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Physical review. B./Physical review. B

Bosonic model of quantum holography

Brian G. Swingle, Michael Winer

Published Mar 21, 2024• 17 citations
Publisher

Abstract

We analyze a model of qubits that we argue has an emergent quantum gravitational description similar to the fermionic Sachdev-Ye-Kitaev (SYK) model. The model we consider is known as the quantum $q$-spin model because it features $q$-local interactions between qubits. It was previously studied as a model of a quantum spin glass, and while we find that the model is glassy for $q=2, q=3$, and likely $q=4$, we also find evidence for previously unexpected SYK-like behavior for the quenched free energy down to the lowest temperatures for $q\ensuremath{\ge}5$. This SYK-like physics includes power-law correlation functions and an extensive low-temperature entropy, so we refer to the model as Spin SYK. The model is generic in that it includes all possible $q$-body couplings, lacks most symmetries, and has no spatial structure, so our results can be construed as establishing a certain ubiquity of quantum holography in systems dominated by many-body interactions. Furthermore, we discuss a generalized family of models that includes Spin SYK and which provably exhibit SYK-like physics in the solvable limit of large local Hilbert space dimension. We also comment on the implications of a bosonic system with SYK-like properties for the study of holography, Hamiltonian complexity, and related topics.

DOI
10.1103/physrevb.109.094206

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We analyze a model of qubits that we argue has an emergent quantum gravitational description similar to the fermionic Sachdev-Ye-Kitaev (SYK) model. The model we consider is known as the quantum qqq-spin model because it features qqq-local interactions between qubits. It was previously studied as a model of a quantum spin glass, and while we find that the model is glassy for q=2,q=3q=2, q=3q=2,q=3, and likely q=4q=4q=4, we also find evidence for previously unexpected SYK-like behavior for the quenched free energy down to the lowest temperatures for q\ensuremath≥5q\ensuremath{\ge}5q\ensuremath≥5. This SYK-like physics includes power-law correlation functions and an extensive low-temperature entropy, so we refer to the model as Spin SYK. The model is generic in that it includes all possible qqq-body couplings, lacks most symmetries, and has no spatial structure, so our results can be construed as establishing a certain ubiquity of quantum holography in systems dominated by many-body interactions. Furthermore, we discuss a generalized family of models that includes Spin SYK and which provably exhibit SYK-like physics in the solvable limit of large local Hilbert space dimension. We also comment on the implications of a bosonic system with SYK-like properties for the study of holography, Hamiltonian complexity, and related topics.
We analyze a model of qubits which we argue has an emergent quantum gravitational description similar to the fermionic Sachdev-Ye-Kitaev (SYK) model. The model we consider is known as the quantum qqq-spin model because it features qqq-local interactions between qubits. It was previously studied as a model of a quantum spin glass, and while we find that the model is glassy for q=2q=2q=2, q=3q=3q=3, and likely q=4q=4q=4, we also find evidence for previously unexpected SYK-like behavior for the quenched free energy down to the lowest temperatures for q≥5q \geq 5q≥5. This SYK-like physics includes power-law correlation functions and an extensive low temperature entropy, so we refer to the model as Spin SYK. The model is generic in that it includes all possible qqq-body couplings, lacks most symmetries, and has no spatial structure, so our results can be construed as establishing a certain ubiquity of quantum holography in systems dominated by many-body interactions. Furthermore, we discuss a generalized family of models which includes Spin SYK and which provably exhibit SYK-like physics in the solvable limit of large local Hilbert space dimension. We also comment on implications of a bosonic system with SYK-like properties for the study of holography, Hamiltonian complexity, and related topics.
A brief survey of some random quantum models with infinite-range couplings is presented, ranging from the quantum Ising model to the Sachdev-Ye-Kitaev model. The Sachdev-Ye-Kitaev model was the first to realize an extensive zero temperature entropy without requiring an exponentially large ground state degeneracy. This phenomenon is closely linked to the absence of a particle-like interpretation of its low energy spectrum--its spectral functions are not those of bosons or fermions but are instead "Planckian", meaning they are universal functions of energy/temperature. A remarkable consequence of these properties is that the SYK model provides an effective low energy theory of non-supersymmetric charged or rotating black holes in 3+1 dimensions, leading to new results on the density of many-body quantum states of such black holes. For applications to non-quasiparticle metallic states of quantum materials, an extension of the SYK model, known as the two-dimensional Yukawa-Sachdev-Ye-Kitaev model, is required. The 2dYSYK model describes quantum phase transitions in metals with spatial inhomogeneity in the position of the quantum critical point. This extension has led to a universal theory of the strange metal state observed in numerous correlated electron compounds, including copper-oxide based high temperature superconductors.
We study a sparse version of the Sachdev-Ye-Kitaev (SYK) model defined on random hypergraphs constructed either by a random pruning procedure or by randomly sampling regular hypergraphs. The resulting model has a new parameter, kkk, defined as the ratio of the number of terms in the Hamiltonian to the number of degrees of freedom, with the sparse limit corresponding to the thermodynamic limit at fixed kkk. We argue that this sparse SYK model recovers the interesting global physics of ordinary SYK even when kkk is of order unity. In particular, at low temperature the model exhibits a gravitational sector which is maximally chaotic. Our argument proceeds by constructing a path integral for the sparse model which reproduces the conventional SYK path integral plus gapped fluctuations. The sparsity of the model permits larger scale numerical calculations than previously possible, the results of which are consistent with the path integral analysis. Additionally, we show that the sparsity of the model considerably reduces the cost of quantum simulation algorithms. This makes the sparse SYK model the most efficient currently known route to simulate a holographic model of quantum gravity. We also define and study a sparse supersymmetric SYK model, with similar conclusions to the non-supersymmetric case. Looking forward, we argue that the class of models considered here constitute an interesting and relatively unexplored sparse frontier in quantum many-body physics.
The Sachdev-Ye-Kitaev (SYK) model may provide us with a good starting point for the experimental study of quantum chaos and holography in the laboratory. Still, the four-local interaction of fermions makes quantum simulation challenging, and it would be good to search for simpler models that keep the essence. In this paper, we argue that the four-local interaction may not be important by introducing a few models that have two-local interactions. The first model is a generalization of the spin-SYK model, which is obtained by replacing the spin variables with SU(d) generators. Simulations of this class of models might be straightforward on qudit-based quantum devices. We study the case of d=3,4,5,6 numerically and observe quantum chaos already for two-local interactions in a wide energy range. We also introduce modifications of spin-SYK and SYK models that have similar structures as the SU(d) model (e.g., H=∑_{p,q}J_{pq}χ_{p}χ_{p+1}χ_{q}χ_{q+1} instead of the original SYK Hamiltonian H=∑_{p,q,r,s}J_{pqrs}χ_{p}χ_{q}χ_{r}χ_{s}), which shows strongly chaotic features although the interaction is essentially two-local. These models may be a good starting point for the quantum simulation of the original SYK model.
We suggest that the holographic principle, combined with recent technological advances in atomic, molecular, and optical physics, can lead to experimental studies of quantum gravity. As a specific example, we consider the Sachdev–Ye–Kitaev (SYK) model, which consists of spin-polarized fermions with an all-to-all complex random two-body hopping and has been conjectured to be dual to a certain quantum-gravitational system. Achieving low-temperature states of the SYK model is interpreted as a realization of a stringy black hole, provided that the holographic duality is true. We introduce a variant of the SYK model, in which the random two-body hopping is real. This model is equivalent to the original SYK model in the large-|NNN| limit. We show that this model can be created in principle by confining ultracold fermionic atoms into optical lattices and coupling two atoms with molecular states via photo-association lasers. This development serves as an important first step towards an experimental realization of such systems dual to quantum black holes. We also show how to measure out-of-time-order correlation functions of the SYK model, which allow for identifying the maximally chaotic property of the black hole.