Simulating the lattice \(SU(2)\) Hamiltonian with discrete manifolds

Simone Romiti \(^{(a)}\)
T. Jakobs\(^{(a)}\), M. Garofalo\(^{(a)}\), T. Hartung\(^{(b)}\), K. Jansen\(^{(c)}\),
J. Ostmeyer\(^{(d)}\), D. Rolfes\(^{(a)}\), C. Urbach\(^{(a)}\)
\(^{(a)}\) University of Bonn (Germany) \(^{(b)}\) Northeastern University (London)
\(^{(c)}\) DESY Zeuthen (Germany) \(^{(d)}\) University of Liverpool

01 Aug 2023

Introduction and theoretical background

Hamiltonian simulations

Schrödinger equation \(\to\) evolution of a system: \[\begin{equation} \label{eq:SchroedingerEquation} i \frac{\partial}{\partial t} | \psi \rangle = H | \psi \rangle \, , \end{equation}\] but the Hilbert space is often infinite dimensional…

  • Truncation of the Hilbert space to a vector space \(\mathcal{V}\) of size \(N\)
  • Operators as matrices on \(\mathcal{V}\)
  • Limit recovered when \(N \to \infty\)

Formalism suited for: tensor networks, quantum devices

Lattice formulation

Gauge invariance: \[\begin{equation} \label{eq:UgaugeTransformation} U_\mu(x) \to V(x) U_\mu(x) V^{-1}(x + \hat{\mu}) \, . \end{equation}\] In the \(A_0 = 0\) gauge we find: \[ H = \frac{g^{2}}{2} \sum_{x, i, a} {(L_i)}_{a}^{2}(x) - \frac{1}{4 g^{2}} \sum_{x, i>j} \, \mathrm{Tr}[{U}_{i j}(x) + {U}_{i j}^\dagger(x) ] \, , \]

\(L_a L_a = R_a R_a\)

\([L_a, U] = - \tau_a U\)

\([R_a, L_b] = U \tau_a\)

Representing the Hilbert space (I)

A basis for the Hilbert space are the Lie algebra irreps (electric basis):

\[\ket{j, m, \mu} \, \, , \, j \in \mathbb{N}/2 \, \, \, |m|,|\mu|<j\]

Clebsh-Gordan expansion

\[\begin{equation} \begin{split} U^{(\alpha,\beta)} \ket{J,m,\mu} =& \sum_{j \in \mathbb{N}/2} \sqrt{\frac{2J+1}{2j+1}} \braket{J,m;\frac{1}{2},\alpha | j,m+\alpha} \\ & \braket{J,\mu;{\frac{1}{2}},\beta|j,\mu+\beta} \, \ket{j,m+\alpha,\mu+\beta} \, . \end{split} \end{equation}\]

This is all fine in an infinite dimensional space, but…

In a finite Hilbert space we have to give up something 🙁:

\[\operatorname{tr}[A,B] = 0\]

Representing the Hilbert space (II)

Clebsh-Gordan truncation

  • Commutation relations ✅
  • Gauss law invariance: \([H, G_a] = 0\) ✅
  • Non unitary links ❌
  • Need penalty term for \(G_a \ket{\psi} = 0\) ❌

Unitary links (our approach)

  • Commutation relations ❌
  • Gauss law breaking: \([H, G_a] \neq 0\) ❌
  • Unitary links ✅
  • \(U\) as gates \(\to\) initial state s.t. \(G_a \ket{\psi} = 0\) ✅

U(1) theory

Continuum limit on the manifold

Points on C are a basis:

\[U \ket{\alpha} = e^{i \alpha} \ket{\alpha}\]

The momenta are simply (abelian group):

  • \(L_a = -i \frac{d}{d \omega}\)

  • \(R_a = +i \frac{d}{d \omega}\)

SU(2) theory

Derivatives on \(S_3\)

Eigenfunctions on \(S_3\) (Wigner D-functions):

\[ D(\theta, \phi, \psi) = e^{i m \phi} d^j_{m,\mu}(\theta) e^{i\mu\psi} \]

\[\begin{align*} {L}_\pm &= e^{\mp i \phi} \left[ \pm \frac{1}{\sin{\theta}} \frac{\partial}{\partial \psi} + \frac{\partial}{\partial \theta} \mp \cot{\theta} \frac{\partial}{\partial \phi} \right] \\ {L}_3 &= -i \frac{\partial}{\partial\phi} \end{align*}\]

\(su(2)\) irreducible representations

\[\begin{eqnarray*} \left(\sum_a R_a^2\right) | j, m, \mu \rangle = \left(\sum_a L_a^2\right) | j, m, \mu \rangle = j(j+1) | j, m, \mu \rangle \\ L_3 | j, m, \mu \rangle = m | j, m, \mu \rangle \\ R_3 | j, m, \mu \rangle = -\mu | j, m, \mu \rangle \\ (L_1 \pm i L_2) | j, m, \mu \rangle = \sqrt{j(j+1) - m (m \pm 1)}| j, m \pm 1, \mu \rangle \\ (R_1 \mp i R_2) | j, m, \mu \rangle = -\sqrt{j(j+1) - \mu (\mu \pm 1)}| j, m, \mu \pm 1 \rangle \end{eqnarray*}\]

Now fix a truncation: \(j \leq q\). We have \(N_q\) states:

\[ N_q = \sum_{j \leq q} (2j+1)^2 = \frac{1}{6}(4q+3)(2q+2)(2q+1) \sim O(q^3) \]

Question: How many eigenstates of \(U\) can I reproduce in the discrete space?

Truncated \(su(2)\) irreps (example)

\[\begin{align*} \label{eq:L3MatrixRepElectricBasis} L_3^{\text{el.}} &= \sum_{j=0}^{q} \sum_{|m| \leq j} | j, m \rangle m \langle j, m| \\ & \, \dot{=} \, \begin{bmatrix} & \cdots & \cdots & 0 \\ \vdots & \ddots & & \\ 0 & \cdots & \begin{bmatrix} \begin{bmatrix}1/2&0\\0&-1/2\end{bmatrix}_{\mu=1/2} & 0 \\ 0 & \begin{bmatrix}1/2&0\\0&-1/2\end{bmatrix}_{\mu=-1/2} \end{bmatrix}_{j=1/2} & 0 \\ 0 & \cdots & 0 & \begin{bmatrix}0\end{bmatrix}_{j=0} \\ \end{bmatrix} \, , \end{align*}\]

Question: How many of these survive after discretizing the \(S_3\)?

Spoiler alert ⚠: It depends on the discretization (see e.g. M. Garofalo - Canonical Momenta in Digitized SU(2) Lattice Gauge Theory)

Frequencies on \(S_3\)

\(S_3\) is a non-abelian manifold \(\to\) \(N_\alpha\) points cannot sample \(N_\alpha\) Fourier modes! (c.f. Shannon-Nyquist theorem)

\(N_\alpha > N_q\) 😤

\[ N_\alpha \geq \begin{cases} (q + 1/2) (4q+1)^2 & q \, \text{half integer}\\ (q + 1) (4q+1)^2 & q \, \text{integer} \end{cases} \]

Physical consequence:

  • \(U^\dagger U = U U^\dagger = 1\) \(\implies\) \(\nexists\) square matrix \(V\) of change of basis between electric and magnetic basis.

Canonical momenta on \(S_3\) partitionings (I)

  • \(V\) is at most rectangular \(\to\) enlarging the space of the first \(N_q\) \(su(2)\) irreps.

  • Presence of extra “garbage states” 🗑️

What is the form of \(V\)? \[\begin{equation*} f(\vec{\alpha}_k) = f(\theta, \phi, \psi) = \sum_{j=0}^{q} \sum_{m,\mu = -j}^{j} {V}^j_{m,\mu}(\vec{\alpha}_k) \hat{f}(j, m, \mu) \end{equation*}\]

Discrete Jacobi Transform

\[\begin{equation} {V}^j_{m,\mu}(\vec{\alpha}_k) = (j + 1/2)^{1/2} \sqrt{\frac{w_s}{N_\phi N_\psi}} D^j_{m,\mu}(\vec{\alpha}_k) \end{equation}\]

  • \(w_s\) Gaussian weights of Legendre polynomials
  • \(V\) of size \(N_\alpha \times N_q\)
  • \(V^\dagger V = 1_{N_q \times N_q}\) (but not \(V V^\dagger = 1_{N_\alpha \times N_\alpha}\))
  • \(\operatorname{dim}[\operatorname{ker}(V^\dagger)] = N_\alpha - N_q\)

Properties of the discrete momenta

\[\begin{equation*} L_a = V \hat{L}_a V^{\dagger} \,\, , \,\, R_a = V \hat{R}_a V^{\dagger} \end{equation*}\]

Properties:

  • Exact Lie algebra: \(if_{abc}\) ✅
  • First \(N_q\) eigenstates \(\ket{j, m, \mu}\) reproduced exactly ✅
  • Commutation relations fulfilled for the first \(N_{q^{'}}=N_{q-1/2}\) irreps ✅

Vacuum and Gauss Law

  • Dense matrices for the momenta (local for \(q \to \infty\)) ❌

  • \(N_\alpha - N_q\) states degenerate with the electric vacuum ❌

    \(\to\) lift with projector \(P_{j>q}\) \(\to\) decoupled ✅

  • \([G_a, H] \neq \vec{0}\) on \(N_\alpha-N_{q^{'}}\) states. ❌

Preliminary results: \(SU(2)\) in 1+1 dimensions

Conclusion

  • We can’t have both unitary links and exact commutation relations on all states:

\[[L_a, U] \, ✅ \implies U U^\dagger = U^\dagger U = 1 \, ❌\]

\[U U^\dagger = U^\dagger U = 1 \, ✅ \implies [L_a, U] \, ❌\]

  • Both unitary and non-unitay links formulations deserve to be considered

  • Unitary links limit the number of faithful represented electric eigenstates

  • Desirable feature: being able to reduce the dimensionality of the space, e.g. constraining the values of the plaquette close to \(1\).

Thank you for your attention!

Backup

Truncated Clebsh-Gordan expansion when \(a \to 0\) (I)

Approaching \(a \to 0\)…

Truncated Clebsh-Gordan exapansion: \(U\) don’t resemble group elements anymore

  • Need to check \(\exists\) critical point

  • It is the same as the continuum theory?

    • what are the residual symmetries?
    • can we exclude nasty operators?

Discrete manifolds when \(a \to 0\) (II)

Approaching \(a \to 0\)…

Unitary links: \(U\) take values in the manifold

  • Same as Lagrangian simulation (finite machine precision \(\to\) not exactly \(SU(2)\))

  • Need to check \(\exists\) 2nd order phase transition at finite \(N\) (Monte Carlo with same partitioning)

  • Check that it has the same step scaling function

\(U(1)\) theory: \(a \to 0\)

\[U_{\mu \nu} \ket{\alpha_{1\to4}} = e^{i \alpha} \ket{\alpha_{1\to4}}\]

In the continuum limit \(a \to 0\) the plaquette approaches \(1\):

\[U_{\mu \nu} = e^{i a^2 F_{\mu \nu} + O(a^3)} = 1 + i a^2 F_{\mu \nu} + \ldots\]

\(\to\) restrict to the corresponding eigenstates gives an effective theory for fine lattices (if correlation length fits the lattice)

S. Romiti - Simulating the lattice \(SU(2)\) Hamiltonian with discrete manifolds