erg_loc

A numerical toolbox to distinguish between ergodic and localized phases in quantum many-body systems.

Prerequisites: Python 3.7.6+, QuSpin 0.3.6+

1. Static Systems
2. Driven Systems
3. Hoffman2 Cluster
4. References

Static Systems

The programs along with their corresponding tools:

• inst_ham -- instantaneous Hamiltonian

  • ener -- disorder-averaged energy spectrum
  • ener_spac -- disorder-concatenated energy spacings
  • ent -- disorder-averaged entanglement entropy spectrum
  • ent_mid* -- disorder-averaged entanglement entropy for a mid state only
  • overlap -- disorder-averaged overlap spectrum
  • exp_val -- disorder-averaged expectation value spectrum
  • spec_func* -- disorder-averaged spectral function

• W_flow -- disorder evolution

  • ener_W_flow -- disorder-averaged energy spectrum against disorder amplitude
  • r_W_flow -- disorder-averaged first moment of the energy spacings against disorder amplitude
  • ent_W_flow -- disorder-averaged entanglement entropy spectrum against disorder amplitude
  • ent_rel_W_flow -- disorder-averaged relative entropy against disorder amplitude

• L_flow -- length evolution

  • ent_L_flow* -- disorder-averaged entanglement entropy spectrum against length
  • ent_mid_L_flow* -- disorder-averaged entanglement entropy of a mid state against length

• t_flow -- time evolution

  • ent_t_flow -- disorder-averaged entanglement entropy of a given state against time
  • numb_fluc_t_flow -- disorder-averaged particle number fluctuations against time
  • overlap_t_flow -- disorder-averaged overlap against time

*mutually exclusive tools

The plot scripts:

• ener_spec -- energy spectrum

python code/inst_ham.py -mod heisenberg -thr 1 -L 8 -bc o -dis 1 -W 0.5

python code/inst_ham.py -mod heisenberg -thr 1 -L 8 -bc o -dis 1 -W 8

• ent_arc -- entanglement entropy arc

python code/inst_ham.py -mod heisenberg -thr 1 -L 12 -bc o -dis 1 -W 0.5

python code/inst_ham.py -mod heisenberg -thr 1 -L 12 -bc o -dis 1 -W 8

• ener_stat -- energy spacing statistics

python code/inst_ham.py -mod heisenberg -Nup 4 -pauli 0 -L 8 -bc o -dis 10000 -W 0.5

python code/inst_ham.py -mod heisenberg -Nup 4 -pauli 0 -L 8 -bc o -dis 10000 -W 8

• overlap_spec -- overlap-eigenenergy spectrum (cf. Fig.3b of [Turner2018])

python code/inst_ham.py -mod pxp -L 18 -bc o

python code/inst_ham.py -mod pxp -L 20 -bc o

• overlap_stat -- overlap statistics (cf. Fig.2b of [Turner2018])

python code/inst_ham.py -mod pxp -L 10 -bc o

python code/inst_ham.py -mod pxp -L 12 -bc o

• eth_arc -- eigenstate thermalization hypothesis arc (cf. Fig.2a of [Turner2018])

python code/inst_ham.py -mod pxp -L 26 -bc o

python code/inst_ham.py -mod pxp -L 28 -bc o

• scar_gap_scal -- scar state energy gap scaling (cf. Fig.4 of [Turner2018])

python code/inst_ham.py -mod pxp -L 10 -bc o

python code/inst_ham.py -mod pxp -L 12 -bc o

• r_W_flow -- first moment of the energy spacings disorder evolution (cf. Fig.3 of [Pal2010])

python code/W_flow.py -mod heisenberg -L 8 -Nup 4 -pauli 0 -bc o -dis 11000 -W_min 0.5 -W_max 12.5 -W_samp 24

• ent_W_flow -- entanglement entropy disorder evolution

python code/W_flow.py -mod heisenberg -L 8 -Nup 4 -pauli 0 -bc o -dis 10000 -W_min 0.5 -W_max 12.5 -W_samp 24

• ent_rel_W_flow -- relative entropy disorder evolution (cf. Fig.2b of [Luitz2015])

python code/W_flow.py -mod heisenberg -L 8 -Nup 4 -pauli 0 -bc o -dis 1000 -W_min 0.2 -W_max 4.9 -W_samp 23

python code/W_flow.py -mod heisenberg -L 10 -Nup 5 -pauli 0 -bc o -dis 1000 -W_min 0.2 -W_max 4.9 -W_samp 23

python code/W_flow.py -mod heisenberg -L 12 -Nup 6 -pauli 0 -bc o -dis 1000 -W_min 0.2 -W_max 4.9 -W_samp 23

• ent_mid_L_flow -- entanglement entropy length evolution (cf. Fig.3c of [Luitz2015])

python code/L_flow.py -mod heisenberg -L_min 12 -L_max 16 -L_samp 3 -Nup_min 6 -Nup_max 8 -pauli 0 -bc o -dis 100 -W 0.5

python code/L_flow.py -mod heisenberg -L_min 12 -L_max 16 -L_samp 3 -Nup_min 6 -Nup_max 8 -pauli 0 -bc o -dis 100 -W 8

• ent_t_flow -- entanglement entropy time evolution (cf. Fig.2 of [Nanduri2014])

With v=0 in the initial Bloch state.

python code/t_flow.py -mod heisenberg -L 6 -J 1 1 0.2 -t_min -1 -t_max 6 -t_samp 100 -dis 100 -bc o -W 0.5

python code/t_flow.py -mod heisenberg -L 6 -J 1 1 0.2 -t_min -1 -t_max 6 -t_samp 100 -dis 100 -bc o -W 10

• numb_fluc_t_flow -- particle number fluctuation time evolution (cf. Fig.2a of [Bardarson2012])

With v=1 in the initial Bloch state.

python code/t_flow.py -mod heisenberg -L 8 -pauli 0 -J 1 1 0 -t_min -2 -t_max 3 -t_samp 50 -dis 1000 -bc o -W 5 -tag ".v_1"

python code/t_flow.py -mod heisenberg -L 8 -pauli 0 -J 1 1 0.2 -t_min -2 -t_max 3 -t_samp 50 -dis 1000 -bc o -W 5 -tag ".v_1"

• overlap_t_flow -- overlap time evolution (cf. Fig.6 of [Turner2018])

python code/t_flow.py -mod pxp -L 24 -bc o -t_min 0 -t_max 30 -t_samp 1000 -tag ".Z2"

python code/t_flow.py -mod pxp -L 24 -bc o -t_min 0 -t_max 30 -t_samp 1000 -tag ".Z3"

python code/t_flow.py -mod pxp -L 24 -bc o -t_min 0 -t_max 30 -t_samp 1000 -tag ".Z4"

• spec_func -- spectral function (cf. Fig.2c of [Turner2018])

python code/inst_ham.py -mod pxp -L 26 -bc o

Driven Systems

The programs along with their corresponding tools:

• inst_U -- instantaneous Floquet unitary

  • q_ener -- disorder-averaged quasi-energy spectrum
  • q_ener_spac -- disorder-concatenated quasi-energy spacings
  • floq_struc -- disorder-averaged Floquet eigenstate structure
  • loc_len -- disorder-averaged localization length per Floquet eigenstate

• N_flow -- Floquet cycle evolution

  • ener_abs_N_flow -- disorder-averaged energy absorbed against N
  • ent_N_flow -- disorder-averaged entanglement entropy against N
  • info_ent_N_flow* -- disorder-averaged Floquet-eigenstate-average information entropy against N

• T_flow -- Floquet time evolution

  • PR_T_flow -- disorder-averaged Floquet-eigenstate-average participation ratio against T

• delta_flow -- drive period evolution

  • q_ener_delta_flow -- disorder-averaged quasi-energy spectrum against delta
  • loc_len_delta_flow -- disorder-averaged Floquet-eigenstate-averaged localization length against delta
  • PR_delta_flow -- disorder-averaged Floquet-eigenstate-average participation ratio against delta
  • ent_delta_flow* -- disorder-averaged entanglement entropy against delta

• L_flow -- length evolution

  • ent_L_flow* -- disorder-averaged entanglement entropy against length
  • ent_mid_L_flow* -- disorder-averaged entanglement entropy of a mid state against length

*mutually exclusive tools

The plot scripts:

• q_ener_spec -- quasi-energy spectrum

python code/inst_U.py -mod ponte2015 -L 8 -pauli 0 -bc o -W 0.5 -h0 2 -T0 7 -T1 1.5

python code/inst_U.py -mod ponte2015 -L 8 -pauli 0 -bc o -W 8 -h0 2 -T0 7 -T1 1.5

• q_ener_stat -- quasi-energy spacing statistics

python code/inst_U.py -mod ponte2015 -L 8 -Nup 4 -pauli 0 -bc o -W 0.5 -h0 2 -T0 7 -T1 1.5 -dis 10000

python code/inst_U.py -mod ponte2015 -L 8 -Nup 4 -pauli 0 -bc o -W 0.5 -h0 2 -T0 7 -T1 1.5 -dis 10000

• floq_struc -- Floquet eigenstate structure (cf. Fig.2a of [Ponte2015])

python code/inst_U.py -mod ponte2015 -L 8 -pauli 0 -bc o -W 0.5 -h0 2 -T0 7 -T1 1.5

python code/inst_U.py -mod ponte2015 -L 8 -pauli 0 -bc o -W 8 -h0 2 -T0 7 -T1 1.5

• ener_abs_N_flow -- energy absorbed N evolution (cf. Fig.2c of [Ponte2015])

python code/N_flow.py -mod ponte2015 -L 8 -pauli 0 -bc o -W 0.5 -h0 2 -T0 7 -T1 1.5 -N 31 -dis 100

python code/N_flow.py -mod ponte2015 -L 8 -pauli 0 -bc o -W 8 -h0 2 -T0 7 -T1 1.5 -N 31 -dis 100

• ent_N_flow -- entanglement entropy N evolution

With v=0 in the initial Bloch state.

python code/N_flow.py -mod spin2021 -L 8 -bc o -W 2 -T0 1 -T1 1 -N 201 -dis 10 -delta 0.9

python code/N_flow.py -mod spin2021 -L 8 -bc o -W 2 -T0 1 -T1 1 -N 201 -dis 10 -delta 0.1

• ent_info_N_flow -- information entropy N evolution (cf. Fig.4 of [DAlessio2014])

python code/N_flow.py -mod spin2021 -L 400 -Nup 1 -bc o -W 2 -T0 1 -T1 1 -N 60 -dis 10 -delta 0.9

python code/N_flow.py -mod spin2021 -L 400 -Nup 1 -bc o -W 2 -T0 1 -T1 1 -N 60 -dis 10 -delta 0.1

• PR_T_flow -- participation ratio T evolution (cf. Fig.2b of [Ponte2015])

python code/T_flow.py -mod ponte2015 -pauli 0 -L 8 -h0 2 -T0 7 -dis 5 -bc o -W 0.5 -T_min 0 -T_max 3 -T_samp 16

python code/T_flow.py -mod ponte2015 -pauli 0 -L 8 -h0 2 -T0 7 -dis 5 -bc o -W 8 -T_min 0 -T_max 3 -T_samp 16

• q_ener_delta_flow -- quasi-energy spectrum delta evolution

python code/delta_flow.py -mod spin2021 -Nup 1 -L 20 -bc o -T0 1 -T1 1 -W 0 -delta_min 0 -delta_max 1 -delta_samp 24

• loc_len_delta_flow -- localization length delta evolution

python code/delta_flow.py -mod spin2021 -dis 10 -Nup 1 -L 200 -bc o -T0 1 -T1 1 -W 1 -delta_min 0 -delta_max 1 -delta_samp 11

• PR_delta_flow -- participation ratio delta evolution

python code/delta_flow.py -mod spin2021 -dis 10 -Nup 1 -L 100 -bc o -T0 1 -T1 1 -W 2 -delta_min 0 -delta_max 1 -delta_samp 21

python code/delta_flow.py -mod spin2021 -dis 10 -Nup 1 -L 200 -bc o -T0 1 -T1 1 -W 2 -delta_min 0 -delta_max 1 -delta_samp 21

python code/delta_flow.py -mod spin2021 -dis 10 -Nup 1 -L 300 -bc o -T0 1 -T1 1 -W 2 -delta_min 0 -delta_max 1 -delta_samp 21

• ent_delta_flow -- entanglement entropy delta evolution

python code/delta_flow.py -mod spin2021 -dis 100 -L 12 -bc o -T0 1 -T1 1 -W 2 -delta_min 0 -delta_max 1 -delta_samp 11

• ent_L_flow -- entanglement entropy length evolution

python code/L_flow.py -mod spin2021 -L_min 6 -L_max 12 -L_samp 4 -bc o -dis 100 -T0 1 -T1 1 -delta 0.1 -W 2

Hoffman2 Cluster

Hoffman2 uses the Univa Grid Engine batch-queueing system on CentOS (Fedora/RHEL).

Programs to install:

• htop -- allows you to view memory and CPU usage
• parallel -- allows you to conveniently parallelize jobs

Edits to ~/.bash_profile:

• export PATH=$PATH:$HOME/local/bin to access personal programs (installed above)
• export PYTHONUNBUFFERED='True' to view output of running python jobs in real time

Modules to load:

• source /u/local/Modules/default/init/modules.sh
• module load anaconda3
• source /u/local/apps/anaconda3/2020.11/etc/profile.d/conda.sh
• conda activate bart

...where the anaconda path is found via...

• conda info | grep -i 'base environment'

Alternatively, you can set up a conda environment in your home directory:

• conda init bash
• conda env create --name bart --file=environment.yml
• conda activate bart

...then, once everything is set-up, you can simply source your bash configuration in the submission script instead:

• source /u/home/b/baandr12/.bash_profile
• source /u/home/b/baandr12/.bashrc

Useful commands:

• module list -- list currently loaded modules
• module avail -- list available modules to load
• module load / module unload -- add or remove modules
• mygroup -- view accessible private resources
• myquota -- view my quota for home and scratch
• qhost -- view all cluster nodes
• qstat -U baandr12 -- view all jobs running on the parts of cluster where I have access
• qstat -u baandr12 -- view the status of my running jobs
• qstat -q bhaumik_pod.q -- status of bhaumik queue
• qstat -j 627506 -- print the information for a particular job
• qsub srun.sh -- submit a script
• qdel 660385 -- delete a job

Private resources:

bhaumik resource group currently has 8 intel-E5-2697 nodes, each with 32 cores and 512GB RAM

• qstat -q *.q@n{6..7}06{1..4} -- view all running jobs on bhaumik nodes (fast)
• qstat -f | sed -n -e '/q@n[6-7]06[1-4]/,/---/ p' -- view the full status of all bhaumik nodes (slow)

References

[Pal2010] "Many-body localization phase transition", by Arijeet Pal and David Huse, PRB 82, 174411 (2010).

[Luitz2015] "Many-body localization edge in the random-field Heisenberg chain", by David Luitz, Nicolas Laflorencie, and Fabien Alet, PRB 91, 081103(R) (2015).

[Nanduri2014] "Entanglement spreading in a many-body localized system", by Arun Nanduri, Hyungwon Kim, and David Huse, PRB 90, 064201 (2014).

[Ponte2015] "Periodically driven ergodic and many-body localized quantum systems" by Pedro Ponte, Anushya Chandran, Zlatko Papić, and Dmitry Abanin, Annals of Physics 353, 196 (2015).

[DAlessio2014] "Long-time Behavior of Isolated Periodically Driven Interacting Lattice Systems" by Luca D’Alessio and Marcos Rigol, PRX 4, 041048 (2014).

[Bardarson2012] "Unbounded growth of entanglement in models of many-body localization" by Jens Bardarson, Frank Pollmann, Joel Moore, PRL 109, 017202 (2012).

[Turner2018] "Quantum scarred eigenstates in a Rydberg atom chain: entanglement, breakdown of thermalization, and stability to perturbations", PRB 98, 155134 (2018).