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by the exponential of the classical action divided by ħ. In general, such an integral Quantum connections is not a well-defined​ mathematical object, because of the uncontrollably large number Sean Hartnoll, Subir Sachdev , Tadashi Takayanagi, Xie Chen, Eva Silverstein of possible configurations. However, in the low-​temperature limit of charged and Julian Sonner black holes, it can be evaluated in the Theoretical high-​energy and condensed-​matter physics share various ideas and near-​horizon region, where the classical tools. New connections between the two have been established through quantum theory shows that the spacetime becomes 1 + 1 dimensional. And the path integral information, providing exciting prospects for theoretical advances and even over these 1 + 1-dimensional​ spacetimes potential experimental studies. Six scientists discuss different directions of and electromagnetic fields maps exactly as research in this inter-disciplinary​ field. a hologram onto the low-temperature​ path integral of the 0 + 1-dimensional​ SYK model. New connections Quantum matter without quasiparticles In recent work, extensions of such mappings have led to advances in our understanding Sean Hartnoll. High-energy​ and Subir Sachdev. The study of compressible of the quantum behaviour of Einsteinian condensed-matter​ physics are organized states of electronic matter without theories of gravity, and insights into the around common fundamental concepts quasiparticle excitations began in the black-hole​ information paradox. such as symmetry breaking and the early 1980s, motivated by observations of renormalization group, and share core the ‘strange metal’ in copper-oxide-​ ​based Gravity and entanglement mathematical machinery including superconductors, and by the quantum Feynman diagrams and topology. This Hall state in the half-filled​ Landau level of Tadashi Takayanagi. The wavefunctions has led to a historically fruitful interface semiconductors in a strong magnetic field. of quantum many-body​ systems are between the two fields. In the last couple When quasiparticles are present, a quantum typically complicated from an algebraic of decades, two new points of connection many-body​ system will relax to equilibrium point of view. However, it is often possible have emerged. in response to an external perturbation in to extract a simple geometrical structure First, holographic duality has a time of the order of the mean collision from a wavefunction, namely quantum established that the classical evolution time between the quasiparticles. Without entanglement. In other words, the ‘skeleton’ of black-hole horizons precisely captures quasiparticles, equilibration proceeds of a many-body​ state is given by a network of the dissipative dynamics of a strongly more rapidly and in a time as short as quantum entanglement. quantum phase of matter. In recent the ‘Planckian time’ ħ/(kTB ), where Quantum entanglement also appears years, this connection has moved beyond T is the temperature of the final state. as a fundamental degree of freedom in simple correlation functions (that Remarkably, there is a solvable model of quantum gravity. The amount of quantum describe the approach to coarse-grained​ non-​quasiparticle quantum dynamics, the entanglement is measured by a quantity thermal equilibrium) to more nuanced SYK model of electrons with all-to-​ all​ and called entanglement entropy. With observables that can probe signatures random interactions. This model displays the framework of holographic duality, of many-body quantum chaos. Tied up Planckian time relaxation and also describes entanglement entropies in conformal field with this shift has been the emergence some other aspects of the physics of the theories (CFTs) turn out to be equal to areas of the Sachdev–Ye–Kitaev (SYK) model. copper oxide superconductors. of minimal surfaces in an anti-de-​ ​Sitter This model shares many of the features A distinct physical system that also space. This holographic entanglement (and limitations) of fully fledged holographic displays Planckian time relaxation is a entropy formula was recently generalized theories, but is both microscopically closer to black hole, and this becomes clear from into CFTs coupled to a dynamical gravity, conventional condensed-matter​ Hamiltonians the quantum temperature of a black hole providing a remarkable explanation for the and under greater technical control. computed by Stephen Hawking in 1974. black hole information paradox. Second, many-body​ quantum In this case, there is a remarkable and Surprisingly, the above two observations entanglement has simultaneously emerged surprisingly close connection between in two different subjects suggest the as an organizing principle in both fields. the SYK model and black holes with a fascinating prospect that gravitational It appears that quantum states supporting net electric charge, the latter described by spacetime can emerge from quantum holographically emergent gravity have quantizing Einstein’s classical theory of entanglement. This idea has been used an entanglement structure that may gravity and Maxwell’s classical theory successfully over the past several years be analogous to that of topologically of electromagnetism. The quantization within the conventional holography non-​trivial phases of matter. Fleshing out is described by a path integral over for a special class of spacetime whose this connection promises to be a source different configurations of spacetime and cosmological constant is negative. I expect of future progress. electromagnetic fields, each weighted this approach will also be crucial when

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trying to tackle holography for general to be possible. All these features suggest that spacetimes, including de Sitter spaces. these models cannot be properly described using a continuous field theory, at least not String theory Fracton models in the usual way. Progress has been made on many Xie Chen. In the past few years, a new aspects, including the discovery of new class of quantum many-​body models, models, connecting some of the features to Quantum Quantum Black-hole called the ‘fracton’ models, has captured systems we are familiar with, and exploring information matter physics the imagination of theorists from the new conceptual ideas to properly capture condensed- matter and the high-​energy the underlying order (for a review, see communities alike. These models, many refs1,2). Yet we are only scratching the of which were discovered as quantum error surface. The exotic physics of the fracton Condensed- correction codes, host a number of exotic model may fundamentally change and matter properties never seen before. They contain extend the way we approach quantum physics point excitations that cannot move on their many-body​ systems, while bringing the own, the ground-​state degeneracy grows quantum-information,​ condensed-matter​ Fig. 1 | Quantum connections. The study of exponentially with system size, and the usual and high-​energy communities closer quantum matter lies at the intersection of four renormalization group transformation does together (Fig. 1). areas of fundamental physics, providing a fertile not apply. ground for the exchange of ideas and tools. Figure Fracton models challenge both A new notion of universality adapted with permission from ref.6, Cambridge condensed-matter​ and high-energy​ theory. University Press. The immobility of the point excitations Eva Silverstein. A classic connection results in slow and non-thermal​ dynamics between high-​energy and condensed-​matter that needs to be better understood and physics concerns the renormalization irrelevant’ phenomena that become characterized. The growing ground-state​ group and the related notion of universality. important in the presence of long times degeneracy indicates an order that is not The details of the microscopic theory do not or large fields. only beyond the symmetry-breaking​ affect the outcome of experiments involving Reversing this flow is generally paradigm but also beyond the conventional low-energy​ probes of a system, enabling fraught with ambiguity, since one cannot topological paradigm. The renormalization a principled focus on a small number of determine the microscopic physics from group transformation needs to be relevant terms in the Lagrangian of the a few relevant terms at low energy. But augmented with non-trivial​ resource states theory. There is an exception: ‘dangerously in a new development3–5, certain irrelevant deformations are solvable and universal in a new sense. These deformations are The contributors specified via a differential equation, through Sean Hartnoll is Associate Professor of Physics at Stanford University. He received his PhD from a step-by-​ ​step procedure. At each step, Cambridge University in 2005 and did postdocs at Cambridge, KITP Santa Barbara and Harvard. the Lagrangian of the theory is deformed by He won the New Horizons prize in 2015. He has co-authored​ a book on ‘Holographic Quantum a certain bilinear of the stress energy tensor Matter’ and works on a range of topics in high energy, gravitational and condensed matter physics. known as TT ; the effect of this tensor on the Subir Sachdev is Herchel Smith Professor of Physics at Harvard University. He was educated energy spectrum is calculable. Joining this at the Indian Institute of Technology, the Massachusetts Institute of Technology and Harvard. to a trajectory including the addition of a His honours include the Dirac Medal and the Onsager Prize. His research on many-particle​ cosmological constant Λ at each step leads quantum entanglement connects to the properties of quantum matter and black holes. to distinct, but equally tractable results4. Tadashi Takayanagi received his PhD from the University of Tokyo in 2002. He worked in Harvard Since any quantum field theory has a stress University and in KITP Santa Barbara as a post-​doctoral fellow. He is currently a professor at energy tensor and a cosmological term, one Yukawa Institute for Theoretical Physics, Kyoto University and has been working on the deep can deform any seed theory in this way: it is connections between quantum gravity and quantum information, especially from the viewpoint of holography in string theory. universal. This approach can be applied to Xie Chen received her PhD from the Massachusetts Institute of Technology in 2012. She was a Miller research fellow at the University of California, Berkeley for two years before joining quantum gravity: the energy spectrum the faculty of the California Institute of Technology in 2014. Dr Chen is interested in studying and other features fit precisely with those non-trivial emergent phenomena in strongly interacting quantum many-body​ systems, such as of finite patches of spacetime, either anti topological order, symmetry anomaly, fracton order and others. de Sitter5 or de Sitter (with Λ). This suggests Eva Silverstein is Professor of Physics at Stanford University. Much of her research connects the a more concrete definition of the gauge/ mathematical structure of string theory to predictions for cosmological observables, with broader gravity duality for more realistic systems implications for primordial cosmology. She and her collaborators developed a framework for than the original anti de Sitter/CFT upgrading the gauge/gravity duality to the realistic case of a positive cosmological constant. correspondence. Entanglement entropy At the interface with condensed matter theory, along with other researchers, she derived calculations for large central charge — examples of non-​Fermi liquids in theoretically controlled systems. many degrees of freedom — agree on both Julian Sonner received his PhD from the University of Cambridge (UK) where he also held a sides of the duality and provide a specific Research Fellowship at Trinity College. After postdoctoral work in London and Cambridge (USA), interpretation of the Gibbons–Hawking– he became a professor of theoretical physics at the University of Geneva. He works on holographic de Sitter entropy. However, many questions duality, most recently on the connection between quantum chaos and black holes, as well as remain about the nature of this theory, quantum simulation of simple holographic dualities. which is currently an active research area.

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Quantum simulation of matter opens up experimental avenues 6University of Geneva, Geneva, Switzerland. to investigate, for example, the relationship ✉e-mail:​ [email protected]; sachdev@ Julian Sonner. The very properties between quantum chaos (information g.harvard.edu; [email protected]​u.ac.jp; [email protected]; [email protected]; that make quantum matter supporting ‘scrambling’) and non-Fermi-​ liquid​ [email protected] holographically emergent gravity so rich transport, as well as building a new path https://doi.org/10.1038/s42254-021-00319-0 and fascinating6 also imply that classical to ‘quantum gravity in the lab’, where the Published online xx xx xxxx computers are ultimately powerless in holographic state of matter serves as an 1. Nandkishore, R. & Hermele, M. Fractons. Annu. Rev. deciphering the underlying quantum analogue gravity system that experimenters Condens. Matter Phys. 10, 295–313 (2019). states. A way to overcome the exponential can manipulate and interrogate in their 2. Pretko, M., Chen, X. & You, Y. Fracton phases of matter. J. Mod. Phys. A 35, 2030003 (2020). 7,8 complexity causing this impasse is to turn laboratory (see refs for two early 3. Smirnov, F. A. & Zamolodchikov, A. B. On space of instead to another quantum system that examples). integrable quantum field theories. Nucl. Phys. B 915, 363–383 (2017). emulates, or ‘simulates’, a many-​body In this way, the holographic language 4. Gorbenko, V., Silverstein, E. & Torroba, G. dS/dS Hamiltonian of interest. This idea — known suggests new interpretations of many-body​ and TT¯. J. High Energy Phys. 03, 085–2019. 5. McGough, L., Mezei, M. & Verlinde, H. Moving the as quantum simulation — has evolved into experiments where, for example, quantum CFT into the bulk with TT¯. J. High Energy Phys. 04, the field of synthetic matter, where highly teleportation is understood in terms of 010 (2018). 6. Zaanen, J., Liu, Y., Sun, Y. W. & Schalm, K. Holographic controllable quantum systems, such as information travelling across wormholes in Duality in Condensed Matter Physics (Cambridge Univ. superconducting qubits or ultra-​cold atomic the gravitational picture9,10. Such insights will Press, 2015). 7. Danshita, I., Hanada, M. & Tezuka, M. Creating and gases, are used to construct the desired guide future fruitful interactions between probing the Sachdev–Ye–Kitaev model with ultracold Hamiltonian from simple ingredients, theory and experiment. gases: towards experimental studies of quantum gravity. Prog. Theor. Exp. Phys. 2017, 083I01 (2017). such as quantum gates and optical lattice 8. García-Álvarez,​ L. et al. Digital quantum simulation of 1 ✉ 2 ✉ potentials. Sean Hartnoll , Subir Sachdev , minimal AdS/CFT. Phys. Rev. Lett. 119, 040501 (2017). Tadashi Takayanagi3 ✉ , Xie Chen4 ✉ , Eva Silverstein5 ✉ 9. Brown, A. R. et al. Quantum gravity in the lab: The convergence of high-energy​ and and Julian Sonner 6 ✉ teleportation by size and traversable wormholes. Preprint at https://arxiv.org/abs/1911.06314 (2019). condensed-matter​ physics has resulted in 1 Stanford Institute for Theoretical Physics, Stanford 10. Nezami, S. et al. Quantum gravity in the lab: new models of many-body​ systems, simple University, Stanford, CA, USA. teleportation by size and traversable wormholes, Part II. Preprint at https://arxiv.org/abs/2102.01064 (2021). enough to lend themselves to a synthetic 2Department of Physics, Harvard University, matter approach, yet intricate enough Cambridge, MA, USA. Competing interests to give rise to holographic descriptions, 3Yukawa Institute for Theoretical Physics, Kyoto The authors declare no competing interests. University, Kyoto, Japan. as exemplified by the Sachdev–Ye–Kitaev Publisher’s note model and its holographic dual, a two- 4California Institute of Technology, Pasadena, CA, USA. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 5 dimensional black hole. The quantum Stanford Institute for Theoretical Physics, Stanford simulation of these holographic states University, Stanford, CA, USA. © Springer Nature Limited 2021

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