Condensed matter and materials physics American Physical Society **99** (2019) 094205

<p>We propose a scaling theory for the many-body localization (MBL) phase transition in one dimension, building on the idea that it proceeds via a “quantum avalanche.” We argue that the critical properties can be captured at a coarse-grained level by a Kosterlitz-Thouless (KT) renormalization group (RG) flow. On phenomenological grounds, we identify the scaling variables as the density of thermal regions and the length scale that controls the decay of typical matrix elements. Within this KT picture, the MBL phase is a line of fixed points that terminates at the delocalization transition. We discuss two possible scenarios distinguished by the distribution of rare, fractal thermal inclusions within the MBL phase. In the first scenario, these regions have a stretched exponential distribution in the MBL phase. In the second scenario, the near-critical MBL phase hosts rare thermal regions that are power-law-distributed in size. This points to the existence of a second transition within the MBL phase, at which these power laws change to the stretched exponential form expected at strong disorder. We numerically simulate two different phenomenological RGs previously proposed to describe the MBL transition. Both RGs display a universal power-law length distribution of thermal regions at the transition with a critical exponent α<sub>c</sub> = 2, and continuously varying exponents in the MBL phase consistent with the KT picture.</p>

## Signatures of information scrambling in the dynamics of the entanglement spectrum

Physical review B: Condensed Matter and Materials Physics American Physical Sociey **100** (2019) 125115

We examine the time evolution of the entanglement spectrum of a small subsystem of a nonintegrable spin chain following a quench from a product state. We identify signatures in this entanglement spectrum of the distinct dynamical velocities (related to entanglement and operator spreading) that control thermalization. We show that the onset of level repulsion in the entanglement spectrum occurs on different timescales depending on the “entanglement energy”, and that this dependence reflects the shape of the operator front. Level repulsion spreads across the entire entanglement spectrum on a timescale that is parametrically shorter than that for full thermalization of the subsystem. This timescale is also close to when the mutual information between individual spins at the ends of the subsystem reaches its maximum. We provide an analytical understanding of this phenomenon and show supporting numerical data for both random unitary circuits and a microscopic Hamiltonian.

## Quantum Hall valley nematics

Journal of Physics: Condensed Matter IOP Publishing **31** (2019) 273001

Two-dimensional electron gases in strong magnetic fields provide a canonical platform for realizing a variety of electronic ordering phenomena. Here we review the physics of one intriguing class of interaction-driven quantum Hall states: quantum Hall valley nematics. These phases of matter emerge when the formation of a topologically insulating quantum Hall state is accompanied by the spontaneous breaking of a point-group symmetry that combines a spatial rotation with a permutation of valley indices. The resulting orientational order is particularly sensitive to quenched disorder, while quantum Hall physics links charge conduction to topological defects. We discuss how these combine to yield a rich phase structure, and their implications for transport and spectroscopy measurements. In parallel, we discuss relevant experimental systems. We close with an outlook on future directions.

## Topological Entanglement Entropy of Fracton Stabilizer Codes

Physical Review B American Physical Society **97** (2018) 125101

Entanglement entropy provides a powerful characterization of two-dimensional gapped topological phases of quantum matter, intimately tied to their description by topological quantum field theories (TQFTs). Fracton topological orders are three-dimensional gapped topologically ordered states of matter that lack a TQFT description. We show that three-dimensional fracton phases are nevertheless characterized, at least partially, by universal structure in the entanglement entropy of their ground-state wave functions. We explicitly compute the entanglement entropy for two archetypal fracton models, the “X-cube model” and “Haah's code,” and demonstrate the existence of a nonlocal contribution that scales linearly in subsystem size. We show via Schrieffer-Wolff transformations that this piece of the entanglement entropy of fracton models is robust against arbitrary local perturbations of the Hamiltonian. Finally, we argue that these results may be extended to characterize localization-protected fracton topological order in excited states of disordered fracton models.

## Many-body localization, symmetry, and topology

Reports on Progress in Physics IOP Publishing **81** (2018) 082501

We review recent developments in the study of out-of-equilibrium topological states of matter in isolated systems. The phenomenon of many-body localization, exhibited by some isolated systems usually in the presence of quenched disorder, prevents systems from equilibrating to a thermal state where the delicate quantum correlations necessary for topological order are often washed out. Instead, many-body localized systems can exhibit a type of eigenstate phase structure wherein their entire many-body spectrum is characterized by various types of quantum order, usually restricted to quantum ground states. After introducing many-body localization and explaining how it can protect quantum order, we then explore how the interplay of symmetry and dimensionality with many-body localization constrains its role in stabilizing topological phases out of equilibrium.

## Strong-disorder renormalization group for periodically driven systems

Physical Review B: Condensed Matter and Materials Physics American Physical Society **98** (2018) 174203

Quenched randomness can lead to robust non-equilibrium phases of matter in periodically driven (Floquet) systems. Analyzing transitions between such dynamical phases requires a method capable of treating the twin complexities of disorder and discrete time-translation symmetry. We introduce a real-space renormalization group approach, asymptotically exact in the strong-disorder limit, and exemplify its use on the periodically driven interacting quantum Ising model. We analyze the universal physics near the critical lines and multicritical point of this model, and demonstrate the robustness of our results to the inclusion of weak interactions.

## Localization-protected order in spin chains with non-Abelian discrete symmetries

Physical Review B American Physical Society **98** (2018) 064203

We study the nonequilibrium phase structure of the three-state random quantum Potts model in one dimension. This spin chain is characterized by a non-Abelian D 3 symmetry recently argued to be incompatible with the existence of a symmetry-preserving many-body localized (MBL) phase. Using exact diagonalization and a finite-size scaling analysis, we find that the model supports two distinct broken-symmetry MBL phases at strong disorder that either break the Z 3 clock symmetry or a Z 2 chiral symmetry. In a dual formulation, our results indicate the existence of a stable finite-temperature topological phase with MBL-protected parafermionic end zero modes. While we find a thermal symmetry-preserving regime for weak disorder, scaling analysis at strong disorder points to an infinite-randomness critical point between two distinct broken-symmetry MBL phases.

## Floquet quantum criticality

Proceedings of the National Academy of Sciences National Academy of Sciences **115** (2018) 9491-9496

We study transitions between distinct phases of one-dimensional periodically driven (Floquet) systems. We argue that these are generically controlled by infinite-randomness fixed points of a strong-disorder renormalization group procedure. Working in the fermionic representation of the prototypical Floquet Ising chain, we leverage infinite randomness physics to provide a simple description of Floquet (multi)criticality in terms of a distinct type of domain wall associated with time translational symmetry-breaking and the formation of “Floquet time crystals.” We validate our analysis via numerical simulations of free-fermion models sufficient to capture the critical physics.

## Recoverable information and emergent conservation laws in fracton stabilizer codes

Physical Review B American Physical Society **97** (2018) 134426

We introduce a new quantity, that we term {\it recoverable information}, defined for stabilizer Hamiltonians. For such models, the recoverable information provides a measure of the topological information, as well as a physical interpretation, which is complementary to topological entanglement entropy. We discuss three different ways to calculate the recoverable information, and prove their equivalence. To demonstrate its utility, we compute recoverable information for {\it fracton models} using all three methods where appropriate. From the recoverable information, we deduce the existence of emergent Z 2 Gauss-law type constraints, which in turn imply emergent Z 2 conservation laws for point-like quasiparticle excitations of an underlying topologically ordered phase.