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arXiv:1703.06890 · Physical review. B./Physical review. B

Approximating the Sachdev-Ye-Kitaev model with Majorana wires

Aaron Chew, Andrew M. Essin, Jason Alicea

Published Sep 29, 2017• 115 citations
arXivHTMLPDF

Abstract

Randomly interacting Majorana fermions governed by the Sachdev-Ye-Kitaev model have recently been shown to exhibit profound links to quantum chaos and black holes. Here, the authors propose a solid-state implementation of the model that exploits hardware components that are now widely pursued for topological quantum computation -- namely, an array of Majorana wires interfaced with a disordered quantum dot that mediates the desired interactions. This work highlights a potential pathway for using table-top condensed-matter systems to probe novel concepts from high-energy physics and quantum information.

DOI
10.1103/physrevb.96.121119

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Randomly interacting Majorana fermions governed by the Sachdev-Ye-Kitaev model have recently been shown to exhibit profound links to quantum chaos and black holes. Here, the authors propose a solid-state implementation of the model that exploits hardware components that are now widely pursued for topological quantum computation -- namely, an array of Majorana wires interfaced with a disordered quantum dot that mediates the desired interactions. This work highlights a potential pathway for using table-top condensed-matter systems to probe novel concepts from high-energy physics and quantum information.
Majorana fermions are the real (in a mathematical sense) counterparts of complex fermions like ordinary electrons. The promise of topological quantum computing has lead to substantial experimental progress in realizing these particles in various synthetic platforms. The realization of Majorana fermions motivates a fundamental question: what phases of matter can emerge if many Majorana fermions are allowed to interact? Here we review recent progress in this direction on the proposed experimental setups, analytical and numerical results on low-dimensional lattice models, and the exactly solvable Sachdev-Ye-Kitaev model. The early progress thus far suggests that strongly correlated phases of matter with Majorana building blocks can exhibit many novel phenomena, such as emergent spacetime supersymmetry, topological order and the physics of black-holes, in condensed matter systems. They may also provide alternative avenues for universal topological quantum computing through the realization of the Fibonacci phase and measurement-based only surface codes.
The Sachdev-Ye-Kitaev (SYK) model is an all-to-all interacting Majorana fermion model for many-body quantum chaos and the holographic correspondence. Here we construct fermionic all-to-all Floquet quantum circuits of random four-body gates designed to capture key features of SYK dynamics. Our circuits can be built using local ingredients in Majorana devices, namely, charging-mediated interactions and braiding Majorana zero modes. This offers an analog-digital route to SYK quantum simulations that reconciles all-to-all interactions with the topological protection of Majorana zero modes, a key feature missing in existing proposals for analog SYK simulation. We also describe how dynamical, including out-of-time-ordered, correlation functions can be measured in such analog-digital implementations by employing foreseen capabilities in Majorana devices.
Recent decades have seen a proliferation of unconventional quasiparticles in condensed matter systems. Majorana fermions, theoretically predicted in several setups and of great interest in topological quantum computation, are a focus of intense research efforts. Higher-spin moments, relevant for both solid-state compounds and cold atoms, are of great theoretical interest as they interpolate between the quantum and classical limits but sometimes show surprising behavior. In the meantime, successful fabrication and characterization of low-dimensional systems have brought new phenomenology and physical insights. In this dissertation I will theoretically explore a few low-dimensional models using Majorana fermions and higher-spin moments as building blocks. I will first discuss a generalized family of the celebrated Sachdev-Ye-Kitaev model, a zero-dimensional all-to-all Majorana model that exhibits non-Fermi liquid behavior and is holographically dual to a black hole. The generalized model has a phase transition between a non-Fermi liquid and a disordered Fermi liquid. Then I will discuss the Heisenberg model with higher spins, with a focus on chaos and information scrambling. Using matrix-product-state-based methods, we are able to obtain numerical results for spin up to 4 and characterize the Lyapunov growth. After that I will discuss a generalization of the Hubbard model to Majorana fermions on the honeycomb lattice. Unlike previous similar models, we find topological phases with (anti-)chiral edge modes for weak interaction. Finally I will show a construction of explicit supersymmetric Majorana model on the kagome lattice, where a family of exact solutions is found and the nature of supersymmetry breaking is explored.
The Sachdev-Ye-Kitaev (SYK) model is zero-dimensional model simulating quantum chaos using interacting Majorana fermions.Previously proposals have been made to realize the SYK model in fermionic systems that can support majorana zero modes. In this work, we simulate the SYK model in a Kitaev spin chain, where zero modes also exist in the ground state from the fermionic representation of spins. By adding other interactions as perturbations, we lift the degenerate majorana zero modes and obatin an effective SYK interactions. Although the majorana fermions are nonlocal in forms of spin operators, we show that the out-of-time-ordered correlator (OTOC) of the majorana fermions can be approximately simulated in forms of spin operators. Our proposal sheds light in simulating the fermonic SKY model and the related physics in purely bosonic systems.
The authors study in detail the quantum mechanical model of NNN Majorana fermions with random interactions of a few fermions at a time (Sachdev-Ye-Kitaev model) in the large NNN limit. At low energies, the system is strongly interacting and an emergent conformal symmetry develops. Performing technical calculations, the authors elucidate a number of properties of the model near the conformal point.
Supersymmetry is a powerful concept in quantum many-body physics. It helps to illuminate ground-state properties of complex quantum systems and gives relations between correlation functions. In this Letter, we show that the Sachdev-Ye-Kitaev model, in its simplest form of Majorana fermions with random four-body interactions, is supersymmetric. In contrast to existing explicitly supersymmetric extensions of the model, the supersymmetry we find requires no relations between couplings. The type of supersymmetry and the structure of the supercharges are entirely set by the number of interacting Majorana modes and are thus fundamentally linked to the model's Altland-Zirnbauer classification. The supersymmetry we uncover has a natural interpretation in terms of a one-dimensional topological phase supporting Sachdev-Ye-Kitaev boundary physics and has consequences away from the ground state, including in q-body dynamical correlation functions.