1. On the spectrum of Hamiltonians in finite dimensions Roberto Oliveira Paraty, August 14 th 2007. Joint with David DiVincenzo and Barbara Terhal @ IBM Watson.

2. In one slide: Ground state energy is hard. Bulk of the spectrum is Gaussian and universal.

3. The setup H = Hamiltonian on a set V of N spin ½ particles. Spec(H) = { 0 · 1 · 2 …}. We will assume Tr(H)=0.  2 (H) = “variance” = Tr(H 2 )/2 N Write H =  X ½ V H X, where H X acts on the spins in X. Can assume Tr(H X H Y )=0 if X  Y.

4. site term H {k} = h  z [k] E.g.: Ising with transverse field H = (bond terms) + (site terms) bond term H {i,j} = J  x  x [i,j] ij k H=  k H {k} +  i~j H {i,j}

5. site term H {k} = h  z [k] E.g.: Ising with transverse field H = (bond terms) + (site terms) bond term H {i,j} = J  x  x [i,j] ij k H=  k H {k} +  i~j H {i,j} +  F H F face term H F

6. Dimensionality assumption 9 metric  and d, C, c,  such that: Radius R center x B(x,R) = {v 2 V :  (v,x) · R} cR d ·  X ½ B(x,R)  2 (H X ) · CR d  X ½ B(x,R) ||H X || · CR d  X ½  B(x,R)  2 (H X ) · CR d- 

7. E.g. nearest neighbor in Z d L 1 norm ¼ R d terms inside ¼ R d-1 at the boundary Radius R

8. Gaussian spectrum Plot a histogram of Spec(H/  (H)). with small but fixed bin size b>0. That will approach a Gaussian as N adiverges, for fixed parameters.

9. A bit more formally There is a probability measure on the line given by H:  H = 2 -N  2 Spec(H)  We show that this measure is approximately normal in the sense that for all a<b, as N grows:  H (a  (H),b  (H)) ! (2  ) -1/2 s a b exp(-t 2 /2)dt

10. Even more formally Recall: strength inside ball ¼ R d, strength across boundary ¼ R d- , with C,c extra parameteres. Then for all a<b, |  H [a  (H),b  (H)]-(2  ) -1/2 s a b exp(-t 2 /2)dt| · D(C,c,d,  ) (Diam(  )) -  d/8

11. A bit less formally again Inside ¼ R d, boundary ¼ R d-   Radius R center x B(x,R) = {v 2 V :  (v,x) · R}  H/  (H) [x,y] ¼ (2  ) -1/2 s x y exp(-t 2 /2)dt

12. A simple case We will now explain a special case of the Theorem. Nearest neighbor interactions on a n x n patch of the planar square lattice (N=n 2 spins). Also assume that all terms in the Hamiltonian have norm of constant order.

13. site term H {k} = h  z [k] Back to that old slide H = (bond terms) + (site terms) bond term H {i,j} = J  x  x [i,j] ij k H=  k H {k} +  i~j H {i,j}

14. What do we do? Main idea: ignore terms acting on red lines (i.e. treat non-interacting systems). Then put them back in via a perturbation argument.

15. Omitting the interactions G =  s G s, subsystems might have high dimension. How does one compute the global spectrum? Answer: convolution of the individual spectral distributions. By the usual Central Limit Theorem,  G  G) is approximately Gaussian as long as some conditions are satisfied.

16. Which conditions? Many terms in the sum. The influence of any given term is small. m ¿ n 1/2 suffices.

17. Putting red lines back in

18. Next step Recall that we have.  H/  (H) = 2 -N  2 Spec(H)   We know that  G/  (G) is approx. gaussian.  G/  (G) = 2 -N  2 Spec(G)   G  We will show that  (H-G) is small. By a perturbation theory argument, this implies that  H/  (H) ¼  G/  (G).

19.  H-G) is small Variance ¼ # of qubits Total # of qubits ¼ n 2 Qubits on red lines ¼ m(n/m) 2 = n 2 /m )  (H-G) 2 ·  (H) 2 /m, small if m À 1

20. To conclude Take some 1 ¿ m ¿ n 1/2 (e.g. m=n 1/3 ). Then  G/  (G) is approx. Gaussian. Also  H/  (H) ¼  G/  (G). So we are done. The key step: m x m boxes have m 2 vertices but only ¼ m vertices on their boundaries.

21. General case Inside ¼ R d, boundary ¼ R d-   Radius R center x B(x,R) = {v 2 V :  (v,x) · R}  H/  (H) [a,b] ¼ (2  ) -1/2 s a b exp(-t 2 /2)dt

22. General result: proof sketch Main idea: break the system into subparts with small total boundary strength. Treat isolated systems via standard CLT. Put the boundary back in via perturbation theory.

23. Conclusions Spectral distribution is approximately Gaussian with std. deviation ¼ N 1/2. This is universal for quantum spin systems in finite dimensional structures when long-range interactions decay fast enough.

24. Further work Bounds are actually weak for many problems. Are there better bounds for specific systems? Fermions? Bosons? Applications?