Quantum Matter

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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Tensor product states are powerful tools for simulating area-law entangled states of many-body systems. The applicability of such methods to the non-equilibrium dynamics of many-body systems is less clear due to the presence of large amounts of entanglement. New methods seek to reduce the numerical cost by selectively discarding those parts of the many-body wavefunction, which are thought to have relatively litte effect on dynamical quantities of interest. We present a theory for the sizes of “backflow corrections”, i.e., systematic errors due to these truncation effects and introduce the dissipation-assisted operator evolution (DAOE) method for calculating transport properties of strongly interacting lattice systems in the high temperature regime. In the DAOE method, we represent the observable as a matrix product operator, and show that the dissipation leads to a decay of operator entanglement, allowing us to capture the dynamics to long times. We benchmark this scheme by calculating spin and energy diffusion constants in a variety of physical models and compare to other existing methods.

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The interplay between spontaneous symmetry breaking and topology can result in exotic quantum states of matter. A celebrated example is the quantum anomalous Hall (QAH) effect, which exhibits an integer quantum Hall effect at zero magnetic field due to topologically nontrivial bands and intrinsic magnetism. In the presence of strong electron-electron interactions, fractional-QAH (FQAH) effect at zero magnetic field can emerge, which is a lattice analog of fractional quantum Hall effect without Landau level formation. In this talk, I will present experimental observation of FQAH effect in twisted MoTe 2 bilayer, using combined magneto- optical and -transport measurements. In addition, we find an anomalous Hall state near the filling factor -1/2, whose behavior resembles that of the composite Fermi liquid phase in the half-filled lowest Landau level of a two-dimensional electron gas at high magnetic field. Direct observation of the FQAH and associated effects paves the way for researching charge fractionalization and anyonic statistics at zero magnetic field.

Reference 1. Observation of Fractionally Quantized Anomalous Hall Effect, Heonjoon Park et al., Nature, https://www.nature.com/articles/s41586-023-06536-0 (2023);
2. Signatures of Fractional Quantum Anomalous Hall States in Twisted MoTe2 Bilayer, Jiaqi Cai et al., Nature, https://www.nature.com/articles/s41586-023-06289-w (2023);
3. Programming Correlated Magnetic States via Gate Controlled Moiré Geometry, Eric Anderson et al., Science, https://www.science.org/doi/full/10.1126/science.adg4268 (2023)

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In this presentation, I will discuss our latest efforts to classify and characterize insulators in two spatial dimensions in the presence of interactions. I will focus on two examples, both on the square lattice. The first example is of bosonic insulators with symmetry G = p4m x K, where K is an internal symmetry group taken to be U(1), SO(3) or Z_N. The second example is of fermionic insulators with G= p4 x U(1)^f. In both scenarios, we propose a set of invariants that reproduce the classification anticipated through real space constructions and topological field theory (TFT). In doing so, we relate these invariants to coefficients appearing in the response action obtained from TFT. For the fermionic case, we propose a map from the non-interacting classification to the interacting classification.

This talk is based on work done with Naren Manjunath and Maissam Barkeshli.

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The single-channel Kondo impurity problem is a classic example of strongly coupled physics. In the Kondo problem, a single magnetic impurity is placed in a metal — the resulting system exhibits interesting properties such as a resistance minimum as a function of temperature. The problem was solved by Wilson’s numerical renormalization group and later by the Bethe ansatz technique. The Bethe ansatz exactly diagonalizes the Kondo hamiltonian for arbitrary impurity spin $s$ and numerically computes the impurity free energy for all temperatures. In this talk, I’ll present an alternate analytic solution for the Kondo problem at large $s$ that builds on recent results in boundary conformal field theory. This solution allows us to access analytically intermediate scales of the Kondo problem at large $s$; our results in this regime agree with the numeric results of the Bethe ansatz.

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Spatial symmetries can enrich the topological classification of interacting quantum matter and endow additional ``weak” topological indices upon systems with non-trivial strong topological invariants (protected by internal symmetries). In this talk, I will discuss the boundary physics of charge conserving systems with a non-zero shift invariant, which is protected by either a continuous U(1) or discrete CN rotation symmetry. In particular, I will discuss an interface between two systems with the same Chern number but distinct shift invariants and show that the interface hosts protected gapless edge modes. For general Abelian topological orders in 2D, I will prove sufficient conditions for gapless edge states protected by continuous rotation symmetry. For the case of discrete rotation symmetries, I will show that the Chern-Simons field theory for systems with gappable edges predicts fractional corner charges. These can also be computed when the system is placed on the two-dimensional surface of a Platonic solid, which relates to the fractional charge bound at disclination defects. Time permitting, I will discuss recent results regarding the relation between the shift and many-body real-space invariants for 2D systems with crystalline symmetry.

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