seminar--机器学习用于寻找Integrable system

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overview

  • Initialization: consider \(d = 3\) and \[ R= \begin{pmatrix} * & 0 & * & 0 & * & 0 & * & 0 & * \\ 0 & * & 0 & * & 0 & * & 0 & * & 0 \\ * & 0 & * & 0 & * & 0 & * & 0 & * \\ 0 & * & 0 & * & 0 & * & 0 & * & 0 \\ * & 0 & * & 0 & * & 0 & * & 0 & * \\ 0 & * & 0 & * & 0 & * & 0 & * & 0 \\ * & 0 & * & 0 & * & 0 & * & 0 & * \\ 0 & * & 0 &*&0&*&0&*&0\\ * & 0 & * & 0 & * & 0 & * & 0 & * \end{pmatrix}, \]

  • Using Neural network solve YBE of this ansatz numerically

  • Look for algebraic variety of integrable model in the vicinity of numerically found Hamiltonian.

  • R-matrix is extracted using hamiltonian.

Quantum integrable spin-chain

Hamiltonian \[ H=\sum_{i=1}^LH_{i,i+1}, \] existence of a tower of mutually commuting charges \[ [\mathbb{Q}_m,\mathbb{Q}_n]=0\mathrm{~.} \] R matrix is a matrix satisfy Yang-Baxter equation \[ R_{ij}(u-v)R_{ik}(u)R_{jk}(v)=R_{jk}(v)R_{ik}(u)R_{ij}(u-v) \] R matrix has property \[ R_{ij}(0)=P_{ij} \] Transfer matrix \([T(u),T(v)]=0\) . \[ T(u)=\mathrm{tr}_a\left(R_{a,L}(u)R_{a,L-1}(u)\ldots R_{a,1}(u)\right)\mathrm{~,} \] Transfer matrix can be expanded using conserved charge \[ \log T(u)=\sum_{n=0}^\infty\mathbb{Q}_{n+1}\frac{u^n}{n!}\mathrm{~.} \] in particular, second charge is hamiltonian \(\mathbb{Q}_2=H\). Hamiltonian density can be generated from R-matrix (P is permutation operator) \[ \begin{aligned} H_{i,i+1} & =R_{i,i+1}^{-1}(u)\frac{d}{du}R_{i,i+1}(u)|_{u=0} \\ & =P_{i,i+1}\frac{d}{du}R_{i,i+1}(u)|_{u=0}, \end{aligned} \] class of equivalence modulo R matrix:

  • similarity transformation \[ (\Omega\otimes\Omega)R(u)(\Omega^{-1}\otimes\Omega^{-1}) \]

  • rescaling transformation \[ u\to cu,\forall c\in\mathbb{C}, \]

  • mutiplication by any scalar holomorphic function \[ R(u)\to f(u)R(u) \]

  • permutation,transpositation and their composition \[ PR(u)P,R(u)^T,PR^T(u)P. \]

families of integrable models as algebraic varieties

define boost operator \[ \mathcal{B}=\sum_{a=-\infty}^\infty aH_{a,a+1} \] higher charges can be generated with boost operator \[ \mathbb{Q}_{r+1}=\left[\mathcal{B},\mathbb{Q}_r\right]. \] denote charge as (L means spin-chains of finite length L) \[ \mathbb{Q}_r^L=\sum_{a=1}^LH_{a,a+1,..,a+r-1}. \] focus on the constraint: (Re'shetikhin condition) \[ [\mathbb{Q}_2^L,\mathbb{Q}_3^L]=0\mathrm{~,} \] This is a necessary condition for quantum integrability.

  • in order for this condition to be non degenerate, minimal length of spin chain should be \(L=4\)
  • if site-space is d-dimensional two-particle hamiltonian has \(d^4\) parameters with \(d^8\) scalar equations.

computational AG

for a polynomial ring \(\mathbb{CP}^{d^4-1}\)。Define reducible ideal by equations \[ I\subset\mathbb{C}[h_{ij}] \] algebraic variety can be decomposed in to irreducible components \[ V(I)=\bigcup_iV(I_i)\mathrm{~,} \] in one irreducible branch, using gauge transformation to set \(d^2-1\) elements in H to be 0。number of none zero element in R-matrix is \(\Pi_R = k<d^4-d^2-1\)

Neural networks

use fully-connected neural network, define matrix norm as \[ ||A|| = \sum_{\alpha\beta}^n|A_{\alpha\beta}| \] Yang-Baxter equation loss \[ \begin{aligned} \mathcal{L}_{YBE} & =||\mathcal{R}_{12}(u_c)\mathcal{R}_{13}(u_a)\mathcal{R}_{23}(u_b) \\ & -\mathcal{R}_{23}(u_b)\mathcal{R}_{13}(u_a)\mathcal{R}_{12}(u_c)||, \end{aligned} \] locality condition \[ \mathcal{L}_{reg}=||\mathcal{R}\left(0\right)-P||. \] once we found a certain hamiltonian from the given class, active repulsion loss \[ \mathcal{L}_{repulsion}=\exp\left(-\|H-H_o\|/\sigma\right), \] loss for Reshetikhin condition \[ \mathcal{L}_{Q_2Q_3}=\max\left|[\mathbb{Q}_2,\mathbb{Q}_3]\right|. \] Anti-collaps (m.c. means mode collapse):

  • prvent every coefficient to be zero \[ \begin{aligned} L_{\mathrm{m.c.}}=\left|\frac{1}{n}\sum_{i,j}|H_{ij}|-1\right| \end{aligned} \]

  • ensure the matrix elements has bose diagonal and off-diagonal \[ \tilde{L}_{\mathrm{m.c.}}=\left|\frac{1}{n_{\mathrm{diag}}}\sum_{i}|H_{ii}|-\lambda_{\mathrm{diag}}\right|+\left|\frac{1}{n_{\mathrm{off-diag}}}\sum_{i\neq j}|H_{ij}|-\lambda_{\mathrm{off-diag}}\right| \]

  • repulsive from known results \[ L_{\mathrm{m.c.}}^{\mathrm{total}}=\tilde{L}_{\mathrm{m.c.}}+\left|\sum_{k}\mu_{k}\left|C_{k}\right|-\lambda_{\mathrm{repul}}\right| \]

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flowchart TB
  A[Sample spectral parameters ua, ub; set uc = ua - ub] --> B
  B[Choose vertex pattern PR, index set of nonzero R entries] --> RNET

  subgraph RNET[R Matrix Net - model each nonzero entry]
    direction TB
    IN[Input: scalar spectral parameter u] --> MLPK[Multiple MLPs produce rk u]
    NOTE1[Each rk has its own MLP; output uses identity activation]
  end

  RNET --> C1[Assemble R12 at uc]
  RNET --> C2[Assemble R13 at ua]
  RNET --> C3[Assemble R23 at ub]

  subgraph LOSSES[Loss computation]
    direction TB
    YBE[Difference between R12 R13 R23 and R23 R13 R12] --> LYBE[LYBE equals residual norm]
    REG[Regularity - constrain R at 0 to permutation matrix P] --> LREG[LREG]
    RESH[Extract Q2 and Q3 from transfer matrix] --> LQ[LQ equals commutator norm]
    MC[Mode collapse guard - separate diagonal vs off diagonal, avoid known families] --> LMC[LMC]
    REP[Repulsion - push away from known H0 neighborhood] --> LREP[LREP]
  end

  C1 --> YBE
  C2 --> YBE
  C3 --> YBE

  LYBE --> SUM
  LREG --> SUM
  LQ --> SUM
  LMC --> SUM
  LREP --> SUM

  SUM[Total loss L equals weighted sum of all loss terms] --> OPT[Optimization - Adam with JAX JIT; learning rate annealed 1e-3 to 1e-8]
  OPT --> OUT[Output - R of u and H seed; obtain H via derivative dR du at u equals 0]

Legend and mapping (no math symbols in the diagram):

  • ua, ub, uc = spectral parameters (u_a, u_b, u_c) with (u_c = u_a - u_b).
  • PR = the vertex pattern (_R), i.e., which entries of the (R) matrix are allowed to be nonzero.
  • rk(u) = scalar functions (r_k(u)) parameterizing the nonzero entries; each has its own MLP.
  • R12, R13, R23 = (R) evaluated on different tensor legs at (uc, ua, ub), respectively.
  • YBE / LYBE = Yang–Baxter equation residual and its norm.
  • REG / LREG = regularity constraint enforcing (R(0) = P) (permutation matrix) and its loss.
  • RESH / LQ = Reshetikhin auxiliary term using the commutator ([Q2, Q3]) and its loss.
  • MC / LMC = mode-collapse guard to avoid trivial or known families.
  • REP / LREP = repulsion term around a known solution (H0).
  • SUM = total loss (L) as a weighted sum of all terms.
  • OPT = optimization with Adam (JAX JIT), annealing learning rate from (10^{-3}) to (10^{-8}).
  • OUT = outputs the learned (R(u)) and the seed Hamiltonian (H), obtained from the derivative (R(u)) at (u=0) via the usual prescription.

find solution

  • from R-matrix net find new numerical solution $H_{seed}

  • solve Reshetikhin condition near \(H_{seed}\) , find a set of solution \[ V=\{H_{\mathrm{num}}^{(k)}\mid1\leq k\leq N\mathrm{~}\}. \]

  • their relation should be a linear transformation

  • denote a solution as a k- vector \(\eta = \{h_{ij}\}\)

  • ma ke N=K solutions to a matrix \[ M=\{\eta^{(1)},\ldots,\eta^{(N)}\}. \]

  • SVD decomposition \[ M = u\sigma v^\dagger \]

  • Zero-eigen value means linear constraint。 （rank = i0-1） \[ R_i:\sum_{\alpha=1}^Kv_{\alpha i}\left.\eta_\alpha=0,\quad\right.\quad i\geq i_0, \]

  • use \(i_0-1\) basis others can be represented by them \[ \eta_\beta=\sum_{i=1}^{i_0-1}\tilde{c}_{\beta i}\eta_i,\quad\beta>i_0-1. \]

  • set the numerical coefficients to integer or rational number, and regard them as precission relation.

  • reconstruct analytical result from function \[ H_{i,i+1}=R_{i,i+1}^{-1}(u)\left.\frac{d}{du}R_{i,i+1}(u)\right|_{u=0}=P_{i,i+1}\left.\frac{d}{du}R_{i,i+1}(u)\right|_{u=0} \] and \[ R_{ij}(0)\sim P_{ij} \]