Quantum Matter

Like ordinary symmetries, non-invertible symmetries can characterize Symmetry-Protected Topological (SPT) phases. In this talk, we will discuss weak SPTs protected by projective non-invertible symmetries. Projective symmetries are ubiquitous in quantum spin models and can be leveraged to constrain their phase diagram and entanglement structure, e.g., Lieb-Schultz-Mattis (LSM) theorems. We will show how, surprisingly, projective non-invertible symmetries do not always imply LSM theorems. We will first discuss a simple, exactly solvable 1+1D quantum spin model in an SPT phase protected by both translation and non-invertible symmetries forming a non-trivial projective algebra. We will then generalize this example to a class of projective non-invertible Rep(G) x G x translation symmetries. For some finite groups G, this projectivity implies an LSM theorem. When it does not, we prove it still provides a constraint through an SPT-LSM theorem: any unique and gapped ground state is necessarily a non-invertible weak SPT state with non-trivial entanglement. [This talk is based on arXiv:2409.18113]

The exactly solvable spin-1/2 Kitaev model on a honeycomb lattice has drawn significant interest, as it offers a pathway to realizing the long-sought after quantum spin liquid. Building upon the Kitaev model, Yao and Lee introduced another exactly solvable model on an unusual star lattice featuring non-abelian spinons. The additional pseudospin degrees of freedom in this model could provide greater stability against perturbations, making this model appealing. However, a mechanism to realize such an interaction in a standard honeycomb lattice remains unknown. I will present a microscopic theory to obtain the Yao-Lee model on a honeycomb lattice by utilizing strong spin-orbit coupling of anions edge-shared between two eg ions in the exchange processes. This mechanism leads to the desired bond-dependent interaction among spins rather than orbitals, unique to our model, implying that the orbitals fractionalize into gapless Majorana fermions and fermionic octupolar excitations emerge. Since the conventional Kugel-Khomskii interaction also appears, the phase diagram including these interactions using classical Monte Carlo simulations and exact diagonalization techniques will be presented. Several open questions will be also discussed.

UTe₂ is an intriguing recently discovered superconductor that exhibits a wide range of exotic properties. Experimental evidence increasingly supports spin-triplet pairing, and under pressure or in magnetic fields, UTe₂ displays multiple superconducting phases, including a remarkable reentrant phase above 40 T. However, conflicting results persist regarding the presence of chiral and time-reversal symmetry breaking. Recent STM measurements have identified a charge density wave in the normal state, which couples with the superconducting state at lower temperatures to form a pair density wave. In this talk, I will provide an overview of the latest developments in understanding the unconventional superconductivity of UTe₂.

The quest for Anderson localization of light is at the center of many experimental and theoretical activities. Cold atoms have emerged as interesting quantum system to study coherent transport properties of light. Initial experiments have established that dilute samples with large optical thickness allow studying weak localization of light, which has been well described by a mesoscopic model. Recent experiments on light scattering with cold atoms have shown that Dicke super- or subradiance occurs in the same samples, a feature not captured by the traditional mesoscopic models. The use of a long range microscopic coupled dipole model allows to capture both the mesoscopic features of light scattering and Dicke super- and subradiance in the single photon limit. I will review experimental and theoretical state of the art on the possibility of Anderson localization of light by cold atoms.

Condensation of topological defects is the foundation of the modern theory of bulk-boundary correspondence, also known as topological holography. In this talk, I discuss string condensation in 3+1D topological orders, which plays a role analogous to anyon condensation in 2+1D topological orders. I will demonstrate through examples how they correspond to 2+1D symmetry enrichd phases, including both gapped and gapless phases. Then I give a detailed analysis of string condensaiton in 3+1D discrete gauge theories. I compute the outcome of the condensation, namely the category of excitations surviving the condensation. The results suggest that a complete topological holography for 2+1D phases can only be established by taking into account all possible ways of condensing strings in the bulk 3+1D topological order.

A computational phase transition in a classical or quantum system is a non-analytic change in behavior of an order parameter which can only be observed with the assistance of a nontrivial classical computation. Such phase transitions, and the computational observables which detect them, play a crucial role in the optimal decoding of quantum error-correcting codes and in the scalable detection of measurement-induced phenomena. We show that computational phase transitions and observables can also provide important physical insight on the phase diagram of a classical statistical physics system, specifically in the context of the dislocation-mediated melting of a two-dimensional antiferromagnetic (AF) crystal. In the solid phase, elementary dislocations disrupt the bipartiteness of the underlying square lattice, and as a result, pairs of dislocations are linearly confined by string-like AF domain walls. It has previously been argued that a novel AF tetratic phase can arise when double dislocations proliferate while elementary dislocations remain bound. We will argue that, although there is no thermodynamic phase transition separating the AF and paramagnetic (PM) tetratic phases, it is possible to algorithmically construct a nonlocal order parameter which distinguishes the AF and PM tetratic regimes and undergoes a continuous computational phase transition. We discuss both algorithm-dependent and "intrinsic" algorithm-independent computational phase transitions in this setting, the latter of which includes a transition in one's ability to consistently sort atoms into two sublattices to construct a well-defined staggered magnetization.

Topological phases of matter offer a promising platform for quantum computation and quantum error correction. Nevertheless, unlike its counterpart in pure states, descriptions of topological order in mixed states remain relatively under-explored. We will give various definitions for replica topological order in mixed states. Similar to the replica trick, our definitions also involve n copies of density matrix of the mixed state. Within this framework, we categorize topological orders in mixed states as either quantum, classical, or trivial, depending on the type of information they encode.

For the case of the toric code model in the presence of decoherence, we associate for each phase a quantum channel and describes the structure of the code space. We show that in the quantum-topological phase, there exists a postselection-based error correction protocol that recovers the quantum information, while in the classical-topological phase, the quantum information has decohere and cannot be fully recovered. We accomplish this by describing the mixed state as a projected entangled pairs state (PEPS) and identifying the symmetry-protected topological order of its boundary state to the bulk topology.

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Hilbert spaces are incomprehensibly vast and rich. Model Hamiltonians are space ships. They could take us to new worlds, such as cold \textit{spin liquid oasis} in hot regions in Hilbert space deserts. Exact decomposition of isotropic Heisenberg Hamiltonian on a Honeycomb lattice into a sum of 3 non-commuting (permuted) Kitaev Hamiltonians, helps us build a degenerate \textit{manifold of metastable flux free Kitaev spin liquid vacua} and vector Fermionic (Goldstone like) collective modes. Our method, \textit{symmetric decomposition of Hamiltonians}, will help design exotic metastable quantum scars and exotic quasi particles, in nonexotic real systems.
G. Baskaran, arXiv:2309.07119

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The code space of a quantum error-correcting code can often be identified with the degenerate ground-space within a gapped phase of quantum matter. We argue that the stability of such a phase is directly related to a set of coherent error processes against which this quantum error-correcting code (QECC) is robust: such a quantum code can recover from adiabatic noise channels, corresponding to random adiabatic drift of code states through the phase, with asymptotically perfect fidelity in the thermodynamic limit, as long as this adiabatic evolution keeps states sufficiently "close" to the initial ground-space. We further argue that when specific decoders -- such as minimum-weight perfect matching -- are applied to recover this information, an error-correcting threshold is generically encountered within the gapped phase. In cases where the adiabatic evolution is known, we explicitly show examples in which quantum information can be recovered by using stabilizer measurements and Pauli feedback, even up to a phase boundary, though the resulting decoding transitions are in different universality classes from the optimal decoding transitions in the presence of incoherent Pauli noise. This provides examples where non-local, coherent noise effectively decoheres in the presence of syndrome measurements in a stabilizer QECC.

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A formulation of the 2-dimensional Ising model on a triangulated Riemann sphere is proposed that converges to the exact conformal field theory (CFT) in the continuum limit. The solution is based on reconciling Regge's simplicial geometry for the Einstein Hilbert action with an Affine map to quantum correlators on the tangent plane. Numerical tests of the 2d Ising sphere and radial quantized phi4 theory on $R x S^2$ are presented.  Extending the method to more general fields theories on curved manifolds is discussed. 

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