An introduction to the SYK model

The simplest large N quantum field theories are vector models [3, 4]. An example is the $O(N)$ vector model, having N scalar fields, $\vec \phi = (\phi^1, \ldots, \phi^N)$, with a quartic $O(N)$ invariant interaction,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle I = \int {\rm d}^d x \left( \frac{1}{2} (\partial \vec {\phi})^2 + \frac{1}{2} \mu^2 \vec {\phi}^{\,2} + \frac{g}{4} ( \vec {\phi} \cdot \vec {\phi})^2 \right). \nonumber \end{align} \tag{ 2.1 }$$

In dimensions 2  <  d  <  4, if one appropriately tunes the bare mass $\mu$, there is an infrared fixed point: the Wilson–Fisher fixed point, describing magnets.

The power of large N is that instead of studying the theory perturbatively in the coupling g, one can reorganize the perturbative expansion, into powers of gN and 1/N. At a given order in 1/N, one is able to compute to all order in gN. For instance, at leading order in 1/N, the only Feynman diagrams contributing to the two-point function of the $O(N)$ vector model are bubbles, see figure 1, all of which are summed by the integral equation,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:1} G(\,p) = \frac{1}{p^2 + \mu^2 + \Sigma(\,p)}, \ \ \ \ \Sigma(\,p) = g N \int {\rm d}^d q\, G(q), \nonumber \end{align} \tag{ 2.2 }$$

where $G(\,p)$ is the momentum space two-point function, and $\Sigma(\,p)$ is the self-energy. The self-energy is independent of the momentum: the only effect of the bubble diagrams is to shift the mass. Defining $m^2 = \mu^2 + \Sigma$, the above Schwinger–Dyson equation becomes,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:m} m^2 = \mu^2 + g N \int {\rm d}^d q \frac{1}{q^2 + m^2}. \nonumber \end{align} \tag{ 2.3 }$$

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An equivalent way of analyzing the $O(N)$ vector model is by introducing an auxiliary Hubbard–Stratonovich field $\sigma$, so as to rewrite the action as,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:HS} I = \int {\rm d}^d x \left( \frac{1}{2} (\partial \vec {\phi})^2 + \frac{1}{2}\mu^2 \vec {\phi}^{\, 2} + \frac{1}{2} \sigma \vec {\phi}^{\, 2} - \frac{\sigma^2}{4 g}\right). \nonumber \end{align} \tag{ 2.4 }$$

Integrating out $\sigma$ gives back the original action. Alternatively, integrating out the $\phi^a$ fields, gives an action involving only $\sigma$,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{I_{\sigma}}{N} =\frac{1}{2} \log \det(- \partial^2 + \mu^2 + \sigma) - \frac{1}{4g N}\int {\rm d}^d x \, \sigma^2. \nonumber \end{align} \tag{ 2.5 }$$

The saddle of $I_{\sigma}$ gives back the Schwinger–Dyson equation for summing bubbles. More generally, one could have introduced a source for $\phi$, and then used the resulting $I_{\sigma}$ to, in principle, compute any correlation function of $O(N)$ invariant operators, to any order in 1/N.

2.1.1. Matrix models.

Having discussed vector models and matrix models, it is natural to consider tensor models, with fields having three or more indices. An example of such a model, for fermions in one dimension and rank-3 tensors, is the Klebanov–Tarnopolsky model [18], a simplification of the Gurau–Witten model [19, 20] (see also [21–24]), with the Lagrangian,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{KT} L = \frac{1}{2} \sum_{a, b, c =1}^N \psi_{a b c} \partial_{\tau} \psi_{a b c } - \frac{g}{4} \sum_{a_1, \ldots, c_2 = 1}^N \psi_{a_1 b_1 c_1} \psi_{a_1 b_2 c_2} \psi_{a_2 b_1 c_2} \psi_{a_2 b_2 c_1}, \nonumber \end{align} \tag{ 2.6 }$$

where the real field $\psi_{a b c}$ transforms in the tri-fundamental representation of O(N)³. Remarkably, the two-point function is dominated by melon diagrams in the limit of large N and fixed $\newcommand{\e}{{\rm e}} J^2\equiv g^2 N^3$. The summation of all melon diagrams is encoded in the Schwinger–Dyson equation, see figure 2,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{SD} G(\omega)^{-1} = - {\rm i}\omega - \Sigma(\omega), \ \ \ \ \ \Sigma(\tau) = J^2 G(\tau)^{3}, \nonumber \end{align} \tag{ 2.7 }$$

where $G(\omega)$ is the Fourier transform of $G(\tau)$. Tensor models, which sum melon diagrams, are more rich, and more difficult, than vector models. They are however, at least in some ways, simpler than matrix models, for which there is no closed set of equations to sum the planar diagrams.

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One challenge in the study of tensor models, beyond the melonic dominance at large N, is the relatively low degree of symmetry: the number of degrees of freedom scales as N³, while the rank of the symmetry group, O(N)³, scales as N². This difficulty is alleviated by the SYK model [2], at the expense of introducing disorder,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle L=\frac{1}{2}\sum_{i=1}^N \chi_i \partial_{\tau} \chi_i -\frac{1}{4!} \sum_{i, j, k, l=1}^N J_{i j k l} \chi_i \chi_j \chi_k \chi_l, \label{SYKL} \nonumber \end{align} \tag{ 2.8 }$$

where the couplings are Gaussian-random, with zero mean, and variance $\langle J_{i j k l } J_{i j k l}\rangle = 6 J^2/N^3$. The leading large N diagrams in SYK are melons; identical to those in the tensor model². The subleading 1/N corrections, as well as the symmetry group, are however different. Taking the partition function, and disorder averaging/integrating out the J_ijkl gives a bilocal, $O(N)$ invariant action³. Introducing bilocal fields $\Sigma(\tau_1, \tau_2)$ and $G(\tau_1, \tau_2)$, with $\Sigma$ acting as a Lagrange multiplier field enforcing $G(\tau_1, \tau_2) = \frac{1}{N}\sum_{i=1}^N\chi_i(\tau_1) \chi_i(\tau_2)$, and integrating out the fermions, one is left with an action for $\Sigma$ and G [2], see also [27–29],

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{Ieff} \frac{I_{\rm ef\,\!f}}{N} = - \frac{1}{2} \log \det\left( \partial_{\tau} - \Sigma\right) + \frac{1}{2} \int {\rm d}\tau_1 {\rm d}\tau_2 \left( \Sigma(\tau_1, \tau_2)G(\tau_1, \tau_2) - \frac{J^2}{4}G(\tau_1, \tau_2)^4\right). \nonumber \end{align} \tag{ 2.9 }$$

Compared with the $O(N)$ vector model, which was captured by an action for a local field $\sigma(x)$, SYK is instead captured by the above action for bilocal fields $\Sigma(\tau_1, \tau_2)$ and $G(\tau_1, \tau_2)$⁴. With this action, the model is in principle solved: instead of the original N fields, there are now only two fields. At infinite N, the theory is classical, with the path integral dominated by the saddle for $\Sigma$ and G, given by (2.7), reflecting the summation of melon diagrams. All higher point correlation functions follow from expanding the action in powers of 1/N. The rest of the notes are devoted to computing these in an explicit form, and understanding their physical consequences.

In this section we discuss some properties of an arbitrary conformal field theory. We first recall the constraints that conformal symmetry places on the form of correlation functions. A one-dimensional conformal field theory, CFT₁, has SL₂(R) symmetry,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \tau \rightarrow \frac{a\tau + b}{c\tau + d}, \ \ \ \ ad-b c=1. \nonumber \end{align} \tag{ 4.1 }$$

Any such transformation is generated by a combination of translations $\tau\rightarrow \tau + a$, dilatations $\tau \rightarrow \lambda \tau$, and inversions $\tau\rightarrow \frac{1}{\tau}$. The symmetry fully fixes the functional form of the two-point and three-point functions,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\mO}{\mathcal{O}} \displaystyle \langle \mO_h(\tau_1) \mO_h(\tau_2) \rangle = \frac{1}{|\tau_{12}|^{2h}}, \ \ \ \ \ \ \ \ \ \langle \mO_1 \mO_2 \mO_3\rangle = \frac{C_{h_1 h_2 h_3}}{|\tau_{12}|^{h_1+h_2-h_3} |\tau_{13}|^{h_1+h_3 - h_2} |\tau_{23}|^{h_2 + h_3 - h_1}}, \label{3pt} \nonumber \end{align} \tag{ 4.2 }$$

where the $\newcommand{\mO}{\mathcal{O}} \mO_i$, shorthand for $\newcommand{\mO}{\mathcal{O}} \mO_{h_i}(\tau_i)$, are primary operators of dimension h_i, and the $C_{h_1 h_2 h_3}$ are structure constants.

To find the functional form of a four-point function, we combine two three-point functions, and integrate over one of the points,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\mO}{\mathcal{O}} \newcommand{\mF}{\mathcal{F}} \displaystyle \label{CPW} \int {\rm d}\tau_0\, \langle \mO_1\mO_2 \mO_{h}(\tau_0)\rangle \langle \mO_{1-h}(\tau_0) \mO_3 \mO_4\rangle = \beta(h, h_{34}) \mF_{1234}^h (\tau_i) + \beta(1-h, h_{12}) \mF_{1234}^{1-h}(\tau_i), \nonumber \end{align} \tag{ 4.3 }$$

forming a conformal partial wave, which captures the exchange of $\newcommand{\mO}{\mathcal{O}} \mO_h$ and its descendants⁷. Explicitly evaluating the integral yields the right-hand side: $\beta(h, h_{34})$ is a ratio of gamma functions, whose explicit form we have not written, and $\newcommand{\mF}{\mathcal{F}} \mF_{1234}^h (\tau_i)$ is identified as the conformal block, and contains a hypergeometric function of the conformally invariant cross ratio of the four times $\tau_1, \ldots, \tau_4$, and depends on the dimensions of the external operators, $h_1, \ldots h_4$, and the dimension h of the exchanged operator. The conformal blocks form a basis, in terms of which one can express a general four-point function,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\mC}{\mathcal{C}} \newcommand{\mO}{\mathcal{O}} \newcommand{\mF}{\mathcal{F}} \displaystyle \label{cBlock} \langle \mO_1 \cdots \mO_4\rangle = \int_{\mC} \frac{{\rm d} h}{2\pi {\rm i}}\, \rho (h) \mF_{1234}^h(\tau_i). \nonumber \end{align} \tag{ 4.4 }$$

The contour $\newcommand{\mC}{\mathcal{C}} \mC$ runs parallel to the imaginary axis, $h = \frac{1}{2} + {\rm i} s$, and in addition has counterclockwise circles enclosing the positive even integers h  =  2n. These are the principal and discrete series, respectively, of SL₂(R)⁸.

As an analogy, this expression for the four-point function is for the conformal group SL₂(R) what the Fourier transform is for the translation group. Specifically, we may write any function of x as,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle f(x) = \int \frac{{\rm d}p}{2\pi}\, f(\,p) {\rm e}^{{\rm i} p x}. \nonumber \end{align} \tag{ 4.5 }$$

Here ${\rm e}^{{\rm i} p x}$ are a complete set of eigenfunctions of the Casimir of the translation group, $\partial_x^2$, while in (4.4), the conformal blocks $\newcommand{\mF}{\mathcal{F}} \mF_{1234}^h(\tau_i)$, with h running over $\newcommand{\mC}{\mathcal{C}} \mC$, are a complete set of eigenfunctions of the SL₂(R) Casimir. Any CFT₁ four-point function is completely specified by an analytic function $\rho(h)$. The poles and residues of $\rho(h)$ set the dimensions and OPE coefficients of the exchanged operators in the four-point function, as one can see by closing the contour in (4.4).

For theories with $O(N)$ symmetry, it is natural to study operators that have definite transformation under the action of $O(N)$. We will be interested in $O(N)$ singlets, such as⁹,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\mO}{\mathcal{O}} \displaystyle \label{ST} \mO_{h}=\frac{1}{N} \sum_{i=1}^N \chi_i\, \partial_{\tau}^{1+2n} \chi_i. \nonumber \end{align} \tag{ 4.6 }$$

We will refer to such an operator as single-trace. One can make more $O(N)$ invariant operators, by taking products. For instance, a double-trace operator is schematically of the form $\newcommand{\mO}{\mathcal{O}} \mO_{h_1} \partial_{\tau}^{2n} \mO_{h_2}$.

We discussed that the fermion two-point function is dominated by melons at large N. Now let us look at the Feynman diagrams contributing to a connected fermion 2k-point function. At leading order, these correlators will scale as N^−(k−1), and will involve each external index occurring in pairs. Connecting two such lines by a propagator gives a Feynman diagram contributing to a $2(k-1)$-point function. Therefore, the Feynman diagrams for a 2k-point function are found by successively cutting melon diagrams.

A single cut gives the four-point function: it scales as 1/N, and is a sum of ladder diagrams, shown in figure 3(a), with the kernel, shown in figure 3(b), adding rungs to the ladder. To perform the sum, one need only find the eigenfunctions and eigenvalues of the kernel. In fact, as a result of SL₂(R) invariance, the eigenfunctions are just the conformal partial waves discussed earlier, labeled by the dimension h of the exchanged operator. In this basis, the sum becomes a geometric sum, and the fermion four-point function is of the form given in (4.4), with $\rho(h)$ [27]¹⁰,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\al}{\alpha} \displaystyle \label{rho} \rho(h) = \mu(h) \frac{\alpha_0}{2} \frac{k(h)}{1 - k(h)}, \ \ \ \ \ \ \ \ k(h) = - \frac{3}{2}\frac{\tan\frac{\pi (h- 1/2)}{2}}{h - \frac{1}{2}}, \nonumber \end{align} \tag{ 4.7 }$$

where $k(h)$ are the eigenvalues of the kernel, while $\mu(h)$ is a simple measure factor and $\newcommand{\al}{\alpha} \alpha_0$ is a constant, which we have not written explicitly¹¹. As mentioned earlier, the poles and residues of $\rho(h)$ give the dimensions and OPE coefficients, c_h, of the exchanged operators. The poles occur at the h for which $k(h)=1$¹². These can be written in the form, $\newcommand{\eps}{\epsilon} \newcommand{\ep}{\varepsilon} \newcommand{\e}{{\rm e}} h = 2 n+1 + 2\Delta + 2\eps_n$, with $\newcommand{\eps}{\epsilon} \newcommand{\ep}{\varepsilon} \newcommand{\e}{{\rm e}} \eps_n$ small for large integer n, and correspond to the dimensions of the single-trace operators (4.6) (in the infrared)¹³.

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The fermion six-point function consists of the diagrams shown in figure 4(a), and can be viewed as three four-point functions glued together. Since the four-point function is expressed in terms of conformal blocks, computing the six-point function is just a matter of gluing together three conformal blocks. The information i the fermion six-point function can be compactly encoded in the three-point functions of the bilinear operators $\newcommand{\mO}{\mathcal{O}} \mO_h$ (4.6), with coefficients $C_{h_1 h_2 h_3}$, whose explicit form is given in [41]: they can be written as $\newcommand{\mI}{\mathcal{I}} C_{h_1 h_2 h_3} = c_{h_1} c_{h_2} c_{h_3} \mI_{h_1 h_2 h_3}$, where $\newcommand{\mI}{\mathcal{I}} \mI_{h_1 h_2 h_3}$ is an analytic function of the h_i, involving gamma functions and the hypergeometric function ${}_4 F_3$ at argument one.

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The fermion eight-point function is built out of more four-point functions glued together. One such contribution is shown in figure 4(b), and its contribution to the bilinear four-point function takes an incredibly simple form [41],

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\mI}{\mathcal{I}} \newcommand{\mC}{\mathcal{C}} \newcommand{\mF}{\mathcal{F}} \displaystyle \label{4ptO} c_{h_1} c_{h_2} c_{h_3} c_{h_4} \int_{\mC} \frac{{\rm d}h}{2\pi {\rm i}} \, \rho (h)\, \mI_{h_1 h_2\, h}\, \mI_{h\, h_3 h_4}\, \mF_{1234}^h(\tau_i). \nonumber \end{align} \tag{ 4.8 }$$

The result is intuitive: the $\newcommand{\mI}{\mathcal{I}} \mI_{h_1 h_2\, h}$ and $\newcommand{\mI}{\mathcal{I}} \mI_{h\, h_3 h_4}$ are the 'interaction vertices' from the two six-point functions, and there is an operator of dimension h exchanged, giving the $\rho(h)$ factor from the intermediate fermion four-point function. A nontrivial consistency check is that the four-point function of $\newcommand{\mO}{\mathcal{O}} \mO_{h_i}$, at order 1/N, should be a sum of conformal blocks of exchanged single-trace operators as well as double-trace operators. Upon closing the contour in (4.8), the poles of $\rho(h)$ give the single-trace blocks, while the poles of $\newcommand{\mI}{\mathcal{I}} \mI_{h_1 h_2 h}$, occurring at $h= h_1 + h_2 +2n$, give the double-trace blocks. It is remarkable that the analytically extended OPE coefficients of the single-trace operators—the $\newcommand{\mI}{\mathcal{I}} \mI_{h_1 h_2 h_3}$—knew that they should have singularities at precisely these locations.

While SYK has a special set of Feynman diagrams, these results for the correlation functions are more general. The simple expression for the fermion four-point function follows from summing ladder diagrams; it is irrelevant that the propagators are built from melons, those only served to give a conformal two-point function¹⁴. The six-point point function is made up of three four-point functions glued together, and in calculating it, it is not relevant that the four-point function was a sum of ladder diagrams. The expression for the eight-point function/bilinear four-point function is valid regardless of the details of how the three fermion four-point functions combine at the 'interaction vertex'; all of this is encoded in $\newcommand{\mI}{\mathcal{I}} \mI_{h_1 h_2 h_3}$.

At low energies, SYK is dominated by the h  =  2 mode, described by the Schwarzian action. This is a result of being nearly conformally invariant. On the AdS₂ side, since Einstein gravity is topological in two-dimensions, it is natural to instead consider Jackiw–Teitelboim dilaton gravity [54, 55]. Dilaton gravity theories naturally arise from compactifying gravity in higher dimensions down to two dimensions, with the dilaton playing the role of the size of the extra dimension. It has been shown that the dilaton theory in AdS₂ is the same as the Schwarzian theory, as a consequence of the pattern of symmetry breaking [32, 56, 57]¹⁵.

Of course, the h  =  2 mode is just the first in the tower of fermion bilinear $O(N)$ singlets, $\newcommand{\mO}{\mathcal{O}} \mO_{h}$, written schematically in (4.6), and the rest of the tower, with $n\geqslant 1$, encode the structure of SYK. As we have discussed, a connected k-point correlation function of the $\newcommand{\mO}{\mathcal{O}} \mO_h$ scales as N^−(k−2)/2. We can write a putative dual field theory in AdS₂,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{Lbulk} \mathcal{L}_{\rm bulk} = \sum_{n=1}^{\infty} \frac{1}{2} (\partial \phi_n)^2 + \frac{1}{2} m_n^2 \phi_n^2 + \frac{1}{\sqrt{N}} \sum_{n, m, k=1}^{\infty}\lambda_{n m k } \phi_n \phi_m \phi_k+ \ldots \nonumber \end{align} \tag{ 5.1 }$$

containing a tower of scalar fields $\phi_n$. As a result of the SL₂(R) isometry of AdS₂, any correlation function of the $\phi_n$, at points extrapolated to the boundary of AdS₂, will take the form of a CFT correlation function. Identifying each $\phi_n$ with an operator $\newcommand{\mO}{\mathcal{O}} \mO_h$, we can appropriately choose the masses and cubic couplings of the $\phi_n$ so as to match to the SYK two-point and three-point functions of the $\newcommand{\mO}{\mathcal{O}} \mO_{h}$, respectively. The masses are related to the dimensions in the standard way, $m_n^2 = h(h- 1)$, while the cubic couplings, in the limit $n, m, k \gg 1$ where they simplify, are [41, 68],

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{lambda} \lambda_{n m k} \approx \frac{(n + m +k)!}{\Gamma(n+m-k+\frac{1}{2}) \Gamma(m+k-n+\frac{1}{2})\Gamma(k+n -m+ \frac{1}{2})}. \nonumber \end{align} \tag{ 5.2 }$$

One could similarly try to appropriately choose the quartic couplings, so as to match the SYK four-point function of bilinears¹⁶. However, in order to have an actual understanding of the AdS dual of SYK, one needs a simple and independently defined bulk theory, something like the string worldsheet action, rather than just a list of couplings. Such a bulk description is presently lacking; it is not obvious one must exist¹⁷.

Two canonical examples of AdS/CFT duality are between $\mathcal{N} =4$ super Yang–Mills in 4 dimensions and string theory in AdS$_5 \times S^5$ [77–79], and between the free/critical vector $O(N)$ model in 3 dimensions and Vasiliev higher spin theory in AdS₄ [80, 81]. A comparison between SYK and these two theories is given in figure 5.

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One hope for SYK has been that, because of its simplicity, it would provide an example of AdS/CFT in which one could fully understand the duality, directly relating the CFT degrees of freedom to the bulk variables. This remains a goal, though achieving it of course requires knowing what the bulk theory is, in order to have a target. Independently of this, SYK has led to a renewed interest in two-dimensional gravity, and the formulation of modern ideas on spacetime and holography in this context, see e.g. [82–91].

There are many variants of SYK, which retain the key feature of dominance of melon diagrams. One natural generalization, which incorporates some of these, is to consider a model which contains f  flavors of fermions, with N_a fermions of flavor a, each appearing q_a times in the interaction, so that the Hamiltonian couples $\mathrm{q}=\sum_{a=1}^{\,f} q_a$ fermions together [28],

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{GFM} L= \frac{1}{2}\sum_{a =1}^{\,f} \sum_{i=1}^{N_a}\chi_i^a\, \partial_{\tau}\chi_i^a +\frac{(i)^{\frac{\mathrm{q}}{2}}}{\prod_{a=1}^{\,f} q_a!} \sum_I J_{I}( \chi_{i_1}^1 \cdots \chi_{i_{q_1}}^1)\cdots (\chi_{j_1}^{\,f} \cdots \chi_{j_{q_f}}^{\,f}), \nonumber \end{align} \tag{ 5.3 }$$

where I is a collective index, $I = i_1,\ldots, i_{q_1}, \ldots, j_{1},\ldots, j_{q_f}$. The coupling J_I is antisymmetric under permutation of indices within any one of the f  families, and is drawn from a Gaussian distribution. This model with one flavor, f   =  1, reduces to the standard SYK model with a q-body interaction, sometimes denoted by SYK_q; further setting q  =  4 gives the canonical SYK model, which has been the focus of these notes¹⁸.

Regarding flavor as a lattice site index, x, and taking sums of the flavored models, one can build lattices of SYK models [109]. One such model, having some features of a strongly correlated metal, is [110],

$$\begin{align} \renewcommand{\dag}{\dagger} \newcommand{\e}{{\rm e}} \newcommand{\re}{{\rm Re}} \displaystyle \label{dot} L = \sum_{x}\sum_i c_{i, x}^{\dagger} \left(\partial_{\tau} - \mu\right) c_{i, x} - \sum_{\langle x x'\rangle} \sum_{i, j} t_{ij, x x'} c_{i, x}^{\dagger} c_{j, x'} - \sum_{x} \sum_{i, j, k, l} J_{i j k l, x}c_{i, x}^{\dagger} c_{j, x}^{\dagger} c_{k, x} c_{l, x}. \nonumber \end{align} \tag{ 5.4 }$$

Here $t_{ij, x x'}$ and J_ijkl,x are again random couplings, and the c_i,x are now complex fermions. A cartoon of this model is shown in figure 6. The model exhibits incoherent metal behavior, with resistivity scaling linearly with temperature, at high temperature, and Fermi liquid behavior at low temperature¹⁹.

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More generally, the fact that SYK is a system without quasiparticles, yet is nevertheless solvable, makes it a valuable tool with which to study transport and chaos [117–125], non-equilibrium dynamics and entanglement [126–129], and eigenstate thermalization [130–132]. There are limitations, however, as neither the all-to-all interactions nor the large N, which are essential to the solvability of SYK, are present in real metals.

Finally, there are a number of topics which we have not discussed, such as: experimental realizations and quantum simulations of SYK [133–139], the zero-temperature entropy (an infinite N artifact) [140–142], studies of the spectral density, the spectral form factor, and connections with random matrix theory [83, 143–153].