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Home , Douglas Stanford

Mimicking event horizons in atomic and solid-state systems

M. Franz1, 2 and M. Rozali1 1Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada V6T 1Z1 2Quantum Matter Institute, University of British Columbia, Vancouver BC, Canada V6T 1Z4 (Dated: August 3, 2018) Holographic quantum matter exhibits an intriguing connection between quantum black holes and more conventional (albeit strongly interacting) quantum many-body systems. This connection is manifested in the study of their thermodynamics, statistical mechanics and many-body . After explaining some of those connections and their significance, we focus on the most promising example to date of holographic quantum matter, the family of Sachdev-Ye-Kitaev (SYK) models. Those are simple quantum mechanical models that are thought to realize, holographically, quantum black holes. We review and assess various proposals for experimental realizations of the SYK models. Such experimental realization offers the exciting prospect of accessing black hole physics, and thus addressing many mysterious questions in , in tabletop experiments.

I. INTRODUCTION tory a specific class of systems that give rise to holo- graphic quantum matter. They are described by the so called Sachdev-Ye-Kitaev (SYK) models [6–8] which are Among the greatest challenges facing modern physical “dual”, in the sense described above, to a 1+1 dimen- sciences is the relation between and sional gravitating ”bulk” containing a black hole. We gravitational physics, described classically by Einstein’s first review some of the background material on black general theory of relativity. Well known paradoxes arise hole thermodynamics, statistical mechanics and its rela- when the two theories are combined, for example when tion to many-body quantum chaos. We then introduce describing event horizons of black holes [1]. Indeed, those the SYK model and discuss some of its important physi- paradoxes arise even though both general relativity and cal properties that furnish connections to holography and quantum mechanics should be well within their regime of black holes. Finally we review proposals for the experi- validity [2]. mental realization of this model using atomic and solid Perhaps the most promising avenue to resolving these state systems and assess critically their prospects and fea- paradoxes is rooted in the holographic principle [3, 4], sibility. We end by describing the outlook and potential which posits that quantum gravity degrees of freedom in significance of this realization of holographic dualities. a (d + 1)-dimensional spacetime bulk can be represented by a non-gravitational many-body system defined on its d-dimensional boundary. By now there are many well- understood examples of such “holographic dualities” pro- viding concrete realizations of that principle. Thus, using such holographic dualities, mysterious questions pertain- ing to gravitational physics in the quantum regime can be addressed in a context which is much better understood, II. BLACK HOLE MATERIALS that of quantum many-body physics. Part of the difficulty in quantum gravity research is the dearth of relevant experiments. For example, experimen- The relation of black hole physics, the hallmark of tal tests of the quantum gravity ideas using astrophysical quantum gravity research, to materials science is ex- black holes are impractical, as the relevant energy scale is tremely surprising. The story begins in early studies of inaccessible. Instead one can turn to holographic quan- quantum gravity in the 1970s. Famously there is some tum matter [5] – this is a class of strongly interacting sys- tension between fundamental concepts of General Rela- tems that occur in certain solid-state or atomic systems, tivity and Quantum Mechanics, and that tension mani- arXiv:1808.00541v1 [cond-mat.str-el] 1 Aug 2018 in which interactions between individual constituent par- fest itself most clearly when black holes of General Rel- ticles are so strong that the conventional quasi-particle ativity are treated as quantum mechanical objects. In description fails completely. Rather, these systems ex- classical gravity, black holes are characterized by hav- hibit properties of holography, in that they behave ef- ing a surface of no return, the so-called event horizon. fectively as black hole horizons in a higher dimensional Any information crossing that surface and falling into the quantum gravitational theory. This opens up the exciting black hole is lost to observers who stay safely away from possibility of testing quantum gravity ideas experimen- the black hole. However, that defining feature of black tally, as holographic quantum matter is more amenable hole is modified quantum mechanically. Indeed, the cel- to experimental study, at least in principle. ebrated work of Hawking [9] established that quantum This review focuses on recent developments aimed at mechanically black holes are no longer black, they glow theoretically understanding and realizing in the labora- with what came to be called the Hawking radiation. 2

A. Black Hole Thermodynamics B quantum system The discovery of Hawking radiation finalized a concep- tual revolution of our understanding of black hole quan- tum mechanics. Hawking radiation is thermal, like that of a black body, so black holes have a temperature. Ear- AdS2 bulk lier Jacob Bekenstein [10] has speculated that the sur- face area of a black hole behaves analogously to an en- tropy (for example it always increases); apparently this was not merely an analogy. Surprisingly black holes, as viewed by an outside observer, behave exactly as thermo- event horizon dynamic systems [11]. They have static properties: en- ergy, charge, entropy, temperature. They have dynamical emergent dimension properties such as viscosity and conductivity. An outside observer who does experiments on black holes views them as chunks of materials with collective properties which obey all the usual relations of thermodynamics. FIG. 1. Schematic representation of the holographic Indeed, there is a wide variety of possibly black holes duality A (0+1) dimensional many-body system, represented in various gravitational theories, in various spacetime di- here as a graphene flake in applied magnetic field, is holo- mensions and with diverse matter content, and corre- graphically equivalent to a black hole in (1+1) anti-de Sitter spondingly a wide variety of black hole “materials”. To space. explain that diversity, one had to wait for a microscopic theory of quantum black holes. C. Box 1: Gauge/Gravity Duality

B. Black Hole Statistical Mechanics The microscopic description of black holes is at its most developed stage in the context of the AdS/CFT In the mid-1990s , a leading candidate correspondence, also known as the gauge-gravity or holo- for a theory of quantum gravity, has reached the level graphic duality. Here AdS refers to anti-de Sitter space of development needed to provide a microscopic statisti- (see below) and CFT denotes conformal field theory. In cal mechanics description of a large class of black holes this set of examples, the black hole is immersed in a grav- [12]. Within string theory, gravitational interactions itational potential (arising due to a negative cosmological emerge naturally within an inherently quantum mechan- constant), which acts to regulate some problematic long ical framework. It became possible then to discuss, in distance behaviour unrelated to the questions we are in- great quantitative detail, static and dynamical properties terested in. When thus regulated, it turns out the micro- of quantum black holes. The flurry of activity in the fol- scopic description is in terms of well-known many-body lowing decade established a description of quantum black concepts, namely gauge theories of the form that con- holes which is, perhaps surprisingly, fairly conventional. stitutes the standard model of particle physics, as well Within this framework black holes are described as quan- as many other interesting systems appearing in the con- tum many-body systems, composed of microscopic con- densed matter context. stituents interacting strongly, to form a quantum liquid Twenty years and many thousands of detailed calcu- whose thermodynamics forms the gravitational descrip- lations later, we now have a very precise understand- tion of the system. In this sense gravity arises from ing of how gauge theories reproduce properties of quan- coarse-graining, exactly like a thermodynamic descrip- tum black holes immersed in AdS spaces. Note that tion of conventional materials. astrophysical black holes are well-described by classical The microscopic description makes it clear that black physics, whereas (small, cold) quantum black holes per- holes obey the rules of quantum mechanics, in particular haps play a role in the early evolution of the universe, they avoid Hawking’s information loss paradox [1]. Nev- but that avenue of investigation has many uncertainties. ertheless, it is still unclear how the paradox is avoided, Through the gauge/gravity duality we now face the ex- in other words how resolution of the paradox can be citing prospect of accessing quantum black holes experi- phrased in a purely gravitational language. Recently this mentally, in a new and extremely surprising context. set of questions was sharpened and cast in the language This precise correspondence and the possibility for ex- of quantum information processing [2]. Understanding perimental realization raise an urgent question, namely: black holes as quantum computers is an exciting current when does a quantum many-body systems describe black direction in quantum gravity, which is expected to pro- holes in a theory of gravity? Which aspects of the physics vide valuable clues on how a gravitational description require a gravitational description? Or, put more pro- emerges for black holes, while avoiding information loss saically: given some material, what would one measure and related puzzles. to discover if it is usefully described as a black hole in a 3

(usually higher dimensional) gravitational theory? Indeed, an elegant calculation in the gravitational con- text reveals that the Lyapunov exponent depends only on the properties of black holes near the event horizon, and D. Many-Body Quantum Chaos is therefore universal in the sense described above, that it takes the same values for all black holes in the con- text of the AdS/CFT duality. Furthermore, that value is The defining property of a black hole is its event hori- 2π zon. To identify quantum black holes in the lab, it is precisely the maximally allowed value, λL = β for quan- useful to identify universal quantities, universal in that tum black holes, realizing the intuition that holographic they only depend on the structure of spacetime near the matter is as strongly coupled as possible. This provides horizon. One such quantity was identified by Shenker an answer to the question posed in Box 1: looking at the and Stanford [13] and later employed by Kitaev, in an manner in which information is scrambled in a quantum attempt to define a quantum mechanical analog of the system can provide a connection to its gravitational de- butterfly effect and the associated Lyapunov exponent, scription as a black hole. In particular, saturating the which characterize that effect in classical chaos. chaos bound is a strong indication of a connection to The classical butterfly effect quantifies the sensitivity quantum black holes and gravitational description. to initial conditions in chaotic systems. Nearby trajecto- As it turns out, one can explicitly construct a simple ries diverge exponentially quickly from each other, with model which has that property, that of maximal scram- a rate that is called the Lyapunov exponent. In quan- bling. This is the so-called Sachdev-Ye-Kitaev (SYK) tum systems trajectories do not exist, but one can still model, which we review presently. discuss the influence of an initial perturbation on sub- sequent measurements. The suggestion was to quantify that dependence by looking at the size of the commutator III. THE SYK MODEL AND ITS LARGE-N [V (0),W (t)], where the operator V (0) provides the initial SOLUTION perturbation and W (t) measures its influence at subse- quent time t. Instead of calculating the expectation value The SYK model describes a system of N interacting of [V (0),W (t)], which defines the linear response of the Majorana . These are particles identical to their system, it turns out it is useful to focus on the second mo- antiparticles originally predicted in the seminal work of ment: C(t) = tr(e−βH [V (0),W (t)]2), which is sometimes Ettore Majorana [17–23] and recently observed in a va- referred to as the out-of-time-order correlator or OTOC. riety of solid-state platforms [24–32], (also see Box 2). It If we choose V,W to be initially commuting Hermitian is defined by the Hamiltonian operators, the quantity C(t) increases exponentially, as the information contained in the perturbation spreads = J χ χ χ χ . (3.1) HSYK ijkl i j k l rapidly throughout the system, a phenomenon referred i

hints at a connection of the model to black holes and string theory. As a practical matter the invariance al- lows one to extract the low-frequency behavior of the propagator, χi χj 1/4 sgn(ωn) Gc(ωn) = iπ (3.6) J ω Jijkl χk | n| and the corresponding spectral functionp

χl 1 1 Ac(ω) = , (3.7) √2π3/4 J ω | | where subscript c denotes the conformalp regime ω J. At high frequencies the behavior must cross over| to|  1/ω FIG. 2. The SYK model. Blue dots represent Majorana in both cases. The absence of a pole in Gc(ωn) indi- fermions χj while the green region illustrates one possible 4- cates the expected non-Fermi liquid behavior of the SYK fermion interaction term in the Hamiltonian (3.1). model. In addition we will see that the characteristic inverse square root singularity of the spectral function could be measurable by various spectroscopies in some of by Eq. (3.1) remains in the strong coupling regime all the the proposed experimental realizations of the model. way to the lowest energies. The conformal structure implied by Eqs. (3.15), has A closely related model formulated with complex also been instrumental in extracting the quantum-chaotic fermions [33–35], also known as the complex Sachdev- properties of the SYK model. It allows for the calculation Ye-Kitaev (cSYK) model or Sachdev-Ye (SY) model [6], of the out-of-time-order correlator C(t) which shows the is defined by the second-quantized Hamiltonian characteristic exponential growth [7, 8] with the maximal 2π permissible Lyapunov exponent λL = , thus confirm- = J c†c†c c µ c†c , (3.4) β HcSYK ij;kl i j k l − j j ing the intuition that the SYK model is indeed holograph- j ijX;kl X ically connected to a black hole. where c† creates a spinless complex fermion, J are Finally we mention various important extensions of the j ij;kl SYK model developed in the recent literature. These zero-mean complex random variables satisfying Jij;kl = J ∗ and J = J = J and µ denotes the include models showing unusual spectral properties [36– kl;ij ij;kl − ji;kl − ij;lk 38], supersymmetry [39], quantum phase transitions of chemical potential. At half filling (µ = 0) the two models an unusual type [40–42], quantum chaos propagation [43– exhibit very similar behavior but the cSYK model shows 45], and patterns of entanglement [46, 47]. more complexity when µ = 0. In this review we will focus, for the most part, on6 the SYK model Eq. (3.1) as it is more straightforward to analyze. We comment on the cSYK model as appropriate – we will see that it A. Box 2: Majorana zero modes and the Kitaev chain might be easier to realize it in a laboratory because it does not require the elusive Majorana particles as basic building blocks. Majorana fermions have been originally introduced in Perhaps the most remarkable and useful property of the context of particle physics as special solutions of the SYK model is that, despite being strongly interact- the celebrated Dirac equation [48] describing relativistic 1 ing, it is exactly solvable in the limit of large N. Specif- quantum mechanics of spin- 2 particles. Unlike ordinary ically, it is possible to write down a simple pair of equa- fermions (e.g. electrons or protons) Majorana particles tions for the averaged fermion propagator lack the distinction between particle and antiparticle [17]. In the elementary particle physics the leading candidate 0 1 0 for a Majorana fermion is the neutrino but the experi- G(τ, τ ) = τ χj(τ)χj(τ ) (3.5) N hT i mental evidence remains inconclusive at present [23]. j X In Majorana fermions can and the corresponding self energy Σ(τ, τ 0), that become appear as emergent particles in various many-body sys- asymptotically exact in the limit N and have a tems. The most prominent of these are topological su- simple solution in the low-frequency limit.→ ∞ This solution perconductors in which Majorana particles often appear is reviewed in Box 3 and leads to some remarkable con- as topologically protected zero modes [19–23]. A canon- clusions. ical example of such a topological superconductor is the The equations governing the large-Nsolution (3.15) Kitaev chain [49] which we now briefly review to illus- display an intriguing time-reparametrization (also re- trate the emergence of Majorana zero modes in a simple ferred to as conformal) invariance at low energies which setting. 5

a χ1 A B C χ’N Σ = + + + …

b

FIG. 4. The large-N solution. Leading Feynman diagrams in the 1/N expansion of the self energy Σ(τ). Solid lines repre- FIG. 3. The Kitaev chain. Blue and red dots represent sent the fermion propagator G(τ) while dashed lines indicate 0 Majorana modes χj and χj associated with site j of the chain. disorder averaging over the pairs of vertices connected by the 0 a) In the topological phase a bond forms between χj and χj+1 line. 0 for each j, leaving χ1 and χN unpaired. b) In the trivial phase 0 χj and χj pair up on the each site leaving behind no unpaired end modes. 0 Dirac fermion operatorc ˜ = (χ1 + iχN )/√2 then it is clear that it costs zero energy to add such ac ˜-particle Consider spinless fermions in a 1D lattice. The sim- to the system. Remarkably, the zero mode is fundamen- plest model of a superconductor in such a setting is de- tally delocalized between the two ends of the chain as scribed by the Hamiltonian illustrated in figure 3. Kitaev [49] noted the possibility of storing quantum information in this delocalized state which would make it resilient to many sources of deco- = t(c†c + h.c.) µc†c + (∆c†c† + h.c.) , H − j j+1 − j j j j+1 herence. Such a possibility also underlies much of the j X  current interest in Majorana zero modes. (3.8) † With a little extra work one can demonstrate that he where cj creates a fermion on site j and satisfies the usual special limit discussed above represents merely a point fermionic anticommutation relation in the stable topological phase of the Hamiltonian (3.8). † The system exhibits gapped bulk with two Majorana c , cj = δij. (3.9) { i } zero modes localized near the ends of the chain when- t and ∆ represents the nearest-neighbour hopping and ever ∆ = 0 and µ < 2 t . When µ > 2 t the chain pairing amplitudes, respectively and µ denotes the chem- is topologically6 trivial| | and| | zero modes| | are absent.| | Var- ical potential. We assume that ∆ is real and consider a ious instances of Majorana zero modes observed experi- chain with N sites and open boundary conditions. mentally in 1D systems [24–30] can all be understood as Following Kitaev [49] we now express each Dirac realizations of the Kitaev chain. fermion cj in terms of two Majorana fermions χj and 0 χj, B. Box 3: The large-N solution 1 1 c = (χ + iχ0 ), c† = (χ iχ0 ). (3.10) j √ j j j √ j j 2 2 − The fermion propagator of the SYK model (3.5) is re- It is easy to check that if the Majorana operators obey lated to the self energy by the standard Dyson equation, the anticommutation algebra given by Eq. (3.5) then Eq. G(ω ) = [ iω Σ(ω )]−1, (3.13) (3.9) is satisfied and the transformation is thus canonical. n − n − n In the Majorana representation the Hamiltonian (3.8) becomes where we have assumed time-translation invariance and β iωnτ defined G(ωn) = 0 dτe G(τ), with Matsubara fre- = i (∆ + t)χ0 χ + (∆ t)χ χ0 µχ χ0 . quencies ωn = πT (2n + 1) and n integer. The solution H j j+1 − j j+1 − j j R j exploits the fact that for large N diagrammatic expan- X  0 (3.11) sion of the self energy Σ(τ, τ ) is dominated by a class To appreciate the physical content of this Hamiltonian it of the so called melonic diagrams which can be summed is useful to consider a special limit in which µ = 0 and up exactly. All other diagrams, including the vertex cor- ∆ = t. We then have simply rections that typically prevent such exact resummations, are suppressed by factors of 1/N and can be neglected. N−1 This can be seen from Fig. 4 which illustrates vari- = 2it χ0 χ , (3.12) H j j+1 ous diagrams that contribute to Σ. The leading mel- j=1 X onic diagram A has two vertices. Upon averaging over where we restored the summation bounds. disorder Eq. (3.3) implies that these contribute a factor 2 2 3 Equation (3.12) is remarkable for what it is lacking: Jijkl J /N . Summing over the three internal legs con- 0 ∼ 3 notice that two operators, χ1 and χN , do not partici- tributes a factor N as each leg can represent one of the pate in the Hamiltonian. They therefore constitute ex- N fermion flavours.∼ The diagram is therefore O(1). It is act “zero modes” of the system. If we construct a new easy to check that all diagrams with melonic insertions, 6 such as the diagram B are O(1). On the other hand di- correlated “strange metals” may realize various higher- agrams representing vertex corrections, such as diagram dimensional generalizations of the SYK model [54–60]. C, have fewer internal legs and therefore become O(1/N) This connection is made on phenomenological grounds and thus negligible at large N. and although very appealing, it remains speculative be- It is straightforward to sum up the O(1) diagrams com- cause of the difficulties encountered in the attempts to prising the self energy – the full series is generated by relate microscopic models governing crystalline solids to simply replacing the bare propagator G (ω ) = 1/iω highly random SYK model (see however Ref. [61] for an 0 n − n by the fully dressed propagator G(ωn) in the leading mel- SYK-like model without randomness). onic diagram A in Fig. 4. One thus obtains Another class of interesting approaches are quantum simulations. An algorithm has been proposed for digital Σ(τ) = J 2G3(τ), (3.14) quantum simulation of both SYK and cSYK models using a quantum computer [62]. First experimental steps along which together with Dyson equation (3.13) determine the this line have already been reported [63]. form of the propagator at large N. In this review we focus on proposals that aim at en- The mean field equations for the bi-local fields G, Σ be- gineering the cSYK or SYK model bottom up, that is, come exact in the large N limit, where we can ignore fluc- by designing systems with requisite degrees of freedom tuations around their saddle point values, which are the and interactions using building blocks that are reason- solution of the Schwinger-Dyson equations. The prop- ably well understood and thus afford a high degree of erty of being mean-field exact is shared by many infinite- control. ranged models, for example the celebrated Sherrington- Kirkpatrick model of spin glasses [50]. Remarkably, the low energy physics of the SYK model is very different A. Three key challenges from most such models, as it avoids replica symmetry breaking and instead stays quantum mechanical to the When thinking about the experimental realization of lowest energies. the SYK model several challenges become immediately At sufficiently low frequencies ωn J the iωn term in apparent. The original SYK model employs Majorana  Eq. (3.13) can be neglected and the resulting system of fermions so the first challenge is to design a physical two equations can be seen to acquire time parametriza- system capable of hosting a large number N of these tion invariance elusive particles. In the language of condensed matter physics one needs Majorana zero modes [19–23] which G(τ, τ 0) [f 0(τ)f 0(τ 0)]1/4G(f(τ), f(τ 0)), → (3.15) have been observed to exist in several platforms includ- Σ(τ, τ 0) [f 0(τ)f 0(τ 0)]3/4Σ(f(τ), f(τ 0)), ing semiconductor quantum wires [24–28], atomic chains → [30], and vortices in topological insulator - superconduc- for an arbitrary smooth function f(τ). This emergent tor interfaces [31, 32]. In these works signatures con- conformal symmetry is a key property of the SYK model sistent with individual localized Majorana modes have which hints at its deep connection to black hole physics been reported but manipulating and assembling them and holography. In fact, it is described by the same into required multi-mode structures remains a consider- low energy action (the so-called Schwarzian action) as able challenge. black holes in asymptotically AdS2 [51]. These low- Once we have an assembly of N Majorana particles the dimensional black holes arise as the near-horizon region second challenge arises. The leading perturbation in such of a large class of charged black holes in 3+1 dimensions a situation, typically, will come from the hybridization (see e.g [52]), thus the Schwarzian action also describes term the low energy dynamics of those more realistic black = i K χ χ (4.1) holes. H2 ij i j i

atoms with ns molecular states, described by the following Hamiltonian with real Gaussian random couplings gs,ij:

ns ˆ † † † † Hm = Â nsmˆ s mˆ s + Âgs,ij mˆ s cˆicˆj mˆ scˆi cˆj . (3.1) s=1 ( i, j ) ⇣ ⌘ By integratingm ˆ s out, we obtain the following effective Hamiltonian, g g ˆ s,ij s,kl † † Heff = Â cˆi cˆj cˆkcˆl. (3.2) s,i, j,k,l ns (A similar model has been considered in [26].) Previously, a similar way of designing a kind of two-body hopping term by means of intermediate two-particle states has been proposed in [27]. g g Let us take n = n = = n µ pn . When n is large enough, s,ij s,kl should become 1 2 ns s s Âs ns ··· 2 Gaussian except for the diagonal elements (i, j)=(k,l) or (i, j)=(l,k) (note that gs,ij is always positive), because it is simply an ns-step random walk for each set of indices (i, j,k,l). In order to improve the behavior of the diagonal elements, we take ns to be even, and set ns =+pnsss for even 2 2 2 3/2 s and ns = pnsss for odd s. We take the variance of gs,ij to be s = sg , with sg /ss = J/(2N) . 3/2 In the following we set ss = sg = J/(2N) . It is not hard to see that the properties needed g g in the real-SYK model are satisfied at n = •. We identify s,ij s,kl defined in this way with s Âs ns 3/2 Jij,kl/(2N) . The model (3.2), which is equivalent to the SYK model at ns = •, can in principle be created in a system of optical lattices loaded with ultracold gases [20]. In the proposed scheme, we utilize the photoassociation and photodissociation processes that coherently convert two atoms into a bosonic molecule in a certain electronic (or hyperfine), vibrational, and rotational state7 [28], and vice versa (Fig. 3). For details, see [20]. ground state of the system becomes a disordered Fermi liquid. Fortunately, vestiges of the SYK physics still sur- vive in the excited states above the crossover energy scale 2 Ec K /J where we have defined the average hybridiza- tion' strength as

2 2 K = NKij. (4.2) The excited states can be probed by performing exper- iments at non-zero temperature T or frequency ω, such that K2/J < T, ω < J. In order to have a reasonably wide range of temperatures or frequencies in which the SYK physics remains observable one therefore needs to ensure that K J. This is the second key challenge fac- ing any experimental realization of the SYK model as a generic system of N Majorana modes wouldFigure instead 3: By introducing have FIG. laser 5. beamscSYK with model appropriate from frequencies, cold atoms. the photoassociationLaser beams and photodissoci- K J. ation processes, which canare be used regarded to couple as the atoms interaction and molecules between sucha molecule that transitions and two atoms in (3.1), can be The final challenge has to do with ensuringintroduced. sufficient occur as described by Hamiltonian (4.3. (Figure copied from Ref. [68]). randomness in coupling constants Jijkl. In order to de- fine the SYK Hamiltonian these constants must be suf- ficiently close to random independent variables with a Gaussian distribution. As we shall see4. such Discussions randomness andN futurestrongly directions localized atomic states filled with Q atoms. can arise from the microscopic disorder that is inherently While the eigenstate wavefunctions spatially overlap in present in solids or else it must be deliberatelyIn engineered this presentation,the we same have well, suggested their orthogonality a path to create forbids a quantum single black particle hole in the laboratory, tunnelings, so that perturbations similar to Eq. (4.1), bi- in the system through fabrication. by experimentally realizing the SYK model. Of course, that we can make a black hole in principle Finally we remark that the first of the challenges dis- linear in fermion operators, can be ignored. The atoms in cussed above is avoided in proposals that aim at realiza- separate wells are also assumed to be completely decou- tion of the cSYK model Eq. (3.4) which employs ordinary pled from each other so we4 can focus on a single well. In complex fermions as basic building blocks and not Ma- addition, two atoms can come together and form bosonic jorana fermions. Indeed these proposals appear to be molecular states through the process of photo-association simpler to implement and are therefore likely to succeed (PA) stimulated by PA lasers of appropriate frequencies, first. As we mentioned the cSYK model at half filling ex- as illustrated in Fig. 5. The Hamiltonian governing such hibits the same holographic duality to a black hole as the a system is

SYK model and it shows additional interesting behavior nms nms † Uss0 † † away from half filling [64]. In solid state systems com- = ν m m + m m 0 m 0 m H s s s 2 s s s s plex fermions will be electrons and will thus carry electric s=1 s0=1 charge. For this reason such systems will be amenable X X † † † to standard transport and spectroscopic measurements. + gs,ij mscicj msc c , (4.3) − i j i,j This confers another advantage over Majorana fermions, X   which are electrically neutral. † † where ms and ci represent the creation operators for molecules and atoms, respectively, νs are molecular state B. cSYK model with ultracold gases energies, Uss0 are molecular interactions and gs,ij denote the amplitudes of PA processes. The latter are assumed Experiments on ultracold atomic gases in optical lat- fully controlable as they depend on frequencies and in- tices have already succeeded in realizing a number of tensities of the PA lasers. theoretical models originally introduced in the context The molecular states can be integrated out from the of condensed matter physics. Notable examples include model defined by Eq. (4.3) and, using perturbation the- the Bose-Hubbard model [65] and the Haldane model ory to second order in amplitudes gs,ij, one obtains the [66]. Owing to their high degree of controllability, lack effective Hamiltonian for the fermions of the form of disorder, and the unique set of physical observables nms experiments on ultracold gases can often answer many gs,ijgs,kl † † eff = ci cjckcl. (4.4) H νs important questions about these models that remain in- i,j,k,l s=1 ! accessible to their condensed matter analogs. X X A proposal to realize the cSYK model with ultra- This clearly has the structure of the cSYK model Eq. cold gases in optical lattices has been given by Dan- (3.4) provided that we identify the sum in the brackets shita, Hanada and Tezuka [67, 68]. The authors envision with Jij;kl. The coupling constants in this realization of fermionic atoms, such as 6Li, confined in deep potential the cSYK model are real (as opposed to complex-valued) wells of an optical lattice. Each well is assumed to have but the authors [67] argue convincingly that for large N 8 this difference does not matter. Also, unlike the canon- B δ1 ical cSYK model, Jij;kl defined by Eq. (4.4) have inter- nal structure and are not in general Gaussian distributed δ3 δ2 random variables. If however gs,ij and νs are taken as A random variables then for a large number of molecular B states nms 1 the distribution of Jij;kl should approach Gaussian by virtue of the central limit theorem (Js are then given by a sum of a large number of identically dis- FIG. 6. Schematic representation of the graphene de- tributed random variables). In this limit, at least, the vice. Irregular shaped graphene flake in applied magnetic system defined by Hamiltonian (4.3) should behave as a field B hosts degenerate Landau levels which can serve as a canonical cSYK model. platform for the cSYK model when interactions are present. Unfortunately, n 1 is also the limit which might Inset: lattice structure of graphene with A and B sublattices ms  be difficult to realize in a laboratory. The chief bottleneck marked and nearest neighbour vectors denoted by δj . (Figure is the number of lasers required to drive the PA tran- copied from Ref. [72]). sitions between the atomic and molecular states. This equals nmsN(N 1)/2 which becomes a very large num- − The root cause for the above behavior is Landau quan- ber in the interesting limit when both nms and N are large. In addition, frequencies of these PA lasers must tization: at the non-interacting level magnetic field reor- be rather accurately tuned to achieve reasonable rates ganizes the electron bands into a discrete set of massively of transitions. It is known from numeical studies that degenerate dispersionless Landau levels (LLs). The effec- tive single-particle Hamiltonian for an electron in a given N & 10 is required to start observing spectroscopic sig- natures characteristic of the cSYK model. It is less clear Landau level is then simply 2 = 0. The proposal of Ref. [72] uses the Landau level asH the starting point to con- how large nms must be for a given N to yield Gaus- sian distributed Js to a good approximation. The cSYK struct the cSYK model. The key remaining challenge, model variant defined by Eq. (4.4) could however exhibit then, is to produce coupling constants Jij;kl that would be all-to-all and sufficiently random. To achieve this interesting behavior even for small nms and a theoretical study of this limit would be of interest. one must include disorder effects. Under normal circum- The authors [67], among other things, devised a pro- stances, however, microscopic disorder tends to broaden LLs, effectively reintroducing nonzero . For this rea- tocol to measure the out-of-time-order correlator C(t) H2 in this sytem, following the general procedure outlined son ordinary 2D electron gas with disorder does not work in Ref. [69]. The protocol involves coupling the atomic as a platform to build the cSYK model. states to a control qubit, annihilating individual atoms The idea is to use instead a flake of graphene with a at different times according to the qubit state, and fi- highly irregular boundary serving as a source of disor- nally evolving the whole system forward and backward der, see Fig. 6. Electrons in graphene are well-known to in time according to the Hamiltonian (4.4). Backward posses “relativistic” energy dispersion described at low time evolution requires flipping the sign of the Hamilto- energies by a massless Dirac Hamiltonian [76]. Landau levels of such a Hamiltonian occur at energies nian which is achieved by flipping the sign of all νs. This in turn can be implemented by changing the detuning of En ~v 2n(eB/~c) (4.5) the PA lasers. Measuring OTOC is clearly a challenging ' ± proposition but first steps have recently been taken using with the characteristic Fermip velocity v and n = 0, 1, . quantum simulators [70, 71]. ··· The n = 0 Landau level, often referred to as LL0, is special in terms of how it responds to disorder. Accord- ing to the famous Aharonov-Casher construction [77] the C. cSYK model in a graphene flake exact degeneracy of quantum states in LL0 is protected against any disorder that respects the chiral symmetry This proposed solid-state realization [72] takes its in- of graphene. This is most simply explained by inspect- spiration from the rich physics of quantum Hall liquids ing the tight-binding model describing electrons in the [73, 74]. It is well known that application of a strong graphene honeycomb lattice, magnetic field to the 2D electron gas (electrons confined † to a thin layer) results in quenching of their quantum H = t (a br δ + h.c.). (4.6) 0 − r + kinetic energy and strong enhancement of the interac- r,δ X tion effects. This is equivalent to the suppression of two- fermion terms relative to the four-fermion interaction Here a† (b† ) denotes the creation operator of the elec- H2 r r+δ terms contained in cSYK. In a large clean sample one tron on the sublattice A (B) of the honeycomb lattice. then obtains the familyH of fractional quantum Hall liquids r extends over the sites in sublattice A while δ denotes characterized by a range of spectacular physical proper- the 3 nearest neighbour vectors (inset Fig. 6), t = 2.7 eV ties including topological order as well as the charge and is the nearest-neighbour tunneling amplitude. The chiral statistics fractionalization [75]. symmetry χ is generated by setting (ar, br) ( ar, br) → − RAPID COMMUNICATIONS

AARON CHEW, ANDREW ESSIN, AND JASON ALICEA PHYSICAL REVIEW B 96, 121119(R) (2017) 9

lengthSYKξ exceeds Hamiltonian the dot size byL.Morequantitatively,wetake coupling N such wires to a disor- (a) γi γ˜i the meandered free 2D path quantumℓmfp L to dot maximize as illustrated randomness in and Fig. the 7. In such a ≪ dimensionlesssetup Majorana conductance zerog modeskF ℓmfp > can1suchthat be shownL<ξ to. spread out 2D disordered Turning on four-fermion interactions= couples the disordered B Majorana wire array from the tips of the individual wires into the quantum quantum dot Majorana modes with typical Jij kl ’s that are smaller than λ the energydot soϵ to that the their next excited wavefunctions state (which are as essentially we will delocal- see isized enhanced across bylevel its entire repulsion area. compared Once to delocalizedδϵtyp). This Majorana separationwavefunctions of scales allows overlap us to first in analyzereal space the disordered and even perfectly wavelocal functions interactions in the noninteracting producelimit all-to-all and then matrix explore elements Jijkl (b) (c) interactionsprecisely projected as required onto the zero-mode by the SYK subspace. Hamiltonian. We next The fact carry out this program using random-matrix theory, which is FIG.Dot 7. SYK model with quantum wires. An array that the quantum dot is disordered means that the wave- of quantum wires with Majorana end modes is coupled toexpected to apply in the above regime [48,49]. levels Random-matrix-theoryfunctions acquire analysis. randomWe spatial model structure the dot as which a in turns a disordered quantum dot. Majorana modes penetrate into translates to randomness in Js. typ λ 2D lattice composed of Ndot N sites hosting fermions the dot and interact with one another thus realizing the SYK ≫ J ca 1,...,Ndot [50]. In terms of physical dot parameters we have model. (Figure copiedMajorana from Ref. [80]). Majorana = Two features2 of this proposed setup deserve note. Ndot (L/ℓmfp) —thatis,thefermionsrepresentdegreesof zero modes zero modes freedom∼First, coarse-grained Majorana on zeroa length modes scale of avoid the order hybridization of the (that meanis, free formation path. The Hamiltonian of bilinear governing terms the in dot-Majorana the Hamiltonian that for all r which has the effect of flipping the sign of thesystemwould is H beH detrimental0 Hint,with toH0 SYK)and Hint bythe virtue free ofand approximate interacting pieces,= respectively.+ We employ a Majorana˜ basis Hamiltonian H0 H0. artificial time-reversal symmetry present in the sys- FIG. 1. (a) Device→ − that approximates the SYK model using and write ca (ηa iη˜a)/2, where ηa is even underT while Making the shape of the flake irregular introduces tem. This= may+ be illustrated most easilyT on the example topological wires interfaced with a 2D quantum dot. The dot mediates η˜a is odd (similarly to γi ,γ˜i ). In terms of randomness that respects χ. The wavefunctions Φ (r) of the Kitaev chain. Its Hamiltonian Eq. (3.8) respects disorder and four-fermion interactions among Majorana modes γ1,...,Nj T T ) [η1 ηN ; γ1 γN ] , )˜ [˜η˜1 η˜N ] , (4) of theinherited states from belonging the wires, while to MajoranaLL0 acquire bilinears random are suppressed spatial =an antiunitary··· dot ··· symmetry= ···whichdot sends cj cj and by an approximate time-reversal symmetry. (b) Energy levels pre T → structure but, importantly, Aharonov-Casher construc-H0 takesi the formi. In the Majorana representation this symmetry hybridization. The dot-Majorana hybridization energy λ is large → − tion [77] guarantees that their energies remain perfectly is expressedi as compared to Nδϵtyp,whereN is the number of Majorana modes T ˜ T 0 M ) degenerate. Under these circumstances one expects the H0 [) ) ] T ˜ . (5) and δϵtyp is the typical dot level spacing; this maximizes leakage = 4 M 0 ) Coulombinto the interaction dot. (c) EnergyV levels(r) to post produce hybridization. random The N andabsorbed all-to- ! − "! " Time-reversal ˜ fixes the zeros above0 but0 allows for a all couplingMajorana modes constants enhanceJ theij; energykl betweenϵ to the next the excited electrons dot state in via LL0. T : χj χj, χj χj, i i, (4.7) level repulsion; four-Majorana interactions occur on a scale J<ϵ. general real-valuedT (Ndot →N) Ndot-dimensional→ − matrix→ −M. This is because Jij;kl are given as matrix elements of V (r) + × (The matrix is not square since we discarded theγ ˜i modes between random-valued wavefunctions Φj(r) which, ac-that trivially decouple.) One can perform a singular-value ensure that commutes with the ground-state fermion parity and it is easy to check that Hamiltonian˜ ˜ (3.10)˜ is indeed cording to simulationsT [72], have support everywhere indecomposition of M by writing ) )′ and ) )′. P iγγ˜,asitmust. invariant.˜ If we now assume= O that from each= O of a = 1 ...N the flake.= Here , denote orthogonal matrices consisting of singular Consider now N topological wires “plugged into” a two- O O T ˜ a vectors,wires i.e., the only matrix the* left-mostM only Majorana has nonzero mode entriesχ1 is coupled Characteristicdimensional (2D) signatures disordered quantum of the cSYKdot [Fig. physics1(a)], such could ≡ O O T alongto the the diagonal. dot, Writing and that)′ the[η′ entireη′ ; γ system′ γ ′ ] continuesand to re- be observed in this system by a host of spectroscopic 1 Ndot 1 N that the Majoranas γ1,...,N that are even under hybridize ˜ = ··· ··· a b T similarlyspect for )˜′,theHamiltonianbecomes, then bilinear terms of the form iKabχ χ are andwith transport the dot while probes. their partnersScanningγ ˜1,...,N tunnelingdecouple completely.spectroscopy 1 1 prohibitedT as they would be odd under the symmetry. measuresThe full the architecture local density continues of to states approximately which is preserve directly re- Ndot T ˜ i provided (i) the dot carries negligible spin-orbit coupling Of course His0 not aϵ trueaη′ η˜′ microscopic, symmetry(6) of the lated to the spectral function and should thus exhibit the T = 2 a a and (ii) the B field orients in the plane of the dot so that physical system soa strictly1 speaking K is not zero. Ref. characteristic inverse square root divergence Eq. (3.7) as = ab orbital effects are absent. Here the setup falls into class BDI, # where[80]ϵa however*aa are the argued nonzero that dot it energies. is an approximate Most impor- symmetry a functionwhich in ofthe biasfree-fermion voltage. limit Two-terminaladmits an integer topologicalconductance ≡ tantly,whichγi′ 1,...,N impliesdrop out that and formKab theare modified likelyN toMajorana be small compared throughinvariant theν systemZ [44,45 and]thatcountsthenumberofMajorana local compressibility, measured = ∈ zero modesto the guaranteed interaction by energysymmetry. scale. via chargezero modes sensing at each techniques, end; interactions should collapse likewise the classification show signa- We are interested in statisticalT properties of the associated turesto ofZ8 the[46, non-FL47]. In essence behavior. our device It is leverages currently nanowires not known to MajoranaThe wave second functions feature in the haspresence to do of with strong the random- hybridization of construct a topological phase with a free-fermion invariant how to measure out-of-time-ordered quantities, such asness.the To make Majorana analytic modes progress with we assume the electronic (for now) levels that present in C(t)ν inN this:Allbilinearcouplings context and thisiM remainsjkγj γk are a forbidden challenge by for allall elements of M in Eq. (5)areindependent,Gaussian- and= thus cannot be generated by the dot under the conditionsT the dot itself. Such hybridization would also be harmful solid-state realizations of cSYK and SYK models. distributedto the random prospects variables of constructing with zero mean the and SYK the same model. Here the specified above. We exploit the resulting N Majorana zero variance, corresponding to the chiral orthogonal ensemble modes to simulate SYK-model physics mediated by disor- [51,52authors]. This form show permits that Cooper a remarkable pairing of dot fermions—aninstance of level repul- der and interactions native to the dot, similar in spirit to inessentialsion detailtakes for place our purposes—and and protects also the does Majorana not enforce modes. Ran- Refs. [21,38]. D. SYK model with semiconductor quantum wiresthe strong-hybridizationdom matrix theory criterion considerationsλ Nδϵtyp.Wewillseethat imply that a generic Figures 1(b) and 1(c) illustrate the relevant parameter the Majorana wave functions nevertheless≫ live almost entirely regime. The dot-Majorana hybridization energy λ satisfies model describing the situation depicted in Fig. 7 has the in thethe dot as spectrum appropriate composed for the latter of regime. the zero-mode manifold (N λ Nδϵtyp,whereδϵtyp denotes the typical dot level spacing. Semiconductor≫ quantum wires with strong spin-orbit The probability density for such a random matrix M This criterion enables the dot to absorb a substantial fraction of states) separatedπ 2 by  TNδtyp/π from all other energy coupling, such as those made of InSb, have emergedis [53] P (M) exp[ 2 Tr(≈M M)]. Because P (M)is all N Majorana zero modes as shown below. The dot’s disor- levels. Here δ8Ndotδϵtypis the typical energy level spacing in as the prime platform for investigating Majorana zero ∝ − typ dered environment then efficiently “scrambles” the zero-mode invariantthe under dot beforeM itT M has˜ ,thesingular-vectormatrices been coupled to the wires. modes. When such wires are “proximitized” (that is, → O O wave functions, although we assume that their localization , ˜ are uniformly distributed over the spaces O(Ndot N) made superconducting by placing them in contact with aO O The key advantage of this proposal is+ that Majorana superconducting material, such as Al) and placed in121119-2 mag- zero modes are now routinely observed in proximitized netic field, Majorana zero modes are predicted to appear semiconductor quantum wires. Assembling a large num- at their ends [78, 79]. There now exists ample experi- ber of such wires to form a device in Fig. 7 is clearly a mental evidence for this effect [24–29] which realizes the significant challenge. However, recent history of this field Kitaev chain paradigm discussed in Box 2. has proven that even tough chalenges can be met when Chew, Essin and Alicea [80] proposed to engineer the the goal is deemed sufficiently important. 10

cludes any bilinear terms from the effective Hamiltonian B describing the zero modes. If interactions are present be- tween the underlying electron degrees of freedom then the leading term in such an effective description is a 4- fermion SYK-like Hamiltonian (3.1). Using a combina- tion of analytical and numerical tools Ref. [82] showed that when the hole is engineered to have a highly ir- regular shape Majorana wavefunctions acquire random SC spatial structure and the screened Coulomb interaction 3D TI between electrons produces essentially random couplings Jijkl, as required to implement the SYK model. In comparison to the quantum wires experimental un- derstanding of the Majorana modes in the Fu-Kane su- FIG. 8. Schematic of the SYK model realization in the Fu-Kane superconductor. A 3D topological insulator perconductor remains significantly less developed. Re- surface has been proximitized by depositing on it a thin film sults reported in [31, 32] have yet to be reproduced by of a superconducting material. An irregular-shaped hole in another group or in another family of materials. This the film serves to trap magnetic flux. For each magnetic flux makes it more difficult to ascertain the prospects for re- quantum trapped there is one Majorana zero mode localized alization of the above proposal. On the other hand once in the hole. the Fu-Kane superconductor has been better understood and characterized, it should be relatively easy to fabri- cate and probe the device proposed in Ref. [82]. Also, E. SYK model at the surface of a 3D topological the fact that bilinear terms can be controlled by tuning a insulator single parameter (the chemical potential µ) confers some advantage over the quantum wire realization, where the Another canonical platform for Majorana zero modes is smallness of such terms relies on an approximate symme- the so-called Fu-Kane superconductor, which arises from try of the system that cannot be easily manipulated. proximitizing the topologically protected surface state of a 3D strong topological insulator (TI). Magnetic vor- tices in the Fu-Kane superconductor have been predicted V. CONCLUSIONS AND OUTLOOK [81] to harbor Majorana zero modes. Signatures consis- tent with this prediction have recently been reported in In this review we focussed on proposals for experimen- Bi2Te3/NbSe2 heterostructures [31, 32]. tal realizations of the cSYK or SYK model. All such pro- Realization of the SYK model in this platform has been posals currently on the table face significant challenges. proposed in Ref. [82]. As with the quantum wires the These include hurdles in materials synthesis, device fab- key challenge in this setting is to assemble N Majorana rication, reproducibility and control. Nevertheless the modes in the same spatial region without producing bilin- field evolves rapidly and this provides hope that some ear hybridization terms. This is achieved by engineering version of the theoretically proposed devices could be ex- a nanoscale hole in the superconducting film on the sur- perimentally realized in the near future. face of a TI. The hole serves to trap multiple vortices, Several measurements designed to make the connec- each binding a Majorana zero mode. The mechanism tion to quantum black holes explicit have been proposed, behind this trapping has to do with the well known phe- and they present their own challenges. Measurement of nomenon of vortex pinning: since the superconducting out-of-time-ordered quantities, important for the identi- order parameter ∆ is suppressed to zero at the core of a fication of many-body chaotic behavior characteristic of vortex it is energetically preferable to position the core in a black hole, constitutes a huge challenge in the atomic a region where ∆ was already suppressed by other effects physics setup and remains an unsolved problem for all (impurities for example). A hole in the superconductor proposed solid state realizations. Spectroscopic or trans- has ∆ = 0 and thus serves as a perfect pinning center. port identification of the emergent black hole physics is When sufficiently large it can trap many vortices, and more straightforward in principle but by no means free thus many Majorana zero modes. of challenge. Hybridization between the zero modes can be avoided Despite these technical hurdles, the key conceptual ad- in this setup by tuning the chemical potential µ of the vances achieved over the past two years outline a clear TI to the so called neutrality point. The topologically roadmap which, if successfully followed, should lead to protected surface state in a TI is described as a single experimental insights to some of the most fundamen- massless Dirac fermion in 2D, familiar from the physics tal mysteries facing modern physics. The realization of graphene [76]. At neutrality µ coincides with the Dirac that a simple, solvable quantum mechanical model ex- point. Importantly the Fu-Kane superconductor acquires hibits fundamental connections to quantum black holes an extra symmetry at this special point [83] that acts ex- has brought quantum gravity into the realm of being po- actly like ˜ defined in Eq. (4.7). This symmetry then ex- tentially experimentally testable. This exciting prospect T 11 advances the decades-long quest, started already by Ein- model is a recent and fast developing subject, see for stein, to reconcile two fundamental theories of nature, example [85–88] for some examples of such attempts to general relativity and quantum mechanics. That deep address quantum gravitational questions using the SYK ideas on the nature of quantum gravity could be ulti- family of models. mately tested in a humble piece of solid replete with dis- order and imperfections reinforces the modern belief in the principle of emergence, that indeed “more is differ- ACKNOWLEDGMENTS ent” [84]. Speculatively, but most importantly, once an experi- The authors are indebted to many colleagues who ment has produced a system which can be reliably identi- helped shaped their understanding of the subject. Of fied as a quantum black hole, one can turn to empirically these special thanks go to I. Affleck, E. Altman, J. Alicea, investigating many subtle questions pertaining to quan- L. Balents, M. Berkooz, A. Chen, F. Haehl, A. Kitaev, tum gravity. The effort to formulate such longstanding C. Li, E. Lantagne-Hurtubise, P. Narayan, D. Pikulin, S. questions in the context of simple quantum mechanical Sachdev, J. Simon and M. Tezuka.

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