# Seiberg, Lattice vs Continuum QFT 

:::{.remark}
QFT: successful but still not mathematically rigorous!
Regularize by discretizing space to a lattice, then the functional integral is well-defined.
Then take a continuum limit taking spacing to zero but fixing lengths, compute correlation functions.
For condensed matter, find low-energy/long-distance limit.
Expect it to be described by an effective continuum field theory.
:::

:::{.remark}
Challenges with just regularizing to a lattice and taking a limit:

- Does the limit exist?
- Is the limit independent of details at finite levels?
- Lattice theory doesn't necessarily capture topological features ($\pi_1$, characteristic classes, Chern-Simons terms, etc).
  These are tied to global symmetries, anomalies.
- Some QFTs (e.g. with self-dual forms or fermions) don't allow putting DOF on the lattice, can't write an action.
- Some QFTs may not have a continuum Lagrangian (e.g. $2,0$ theory), so can't do a lattice Lagrangian.
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:::{.remark}
Opposite problem: given an actual physical lattice, find the low energy continuum model.
Often solvable, so can couple with whatever, but continuum limits have divergence issues.
E.g. $XY\dash$plaquette, fracton models.
Some issues:

- Separate symmetric group elements for different subspaces
- Observables vary at lattice scale $a$, and can become discontinuous in the limit
- Infinite ground state degeneracy in the limit
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:::{.remark}
Very well understood model: $1+1$ boson.
Take a 2d lattice with periodic boundary conditions, action 
\[
S = -\beta \sum \cos(\Delta_\mu \phi)
,\]
where $e^{i\phi}$ are phases on the lattice points.
Sum over "links" (edge..?), $\Delta_\mu$ is like a discrete derivative.
Take limit to get
\[
S = {\beta\over 2}\int \qty{ \dd{\phi}{\mu} }^2 \dtau \dx
.\]
A symmetry emerges in the continuum: the winding number of $\phi$, and a certain 't Hooft anomaly.
How much is present in the original lattice?
:::

:::{.remark}
Try to make those symmetries appear on the lattice.
Idea: replace $\cos$ by a linear function, add a correction term that sums over plaquettes that is roughly curvature, scale by a constant that forces curvature to be zero.
See *Villain form*.
This kills the vorticity, which helps make the winding number show up on the lattice.
Use Poisson resummation to show self-duality.

Suspected causes of anomalies: infinities, fermions, issues with invariance of measure.
But none are present here!
:::

:::{.remark}
A slightly more complex model: $XY\dash$plaquette in $2+1$ dimensions.
Put phases $e^{i\phi}$ on nodes with action
\[
S = -\beta_0 \sum_{\text{edges}} \cos(\delta_{\tau} \phi) - \beta \sum_{\text{plaqs}} \cos(\Delta_x \Delta_Y \phi)
.\]
Has a global $\U_1$ symmetry, take continuum limit to get
\[
S = \int \dtau \dx \dy \, {\mu_0\over 2 }\qty{ \dd{\phi}{\tau} }^2 + {1\over 2\mu}\qty{ {\partial^2 \over \partial_x \partial_y} \phi }^2
.\]

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:::{.remark}
Other questions to ask to compare lattices to continuum limit:

- What are the (operator) spectra of the Hamiltonians?
- What are the correlation functions?

Importantly: limits don't commute!
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:::{.remark}
Interesting generalizations:

- Replace $\U_1$ with $C_n$
- Go to $3+1$ dimensions
- Look at subsystem symmetries

Modify the Villain term, compute the spectra and correlation functions to study.
QCD: infinite number of states (vs finite number of states in these kinds of models?)
There's a difference between global and gauge symmetries, exotic models land somewhere in between.
See UV/IR mixing (short/long distances).
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