Quantum phases of matter

Classical machine learning has succeeded in the prediction of both classical and quantum phases of matter. Notably, kernel methods stand out for their ability to provide interpretable results, relating the learning process with the physical order parameter explicitly. Here we exploit quantum kernels instead. They are naturally related to the fidelity, and thus it is possible to interpret the learning process with the help of quantum information tools. In particular, we use a support vector machine (with a quantum kernel) to predict and characterize second-order quantum phase transitions. We explain and understand the process of learning when the fidelity per site (rather than the fidelity) is used. The general theory is tested in the Ising chain in transverse field. We show that for small-sized systems, the algorithm gives accurate results, even when trained away from criticality. Besides, for larger sizes we confirm the success of the technique by extracting the correct critical exponent $\ensuremath{\nu}$. Finally, we present two algorithms, one based on fidelity and one based on the fidelity per site, to classify the phases of matter in a quantum processor.

Phases of matter are sharply defined in the thermodynamic limit. One major challenge of accurately simulating quantum phase diagrams of interacting quantum systems is due to the fact that numerical simulations usually deal with the energy density, a local property of quantum wavefunctions, while identifying different quantum phases generally rely on long-range physics. In this paper we construct generic fully symmetric quantum wavefunctions under certain assumptions using a type of tensor networks: projected entangled pair states, and provide practical simulation algorithms based on them. We find that quantum phases can be organized into crude classes distinguished by short-range physics, which is related to the fractionalization of both on-site symmetries and space-group symmetries. Consequently, our simulation algorithms, which are useful to study long-range physics as well, are expected to be able to sharply determine crude classes in interacting quantum systems efficiently. Examples of these crude classes are demonstrated in half-integer quantum spin systems on the kagome lattice. Limitations and generalizations of our results are discussed.

Finding suitable indicators for characterizing quantum phase transitions plays an important role in understanding different phases of matter. It is especially important for fracton phases where a combination of topology and fractionalization leads to exotic features not seen in other known quantum phases. In this paper, we consider the above problem by studying phase transition in the X-cube model in the presence of a nonlinear perturbation. Using an analysis of the ground state fidelity and identifying a discontinuity in the global entanglement, we show there is a first-order quantum phase transition from a type-I fracton phase with a highly entangled nature to a magnetized phase. Accordingly, we conclude that the global entanglement, as a measure of the total quantum correlations in the ground state, can well capture certain features of fracton phase transitions. Then, we introduce a nonlocal order parameter in the form of a foliated operator which can characterize the above phase transition. We particularly show that such an order parameter has a geometric nature which captures specific differences of fracton phases with topological phases. Our study is specifically based on a well-known dual mapping to the classical plaquette Ising model where it shows the importance of such dualities in studying different quantum phases of matter.

Quantum phases of matter are resources for notions of quantum computation. In this work, we establish a new link between concepts of quantum information theory and condensed matter physics by presenting a unified understanding of symmetry-protected topological (SPT) order protected by subsystem symmetries and its relation to measurement-based quantum computation (MBQC). The key unifying ingredient is the concept of quantum cellular automata (QCA) which we use to define subsystem symmetries acting on rigid lower-dimensional lines or fractals on a 2D lattice. Notably, both types of symmetries are treated equivalently in our framework. We show that states within a non-trivial SPT phase protected by these symmetries are indicated by the presence of the same QCA in a tensor network representation of the state, thereby characterizing the structure of entanglement that is uniformly present throughout these phases. By also formulating schemes of MBQC based on these QCA, we are able to prove that most of the phases we construct are computationally universal phases of matter, in which every state is a resource for universal MBQC. Interestingly, our approach allows us to construct computational phases which have practical advantages over previous examples, including a computational speedup. The significance of the approach stems from constructing novel computationally universal phases of matter and showcasing the power of tensor networks and quantum information theory in classifying subsystem SPT order.

A quantum phase of matter can be understood from the symmetry of the system's Hamiltonian. The system symmetry along the time axis has been proposed to show a new phase of matter referred as discrete-time crystals (DTCs). A DTC is a quantum phase of matter in non-equilibrium systems, and it is also intimately related to the symmetry of the initial state. DTCs that are stable in isolated systems are not necessarily resilient to the influence from the external reservoir. In this paper, we discuss the dynamics of the DTCs under the influence of an environment. Specifically, we consider a non-trivial situation in which the initial state is prepared to partly preserve the symmetry of the Liouvillian. Our analysis shows that the entire system evolves towards a DTC phase and is stabilised by the effect of dephasing. Our results provide a new understanding of quantum phases emerging from the competition between the coherent and incoherent dynamics in dissipative non-equilibrium quantum systems.

One of the main goals of physics is to identify and characterize the possible phases of matter, as well as the transitions between these phases. In this chapter we are concerned with theoretical and computational aspects of quantum phases and quantum phase transitions involving extended quantum manybody systems at zero temperature.

In this paper, we present a quantum information framework for the entanglement behavior of the low-energy quasiparticle (QP) excitations in various quantum phases in one-dimensional (1D) systems. We first establish an exact correspondence between the correlation matrix and the QP entanglement Hamiltonian for free fermions and find an extended in-gap state in the QP entanglement Hamiltonian as a consequence of the position uncertainty of the QP. A more general understanding of such an in-gap state can be extended to a Kramers theorem for the QP entanglement Hamiltonian, which also applies to strongly interacting systems. Further, we present a set of ubiquitous entanglement spectrum features, dubbed entanglement fragmentation, conditional mutual information, and measurement-induced nonlocal entanglement for QPs in 1D symmetry protected topological phases. Our result thus provides another framework to identify different phases of matter in terms of their QP entanglement.

Electrons in clean macroscopic samples of graphene exhibit an astonishing variety of quantum phases when strong perpendicular magnetic field is applied. These include integer and fractional quantum Hall states as well as symmetry broken phases and quantum Hall ferromagnetism. Here we show that mesoscopic graphene flakes in the regime of strong disorder and magnetic field can exhibit another remarkable quantum phase described by holographic duality to an extremal black hole in two-dimensional anti-de Sitter space. This phase of matter can be characterized as a maximally chaotic non-Fermi liquid since it is described by a complex fermion version of the Sachdev-Ye-Kitaev model known to possess these remarkable properties.

Abstract2D phases of matter have become a new paradigm in condensed matter physics, bringing in an abundance of novel quantum phenomena with promising device applications. However, realizing such quantum phases has its own challenges, stimulating research into non‐traditional methods to create them. One such attempt is presented here, where the intrinsic crystal anisotropy in a “fractional” perovskite, EuxTaO3 (x = 1/3 − 1/2), leads to the formation of stacked layers of quasi‐2D electron gases, despite being a 3D bulk system. These carriers possess topologically non‐trivial spin textures, indirectly controlled by an external magnetic field via proximity effect, making it an ideal system for spintronics, for which several possible applications are proposed. An anomalous Hall effect with a non‐monotonic dependence on carrier density is shown to exist, signifying a shift in band topology with carrier doping. Furthermore, quantum oscillations in charge conductivity and oscillating thermoelectric properties are examined and proposed as routes to experimentally demonstrate the quasi‐2D behavior.

State-of-the-art machine learning techniques promise to become a powerful tool in statistical mechanics via their capacity to distinguish different phases of matter in an automated way. Here we demonstrate that convolutional neural networks (CNN) can be optimized for quantum many-fermion systems such that they correctly identify and locate quantum phase transitions in such systems. Using auxiliary-field quantum Monte Carlo (QMC) simulations to sample the many-fermion system, we show that the Green’s function holds sufficient information to allow for the distinction of different fermionic phases via a CNN. We demonstrate that this QMC + machine learning approach works even for systems exhibiting a severe fermion sign problem where conventional approaches to extract information from the Green’s function, e.g. in the form of equal-time correlation functions, fail.

Quantum phases of matter are conventionally characterized in terms of their (local and nonlocal) correlation functions. The advent of quantum information processing architectures, however, prompts one to wonder if the correlations inherent in phases of matter could be harnessed to gain quantum advantage in some tasks. This provides an alternative way of thinking about correlations and phases, complementary to the traditional condensed matter perspective, but intimately related to the perspective on generalized Bell tests developed by Mermin. A crisp example is provided by the 'parity game' of Brassard, Broadbent and Tapp. We begin by discussing the parity game with three players: Alice (A), Bob (B), and Charlie (C), who are not allowed to communicate with each other during the course of the game. The three players receive input bits a, b and c respectively, with c = (a + b) mod 2. Each player is asked to

The Hall effects comprise one of the oldest but most vital fields in condensed matter physics, and they persistently inspire new findings, such as quantum Hall effects and topological phases of matter. The recently discovered nonlinear Hall effect is a new member of the family of Hall effects. It is characterized as a transverse Hall voltage in response to two longitudinal currents in the Hall measurement, but it does not require time-reversal symmetry to be broken. It has deep connections to symmetry and topology and, thus, opens new avenues by which to probe the spectral, symmetry and topological properties of emergent quantum materials and phases of matter. In this Perspective, we present an overview of the recent progress regarding the nonlinear Hall effect. We discuss the open problems, the prospects of the use of the nonlinear Hall effect in spectroscopic and device applications, and generalizations to other nonlinear transport effects. The recent measurement of a nonlinear Hall effect has provided a new way to probe the spectral, symmetry and topological properties of quantum materials. This Perspective discusses the open questions around this new effect and potential applications.

This modern text describes the remarkable developments in quantum condensed matter physics following the experimental discoveries of quantum Hall effects and high temperature superconductivity in the 1980s. After a review of the phases of matter amenable to an independent particle description, entangled phases of matter are described in an accessible and unified manner. The concepts of fractionalization and emergent gauge fields are introduced using the simplest resonating valence bond insulator with an energy gap, the Z2 spin liquid. Concepts in band topology and the parton method are then combined to obtain a large variety of experimentally relevant gapped states. Correlated metallic states are described, beginning with a discussion of the Kondo effect on magnetic impurities in metals. Metals without quasiparticle excitations are introduced using the Sachdev-Ye-Kitaev model, followed by a discussion of critical Fermi surfaces and strange metals. Numerous end-of-chapter problems expand readers' comprehension and reinforce key concepts.

Due to their inherent capabilities of capturing non-local dependencies, Transformer neural networks have quickly been established as the paradigmatic architecture for large language models and image processing. Next to these traditional applications, machine learning (ML) methods have also been demonstrated to be versatile tools in the analysis of image-like data of quantum phases of matter, e.g. given snapshots of many-body wave functions obtained in ultracold atom experiments. While local correlation structures in image-like data of physical systems can reliably be detected, identifying phases of matter characterized by global, non-local structures with interpretable ML methods remains a challenge. Here, we introduce the correlator Transformer (CoTra), which classifies different phases of matter while at the same time yielding full interpretability in terms of physical correlation functions. The network’s underlying structure is a tailored attention mechanism, which learns efficient ways to weigh local and non-local correlations for a successful classification. We demonstrate the versatility of the CoTra by detecting local order in the Heisenberg antiferromagnet, and show that local gauge constraints in one- and two-dimensional lattice gauge theories can be identified. Furthermore, we establish that the CoTra reliably detects non-local structures in images of correlated fermions in momentum space (Cooper pairs) and that it can distinguish percolating from non-percolating images.

We consider two related tasks: (a) estimating a parameterisation of a given Gibbs state and expectation values of Lipschitz observables on this state; (b) learning the expectation values of local observables within a thermal or quantum phase of matter. In both cases, we present sample-efficient ways to learn these properties to high precision. For the first task, we develop techniques to learn parameterisations of classes of systems, including quantum Gibbs states for classes of non-commuting Hamiltonians. We then give methods to sample-efficiently infer expectation values of extensive properties of the state, including quasi-local observables and entropies. For the second task, we exploit the locality of Hamiltonians to show that M local observables can be learned with probability 1 − δ and precision ε using N=OlogMδepolylog(ε−1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N={\\mathcal{O}}\\left(\\log \\left(\\frac{M}{\\delta }\\right){e}^{{\\rm{polylog}}({\\varepsilon }^{-1})}\\right)$$\\end{document} samples — exponentially improving previous bounds. Our results apply to both families of ground states of Hamiltonians displaying local topological quantum order, and thermal phases of matter with exponentially decaying correlations.