arXiv subjects

Naren Manjunath

Publications and source records attributed to Naren Manjunath.

At least 19 recordsLinked to original sources

arXiv2026

Crystalline topological invariants in quantum many-body systems

Crystalline symmetries give rise to topological invariants that can distinguish quantum phases of matter. Understanding these in strongly interacting systems is an ongoing research direction requiring non-perturbative methods. Recent developments have demonstrated that even classic models, like the Harper-Hofstadter model of free fermions on a lattice in a magnetic field, yield a host of crystalline symmetry protected topological invariants. Here we review some of these developments, focusing mainly on how to characterize, classify, and detect invariants arising from lattice translation and rotation symmetries along with charge conservation in two-dimensional systems, including integer and fractional Chern insulators.

cond-mat.str-el↗

arXiv2026

Universal quantum computation with group surface codes

We introduce group surface codes, which are a natural generalization of the $\mathbb{Z}_2$ surface code, and equivalent to quantum double models of finite groups with specific boundary conditions. We show that group surface codes can be leveraged to perform non-Clifford gates in $\mathbb{Z}_2$ surface codes, thus enabling universal computation with well-established means of performing logical Clifford gates. Moreover, for suitably chosen groups, we demonstrate that arbitrary reversible classical gates can be implemented transversally in the group surface code. We present the logical operations in terms of a set of elementary logical operations, which include transversal logical gates, a means of transferring encoded information into and out of group surface codes, and preparation and readout. By composing these elementary operations, we implement a wide variety of logical gates and provide a unified perspective on recent constructions in the literature for sliding group surface codes and preparing magic states. We furthermore use tensor networks inspired by ZX-calculus to construct spacetime implementations of the elementary operations. This spacetime perspective also allows us to establish explicit correspondences with topological gauge theories. Our work extends recent efforts in performing universal quantum computation in topological orders without the braiding of anyons, and shows how certain group surface codes allow us to bypass the restrictions set by the Bravyi-K{\"o}nig theorem, which limits the computational power of topological Pauli stabilizer models.

quant-ph↗

arXiv2026

Quantum criticality at strong randomness: a lesson from anomaly

Quantum criticality in the presence of strong quenched randomness remains a challenging topic in modern condensed matter theory. We show that the topology and anomaly associated with average symmetry can be used to predict certain nontrivial universal properties. Our focus is on systems subject to average Lieb--Schultz--Mattis constraints, where lattice translation symmetry is preserved only on average, while on-site symmetries remain exact. We argue that in the absence of spontaneous symmetry breaking and intrinsic topological order, the system must exhibit critical correlations of local operators in two distinct ways: (i) for some operator $O_e$ charged under exact symmetries, the first absolute moment correlation $\overline{|\langle O_e(x)O^{\dagger}_e(y)\rangle|}$ decays slowly; and (ii) for some operator $O_a$ charged under average symmetries, the first-moment correlation $\overline{\langle O_a(x)O^{\dagger}_a(y)\rangle}$ decays slowly. We verify these predictions in a few examples: the random-singlet Heisenberg spin chain in one dimension, and the disordered free-fermion critical states in symmetry class BDI in one and two dimensions. Surprisingly, even for these well-studied systems, our anomaly-based argument reveals critical correlations overlooked in previous literature. We also discuss the experimental feasibility of measuring these critical correlations.

cond-mat.dis-nn↗

arXiv2026

In search of diabolical critical points

A phase transition is an example of a ``topological defect'' in the space of parameters of a quantum or classical many-body systems. In this paper, we consider phase diagram topological defects of higher codimension. These have the property that equilibrium states undergo some kind of non-trivial winding as one moves around the defect. We show that such topological defects exist even in classical statistical mechanical systems, and describe their general structure in this context. We then introduce the term ``diabolical critical point'' (DCP), which is a higher-codimension analog of a continuous phase transition, with the proximate phases of matter replaced by the non-trivial winding of the proximate equilibrium states. We propose conditions under which a system can have a stable DCP. We also discuss some examples of stable DCPs in (1+1)-dimensional quantum systems.

cond-mat.str-el↗

arXiv2025

Invariants for (2+1)D bosonic crystalline topological insulators for all 17 wallpaper groups

We study bosonic symmetry-protected topological (SPT) phases in (2+1) dimensions with symmetry $G = G_{\text{space}}\times K$, where $G_{\text{space}}$ is a general wallpaper group and $K=\text{U}(1),\mathbb{Z}_N, \text{SO}(3)$ is an internal symmetry. In each case we propose a set of many-body invariants that can detect all the different phases predicted from real space constructions and group cohomology classifications. They are obtained by applying partial rotations and reflections to a given ground state, combined with suitable operations in $K$. The reflection symmetry invariants that we introduce include `double partial reflections', `weak partial reflections' and their `relative' or `twisted' versions which also depend on $K$. We verify our proposal through exact calculations on ground states constructed using real space constructions. We demonstrate our method in detail for the groups p4m and p4g, and in the case of p4m also derive a topological effective action involving gauge fields for orientation-reversing symmetries. Our results provide a concrete method to fully characterize (2+1)D crystalline topological invariants in bosonic SPT ground states.

cond-mat.str-el↗

arXiv2025

Detection of 2D SPT phases under decoherence

We propose a bulk order parameter for extracting symmetry-protected topological (SPT) invariants of quantum many-body mixed states on a two dimensional lattice using partial symmetries. The procedure builds on the partial symmetry order parameter recently developed by some of the authors to study SPT phases of pure states and adapts them to the decohered setting. For a symmetry $G = E \times A$ where $E$ is a strong symmetry and $A$ is a weak symmetry, we show that the partial symmetry order parameter detects SPT invariants jointly protected by $E$ and $A$. We demonstrate this explicitly using a class of mixed states obtained from CZX-type models with $\mathbb{Z}_2\times\mathbb{Z}_2$ symmetry and subjecting them to noise that weakens one of the $\mathbb{Z}_2$ symmetries. We also comment on the practical detection of SPT invariants in quantum simulators through randomized measurements.

cond-mat.str-el↗

arXiv2025

Symmetry constrained field theories for chiral spin liquid to spin crystal transitions

We consider the spin rotationally invariant Kalmeyer-Laughlin chiral spin liquid (CSL) in systems with broken time-reversal symmetry and explore symmetry constraints on possible conventional spin crystal states accessible via a direct transition. These constraints provide a framework to identify topological invariants of the magnetically ordered state. We show that the existence of a direct transition from a CSL requires a precise compatibility condition between the topological invariants of the ordered state and the anomaly of the CSL. The lattice symmetries also constrain the functional form of the low-energy theory to describe these transitions. This allows us to construct explicit Chern-Simons-matter field theories for the transition into a class of noncoplanar orders identified as candidates directly accessible from the CSL, including the octahedral spin crystal on the kagom\'e lattice, and the tetrahedral order on the triangular and honeycomb lattice. These transitions can either be described using coupled fractionalized $ \mathbb{CP}^1 $ theories or fractionalized matrix principal chiral models. We also discuss extensions to more general magnetic ordering transitions out of the CSL.

cond-mat.str-el↗

arXiv2025

Detection of 2D SPT Order with Partial Symmetries

A method of using partial symmetries to distinguish two dimensional symmetry protected topological (SPT) phases of on-site, unitary symmetries is proposed. This novel order parameter takes a wavefunction, such as a ground state of a lattice model, and detects its SPT invariants as expectation values of finitely supported operators, without the need for flux insertion. The construction exploits the rotational symmetry of the lattice to extract on-site SPT invariants, building upon prior work on probing crystalline SPT phases with partial rotations. The method is demonstrated by computing the order parameter analytically on group cohomology models and numerically on a family of states interpolating between the CZX state and a trivial state. Its robustness is suggested by interpreting partial symmetries as generating the topological partition functions of lens spaces.

cond-mat.str-el↗

arXiv2024

Anomalous continuous symmetries and quantum topology of Goldstone modes

We consider systems in which a continuous symmetry $G$, which may be anomalous, is spontaneously broken to an anomaly-free subgroup $H$ such that the effective action for the Goldstone modes contains topologically non-trivial terms. If the original system has trivial $G$ anomaly, it is known that the possible topological terms are fully determined by SPT or SET invariants of the residual $H$ symmetry. Here we address the more general setting in which the $G$ symmetry has an anomaly. We argue that in general, the appropriate concept to consider is the "compatibility relation" between the Goldstone invariants and the $G$ anomaly. In the case where the Goldstone modes can be gapped out to obtain invertible families (i.e. without any topological order), we give an explicit mathematical scheme to construct the desired compatibility relation. We also address the case where gapping out the Goldstone modes leads to a family of topologically ordered states. We discuss several examples including the canonical Thouless pump, the quantum Hall ferromagnet, pumps arising from breaking $\text{U}(1)$ symmetry at the boundary of topological insulators in two and three dimensions, and pumps classified by the higher Chern number.

cond-mat.str-el↗

arXiv2024

Crystalline invariants of fractional Chern insulators

In the presence of crystalline symmetry, topologically ordered states can acquire a host of symmetry-protected invariants. These determine the patterns of crystalline symmetry fractionalization of the anyons in addition to fractionally quantized responses to lattice defects. Here we show how ground state expectation values of partial rotations centered at high symmetry points can be used to extract crystalline invariants. Using methods from conformal field theory and G-crossed braided tensor categories, we develop a theory of invariants obtained from partial rotations, which apply to both Abelian and non-Abelian topological orders. We then perform numerical Monte Carlo calculations for projected parton wave functions of fractional Chern insulators, demonstrating remarkable agreement between theory and numerics. For the topological orders we consider, we show that the Hall conductivity, filling fraction, and partial rotation invariants fully characterize the crystalline invariants of the system. Our results also yield invariants of continuum fractional quantum Hall states protected by spatial rotational symmetry.

cond-mat.str-el↗

arXiv2023

Characterization and classification of interacting (2+1)D topological crystalline insulators with orientation-preserving wallpaper groups

While free fermion topological crystalline insulators have been largely classified, the analogous problem in the strongly interacting case has been only partially solved. In this paper, we develop a characterization and classification of interacting, invertible fermionic topological phases in (2+1) dimensions with charge conservation, discrete magnetic translation and $M$-fold point group rotation symmetries, which form the group $G_f = \text{U}(1)^f \times_{\phi} [\mathbb{Z}^2\rtimes \mathbb{Z}_M]$ for $M=1,2,3,4,6$. $\phi$ is the magnetic flux per unit cell. We derive a topological response theory in terms of background crystalline gauge fields, which gives a complete classification of different phases and a physical characterization in terms of quantized response to symmetry defects. We then derive the same classification in terms of a set of real space invariants $\{\Theta_{\text{o}}^\pm\}$ that can be obtained from ground state expectation values of suitable partial rotation operators. We explicitly relate these real space invariants to the quantized coefficients in the topological response theory, and find the dependence of the invariants on the chiral central charge $c_-$ of the invertible phase. Finally, when $\phi = 0$ we derive an explicit map between the free and interacting classifications.

cond-mat.str-el↗

arXiv2023

Complete crystalline topological invariants from partial rotations in (2+1)D invertible fermionic states and Hofstadter's butterfly

The theory of topological phases of matter predicts invariants protected only by crystalline symmetry, yet it has been unclear how to extract these from microscopic calculations in general. Here we show how to extract a set of many-body invariants $\{\Theta_{\text{o}}^{\pm}\}$, where ${\text{o}}$ is a high symmetry point, from partial rotations in (2+1)D invertible fermionic states. Our results apply in the presence of magnetic field and Chern number $C \neq 0$, in contrast to previous work. $\{\Theta_{\text{o}}^{\pm}\}$ together with $C$, chiral central charge $c_-$, and filling $\nu$ provide a complete many-body characterization of the topological state with symmetry group $G = \text{U}(1) \times_\phi [\mathbb{Z}^2 \rtimes \mathbb{Z}_M]$. Moreover, all these many-body invariants can be obtained from a single bulk ground state, without inserting additional defects. We perform numerical computations on the square lattice Hofstadter model. Remarkably, these match calculations from conformal and topological field theory, where $G$-crossed modular $S, T$ matrices of symmetry defects play a crucial role. Our results provide additional colorings of Hofstadter's butterfly, extending recently discovered colorings by the discrete shift and quantized charge polarization.

cond-mat.str-el↗

arXiv2023

Rotational Symmetry Protected Edge and Corner States in Abelian topological phases

Spatial symmetries can enrich the topological classification of interacting quantum matter and endow systems with non-trivial strong topological invariants (protected by internal symmetries) with additional "weak" topological indices. In this paper, we study the edge physics of systems with a non-trivial shift invariant, which is protected by either a continuous $\text{U}(1)_r$ or discrete $\text{C}_n$ rotation symmetry, along with internal $\text{U}(1)_c$ charge conservation. Specifically, we construct an interface between two systems which have the same Chern number but are distinguished by their Wen-Zee shift and, through analytic arguments supported by numerics, show that the interface hosts counter-propagating gapless edge modes which cannot be gapped by arbitrary local symmetry-preserving perturbations. Using the Chern-Simons field theory description of two-dimensional Abelian topological orders, we then prove sufficient conditions for continuous rotation symmetry protected gapless edge states using two complementary approaches. One relies on the algebraic Lagrangian sub-algebra framework for gapped boundaries while the other uses a more physical flux insertion argument. For the case of discrete rotation symmetries, we extend the field theory approach to show the presence of fractional corner charges for Abelian topological orders with gappable edges, and compute them in the case where the Abelian topological order is placed on the two-dimensional surface of a Platonic solid. Our work paves the way for studying the edge physics associated with spatial symmetries in symmetry enriched topological phases.

cond-mat.str-el↗

arXiv2022

Fractional disclination charge and discrete shift in the Hofstadter butterfly

In the presence of crystalline symmetries, topological phases of matter acquire a host of invariants leading to non-trivial quantized responses. Here we study a particular invariant, the discrete shift $\mathscr{S}$, for the square lattice Hofstadter model of free fermions. $\mathscr{S}$ is associated with a $\mathbb{Z}_M$ classification in the presence of $M$-fold rotational symmetry and charge conservation. $\mathscr{S}$ gives quantized contributions to (i) the fractional charge bound to a lattice disclination, and (ii) the angular momentum of the ground state with an additional, symmetrically inserted magnetic flux. $\mathscr{S}$ forms its own `Hofstadter butterfly', which we numerically compute, refining the usual phase diagram of the Hofstadter model. We propose an empirical formula for $\mathscr{S}$ in terms of density and flux per plaquette for the Hofstadter bands, and we derive a number of general constraints. We show that bands with the same Chern number may have different values of $\mathscr{S}$, although odd and even Chern number bands always have half-integer and integer values of $\mathscr{S}$ respectively.

cond-mat.str-el↗

arXiv2022

Quantized charge polarization as a many-body invariant in (2+1)D crystalline topological states and Hofstadter butterflies

We show how to define a quantized many-body charge polarization $\vec{\mathscr{P}}$ for (2+1)D topological phases of matter, even in the presence of non-zero Chern number and magnetic field. For invertible topological states, $\vec{\mathscr{P}}$ is a $\mathbb{Z}_2 \times \mathbb{Z}_2$, $\mathbb{Z}_3$, $\mathbb{Z}_2$, or $\mathbb{Z}_1$ topological invariant in the presence of $M = 2$, $3$, $4$, or $6$-fold rotational symmetry, lattice (magnetic) translational symmetry, and charge conservation. $\vec{\mathscr{P}}$ manifests in the bulk of the system as (i) a fractional quantized contribution of $\vec{\mathscr{P}} \cdot \vec{b} \text{ mod 1}$ to the charge bound to lattice disclinations and dislocations with Burgers vector $\vec{b}$, (ii) a linear momentum for magnetic flux, and (iii) an oscillatory system size dependent contribution to the effective 1d polarization on a cylinder. We study $\vec{\mathscr{P}}$ in lattice models of spinless free fermions in a magnetic field. We derive predictions from topological field theory, which we match to numerical calculations for the effects (i)-(iii), demonstrating that these can be used to extract $\vec{\mathscr{P}}$ from microscopic models in an intrinsically many-body way. We show how, given a high symmetry point $\text{o}$, there is a topological invariant, the discrete shift $\mathscr{S}_{\text{o}}$, such that $\vec{\mathscr{P}}$ specifies the dependence of $\mathscr{S}_{\text{o}}$ on $\text{o}$. We derive colored Hofstadter butterflies, corresponding to the quantized value of $\vec{\mathscr{P}}$, which further refine the colored butterflies from the Chern number and discrete shift.

cond-mat.str-el↗

arXiv2022

Non-perturbative constraints from symmetry and chirality on Majorana zero modes and defect quantum numbers in (2+1)D

In (1+1)D topological phases, unpaired Majorana zero modes (MZMs) can arise only if the internal symmetry group $G_f$ of the ground state splits as $G_f = G_b \times \mathbb{Z}_2^f$, where $\mathbb{Z}_2^f$ is generated by fermion parity, $(-1)^F$. In contrast, (2+1)D topological superconductors (TSC) can host unpaired MZMs at defects even when $G_f$ is not of the form $G_b \times \mathbb{Z}_2^f$. In this paper we study how $G_f$ together with the chiral central charge $c_-$ strongly constrain the existence of unpaired MZMs and the quantum numbers of symmetry defects. Our results utilize a recent algebraic characterization of (2+1)D invertible fermionic topological states, which provides a non-perturbative approach based on topological quantum field theory, beyond free fermions. We study physically relevant groups such as $\mathrm{U}(1)^f\rtimes H,\mathrm{SU}(2)^f \times H, \mathrm{U}(2)^f\rtimes H $, generic Abelian groups, as well as more general compact Lie groups, antiunitary symmetries and crystalline symmetries. We present an algebraic formula for the fermionic crystalline equivalence principle, which gives an equivalence between states with crystalline and internal symmetries. In light of our theory, we discuss several previously proposed realizations of unpaired MZMs in TSC materials such as Sr$_2$RuO$_4$, transition metal dichalcogenides and iron superconductors, in which crystalline symmetries are often important; in some cases we present additional predictions for the properties of these models.

cond-mat.str-el↗

arXiv2022

Classification of (2+1)D invertible fermionic topological phases with symmetry

We provide a classification of invertible topological phases of interacting fermions with symmetry in two spatial dimensions for general fermionic symmetry groups $G_f$ and general values of the chiral central charge $c_-$. Here $G_f$ is a central extension of a bosonic symmetry group $G_b$ by fermion parity, $(-1)^F$, specified by a second cohomology class $[ω_2] \in \mathcal{H}^2(G_b, \mathbb{Z}_2)$. Our approach proceeds by gauging fermion parity and classifying the resulting $G_b$ symmetry-enriched topological orders while keeping track of certain additional data and constraints. We perform this analysis through two perspectives, using $G$-crossed braided tensor categories and Spin$(2c_-)_1$ Chern-Simons theory coupled to a background $G$ gauge field. These results give a way to characterize and classify invertible fermionic topological phases in terms of a concrete set of data and consistency equations, which is more physically transparent and computationally simpler than the more abstract methods using cobordism theory and spectral sequences. Our results also generalize and provide a different approach to the recent classification of fermionic symmetry-protected topological phases by Wang and Gu, which have chiral central charge $c_- = 0$. We show how the 10-fold way classification of topological insulators and superconductors fits into our scheme, along with general non-perturbative constraints due to certain choices of $c_-$ and $G_f$. Mathematically, our results also suggest an explicit general parameterization of deformation classes of (2+1)D invertible topological quantum field theories with $G_f$ symmetry.

cond-mat.str-el↗

arXiv2020

Classification of fractional quantum Hall states with spatial symmetries

Fractional quantum Hall (FQH) states are examples of symmetry-enriched topological states (SETs): in addition to the intrinsic topological order, which is robust to symmetry breaking, they possess symmetry-protected topological invariants, such as fractional charge of anyons and fractional Hall conductivity. In this paper we develop a comprehensive theory of symmetry-protected topological invariants for FQH states with spatial symmetries, which applies to Abelian and non-Abelian topological states, by using a recently developed framework of $G$-crossed braided tensor categories ($G\times$BTCs) for SETs. We consider systems with $U(1)$ charge conservation, magnetic translational, and spatial rotational symmetries, in the continuum and for all $5$ orientation-preserving crystalline space groups in two dimensions, allowing arbitrary rational magnetic flux per unit cell, and assuming that symmetries do not permute anyons. In the crystalline setting, applicable to fractional Chern insulators and spin liquids, symmetry fractionalization is fully characterized by a generalization to non-Abelian states of the charge, spin, discrete torsion, and area vectors, which specify fractional charge, angular momentum, linear momentum, and fractionalization of the translation algebra for each anyon. The topological response theory contains $9$ terms, which attach charge, linear momentum, and angular momentum to magnetic flux, lattice dislocations, disclinations, corners, and units of area. Using the $G\times$BTC formalism, we derive the formula relating charge filling to the Hall conductivity and flux per unit cell; in the continuum this relates the filling fraction and the Hall conductivity without assuming Galilean invariance. We provide systematic formulas for topological invariants within the $G\times$BTC framework; this gives, for example, a new categorical definition of the Hall conductivity.

cond-mat.str-el↗