1. Tensor Network Simulations of QFT in Curved Spacetime
Adam GM Lewis, Qi Hu, Guifre Vidal
Perimeter institute for theoretical physics, Waterloo, On, CA

2. Motivation: Simulate Semiclassical Gravity
-Numerical integration of the Einstein equations is now a pretty mature field – LIGO, etc.
(SXS Collab)
-Another situation where quantitative information might be useful: semiclassical gravity.
-Inflation? Black hole evaporation? Black hole formation? With interactions?
Chakraborty, Sumanta & Lochan, Kinjalk. (2017). Black Holes: Eliminating Information or Illuminating New Physics?. Universe /universe

3. Numerical Relativity
Foliate spacetime to get initial value problem.
EFEs -> PDE system evolving hypersurface-local quantities.
Diff. symmetry becomes a literal gauge symmetry.
We can project other equations onto the hypersurfaces just as well.
"Exploring New Physics Frontiers Through Numerical Relativity"
Vitor Cardoso and Leonardo Gualtieri and Carlos Herdeiro and Ulrich Sperhake 

4. Numerical QFTCS?
"Exploring New Physics Frontiers Through Numerical Relativity"
Vitor Cardoso and Leonardo Gualtieri and Carlos Herdeiro and Ulrich Sperhake 

5. Numerical Semiclassical Gravity?
Your Favourite Theory of Gravity

6. Hilbert space is infinite!
My laptop is finite.
The target expectation values aren’t even defined!

7. How to represent an infinite state?
As limit of sequence of lattice states.
Same problem occurs classically.
𝜑 1
𝜑 2
𝜑 3
𝜑 1
𝜑 2
𝜑 3
𝜑 4
𝜑 5
𝜑 6
𝜑 1
𝜑 2
𝜑 3
𝜑 4
𝜑 5
𝜑 6
𝜑 7
𝜑 8
𝜑 9
𝜑 10
𝜑 11
𝜑 12
𝜑(𝑥)

8. But… Exponential Hilbert Space of N qubits
State Space of N Transistors

9. But unlike the classical case…
Hilbert Space of N qubits
Ground States of
Gapped 1D Local Hamiltonians
Exponential
State Space of N Transistors

10. But… Exponential Polynomial Hilbert Space of N qubits MPS Ansatz
State Space of N Transistors
Ground States of
Gapped 1D Local Hamiltonians
MPS Ansatz

11. MPS Ansatz Sparse data structure for many-body quantum states.
Polynomial
State Space of N Transistors
Ground States of
Gapped 1D Local Hamiltonians
MPS Ansatz
Sparse data structure for many-body quantum states.
Complexity exponential in entanglement rather than system size.
Can apply local and nearest-neighbor operators (time evolution).
Limited real-time evolution possible.
Prepares for near-term quantum simulators.

12. Theory Choice 1+1 Massive Fermions Interactions ok! Field ops.
Ground/Thermal States of
Gapped 1D Local Hamiltonians
Field ops.
Staggered fermions.
Equivalent spin chain.

13. Hypersurface-local Hamiltonian
Stagger

14. Strategy
-Prepare initial state e.g. by imaginary time evolution with this H.
-If real time evolution desired, simulate it once again using this H.
-Compute expectation values of discretized operators.
-Take continuum limit.
-Already successfully applied to Schwinger model (1+1 QED)
-See e.g. works by Karl Jansen, MC Banuls, PhD thesis of Kai Zapp…

15. Defining operators?
-Need a principled way to subtract divergent terms!
-In flat spacetime a heuristic fit Ansatz works because you only care about differences.

16. Covariant Point-Splitting

17. Covariant Point-Splitting

18. Covariant Point-Splitting

19. Hadamard Regularization
State Independent
Compute from field eqns
Regularized Bispinor

20. Hadamard Regularization

22. Note the lattice T00 and T11 will be the same operator with m=0!
Now we have a defined quantity to approximate on the lattice.

23. Quick Breather We want to simulate QFT on curved backgrounds.
Using staggered fermions, we can express the QFT as the continuum limit of a lattice theory.
Using matrix product states, we can efficiently represent the states of that lattice theory.
Using Hadamard regularization, we can extract meaningful finite numbers from the limits thus obtained.

24. Extracting the Hadamard Form
-Now that we can compute the Hadamard continuum counterterms, we need to take the continuum limits such that we can apply them.

25. Extracting the Hadamard Form
Flat spacetime result.
𝜑 1
𝜑 2
𝜑 3
𝜑 4
𝜑 5
𝜑 6
𝜑 7
𝜑 8
𝜑 9
𝜑 10
𝜑 11
𝜑 12

26. Extracting the Hadamard Form
Flat spacetime result.
𝜑 1
𝜑 2
𝜑 3
𝜑 4
𝜑 5
𝜑 6
𝜑 7
𝜑 8
𝜑 9
𝜑 10
𝜑 11
𝜑 12

27. Extracting the Hadamard Form
Flat spacetime result.
𝜑 1
𝜑 2
𝜑 3
𝜑 4
𝜑 5
𝜑 6
𝜑 7
𝜑 8
𝜑 9
𝜑 10
𝜑 11
𝜑 12

28. Analytically Diagonalize:
Must hold fixed

29. Analytically Diagonalize:
Must hold fixed
𝜑 1
𝜑 2
𝜑 3
𝜑 1
𝜑 2
𝜑 3
𝜑 4
𝜑 5
𝜑 6
𝜑 1
𝜑 2
𝜑 3
𝜑 4
𝜑 5
𝜑 6
𝜑 7
𝜑 8
𝜑 9
𝜑 10
𝜑 11
𝜑 12

30. Analytically Diagonalize:
Must hold fixed
𝜑 1
𝜑 2
𝜑 3
𝜑 1
𝜑 2
𝜑 3
𝜑 4
𝜑 5
𝜑 6
𝜑 1
𝜑 2
𝜑 3
𝜑 4
𝜑 5
𝜑 6
𝜑 7
𝜑 8
𝜑 9
𝜑 10
𝜑 11
𝜑 12

31. Hawking-Hartle Vacuum
In a spacetime with a “bifurcate Killing horizon” there is a unique state that is Hadamard everywhere, named the Hawking-Hartle-(Israel?) vacuum.
In static spacetimes you get it from a Euclidean path integral along the analytic continuation from the right to the left wedge. Topological arguments assign this state an “Unruh” temperature.
In other words we prepare a thermal state at the given Unruh temperature using the Hamiltonian of the Killing observers.

32. Hawking-Hartle Vacuum - Minkowski
In Minkowski spacetime the Hawking-Hartle vacuum is the Minkowski vacuum.
Can we prepare it using the Hamiltonian in the Rindler wedge?

33. Hawking-Hartle Vacuum - Minkowski
In Minkowski spacetime the Hawking-Hartle vacuum is the Minkowski vacuum.
The discretized conformally-flat Hamiltonian of the Killing observers is:
Using tensor networks* we can prepare a thermal state of this Hamiltonian at the Unruh temperature and check that it is indeed the Minkowski vacuum.

39. Conformal Vacuum
-With m=0 the “conformal vacuum” obtained by conformally transforming the Minkowski G(x, x’) is a Hadamard state.
-It is “+ frequency with respect to the conformal Killing vector”; i.e. it diagonalizes the Hamiltonian of the conformally flat metric.
-Can we recover the analytic result?

40. Conformal Vacuum- FLRW
-The state is the same!
-But:
-So the correct Hadamard prop. is “recovered” trivially in this case.

41. Conformal Vacuum- Static
-We get the same result from the Hadamard solution – the renormalized value is zero.
-The lattice op is NOT the same, so for results to agree Hamiltonian must prepare the correct state.

42. Conformal Vacuum- Test in AdS

43. Thanks! (hire me)