A Hunt for
Phase Space

Sagnik Ghosh •Group Retreat• Die Ebernburg  •January 15, 2024

Order paramerters:

 

 

 

1. avg level spacing ratio

 

2. Envelope of Z_iZ_j

 

3. SFF

 

4. Statistics of Lazadires diagonals

 

5. Half chain entanglement entropy

Recall: Model

Parameter 1. Level spacing ratio:

Half chain EE: Motivation: PRR L032016

Model:
 

  • Z2 symmetric random interacting Majorana chains at high energies.
  • Note the NNN term.
  • Two topologically different distinct MBL phases. Consequence of avalanche theory.

Shift Invert ED:


Explored the infinite temp. phase diagram. Kramers-Wanier duality [31] leads to symmetric phase boundaries .

 

 

Consequences:

 

MBL-SG phase exhibits cat states for all energies with a global double degeneracy of the MBL spectrum in the two parity sectors.

 

Writing H as Majoranas:

The original Hamiltonian becomes:

Randomness: Assume Box distribution for both couplings J and fields h,

 

with          

Numerics: Shift invert ED of IM chains upto L=16.
 

Computed ~100 midspectrum eigenpairs of each parity sector for ~500-1000 independennt random samples.

Phase Diagram:

Algorithm:

Algorithm:

Algorithm:

Algorithm:

Algorithm:

SV_{L/2}= \sum_i -\lambda_i \log_e \lambda_i

The entanglement entropy is then computed as the Von-Neumann entropy of this matrix

Behaviour:

Scaling with size:

Behaviour in the pure MBL phases with θ

Finite size scaling:

Phase space:

Phase space:Till now

Questions!

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