Quantum Matter

Recent studies of highly frustrated antiferromagnets (AFMs) have demonstrated the qualitative impact of virtual, longer-range singlet excitations on the effective RVB tunneling parameters of the low energy sector of the problem [1,2]. Here, I will discuss the current state of affairs on the RVB description of the spin-1/2 kagome AFM, and present new results that settle a number of issues in this problem [3].

[1] I. Rousochatzakis, Y. Wan, O. Tchernyshyov, and F. Mila, PRB 90,

100406(R) (2014)

[2] A. Ralko and I. Rousochatzakis, PRL 115, 167202 (2015) [3] in preparation.

We consider the problem of certifying entanglement and nonlocality in one-dimensional translation-invariant (TI) infinite systems when just averaged near-neighbor correlators are available. Exploiting the triviality of the marginal problem for 1D TI distributions, we arrive at a practical characterization of the near-neighbor density matrices of multi-separable TI quantum states. This allows us, e.g., to identify a family of separable two-qubit states which only admit entangled TI extensions. For nonlocality detection, we show that, when viewed as a vector in R^n, the set of boxes admitting an infinite TI classical extension forms a polytope, i.e., a convex set defined by a finite number of linear inequalities. Using DMRG, we prove that some of these inequalities can be violated by distant parties conducting identical measurements on an infinite TI quantum state. Both our entanglement witnesses and our Bell inequalities can be used to certify entanglement and nonlocality in large spin chains (namely, finite, and not TI chains) via neutron scattering.

Our attempts at generalizing our results to TI systems in 2D and 3D lead us to the virtually unexplored problem of characterizing the marginal distributions of infinite TI systems in higher dimensions. In this regard, we show that, for random variables which can only take a small number of possible values (namely, bits and trits), the set of nearest (and next-to-nearest) neighbor distributions admitting a 2D TI infinite extension forms a polytope. This allows us to compute exactly the ground state energy per site of any classical nearest-neighbor Ising-type TI Hamiltonian in the infinite square or triangular lattice. Remarkably, some of these results also hold in 3D.

In contrast, when the cardinality of the set of possible values grows (but remaining finite), we show that the marginal nearest-neighbor distributions of 2D TI systems are not described by a polytope or even a semi-algebraic set. Moreover, the problem of computing the exact ground state energy per site of arbitrary 2D TI Hamiltonians is undecidable.

Quantum triangles can work as interferometers. Depending on their geometric size and interactions between paths, “beats” and/or “steps”

patterns are observed. We show that when inter-level distances between level positions in quantum triangles periodically change with time, formation of beats and/or steps no longer depends only on the geometric size of the triangles but also on the characteristic frequency of the transverse signal. For large-size triangles, we observe the coexistence of beats and steps for moderated frequencies of the signal and for large frequencies a maximum of four steps instead of two as in the case with constant interactions are observed.

Small-size triangles also revealed counter-intuitive interesting dynamics for large frequencies of the field: unexpected two-step patterns are observed. When the frequency is large and tuned such that it matches the uniaxial anisotropy, three-step patterns are observed.

We have equally observed that when the transverse signal possesses a static part, steps maximize to six. These effects are semi-classically explained in terms of Fresnel integrals and quantum mechanically in terms of quantized fields with a photon-induced tunneling process. Our expressions for populations are in excellent agreement with the gross temporal profiles of exact numerical solutions. We compare the semi-classical and quantum dynamics in the triangle and establish the conditions for their equivalence.

A central theme of modern condensed matter physics is the study of topological quantum matter enabled by quantum mechanics, which provides a further "topological" twist to the classical theory of ordered phases. These quantum-entangled phases of matter such as fractional quantum Hall phases, spin liquids, and some non-Fermi liquids, are typically strongly-correlated and thus cannot be studied within conventional perturbative approaches. Because of the spectacular emergent phenomena as well as their potential for realistic applications, there has been much recent interest in exploring the physics of these exotic phases. In this talk, I show that the powerful methods of quantum field theory, namely quantum anomaly and duality, can expose the global landscape in parameter space of these gapped and gapless topological quantum phases and lead to several novel insights on these phases. As a demonstration of this principle, we study clean fractional quantum Hall transitions, composite Fermi liquids, and the surface of fractional topological insulators. Despite long and storied histories, these three systems are at the frontier of our knowledge of two and three dimensional topological phases. I show that the non-perturbative approach for these systems, i.e., the duality, sheds some new light on these systems and allows us to resolve some longstanding puzzles, which have not been clear previously. Furthermore, it uncovers novel physics of these intrinsically strongly-correlated phases of matter.

The quest for quantum spin liquids is an important enterprise in strongly correlated physics, yet candidate materials are still few and far between. Meanwhile, the classical front has had far more success, epitomized by the exceptional agreement between theory and experiment for a class of materials called spin ices. It is therefore natural to introduce quantum fluctuations into this well-established classical spin liquid model, in the hopes of obtaining a fully quantum spin liquid state.

The spin-flip excitations in spin ice fractionalize into pairs of effective magnetic monopoles of opposite charge. Quantum fluctuations have a parametrically larger effect on monopole motion than on the spin ice ground states so the leading manifestations of quantum behavior appear when monopoles are present. We study magnetic monopoles in quantum spin ice, whose dynamics is induced by a transverse field term. For this model, we find a family of extensively degenerate excited states, that make up an approximately flat band at the classical energy of the nearest neighbor monopole pair. These so-called mesonic states are exact up to the splitting of the spin ice ground state manifold. In my talk I will discuss their construction and properties that may be relevant in neutron scattering experiments on quantum spin ice candidates.

Topological quantum computing requires phases of matter which host fractionalized excitations that are neither bosons nor fermions. I will present a new route toward realizing such fractionalized phases of matter by literally building on existing topological phases. I will first discuss how existing topological phases, when interfaced with other systems, can exhibit a “topological proximity effect” in which nontrivial topology of a different nature is induced in the neighboring system. Then, I will show how this enables a new entanglement based technique (the “topological bootstrap”) for upgrading topological phases from the integer into the fractional quantum Hall variety. Finally, I will highlight the rich phenomenology of systems with interacting Majorana modes. Such systems can exhibit physics ranging from black hole scrambling to supersymmetry and from alternative surface code architectures to topological phases in three dimensions with completely immobile excitations. I will discuss my plans for understanding both general properties and specific models of such fascinating systems.

A closed quantum system is ergodic and satisfies equilibrium statistical physics when it completely loses local information of its initial condition under time evolution, by 'hiding' the information in non-local properties like entanglement. In the last decade, a flurry of theoretical work has shown that ergodicity can be broken in an isolated, quantum many-body system even at high energies in the presence of disorder, a phenomena known as many-body localization (MBL). In this novel phase of matter, highly excited states of an interacting system can serve as quantum memory and even protect exotic forms of quantum order. Recent claims of experimental observation of MBL in two dimensions using ultra-cold atoms has further raised a plethora of intriguing questions.

In one dimension, the strongly localized regime is described in terms of quasi-local integrals of motion, also known as l-bits. Based on this picture I will describe an efficient tensor network method to efficiently represent the entire spectrum of fully many-body localized systems. This ansatz is also successful at capturing features of the MBL to thermal transition. For higher dimensions, I will develop a refined phenomenology of MBL in terms of l*-bits which are only 'approximately' conserved, based on the stability of the localized phase to perturbations on the boundary. I will conclude with a bird's-eye view of some of the open problems in this rapidly growing field.

In this talk we propose a Hamiltonian approach to 2+1D gapped topological phases on an open surface with boundary. The bulk part is

(Levin-Wen) string-net models arising from a unitary fusion category (can be viewed as Hamiltonian approach to extended Turaev-Viro TQFT), while the boundary Hamiltonian is constructed using any Frobenius algebra in the input category. The combined Hamiltonian is exactly solvable and gives rise to a gapped energy spectrum which is topologically protected.

Our boundary Hamiltonians can be used to characterize and classify boundary conditions that give rise to gapped topological phase. We study the ground states and boundary excitations. Particularly, we show a correspondence between elementary excitations and the ground states on a cylinder system. Both are characterized by the category of bimodules over the Frobenius algebra that defines the  boundary Hamiltonian.

Near a quantum-critical point in a metal strong fermion-fermion interaction mediated by a soft collective boson gives rise to incoherent, non-Fermi liquid behavior. It also often gives rise to superconductivity which masks the non-Fermi liquid behavior. We analyze the interplay between the tendency to pairing and fermionic incoherence for a set of quantum-critical models with effective dynamical interaction between low-energy fermions. We argue that superconducting Tc is non-zero even for strong incoherence and/or weak interaction due to the fact that the self-energy from dynamic critical fluctuations vanishes for the two lowest fermionic Matsubara frequencies $\omega_m = \pm \pi T$. We obtain the analytic formula for T_c which reproduces well earlier numerical results, including the ones  for the  electron-phonon model at vanishing Debye frequency.

Dicke's seminal 1954 paper introduced the notion of `superradiance' in a system of spins coupled to a common photon mode.

Certain quantum states of the spins dominate the radiation process so that the spins radiate coherently. Dicke's original thought experiment has recently been recreated in the lab using cavity-QED setups with two spins. I will explore extending this experiment to $N$ spins and show that the radiation process naturally gives rise to entangled states. This suggests a new experimental tool to create multi-particle entanglement in the lab. In particular, a null-observation (non-observation of emitted photon) can be used to collapse the wavefunction onto a dark state. Remarkably, this dark state has resonating valence bond character. We show that the probability of collapse onto RVB state scales as $N^{-1}$, making it possible to generate entangled states of more than 20 spins.

Reference: arXiv:1609.04853

Topological states of matter are of of fundamental interest in contemporate condensed matter physics. Today, the fractional Quantum Hall effect remains the only known experimental system expected to exhibit intrinsic topological order. The question remains whether also different systems might stabilize this kind of ordering. Chiral spin liquids are an analogue of Fractional Quantum Hall Effect wave functions for spin systems. These wavefunctions have been envisioned in 1987 but only very recently several simple frustrated quantum spin models have been proposed realizing this physics. In this talk we will introduce chiral spin liquids, discuss their relation to the Fractional Quantum Hall effect and present our recent numerical studies that provide conclusive evidence for the emergence of this exotic state of matter in extended frustrated Heisenberg models. In the course of these projects novel algorithms and techniques for large scale Exact Diagonalization were developed and applied. We briefly discuss these techniques and show that they allow for sparse diagonalizations of Heisenberg systems up to 48-50 spin-1/2 particles.

Since its first discovery in 1986, high-Tc superconductors have been attracting constant interests and meticulous efforts from both theorists and experimentalists, not merely due to its large transition temperature, but also because it offers a well characterized laboratory for the study of exotic phenomena such as quantum criticality, non-Fermi liquid behavior, and intertwined orders. One pressing question in the field is the role played by disorder:

inevitable in real materials, disorder is able to fundamentally alter the properties of the system under certain circumstances. In this talk, we will discuss from a theoretical point of view, how different types of disorders affect various electronic orders in copper-based high-Tc superconductors. We also discuss the implications in experiments, and investigate the possibility of generalizing our conclusions to a variety of other physical systems.