Emergent large N matrices from a nonlocal spin system Victoria Martin Stanford University July Based on: arXiv: [hep-th] By: Dionysios Anninos, Sean Hartnoll, Liza Huijse, VM Holographic duality for condensed matter physics KITPC Beijing

Dionysios Anninos, IASLiza Huijse, SITPSean Hartnoll, SITP Emergent large N matrices from a nonlocal spin system

Introduction Large NxN MatricesEmergent Spacetime For example, [hep-th/ ]

Introduction Large NxN MatricesEmergent Spacetime

Introduction Large NxN MatricesEmergent Spacetime Quantum Spin System

Introduction Large NxN MatricesEmergent Spacetime Quantum Spin System Why? 1) Use matrix integral techniques to study spin systems 2) Understand emergent locality, etc using known results of quantum information 3) Study nontrivial emergence of continuous symmetry (SO(N)) from discrete 4) Study novel phase transition involving topological order, not symmetry breaking

Outline Introduce the system and its symmetries Emergence of matrix integral with SO(N) symmetry Solving the matrix integral; Results Monte Carlo verification Discussion

The system: A nonlocal Ising model N N Note: 1)We will study the ferromagnetic case, λ<0 2)We use the trace structure to make connection with bosonic matrices later

The system: A nonlocal Ising model N N

N N Discrete symmetries: 1)S N x S N – Permuting rows or columns 2)Z 2 N x Z 2 N – Flipping all spins in a row or column

Finite Temperature Partition Function

H-S Field

Finite Temperature Partition Function H-S Field Denef, arXiv: [hep-th]

Finite Temperature Partition Function

Partition Function: Focus on z(Q)

H-S Vector

Partition Function: Focus on z(Q) H-S Vector

Partition Function: Focus on z(Q)

We see: From: We see: From:

Partition Function: Focus on z(Q) Rescale z(Q): This removes part of the trace of Q: Cugliando, Kurchan, Parisi, Ritort [cond-mat/ ]

Partition Function: Focus on z(Q) Rescale z(Q): This removes part of the trace of Q: Cugliando, Kurchan, Parisi, Ritort [cond-mat/ ] cf:

Solving the Matrix Integral Rotate eigenvalue basis: Brezin, Itzykson, Parisi, Zuber Note:

Solving the Matrix Integral Introduce eigenvalue distribution: EOM in large N limit: Brezin, Itzykson, Parisi, Zuber 1978.

Solving the matrix integral Evidence of metastability

Solving the matrix integral Evidence of metastability

Monte Carlo First order phase transition

Monte Carlo For a first order phase transition, the canonical distribution is doubly peaked near the transition temperature. The transition temperature occurs when the peaks are the same height. Canonical distributionDensity of states Matrix integral saddleGround states

Monte Carlo

Order and Disorder Low temperature saddle -> ordered phaseMatrix integral saddle -> disordered phase

Ferromagnetic vs Antiferromagnetic model 1)High temperature phase related to a matrix integral 2)Low temperature phase exhibits frustration and glassiness 3)Existence of phase with disconnected eigenvalue distribution Cugliando, Kurchan, Parisi, Ritort 1994 AF F

Conclusions

Future Directions