Seeking exact solutions
Integrability,
from Newton's cradle to Gibbs' grave

Pierre et Marie Curie Colloquium

Paris, 24 September 2026

Jean-Sébastien Caux

Institute of Physics
University of Amsterdam

Plan of the talk

Isaiah Berlin, hedgehogs and foxes

1909-06-06/1997-11-05

The fox knows many things...

... but the hedgehog knows one big thing.
(Archilochus)

Some "big questions"

  • What does integrability even mean?
  • Classical and quantum integrability: different beasts
  • Do integrable systems behave "normally"?
  • As models: the exception rather than the rule
  • Can integrability predict equilibrium properties?
  • Difficult to calculate anything exactly
  • Can integrability shed light on out-of-equilibrium issues?
  • Tough to get started... too complicated

Why is 1d special?

  • Particles "can't avoid each other"
    interaction effects are automatically saturated
  • Perturbation theory fails
    Fermi surface is fully nested, susceptibilities diverge, Fermi liquid "jams up"

What can be done?

  • Brute force resummations (Dzyaloshinskii-Larkin 1974)
  • Low-energy phenomenology:
    the Luttinger liquid (Haldane 1981)
  • Numerics (everybody)
  • Exact methods: integrability (Bethe 1931)

Integrability

fundamental notions
and models

⮕ integrability.org online lecture notes

Early days of (quantum) magnetism

The problem (1920's): why ferromagnetism?

Early attempt: Ising (actually: Lenz) ⮕ inconclusive

The Heisenberg Model

\[ H = J \sum_{j} {\boldsymbol S}_j \cdot {\boldsymbol S}_{j+1} \]

⮕the "most quantum" model

The Bethe Ansatz (1931)

1906-07-02/2005-03-06

Hamiltonian: $H = \int dx {\cal H}(x)$

"Reference state": FM state, vacuum, ...

"Particles": down spins, atoms, ...

Eigenstates with fixed # of particles:

\[ \Psi_N (\{ x \};\!\{ k \}) = \sum_{P \in S_N} \!(-1)^{[P]} A_P ~e^{i \sum_j k_{\scriptscriptstyle P_j} x_j} \]

Free waves combined with tuned amplitudes, with quasimomenta obeying Pauli-like exclusion

Quantization: Bethe equations

Imposing periodicity of the wavefunction

\[ \Psi (\ldots, x_j + N, \ldots) = \Psi (\ldots, x_j, \ldots) \]
↧ $k \equiv k(\lambda) = \!\pi \!- \!2\,\mbox{atan}2\lambda$
\[ \left[\frac{\lambda_j + i/2}{\lambda_j - i/2}\right]^N = \prod_{k \neq j} \frac{\lambda_j - \lambda_k + i}{\lambda_j - \lambda_k + i} \]

Fully coupled set of equations

Analytic solution? only in exceptional circumstances

State classification

More "refined" (branch cut-aware) form of Bethe equations: log form, with $\phi_n(\lambda) \equiv 2\,\mbox{atan}\frac{2\lambda}{n}$

\[ N \phi_1 (\lambda_j) - \sum_k \phi_2 (\lambda_j - \lambda_k) = 2\pi I_j \]

with $I_j$ being quantum numbers, $\{ I \}$ labelling a state

◯ ⬤ ◯ ⬤ ⬤ ◯ ⬤ ⬤ ⬤ ◯ ⬤ ⬤ ◯ ⬤ ⬤ ◯
\[ \{ I \} ~\substack{\text{Bethe}\\ \text{equations}\\ \leftrightarrow} ~\{ \lambda \} ~\substack{\text{Bethe}\\ \text{Ansatz}\\ \leftrightarrow} ~\Psi \]

The Lieb-Liniger Model

$ H = \int_0^L dx \left[ \partial_x \Psi^{\dagger} \partial_x \Psi + c \Psi^{\dagger} \Psi^{\dagger} \Psi \Psi \right] $

Eigenstates: Bethe Ansatz

\[ \begin{align} \Psi_N(\{ x \} | \{ \lambda \}) &= \prod_{N \geq j_1 > j_2 \geq 1} sgn(x_{j_1} - x_{j_2}) sgn (\lambda_{j_1} - \lambda_{j_2}) \times \\ &\times \sum_{P \in \pi_N} (-1)^{[P]} e^{i \sum_{j=1}^N \lambda_{P_j} x_j + \frac{i}{2} \sum_{N \geq j_1 > j_2 \geq 1} sgn(x_{j_1} - x_{j_2}) \phi (\lambda_{P_{j_1}} - \lambda_{P_{j_2}})} \end{align} \]

in which $\phi(\lambda) \equiv 2\,\mbox{atan}(\lambda/c)$

The Lieb-Liniger Model

Rapidities must obey the Bethe equations:

$ e^{\mathrm{i}\lambda_j L} = \prod_{\substack{{\tiny l=1} \\ {\tiny (l\neq j)}}}^{\tiny N} \frac{ \lambda_j - \lambda_l + \mathrm{i}c}{\lambda_j - \lambda_l - \mathrm{i}c} $

Eigenvalues of conserved charges:

$ Q_n\left(\{\lambda\}_N\right) = \sum_{j=1}^N \lambda_j^n, \quad n = 1,2,\ldots $

with in particular $P = Q_1, E = Q_2$.

The Lieb-Liniger Model

Description using the Algebraic Bethe Ansatz

\[ \tau(\lambda) = tr T(\lambda), \hspace{5mm} T(\lambda) = \left( \begin{array}{cc} A(\lambda) & B(\lambda) \\ C(\lambda) & D(\lambda) \end{array} \right), \hspace{5mm} | \Psi_N \left( \left\{ \lambda \right\}_N \right \rangle = \prod_j B(\lambda_j) | 0 \rangle \]

Matrix elements of physical operators known from:
  • Gaudin formula for state norm

    \[ \langle \{ \lambda \} | \{ \lambda \} \rangle = c^N \prod_{j > k} \frac{\lambda_{jk}^2 + c^2}{\lambda_{jk}^2} \mbox{det} \left[ \delta_{jk} \left(L + \sum_l K_{jl} \right) - K_{jk} \right], \hspace{10mm} K_{jk} = \frac{2c}{\lambda_{jk}^2 + c^2} \]

  • solution of inverse problem $A, B, C, D \leftrightarrow \psi(x), \psi^\dagger(x)$
  • Slavnov formula for $\langle 0 | \prod_j C(\lambda_j) \prod_k B(\mu_k) | 0 \rangle$

Integrability
in
equilibrium

Equilibrium properties

  • Statics (partition function):

    ⮕ Thermodynamic Bethe Ansatz (1967+)

  • Dynamics:

    \[ {\cal S}^{a \bar{a}} (k, \omega) \equiv \frac{1}{N} \sum_{j, j'} e^{-i k(j-j')} \int_{-\infty}^\infty dt e^{i\omega t} \langle \frac{1}{2} \left[ {\cal O}^a_j (t), ({\cal O}^a_{j'} (0))^\dagger \right] \rangle \]

    (as far as I'm concerned) ⮕ ABACUS

Dynamical correlation functions
from (algebraic) Bethe Ansatz

ABACUS (post Achilles)

Abacus_Explorer.png

Equilibrium Correlations
in Spin Chains

The structure factor
of the Heisenberg chain

Heisenberg DSF: experimental correspondence

KCuF3
(Lake et al, PRL 2013)
CuSO4
(Mourigal et al, Nat Phys 2013)

Equilibrium Correlations
in Quantum Gases

Dynamical structure factor
from (algebraic) Bethe Ansatz



The ABACUS algorithm


Dynamical structure factor from Bethe Ansatz

extension to finite T

M. Panfil and J-SC, PRA 89 (2014)

Lieb-Liniger using cold atoms (I)

Lieb-Liniger using cold atoms (II)

Integrability
out of
equilibrium

Out-of-equilibrium: why?

Why NOT?

  • Real systems equilibrate very rapidly
  • Experimentally: tough to control
  • Theoretically: can't do it - toolbox too limited

Why INDEED?

  • Real isolated systems can equilibrate slowly
  • Novel states & properties
  • Experimentally: frontier of current capabilities
  • Theoretically: the toughest challenge around?

Mechanical
motivation

Original experimental motivation

Quantum version of Newton's cradle

Kinoshita, Wenger, Weiss (2006)

Ergodicity: broken

Out of equilibrium: what? how?

Probed
  • Neutron scattering
  • Bragg spectroscopy
  • Transport
  • Linear response
  • Traditional methods
  • CFT, Bosonization, ...
Pulsed
  • Magnetic resonance
  • Photoemission
  • Ultrafast spectr.
  • Free models
  • Numerics
  • Nonlinear response theory
Quenched
  • Feshbach resonances
  • Integrability
  • Hydrodynamics ($t\rightarrow \infty$)
  • GGE ($t = \infty$)
  • Quench Action ($\forall t$)
Driven
  • Masers/lasers
  • Resonators
  • Free models
  • Floquet theory
  • Numerics
  • Despair?

Let's

Interaction quench in Lieb-Liniger

We will consider a sudden switch of the interaction:

$ c (t) = \left\{ \begin{array}{ll} c_i & t < 0, \\ c_f & t > 0 \end{array} \right. $

starting in ground/(any eigen)state of $H(c_i)$ at $t < 0$

For many reasons, this is a complicated problem...

  • GGE logic fails, charges have $\infty$ exp. values
  • approaches using regularizations (q-bosons, extended charges, ...): no complete solution

The strategy

We write the initial state in the basis of the eigenstates of $H(c_f)$, denoted as $\left\{ |\{\lambda\}^{(n)}_N\rangle \right\}, n = 0, 1, ...$

The initial state is obtained as a linear decomposition weighed by exact overlaps:

$ |\Psi_i\rangle = \sum_{n=0}^\infty |\{\lambda\}^{(n)}_N\rangle \underbrace{\langle \{\lambda\}^{(n)}_N | \Psi_i\rangle}_{\text{the overlaps}} $

Exact time-evolved state:

$ |\Psi_i(t)\rangle \equiv e^{-iH(c_f)t} |\Psi_i\rangle = \sum_{n=0}^\infty e^{-\mathrm{i}E(\{\lambda\}^{(n)}_N)t} |\{\lambda\}^{(n)}_N\rangle \langle \{\lambda\}^{(n)}_N | \Psi_i\rangle $

The Quench Action

Setup

The initial state is decomposed in the basis of post-quench Hamiltonian eigenstates:

$ |\Psi (t=0) \rangle = \sum_{\{ I \}} e^{-S^\Psi_{\{ I \}}} | \{ I \} \rangle, $

with overlaps encoded in the coefficients

$ S^\Psi_{\{ I \}} = -\ln \langle \{ I \} | \Psi (t=0) \rangle ~\in ~\mathbb{C} $

Exact time-dependend expectation values:

$ \bar{\cal O} (t) = \frac{ \sum_{\{ I^l \} } \sum_{\{ I^r\}} e^{-(S^\Psi_{\{ I^l\}})^* - S^\Psi_{\{ I^r\}} + i (\omega_{\{ I^l\}} - \omega_{\{ I^r\}})t} \langle \{ I^l \}| {\cal O} | \{ I^r \} \rangle}{\sum_{\{ I \}} e^{-2\Re S^\Psi_{\{ I \}}}} $

Going to the thermodynamic limit

$\sum$ over eigenstates: replaced by functional integrals

$ \lim_{Th,reg} \sum_{\{ \rho_i \}} (...) = \int_{\rho_{sm} \in C^\infty} D\rho_{sm} (...) $

Normalization sum (trivially 1) represented as

$ \lim_{Th,reg} \langle \Psi (t) | \Psi (t) \rangle = \int D\rho_{sm} ~ e^{-S^Q[\rho_{sm}]} $

where the Quench Action is defined as

$ S^{Q}[\rho] = S^o[\rho] - S^{YY}[\rho] $

QA: just like stat mech but overlap $\leftrightarrow$ (pseudo-)energy

Time-dependent expectation values

In ThLim, saddle-point evaluation $\left. \frac{\delta S^Q[\rho]}{\delta \rho} \right|_{\rho_{sp}} = 0$ exactly gives full time evolution of observables:

$ \begin{align} \lim_{Th,reg} \bar{\cal O} (t) = \lim_{Th,reg} \frac{1}{2} \sum_{\bf e} \left[ e^{-\delta S_{\bf e}[\rho_{sp}] - i \omega_{\bf e}[\rho_{sp}] t} \langle \rho_{sp} | {\cal O} | \rho_{sp}; {\bf e} \rangle \right. \nonumber \\ \left. + e^{-\delta S^*_{\bf e}[\rho_{sp}] + i \omega_{\bf e}[\rho_{sp}] t} \langle \rho_{sp}; {\bf e} | {\cal O} | \rho_{sp} \rangle \right] \end{align} $

provided the
operator ${\cal O}$ is

  • weak
  • smooth
  • thermodynamically finite

The BEC to Lieb-Liniger quench

Initial state: ground state of free bosons

$ | \Psi_0 \rangle = \frac{1}{\sqrt{L^N N!}} \left(\psi_{k=0}^\dagger\right)^N | 0 \rangle $

Exact overlaps with finite $c$ Lieb-Liniger states:

$ \langle \Psi_0 | \{ \lambda \}_{N/2} \cup \{ -\lambda \}_{N/2} \rangle = \left[ \frac{(cL)^{-N} N!}{\mbox{det}_N G_{jk} }\right]^{1/2} \frac{\mbox{det}_{N/2} G^Q_{jk}}{\prod_{j=1}^{N/2} \frac{\lambda_j}{c} \left[ \frac{\lambda_j^2}{c^2} + \frac{1}{4} \right]^{1/2}} $

in which $G_{jk}$ is the Gaudin matrix

$ G_{jk} = \delta_{jk} \left[ L + \sum_{l=1}^N K (\lambda_j, \lambda_l) \right] - K(\lambda_j, \lambda_k) $

Exact solution using Quench Action

$ S^Q = L\int_0^\infty d\lambda \rho(\lambda) \log\left(\frac{\lambda^2}{c^2} \left( \frac{1}{4} + \frac{ \lambda^2}{c^2} \right) \right) + S^{YY} $

Saddle-point equation:

$ \ln \eta(\lambda) = g(\lambda) - h - \int_{-\infty}^\infty \frac{d\lambda'}{2\pi} K (\lambda - \lambda') \ln \left[ 1 + \eta^{-1}(\lambda') \right], $

with driving term $ g(\lambda) = \ln \left[ \frac{\lambda^2}{c^2} \left( \frac{\lambda^2}{c^2} + \frac{1}{4} \right) \right] $

Analytic solution for steady state:

$ \begin{align} \rho_{sp} (\lambda) &= -\frac{\gamma}{4\pi} \frac{1}{1 + a(\lambda)} \frac{\partial a(\lambda)}{\partial \gamma},\\ a(\lambda) &\equiv \frac{2\pi}{\frac{\lambda}{n} \sinh \frac{2\pi \lambda}{c}} I_{1 - 2i\frac{\lambda}{c}} \left(\frac{4}{\sqrt{\gamma}}\right) I_{1 + 2i\frac{\lambda}{c}} \left(\frac{4}{\sqrt{\gamma}}\right) \end{align} $

Exact solution using Quench Action

Steady state distribution: non-thermal
J. De Nardis, B. Wouters, M. Brockmann & J-SC, PRA 89, 2014

Exact solution using Quench Action

Steady state distribution: non-thermal

Generalized Hydrodynamics

Quenches from spatially inhomogeneous states

  • B. Bertini, M. Collura, J. De Nardis and M. Fagotti, PRL 117, 207201 (2016)
  • O. A. Castro-Alvaredo, B. Doyon and T. Yoshimura, PRX 6, 041065 (2016)
  • B. Doyon and T. Yoshimura, SciPost Phys. 2, 014 (2017)


After initial dephasings: 'hydrodynamic' evolution described by local GGE

$\partial_t \rho (\lambda) + \partial_x (v^{\tiny \mbox{eff}}(\lambda) \rho (\lambda)) = 0$

with effective velocities determined by the dressing equation

$v^{\tiny \mbox{eff}}(\lambda) = v^{\tiny \mbox{gr}}(\lambda) + \int d\lambda' \frac{\varphi(\lambda, \lambda')}{p'(\lambda)} \rho(\lambda') \left(v^{\tiny \mbox{eff}}(\lambda') - v^{\tiny \mbox{eff}}(\lambda)\right)$

Effective dynamics:
"flea gas" of scattering quasiparticles

Flea gas GHD
for the Quantum Newton's Cradle

JSC, B. Doyon, J. Dubail, R. Konik and T. Yoshimura, SciPost Phys. 6, 070 (2019)

Short time scales:

Flea gas GHD
for the Quantum Newton's Cradle

JSC, B. Doyon, J. Dubail, R. Konik and T. Yoshimura, SciPost Phys. 6, 070 (2019)

Longer time scales:

Generalized Hydrodynamics with space/time-dependent potentials

Phys. Rev. Lett. 123, 130602 (2019)

Adiabatic formation of bound states in the 1d Bose gas

Phys. Rev. B 103, 165121 (2021)

Observing Bethe strings in an attractive Bose gas far from equilibrium

Milena Horvath, Alvise Bastianello, Sudipta Dhar, Rebekka Koch, Yanliang Guo,
Jean-Sébastien Caux, Manuele Landini, and Hanns-Christoph Nägerl, Nat Commun (2026)

Observing Bethe strings in an attractive Bose gas far from equilibrium

Milena Horvath, Alvise Bastianello, Sudipta Dhar, Rebekka Koch, Yanliang Guo,
Jean-Sébastien Caux, Manuele Landini, and Hanns-Christoph Nägerl, Nat Commun (2026)

Quasi-solitons

Solitons (classical)

John Scott Russell:
Solitary Wave of Translation (1834)

Solitons (classical)

(Boussinesq-)Korteweg-de Vries

\[ \partial_t u + u \partial_x u + \delta^2 \partial_x^3 u = 0 \]

Fermi-Pasta-Ulam(-Tsingou):
no ergodicity!

Zabusky-Kruskal (1965):
concept of a soliton

⮕ Classical Inverse Scattering

Lieb-Liniger Type II quasi-solitons

A. Sarvi, C. Vianello and JSC, tbp

Strategy:
  • start with the ground state
  • punch holes in small $k$ window
  • combine multiple states into wavepacket

Expectation: finite, but looooong lifetime

⮕ classical-like dynamics from quantum unitary time evolution

Lieb-Liniger Type II quasi-solitons

A. Sarvi, C. Vianello and JSC, tbp

Local Dynamics
in
Quantum Spin Chains

Quantum wake dynamics in Heisenberg

A. Scheie, P. Laurell, B. Lake, S. E. Nagler, M. B. Stone, JSC, D. A. Tennant
Nature Communications 13, 5796 (2022)

  • Inelastic neutron scattering on $K Cu F_3$
  • Really accurate data in momentum/frequency space enabling...
  • Fourier transform to real space / time
  • Look specifically at high-energy features

Quantum wake dynamics in Heisenberg

A. Scheie, P. Laurell, B. Lake, S. E. Nagler, M. B. Stone, JSC, D. A. Tennant
Nature Communications 13, 5796 (2022)

  • Quantum wake:
    a local, coherent, long-lived, quasiperiodically oscillating magnetic state emerging out of the distillation of propagating excitations following a local quantum quench

Quantum wake dynamics in Heisenberg

Quantum wake dynamics in Heisenberg

Quantum wake dynamics in Heisenberg

Selectively (patch-wise) Fourier transforming: $k=\pi/2$ (mod $2\pi$) states are responsible for the quantum wake dynamics

Beyond the reach of any low-energy method

Not mentioned here

(a selection out of many more)

  • Renormalization from integrability
  • (Nonlinear) Luttinger liquids
  • Lessons for field theory
  • "Bethe liquid theory"

Challenges and Opportunities

  • Detailed phenomenology for quantum gases
  • Transport: (sub/super)diffusion
  • Ultrafast magnetism
  • Central spin / NV centers
  • Floquet dynamics
  • Driven dissipative systems
  • Quantum circuits

Merci !