We consider a \(U(1)\) gauge theory in \(2+1\) dimensions and regularize it on a asymmetric lattice (Degrand and DeTar (2006)):

\[ S = \frac{\beta}{\xi} \sum_{x} \sum_{\nu} \left[1 - \frac{1}{N_c} \operatorname{ReTr} (P_{0\nu}(x)) \right] + \beta \xi \sum_{x} \sum_{j<i} \left[1 - \frac{1}{N_c} \operatorname{ReTr} (P_{ij}(x)) \right] \] where \(P_{\mu \nu}\) is the standard Wilson Plaquette (Gattringer and Lang (2009)): \[ P_{\mu \nu}(x) = U_\mu(x) U_{\nu}(x+\hat{\mu}) U_{\mu}^{\dagger}(x + \hat{\nu}) U_{\nu}^{\dagger}(x) %\quad (\mu,\nu=0,..,d-1) \quad . \] At the boundary we impose periodic boundary conditions. \(\beta=\frac{2 N_c}{g^2}\) and \(\xi\) is the bare anisotropy.

Due to its group structure, this theory is often referred as (here anisotropic) QED \(2+1\) or QED\(_3\).

It is relevant for some condensed matter systems (see e.g. Kosiński et al. (2012)).

It resembles QCD (in \(3+1\) dimensions), showing:

• dynamical mass generation (Maris and Lee (2003))
• confinement (DeGrand (2019)).

• It is studied with Monte Carlo techniques : \[\mathcal{Z} = \int \mathcal{D} \phi e^{ - S[\phi]} = \int \mathcal{D} \phi e^{ - \sum_{x} \mathcal{L}[\phi](g_i^L)} \]

  software: https://github.com/urbach/su2

• There exist Hamiltonian formulations (Clemente et al. (2022)) suited to run on NISQ (Noisy Intermediate-Scale Quantum) devices in the near future. \[U(t) = e^{-i H(g_i^H) t}\]

We want to combine Lagrangian and Hamiltonian results

\(\to\) required non-perturbative matching of the couplings \(g_i\).

A little background…

Lattice correlators are defined as:

\[ C_T(t) = \langle O_1(t) O_2(0) \rangle_T = \frac{\operatorname{Tr}[e^{- H(T-t) } O_1 e^{- H t} \, O_2]}{\operatorname{Tr} [e^{- H T}] } \\ = \frac{\sum_n \langle{n}| e^{- H(T-t) } O_1 e^{- H t} \, O_2 |{n}\rangle }{\sum_{n} \langle{n}| e^{- H T} |{n}\rangle} \]

• We insert \(N\) identity operators for each timeslice \(t_k = \epsilon k \, (\text{aka} \quad a_t k)\): \[\mathbb{1} = \sum_{m} \frac{1}{2 E_m} |m \rangle \langle m |\]

• After some algebra we find:

\[C_T(t) = \frac{\int \mathcal{D} \phi \, e^{-S_T[\phi]} O_1(t) O_2(0) }{\int \mathcal{D} \phi \, e^{-S[\phi]}}\] where \(S_T\) is the discretized action (but with different lattice spacings in the time and space directions!).

• Now one says that in the limit \(a_t, a_s \to 0\) spacetime symmetry is recovered. \(\implies\) we define the path integral with \(a_t = a_s\).

• The limit \(a_t \to 0 \, , \, a_s \neq 0\) is the limit to the lattice Hamiltonian.

Steps:

• Calculation of \(n\) observables \(O_i\) in the two formalisms
• Lagrangian \(\to\) Monte Carlo
• Hamiltonian \(\to\) tensor network, quantum simulation, …

• In the action we introduce an anisotropy \(\xi\) \(\implies\) different temporal and spatial lattice spacings: \[a_t \neq a_s\]

• Find the limit \(a_t \to 0\) of the \(O_i\)

• Find the couplings \(g^H_i\) that match this limit of the Lagrangian

• Small \(\beta\) region: harder determination of \(\xi_R\) from the static potential

• Wilson flow renormalizes fields and the determination of \(\xi_R\) from the Wilson flow is known to work in QCD (Borsányi et al. (2012))

Main idea

• Compute observables at fixed \(a_s\):
  • \[\tau_0^2 E(\tau_0) = c\]
  • vary \(\xi\) and find the \(\beta\) s.t. \(a_s\) is constant
  • Find \(\xi_R = \xi(\tau_0)\) for each ensemble
  • Go to \(\xi_R \to 0\)
• Extrapolate to \(a_t \to 0\) through \(\xi_R = a_t/a_s\)

Here we consider the pure gauge theory and want to match the 1st eigenvalue (mass gap). This is the lightest state of the theory (Loan and Ying (2006)): glueball \(0^{--}\)

Numerical studies in \(U(1)\) (Athenodorou and Teper (2019)) and \(SU(N)\) (Teper (1998)) theories find that the glueball \(0^{--}\) is the lightest state.

Interpretation

In the continuum limit we expect a theory of free screened photons of mass \(m_D=m^{gs}_{0^{--}}\) (Athenodorou and Teper (2019)):

\[ m^{ex1}_{0^{--}} = \frac{3}{2} m^{gs}_{0^{++}} = 3 m^{gs}_{0^{--}} \]

\[ \phi_i(x) = \operatorname{Tr} \left[ \prod_{k \in \mathcal{L}_i} U_{\mu_k}(x_k+\hat{e}_k) \right] \quad , \, x_1=x_n=x \]

Combination of \(\phi_i\) with its transform under parity \(\phi^{(P)}\) (Gattringer and Lang (2009)) and taking their real/imaginary part: \[ \phi_i^{PC} = \begin{pmatrix} \operatorname{Re} \\ \operatorname{Im} \end{pmatrix} \left[ \phi_i \pm \phi_i^{(P)} \right] \quad , \]

• Finally, we project to \(\vec{p}=\vec{0}\): \[ \varphi_i^{PC}(t, \vec{0}) = \frac{1}{V} \sum_{\vec{x}} \phi_i^{PC}(t, \vec{x}) \\ = \frac{1}{L^2} \frac{2}{(d-1)(d-2)} \sum_{\vec{x}} \sum_{j<i} \operatorname{Re Tr} [ U_{ij}(t, \vec{x}) ] \]

The glueball correlators are build averaging over all possible timeslices:

\[ C_{r_i r_j}^{PC} (t) = \frac{1}{T} \sum_{\tau=1}^{T} \langle \varphi_{r_i}^{PC}(t+\tau) \varphi_{r_j}^{PC}(\tau) \rangle \]

The large-time behavior is: \[ C_{r_i r_j}^{PC} (t) \to v_{r_i r_j} + A_{r_i r_j} (e^{-E_0^{PC} t} \pm e^{-E_0^{PC} (T-t)}) \] where \(v_{r_i r_j}=\langle \varphi_{r_i}^{PC}(0) \varphi_j^{PC}(0) \rangle\).

The VEV is subtracted exactly as \(C_{\text{sub}}(t) = C(t)-C(t+1)\). Combining the \(C_{r_i r_j}\) we can apply the GEVP.

(work in progress…)

Lagrangian and Hamiltonian formalism are equivalent at zero temporal lattice spacing: \({a_t \to 0}\).

We write the action as:

\[S = \beta_t S_t + \beta_s S_s\] - \(\beta_t = \beta/\xi\) temporal coupling (electric field)

• \(\beta_s=\beta \xi\) spatial coupling (magnetic field)

\(\beta_t \neq \beta_s\) \(\to\) anisotropy of the lattice: we have a temporal and spatial lattice spacings, \(a_t\) and \(a_s\).

\(\xi \neq 1\)

How to go to \(a_t \to 0\) ?

Need to compute the renormalized anisotropy:

\[\xi_R = \frac{a_t}{a_s} \neq \xi\] and extrapolate along \(a_s = \text{const.}\)

How to compute \(\xi_R\)?

• Static potential \(V(r)\) between \(q\bar{q}\) pairs at distance \(r\).

• Wilson flow: \(\xi(t)\) fixed at some \(t_0/a_s^2\) along the flow.

In \(2+1 \, d\) the pair \(e^+ e^{-}\) at distance \(r\) has potential (Clemente et al. (2022)): \(V(r) = V_0 + \alpha \log{(r)} + \sigma r\)

Anisotropy

\[a_t V(a_s x) = \xi_R a_s V(a_s x)\] \(V\) is extracted from the \(t \gg 1\) behavior of Wilson loop expectation values (\(r = |x|\)):

\[ \frac{W_{ts}(x, t+1)}{W_{ts}(x, t)} \xrightarrow{t \to \infty} e^{- a_t \, V(r) } \]

\[ \frac{W_{ss}(x, y+1)}{W_{ss}(x, y)} \xrightarrow{y \to \infty} e^{- a_s \, V(r) } \]

Wilson flow evolution (Lüscher (2010)):

\[ \dot{V}_\tau(x, \mu) = - \frac{1}{\beta} \{ \nabla_{\mu}(x) S_G[V_\tau] \} \, V_\tau(x, \mu) , \quad V_0(x, \mu) = U(x, \mu)\]

where

\[\nabla_{\mu}(x) f(U) = -i T^a \frac{d}{d \omega} f \left(e^{i \omega T^a} U \right) |_{\omega=0}\]

Main properties:

• For any \(t > 0\) the fields are renormalized.

\(\to\) we can extract \(\xi_R\) from the flow evolution of \(E_{ts}\) and \(E_{ss}\)

• Perturbative calculation: \[\langle E \rangle \propto t^{-D/2}\] \(\to\) choose \(\tau_0\) in the non perturbative region of the flow.

The energy density of the system is:

\[ \mathcal{E} \propto \sum_{i \neq j} F_{ij}^2 + 2 \sum_{i} F_{0i}^2 = (d-1)(d-2) E_{ss} + 2 (d-1) E_{ts}\]

In the continuum limit \(E_{ss} = E_{ts} = \bar{E}\)

On the lattice we compute: \[ E_{ss}^{LAT} = a_s^4 (d-1) (d-2) E_{ss} \, , \quad E_{ts}^{LAT} = 2 a_t^2 a_s^2 (d-1) E_{ts} \] Renormalized anisotropy: \[ {\xi^{LAT}}^2_R = \frac{d-2}{2} \frac{\langle E_{ts}^{LAT} \rangle}{\langle E_{ss}^{LAT} \rangle} =\frac{d-2}{2} \frac{a_t^2 a_s^2 \langle E_{ts} \rangle}{a_s^4 \langle E_{ss} \rangle}\]

In the limit \(a_s \to 0\) we have: \(\xi^{LAT}_R \to \frac{a_t}{a_s}\).

• Complete determination of the glueball spectrum

• Determine \(\xi_R\) from the Wilson flow at small \(\beta\)

  • Expected better precision with respect to static potential
  • \(V(a_s) - V(a_s\sqrt{2})\) available for the matching at small volume

• Inclusion of staggered fermions (in progress): \[\mathcal{L}_F = \sum_{f} \sum_{x,y} \bar{\psi}(x) D(x | y) \psi(y)\]

Backup slides following

(work in progress…)

(work in progress…)