SYK Conformal Regime and Schwarzian Matching

Starting from the bilocal saddle, this page identifies the reparametrization fluctuation whose conformal ladder-kernel eigenvalue is one, computes how the ultraviolet kinetic term shifts that eigenvalue, and packages the result in a Schwarzian action. For standard even-$q\ge4$ Majorana SYK at large $N$ and $\beta J\gg1$, the coefficient extracted from the kernel must equal the coefficient of the low-temperature heat capacity. That equality is a quantitative infrared test; it neither fixes the entropy intercept nor reconstructs a finite-$N$ spectrum.

Required background. SYK Bilocal Collective Fields as Near-AdS₂ Data supplies the no-minus Green-function convention, disorder normalization, full saddle equations, fluctuation kernel, and $q=4$ numerical data. JT Gravity and the Schwarzian Boundary Mode supplies the independent gravitational coefficient $C_{\rm JT}=\varphi_r/(8\pi G_2)$.

Helpful background. SYK Models and Local Quantum Criticality develops the many-body model. Nearly AdS₂ Effective Theory Beyond the Leading Schwarzian explains how additional irrelevant operators limit the match.

Scope. The explicit calculation is the leading one-copy, disorder-averaged saddle, with $N\to\infty$ taken before the low-temperature limit. A quenched or fixed-realization claim needs additional evidence. The derivation proceeds from the conformal saddle to the unit kernel mode, its finite-coupling lift, the Schwarzian coefficient, and an independent thermodynamic check.

Reading route. Follow the conformal orbit to the unit kernel mode, its ultraviolet lift, the Schwarzian action, and the thermodynamic match. The last sections delimit the JT comparison and the three breakdown scales before the solved exercises.

Evidence reviewed: 29 August 2026.

The infrared saddle has a reparametrization orbit

Use the coupling $J$ fixed by

$$\overline{J_{i_1\cdots i_q}^{\,2}} =\frac{(q-1)!J^2}{N^{q-1}},$$

and define the second common coupling convention by

$$\mathcal J =\frac{\sqrt q}{2^{(q-1)/2}}J.$$

For $q=4$, $\mathcal J=J/\sqrt2$. Keeping both symbols visible will prevent a factor-$\sqrt2$ error later.

At strong coupling, drop the kinetic term in the Schwinger–Dyson equations. The conformal equations are

$$\begin{aligned} G_c\circ\Sigma_c&=-\mathbf 1,\\ \Sigma_c(\tau_1,\tau_2)&=J^2G_c(\tau_1,\tau_2)^{q-1}. \end{aligned}$$

Under a monotone reparametrization $f$, they are invariant if

$$\begin{aligned} G_c(\tau_1,\tau_2)&\longmapsto [f'(\tau_1)f'(\tau_2)]^\Delta G_c(f(\tau_1),f(\tau_2)),\\ \Sigma_c(\tau_1,\tau_2)&\longmapsto [f'(\tau_1)f'(\tau_2)]^{(q-1)\Delta} \Sigma_c(f(\tau_1),f(\tau_2)). \end{aligned}$$

The convolution measure supplies one inverse Jacobian. Covariance therefore requires $q\Delta=1$, so the fermion dimension is

$$\Delta=\frac1q.$$

On the line, write the dimensionless normalization as $b_\chi$:

$$G_c(\tau) =b_\chi\frac{\operatorname{sgn}\tau} {\lvert J\tau\rvert^{2/q}}, \qquad \pi b_\chi^q =\left(\frac12-\frac1q\right) \tan\frac{\pi}{q}.$$

For $q=4$, $b_\chi^4=1/(4\pi)$. The normalization follows by Fourier transforming the power law and imposing $G_c\Sigma_c=-1$; it is not fixed by scaling alone. These steps are derived in Maldacena and Stanford 2016, § 2.3, Eqs. (2.7)–(2.11).

The fixed line-to-circle map

$$F(\tau)=\tan\frac{\pi\tau}{\beta}$$

gives, for $0<\tau<\beta$,

$$G_{c,\beta}(\tau) =b_\chi \left[ \frac{\pi}{\beta J\sin(\pi\tau/\beta)} \right]^{2/q},$$

extended antiperiodically. This formula requires $\beta J\gg1$ and time separations far from both ultraviolet endpoints:

$$J^{-1}\ll\tau, \qquad J^{-1}\ll\beta-\tau.$$

Those conditions justify dropping the derivative. The operative semiclassical condition derived below is

$$\frac{C_{\rm SYK}}{\beta} =\frac{N\alpha_S^{(J)}}{\beta J}\gg1.$$

It is often abbreviated as $1\ll\beta J\ll N$, but the shorthand hides the fixed-$q$ coefficient $\alpha_S^{(J)}$ and is not part of the conformal power-counting step.

Now distinguish $F$ from the dynamical circle diffeomorphism $f$:

$$f(\tau+\beta)=f(\tau)+\beta, \qquad f'(\tau)>0.$$

The reparametrized thermal family is obtained from $F\circ f=\tan[\pi f(\tau)/\beta]$. Projective transformations of $F\circ f$ leave the conformal correlator unchanged, so distinct configurations are circle diffeomorphisms modulo $\mathrm{PSL}(2,\mathbb R)$. The literature often abbreviates the stabilizer as $\mathrm{SL}(2,\mathbb R)$.

The weight-two ladder mode is exactly flat

The conformal ladder kernel acting on antisymmetric bilocal fluctuations is

$$K_c(\tau_1,\tau_2;\tau_3,\tau_4) =-(q-1)J^2 G_c(\tau_{13})G_c(\tau_{24}) G_c(\tau_{34})^{q-2}.$$

Consider $f(\tau)=\tau+\varepsilon(\tau)$, where $\varepsilon$ has units of time. To first order,

$$\delta_\varepsilon G_c =\left[ \Delta(\varepsilon_1'+\varepsilon_2') +\varepsilon_1\partial_1 +\varepsilon_2\partial_2 \right]G_c.$$

Differentiate the conformal saddle equation along this orbit. Since

$$\delta_\varepsilon\Sigma_c =(q-1)J^2G_c^{q-2}\delta_\varepsilon G_c,$$

the variation reduces to

$$(1-K_c)\,\delta_\varepsilon G_c=0.$$

Thus every nontrivial infinitesimal reparametrization is a unit-eigenvalue fluctuation of $K_c$. In the bilocal Hessian, the corresponding factor $K_c^{-1}-1$ vanishes. This is the weight-$h=2$ channel that was only anticipated on the preceding page; the exact derivation is Maldacena and Stanford 2016, § 3.3.1, Eqs. (3.108)–(3.113).

Put $\theta=2\pi\tau/\beta$ and separate the dimensionful shift from its dimensionless Fourier field:

$$\varepsilon(\tau) =\frac{\beta}{2\pi}\eta(\theta), \qquad \eta(\theta)=\sum_{n\in\mathbb Z}\eta_ne^{in\theta}.$$

The modes $n=0,\pm1$ generate the projective stabilizer and give $\delta_\varepsilon G_c=0$. They are not physical zero-action fluctuations to integrate over. The soft orbit begins at $\lvert n\rvert=2$:

$$\operatorname{Diff}^{+}(S^1)/\mathrm{PSL}(2,\mathbb R) \quad\Longleftrightarrow\quad \eta_n,\ \lvert n\rvert\ge2.$$

This distinction matters: an eigenvalue equal to one explains the conformal divergence of $(1-K_c)^{-1}$, but it does not yet determine the energy scale or coefficient that regulates it.

The ultraviolet term lifts the unit eigenvalue

Restore the kinetic term and solve the full saddle to first order away from the conformal limit. On the thermal circle, choose $-\pi\le\theta\le\pi$. The following expansion is local to the infrared region and is not uniform at coincident points; it requires

$$\beta\mathcal J\, \left\lvert2\sin\frac\theta2\right\rvert\gg1.$$

There the leading correction has the form

$$\frac{G(\theta)-G_c(\theta)}{G_c(\theta)} =-\frac{\alpha_G}{\beta\mathcal J} f_0(\theta) +O\!\left((\beta\mathcal J)^{-2}\right),$$

where

$$f_0(\theta) =2+\frac{\pi-\lvert\theta\rvert} {\tan(\lvert\theta\rvert/2)}.$$

The shape is fixed by the infrared equations, but $\alpha_G$ is a UV-to-IR matching coefficient. Its value requires the full saddle or equivalent microscopic data. For fixed Fourier index with $\lvert n\rvert/(\beta\mathcal J)\ll1$, perturbing the exact kernel by this corrected propagator gives

$$k(2,n) =1-\frac{\alpha_K}{\beta\mathcal J}\lvert n\rvert +O\!\left((\beta\mathcal J)^{-2}\right),$$

with

$$\begin{aligned} \alpha_K&=-qk_c'(2)\alpha_G,\\ \alpha_0&= \frac{2\pi q} {(q-1)(q-2)\tan(\pi/q)}. \end{aligned}$$

The factor $\lvert n\rvert$ is the leading explicit-breaking lift. Combining it with the $\lvert n\rvert(n^2-1)$ normalization of the $h=2$ eigenfunction produces $n^2(n^2-1)$ in the inverse soft propagator. The calculation and its UV sensitivity are explicit in Maldacena and Stanford 2016, § 3.3.2, Eqs. (3.124)–(3.130), and Appendix E.

For $q=4$, the full Schwinger–Dyson solution gives

$$\alpha_G\simeq0.1872, \qquad \alpha_0=\frac{4\pi}{3}, \qquad -k_c'(2)=\pi+\frac23,$$

and hence

$$\alpha_K\simeq2.8516$$

when the eigenvalue shift is written using $\mathcal J$.

There is a useful reproducibility check against the full $q=4$ saddle on the preceding page. Since $f_0(\pi)=2$, its midpoint data define

$$\alpha_G^{\rm(mid)}(\beta J) =\frac{\beta\mathcal J}{2} \left[ 1-\frac{G(\beta/2)}{G_c(\beta/2)} \right].$$

At $\beta J=50$, the recorded values $G(\beta/2)=0.131678$ and $G_c(\beta/2)=0.133134$ give $\alpha_G^{\rm(mid)}=0.1933$. This is $3.3\%$ above the asymptotic $0.1872$. The grid shift is only $3.02\times10^{-8}$, so the visible discrepancy is finite-coupling truncation, not solver convergence.

The repository checker reruns the full saddle on a smaller $M=512$ audit grid. At $\beta J=20,50,100$, it obtains $\alpha_G^{\rm(mid)}=0.2034,0.1933,0.1900$, respectively, approaching $0.1872$ monotonically; the last Dyson residual is $1.2\times10^{-12}$. These values diagnose convergence rather than independently determine $\alpha_G$. Invoke scripts/benchmark-syk-schwarzian-soft-mode.py with a Python 3 interpreter; --skip-saddle runs only the analytic normalization and thermodynamic checks.

The lifted mode becomes the Schwarzian

Inserting the lifted $h=2$ channel into the bilocal quadratic action gives

$$\frac{I_{\rm soft}^{(2)}}{N} =\frac{\widehat\alpha_S}{2\mathcal J} \int_0^\beta d\tau\, \left[ (\varepsilon'')^2 -\left(\frac{2\pi}{\beta}\right)^2(\varepsilon')^2 \right],$$

where

$$\widehat\alpha_S =\frac{\alpha_K}{6q^2\alpha_0}.$$

Using the dimensionless Fourier coefficients $\eta_n$ defined above, this is

$$I_{\rm soft}^{(2)} =\frac{2\pi^2C_{\rm SYK}}{\beta} \sum_{\lvert n\rvert\ge2} n^2(n^2-1)\lvert\eta_n\rvert^2.$$

Before quotienting, the unrestricted quadratic operator has exactly three null directions, $n=0,\pm1$, as projective invariance requires. The displayed sum already omits them.

The kernel calculation fixes this quadratic action. At general finite $q$, locality, the derivative expansion, and projective invariance select the leading nonlinear completion

$$I_{\rm Sch}[f] =-C_{\rm SYK} \int_0^\beta d\tau\, \left\{ \tan\frac{\pi f(\tau)}{\beta},\tau \right\},$$

where

$$\{u,\tau\} =\frac{u'''}{u'} -\frac32\left(\frac{u''}{u'}\right)^2.$$

The coefficient dictionary is

$$\begin{aligned} C_{\rm SYK} &=\frac{N\widehat\alpha_S}{\mathcal J} =\frac{N\alpha_S^{(J)}}{J},\\ \alpha_S^{(J)} &=\frac{J}{\mathcal J}\widehat\alpha_S. \end{aligned}$$

Maldacena and Stanford obtain the lifted quadratic action and motivate this completion Maldacena and Stanford 2016, § 4, Eqs. (4.174)–(4.180). Kitaev and Suh formulate the UV/IR matching and derive the leading nonlocal correction Kitaev and Suh 2018, §§ 1 and 6. A direct microscopic nonlinear derivation is now available in the controlled large-$p$ limit, where $p$ is the interaction order; it is not a theorem for finite $q=4$ Bucca and Mezei 2025, §§ 2.1 and 4–5.

For $q=4$,

$$\begin{aligned} \widehat\alpha_S&\simeq0.007091, &\quad&\text{coefficient relative to }\mathcal J,\\ \alpha_S^{(J)}&\simeq0.010029, &&\text{coefficient relative to }J,\\ C_{\rm SYK}&\simeq0.01003\,\frac NJ. \end{aligned}$$

Quoting a bare number called $\alpha_S$ without naming its coupling convention is therefore incomplete.

Thermodynamics closes the coefficient loop

At the thermal saddle $f(\tau)=\tau$,

$$\left\{ \tan\frac{\pi\tau}{\beta},\tau \right\} =\frac{2\pi^2}{\beta^2}.$$

The one-copy low-temperature partition function, written as $Z_{\rm a}\equiv\overline{Z_J}$, is consequently

$$\log Z_{\rm a} =-\beta E_\star +S_{\star,{\rm a}} +\frac{2\pi^2C_{\rm SYK}}{\beta} +\cdots .$$

Here $S_{\star,{\rm a}}$ is the large-$N$ entropy intercept of the annealed saddle, with $N\to\infty$ taken before $T\to0$. It is not a claim of an exactly degenerate finite-$N$ ground space. Differentiation gives three linked observables:

$$\begin{aligned} E-E_\star&=2\pi^2C_{\rm SYK}T^2,\\ S-S_{\star,{\rm a}}&=4\pi^2C_{\rm SYK}T,\\ C_V&=4\pi^2C_{\rm SYK}T. \end{aligned}$$

For $q=4$, the independent full-saddle thermodynamic result is

$$\frac{C_V}{NT} =\frac{4\pi^2C_{\rm SYK}}N \simeq\frac{0.39592}{J},$$

equivalently

$$\frac{E-E_\star}{N} \simeq0.19796\,\frac{T^2}{J}.$$

Agreement with $C_{\rm SYK}$ inferred from the lifted kernel is the promised normalization check Maldacena and Stanford 2016, § 5, Eqs. (5.181)–(5.182). It tests the slope, not the entropy intercept.

At order $N^0$, integrating the quantum soft mode adds the characteristic $-\tfrac32\log(\beta J)$ term to $\log Z$. The exact Schwarzian path integral is one-loop exact and has the continuous form

$$Z_{\rm Sch}(\beta) \propto\beta^{-3/2} \exp\!\left(\frac{2\pi^2C_{\rm SYK}}{\beta}\right),$$

up to convention-dependent normalization Stanford and Witten 2017, § 2.4. This quantum result improves the classical saddle; it still does not resolve the discrete levels of one finite-$N$ realization.

Matching one coupling to JT gravity

JT gravity has the same leading boundary action with

$$C_{\rm JT}=\frac{\varphi_r}{8\pi G_2}.$$

The quantitative infrared matching condition is simply

$$C_{\rm SYK}=C_{\rm JT}.$$

When a parent near-extremal black hole determines $\varphi_r/G_2$ independently, this is a nontrivial comparison of thermodynamic slopes. In bottom-up JT, it is a parameter identification. The common statement is therefore precise but limited: SYK and JT can share the leading effective theory $\operatorname{Diff}^{+}(S^1)/\mathrm{PSL}(2,\mathbb R)$ after matching one coupling Maldacena, Stanford, and Yang 2016, § 3.1.

This equality does not identify $S_{\star,{\rm a}}$ with the JT topological entropy, construct a microscopic Hilbert-space isomorphism, decide whether gravitational observables are ensemble averaged, or match the tower of nonsoft bilinear modes. A comparison of soft exchange in four-point functions additionally needs a specified matter bilocal with matched dimension and normalization; pure JT alone has no SYK fermion operator. The fuller claim boundary belongs to Near-AdS₂, SYK, and the Duality Interface, while JT Correlators, Bilocals, and Chaos develops the gravitational correlators.

The approximation changes character at three scales

Let $E$ denote the characteristic external energy, of order $\max(T,\lvert\omega\rvert)$. Three different expansions must not be compressed into one error term.

1. Finite-coupling expansion. The leading correction to the conformal two-point function is $O(E/J)$. Higher powers, nonsoft bilocal modes, and the nonlocal correction to the soft action become progressively important. At $E\sim J$, neither the power law nor a Schwarzian-only description is controlled; the full Schwinger–Dyson equations remain valid.

2. Soft-loop expansion. The semiclassical parameter is not a uniform $1/N$ but

   $$\frac1{C_{\rm SYK}E} \sim\frac{J}{N\alpha_S^{(J)}E}.$$

   When $C_{\rm SYK}E\gg1$, the classical soft saddle is reliable. Near $E\sim1/C_{\rm SYK}=J/[N\alpha_S^{(J)}]$—parametrically of order $J/N$ at fixed $q$—soft fluctuations become strong. This is the quantum-Schwarzian regime, not a failure of the Schwarzian effective theory.

3. Sample-specific discreteness. A fixed finite-$N$ Hamiltonian has a mean spacing of order $Je^{-S(E)}$, parametrically much smaller than $J/N$ in a many-body window. Once the requested energy resolution reaches that spacing, or times reach the Heisenberg scale, a smooth Schwarzian density cannot reproduce individual levels or recurrences. The next page, Random Matrices, Spectral Statistics, and Ensemble Questions, begins there.

There is also a theory-space failure test. An irrelevant operator with $1<\Delta_{\mathcal O}<3/2$ can generate a nonlocal term that dominates the Schwarzian at low temperature. Standard SYK has no operator in this dangerous range, but approximate reparametrization symmetry alone does not exclude it in another model Maldacena, Stanford, and Yang 2016, Appendix D.

The strongest statement surviving all adversarial tests is this: the normalized large-$N$, strong-coupling SYK saddle has a lifted $h=2$ sector whose leading local action and thermodynamic slope match a Schwarzian theory. It does not follow that the Schwarzian reconstructs the ultraviolet Hamiltonian, remains semiclassical at every infrared scale, or resolves a fixed sample’s exact spectrum.

Common pitfalls

Using one symbol for two maps. $F(\tau)=\tan(\pi\tau/\beta)$ is the fixed line-to-circle map. $f$ is the dynamical circle diffeomorphism. The Schwarzian acts on $F\circ f$.

Integrating over the stabilizer. The modes $n=0,\pm1$ leave the conformal correlator unchanged. They are divided out as $\mathrm{PSL}(2,\mathbb R)$, not counted as fluctuating zero modes.

Mixing $J$ and $\mathcal J$. At $q=4$, $\mathcal J=J/\sqrt2$. The numbers $0.007091$ and $0.010029$ describe the same action in different coupling conventions.

Matching an intercept from a slope. $C_{\rm SYK}=C_{\rm JT}$ matches the $T^2$ energy and linear-$T$ entropy corrections. It does not match the constant entropy terms.

Calling $J/N$ a level spacing. That is the parametric quantum-soft crossover. A many-body level spacing is exponentially smaller.

Treating the nonlinear completion as a finite-$q$ microscopic theorem. The finite-$q$ kernel directly fixes the quadratic action. The leading nonlinear Schwarzian follows from the local effective-theory construction; the currently controlled microscopic nonlinear derivation is restricted to the large-interaction-order limit.

Exercises

1. Scaling dimension and normalization

Derive $\Delta=1/q$ from reparametrization covariance, then verify $b_\chi^4=1/(4\pi)$ at $q=4$.

Solution

The convolution $G_c\circ\Sigma_c$ contains one integration measure. Under $\tau\mapsto f(\tau)$, the two kernels contribute weights $\Delta+(q-1)\Delta=q\Delta$ at the integrated point, while $d\tau$ contributes one inverse power. Covariance requires

$$q\Delta-1=0,$$

so $\Delta=1/q$. At $q=4$,

$$\pi b_\chi^4 =\left(\frac12-\frac14\right) \tan\frac\pi4 =\frac14,$$

hence $b_\chi^4=1/(4\pi)$.

2. The unit kernel eigenvalue

Differentiate the conformal saddle equation along $f=\tau+\varepsilon$ and show that $K_c\delta_\varepsilon G_c=\delta_\varepsilon G_c$. Why do $n=0,\pm1$ not give physical bilocal fluctuations?

Solution

Vary $G_c\circ\Sigma_c=-\mathbf1$:

$$\delta G_c\circ\Sigma_c +G_c\circ\delta\Sigma_c=0, \qquad \delta\Sigma_c =(q-1)J^2G_c^{q-2}\delta G_c.$$

Right-convolve the first equation with $G_c$ and use $\Sigma_c\circ G_c=-\mathbf1$:

$$-\delta G_c +G_c\circ\delta\Sigma_c\circ G_c=0, \qquad \delta G_c=G_c\circ\delta\Sigma_c\circ G_c.$$

In components, the last factor satisfies $G_c(\tau_4,\tau_2)=-G_c(\tau_2,\tau_4)$. Substituting $\delta\Sigma_c=(q-1)J^2G_c^{q-2}\delta G_c$ therefore supplies the minus sign in the displayed definition of $K_c$. Hence

$$(1-K_c)\delta_\varepsilon G_c=0.$$

For $n=0,\pm1$, the reparametrization is an infinitesimal projective transformation of the thermal coordinate. The conformal two-point function is invariant under those transformations, so $\delta_\varepsilon G_c=0$ itself. They are stabilizer directions, not nonzero bilocal eigenvectors.

3. Quadratic expansion of the Schwarzian

Let

$$f(\tau)=\tau+\frac{\beta}{2\pi} \eta\!\left(\frac{2\pi\tau}{\beta}\right).$$

Expand $I_{\rm Sch}$ to second order and recover the factor $n^2(n^2-1)$.

Solution

Use the composition identity

$$\left\{\tan\frac{\pi f}{\beta},\tau\right\} =\{f,\tau\} +2\left(\frac{\pi}{\beta}\right)^2(f')^2.$$

After expanding $f=\tau+\delta f$ and integrating by parts, the linear term vanishes and

$$I_{\rm Sch}^{(2)} =\frac{C_{\rm SYK}}2 \int_0^\beta d\tau\, \left[ (\delta f'')^2 -\left(\frac{2\pi}{\beta}\right)^2 (\delta f')^2 \right].$$

With $\eta(\theta)=\sum_n\eta_ne^{in\theta}$, orthogonality gives

$$I_{\rm Sch}^{(2)} =\frac{2\pi^2C_{\rm SYK}}{\beta} \sum_n n^2(n^2-1)\lvert\eta_n\rvert^2.$$

It vanishes precisely at $n=0,\pm1$.

4. The q = 4 coefficient dictionary

Starting from $\alpha_G=0.1872$, compute $\alpha_K$, $\widehat\alpha_S$, and $\alpha_S^{(J)}$.

Solution

At $q=4$,

$$-k_c'(2)=\pi+\frac23, \qquad \alpha_0=\frac{4\pi}{3}.$$

Therefore

$$\alpha_K =4\left(\pi+\frac23\right)(0.1872) \simeq2.8516.$$

Next,

$$\widehat\alpha_S =\frac{2.8516}{6(4)^2(4\pi/3)} \simeq0.007091.$$

Because $\mathcal J=J/\sqrt2$,

$$\alpha_S^{(J)} =\sqrt2\,\widehat\alpha_S \simeq0.010029.$$

Thus $4\pi^2\alpha_S^{(J)}\simeq0.39592$.

5. One coefficient, three thermodynamic observables

Differentiate the low-temperature partition function to obtain the energy, entropy, and heat capacity corrections. Explain why the calculation does not determine the entropy intercept.

Solution

Start with

$$\log Z =-\beta E_\star+S_\star +\frac{2\pi^2C}{\beta}.$$

Then

$$E=-\partial_\beta\log Z =E_\star+\frac{2\pi^2C}{\beta^2}.$$

Using $S=\log Z+\beta E$ and $C_V=\partial E/\partial T$ gives

$$S-S_\star=4\pi^2CT, \qquad C_V=4\pi^2CT.$$

The Schwarzian term determines only the temperature-dependent part. The constant $S_\star$ comes from other saddle data and from the declared order of limits.

6. Diagnose two infrared crossovers

Suppose $C_{\rm SYK}=0.01003N/J$, $N=10^4$, and the entropy in a low-energy window is $S(E)=20$. Estimate the quantum-soft scale and the mean level spacing. Which description changes at each scale?

Solution

The quantum-soft crossover follows from $C_{\rm SYK}E\sim1$:

$$E_{\rm qSch} \sim\frac1{C_{\rm SYK}} \simeq9.97\times10^{-3}J.$$

This is $1/[N\alpha_S^{(J)}]$ times $J$. It scales as $J/N$ at fixed $q$, with the $q=4$ coefficient retained rather than hidden in the order symbol.

The rough level spacing is

$$\delta E\sim Je^{-S(E)} =Je^{-20} \simeq2.1\times10^{-9}J.$$

Near $E_{\rm qSch}$, the classical reparametrization saddle must be replaced by the quantum Schwarzian path integral. Near $\delta E$, even its smooth spectral density is insufficient and sample-specific levels are required. The two scales are physically distinct.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Bucca, Marta, and Márk Mezei. “Nonlinear Soft Mode Action for the Large-$p$ SYK Model.” Journal of High Energy Physics 2025, 89 (2025). DOI. Open PDF.
• Kitaev, Alexei, and S. Josephine Suh. “The Soft Mode in the Sachdev–Ye–Kitaev Model and Its Gravity Dual.” Journal of High Energy Physics 2018, 183 (2018). DOI. Open PDF.
• Maldacena, Juan, and Douglas Stanford. “Remarks on the Sachdev–Ye–Kitaev Model.” Physical Review D 94, 106002 (2016). DOI. Open PDF.
• Maldacena, Juan, Douglas Stanford, and Zhenbin Yang. “Conformal Symmetry and Its Breaking in Two-Dimensional Nearly Anti-de Sitter Space.” Progress of Theoretical and Experimental Physics 2016, 12C104 (2016). DOI. Open PDF.
• Stanford, Douglas, and Edward Witten. “Fermionic Localization of the Schwarzian Theory.” Journal of High Energy Physics 2017, 008 (2017). DOI. Open PDF.