Quantum Matter

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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Weak measurements can be viewed as a soft projection that interpolates between an identity operator and a projection operator, and can induce an effective central charge distinct from the unmeasured CFT. In the first part, I will discuss the effect of measurement and postselection on the critical ground state of a Luttinger liquid theory. Depending on the Luttinger parameter K, the effect of measurement is irrelevant, marginal, or relevant, respectively. When the measurement is marginal, and we find a critical state whose entanglement entropy exhibits a logarithmic behavior with a continuous effective central charge as a function of measurement strength. Inspired by this result, in the second part, I will discuss a holographic description of the weak measurement. The weak measurement is modeled by an interface brane, separating different geometries dual to the post-measurement state and the unmeasured CFT. In an infinite system, the weak measurement is related to ICFT via a spacetime rotation. We find that the holographic entanglement entropy with twist operators located on the defect is consistent in both calculations for ICFT and weak measurements. In a finite system, the weak measurement can lead to a rich phase diagram, in which the post-measurement geometry can realize a Python’s lunch.

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The Levin-Wen model is known to produce a vast array of topological phases of matter. Among these theories are gauge theories such as the twisted quantum double. In this talk, we will show that the Levin-Wen model is itself a gauge theory. In particular, given a unitary fusion category C, we construct a globally tube algebra (Tube(C)) symmetric lattice model and gauge this symmetry to produce the Levin-Wen model with anyons described by the Drinfeld center Z(C). This construction endows the terms of the Levin-Wen Hamiltonian with the interpretation of flux and charge operators for the Tube(C) gauge symmetry. Furthermore, this construction gives a gauge theoretic interpretation to the mathematical fact that the category of representations of Tube(C) is equivalent to Z(C). To demonstrate this new class of Tube(C) symmetric theories, we will explicitly explore the case where C is the Fibonacci category Fib. We will write down the ungauged Tube(Fib) symmetric theory, compute the symmetry action, and show how to gauge the Tube(Fib) global symmetry to produce the double Fibonacci Levin-Wen model.

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In this talk I will use matrix product states (MPS) to study topological families of gapped ground states in one spatial dimension. To such families I will describe how to associate a gerbe, a mathematical structure which generalizes the line bundle associated to gapped ground states in 0d. Nontriviality of the gerbe represents an obstruction to representing the family of ground states with an MPS tensor that is continuous everywhere over parameter space. I will illustrate these constructions using an exactly solvable topological family which exhibits the key physics in a simple manner.

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Zoom link https://pitp.zoom.us/j/91497524520?pwd=MkFHSW9PeGlMb2lFOTR3Qmo5clU0dz09

It is known that the classical O(N) model in dimension d > 3 at its bulk critical point admits three boundary universality classes: the ordinary, the extraordinary and the special. The extraordinary fixed point corresponds to the bulk transition occurring in the presence of an ordered boundary, while the special fixed point corresponds to a boundary phase transition between the ordinary and the extra-ordinary classes. While the ordinary fixed point survives in d = 3, it is less clear what happens to the extraordinary and special fixed points when d = 3 and N is greater or equal to 2. I'll show that formally treating N as a continuous parameter, there exists a critical value Nc > 2 separating two distinct regimes. For N < Nc the extra-ordinary fixed point survives in d = 3, albeit in a modified form: the long-range boundary order is lost, instead, the order parameter correlation function decays as a power of log r. For N > Nc there is no fixed point with order parameter correlations decaying slower than power law. I'll discuss how these findings compare to recent Monte-Carlo studies of classical and quantum spin models.

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Zoom link https://pitp.zoom.us/j/97209122334?pwd=UHQ2OXR4bnVZREV0SlJOYXphWjh0QT09