Entanglement entropy of highly excited states

Introduction

We are interested in calculating the entanglement entropy for states at the middle of the spectrum, which sparse solvers do not do well (though it is ameliorated by transformations like shift-invert mode). Instead, I will obtain states via dense diagonalization

Program

Calculate wavefunctions at center of spectrum using dense solver
Repeat entropy calculations as in part 2

import numpy as np
import pandas as pd
import scipy.linalg as la
import matplotlib.pyplot as plt
%matplotlib inline

from ph121c_lxvm import tfim, basis, tests, measure, data

%%time
entropies = {
    'L' : [],
    'h' : [],
    'l' : [],
    'S' : [],
    'bc': [],
}

for oper_params in tests.tfim_sweep(
    L = list(range(4,15)),
    h = [0.3],
    bc= ['c'],
):
    job = dict(
        oper=tfim.z.H_dense,
        oper_params=oper_params,
        solver=la.eigh,
        solver_params={ 
            'subset_by_index' : list(np.array([-2, 3]) + 2 ** (oper_params['L'] - 1)),
        },
    )
    evals, evecs = data.jobs.obtain(**job)
        
    es = evecs[:, 5]
    print('using state with eigenvalue:', evals[5])
    A = []
    for l in range(oper_params['L']-1):
        A.append(l)
        entropies['S'].append(
            measure.entropy.entanglement(
                basis.schmidt.values(es, A, oper_params['L'])
            )
        )
        entropies['l'].append(l)
        entropies['L'].append(oper_params['L'])
        entropies['h'].append(oper_params['h'])
        entropies['bc'].append(oper_params['bc'])

df = pd.DataFrame(entropies)

using state with eigenvalue: 0.6
using state with eigenvalue: -0.255107460434849
using state with eigenvalue: 1.4196736589527976
using state with eigenvalue: 0.7000000000000011
using state with eigenvalue: 1.3541254488577767e-16
using state with eigenvalue: -0.5651924049830526
using state with eigenvalue: 0.5813865554288129
using state with eigenvalue: 0.5809819774954872
using state with eigenvalue: 1.227076422501945e-15
using state with eigenvalue: -0.5150533998116924
using state with eigenvalue: 0.00034525367950669615
CPU times: user 236 ms, sys: 12 ms, total: 248 ms
Wall time: 53.7 ms

def plot_script(df, bc):
    """Make the display plots."""
    L = sorted(set(df.L))
    h = sorted(set(df.h))
    w = 2

    fig_l, axes_l = plt.subplots(len(L)//w+len(L)%w, w)
    for i, row in enumerate(axes_l):
        for j, ax in enumerate(row):
            if w*i + j < len(L):
                for s in h:
                    sub = df[(df.h==s) & (df.L==L[w*i+j]) & (df.bc==bc)]
                    ax.plot(sub.l.values, sub.S.values, label='h='+str(s))
                ax.set_xlabel(f'$l$ at $L={L[w*i+j]}$')
                ax.set_ylabel('$S$')
                handles, labels = ax.get_legend_handles_labels()
            else:
                ax.set_axis_off()
                ax.legend(handles, labels, loc='center')
    st = fig_l.suptitle('Entanglement entropies')
    fig_l.set_size_inches(5, 6)
    fig_l.tight_layout()
    st.set_y(.95)
    fig_l.subplots_adjust(top=.9)

    fig_L, ax_L = plt.subplots()
    for s in h:
        sub = df[(df.h==s) & (df.l==df.L//2) & (df.bc==bc)]
        ax_L.plot(sub.L.values, sub.S.values, label='h='+str(s))
    ax_L.set_title('Entanglement entropy at half-subsystem')
    ax_L.set_xlabel('$L$')
    ax_L.set_ylabel('$S$')
    ax_L.legend()
    fig_L.set_size_inches(3, 3)

    return (fig_l, fig_L)

%%capture
figs = plot_script(df, 'c')

Results

The main goal is to observe a different scaling of entropy for a highly-excited state which is not a band state. In fact this is what we observe for the Hamiltonian in the ferromagnetic phase and with periodic boundary conditions.

figs[0]

png

It really is different behavior than the constant for the area law in the band states. From smaller $L=4$ to larger systems, $L=14$, the same pattern emerges where the entanglement entropy has a maximum at $\ell = L/2$ and appears to decay linearly away from that maximum.

Next we how a summary of the growth of the entanglement entropy at half subsystems for various $L$:

figs[1]

png

Discussion

Instead of observing a constant entanglement entropy with respect to subsystems, as in the extremal eigenstates, the states at the center of the spectrum display new behaviors: increasing entropy with respect to system size and also with respect to subsystem size. One might expect that the entropy increases at equipartition, and indeed, this is what we are observing. From this perspective, the excited states are more random than those at the bands.