Hamiltoniano para semimetales de Weyl tipo I - Koshino
#

El objetivo general de este notebook es explorar el Hamiltoniano presentado en el Review[1]

Los conceptos a introducir serán:

  • Hamiltoniano para un semimetal del Weyl y uno de punto nodal

  • Relación de dispersión generada por este tipo de materiales

A diferencia de los hamiltonianos explorados anteriormente, este es de una matriz de \(4x4\) porque considera al término de masa. Adicionalmente, este notebook se enfocara sólo en la fase correspondiente al semimemtal de Weyl tipo I.

Fuente: Mikito Koshino and Intan Fatimah Hizbullah Phys. Rev. B 93, 045201 – Published 4 January 2016; Erratum Phys. Rev. B 99, 209903 (2019)

Hamiltoniano del sistema
#

(4)#\[\begin{eqnarray} H(p) = \left[ \begin{array}{cc} m\sigma_0 + b\sigma_z & v\sigma \cdot p\\ v\sigma \cdot p & -m\sigma_0 + b\sigma_z \end{array} \right] \end{eqnarray}\]

Para poder computar los cálculos en la librería de PythTB[2] es necesario expresar a los momentos \(p_i=\sin(k_i)\) y consecuentemente, se puede expresar al \(\sin\) en su forma exponencial.

Fuente:S. Coh and D. Vanderbilt, Python Tight Binding (PythTB) — PythTB 1.7.2 documentation (2016)

Expresión de la forma exponencial#

(5)#\[\begin{eqnarray} H(k) = \left[ \begin{array}{cc} m+b & 0 &v\hbar (\frac{e^{ik_za}-e^{-ik_za}}{2i}) & v\hbar (\frac{e^{ik_xa}-e^{-ik_xa}}{2i})- i\hbar v (\frac{e^{ik_ya}-e^{-ik_ya}}{2i}) \\ 0 & m-b & v\hbar (\frac{e^{ik_xa}-e^{-ik_xa}}{2i})+ i\hbar v (\frac{e^{ik_ya}-e^{-ik_ya}}{2i}) & v\hbar (\frac{e^{ik_za}-e^{-ik_za}}{2i}) \\ v\hbar (\frac{e^{ik_za}-e^{-ik_za}}{2i}) & v\hbar (\frac{e^{ik_xa}-e^{-ik_xa}}{2i})- i\hbar v (\frac{e^{ik_ya}-e^{-ik_ya}}{2i}) & -m+b & 0\\ v\hbar (\frac{e^{ik_xa}-e^{-ik_xa}}{2i})+ i\hbar v (\frac{e^{ik_ya}-e^{-ik_ya}}{2i})& v\hbar (\frac{e^{ik_za}-e^{-ik_za}}{2i}) &0 & -m-b \end{array} \right] \end{eqnarray}\]

Acerca de la notación:

  • \(b\): es el campo intrinseco de Zeeman

  • \(v\): es el parametro de velocidad.

  • \(m\): es la masa

Hide code cell source 
import glob
from matplotlib.ticker import (MultipleLocator,
                               FormatStrFormatter,
                               AutoMinorLocator)
from pylab import *
from pythtb import *
Hide code cell source 
mpl.rcParams.update({'font.size': 22, 'text.usetex': True})
mpl.rcParams.update({'axes.linewidth':1.5})
mpl.rcParams.update({'axes.labelsize':'large'})
mpl.rcParams.update({'xtick.major.size':12})
mpl.rcParams.update({'xtick.minor.size':6})
mpl.rcParams.update({'ytick.major.size':12})
mpl.rcParams.update({'ytick.minor.size':6})
mpl.rcParams.update({'xtick.major.width':1.5})
mpl.rcParams.update({'xtick.minor.width':1.0})
mpl.rcParams.update({'ytick.major.width':1.5})
mpl.rcParams.update({'ytick.minor.width':1.0})
Hide code cell source 
m = 0.7
b = 1.0
v = 1.0
a = 1.0
ħ = 1.0

##################----------------Inicia TB----------------##################
lat= [[a,0,0],[0,a,0],[0,0,a]]
orb= [[1/2,1/2,1/2],[0,0,0],[0,0,0],[1/2,1/2,1/2]] #H, solo los sitios A[000] B1/2[111]

PWSM = tb_model(3,3,lat,orb)


#FUERA DE LA DIAGONAL

#(0,2)=vhsink_z
PWSM.set_hop(+v*ħ/(2*1J), 0, 2, [0,0,1] )
PWSM.set_hop(-v*ħ/(2*1J), 0, 2, [0,0,-1])
#(1,3)=-vhsink_z
PWSM.set_hop(-v*ħ/(2*1J) ,1 , 3, [0,0,1])
PWSM.set_hop(+v*ħ/(2*1J) ,1 , 3, [0,0,-1])

#(0,3)X=vh sink_x-hiv sinky
PWSM.set_hop(+v*ħ/(2*1J) ,0 , 3, [+1,0,0])
PWSM.set_hop(-v*ħ/(2*1J),0 , 3, [-1,0,0])
#(0,3)Y=vh sink_x-hiv sinky
PWSM.set_hop(-v*ħ/2 ,0 , 3, [0,+1,0])
PWSM.set_hop(+v*ħ/2 ,0 , 3, [0,-1,0])

#(1,2)X=vh sink_x+hiv sinky
PWSM.set_hop(+v*ħ/(2*1J) ,1 , 2, [+1,0,0])
PWSM.set_hop(-v*ħ/(2*1J) ,1 , 2, [-1,0,0])
#(1,2)Y=vh sink_x+hiv sinky
PWSM.set_hop(+v*ħ/2 ,1 , 2, [0,+1,0])
PWSM.set_hop(-v*ħ/2 ,1 , 2, [0,-1,0])

#DIAGONAL
#m+b
PWSM.set_onsite([m+b,m-b,-m+b,-m-b]) # No hay hooping, es energia

###############---------Inicia extraccion de información-----------##############
k=[[0,0.0,0],[0,0,0.25],[0,0,0.5]]#Punto por los cuales que quiero que pase. Son los punto de al simetria
                                 #unidades en unidades de V de red 1=2pi/a

                                 #unidades en unidades de V de red 1=2pi/a
k_label=["-Z",r"$\Gamma$","Z"]
(k_vec,k_dist,k_node)=PWSM.k_path(k,1001,report=False)
Ek,U=PWSM.solve_all(k_vec,eig_vectors=True)

#######################---Graficos----######################
 
fig,ax  = plt.subplots(ncols=2,nrows=1,figsize=(12,6),
                       gridspec_kw = {'wspace':0.4, 'hspace':0, 'width_ratios': [1, 1]})
ax[0].plot(k_dist,Ek[1,:],zorder=1,lw=1.5)
ax[0].plot(k_dist,Ek[2,:],zorder=1,lw=1.5)
ax[0].set_xticks(k_node) # puntos de alta simetria, valor que correspomde en k_path
ax[0].set_xticklabels(k_label )
# ax[0].set_yticks([-2,0.,2])
# xsticklabels(k_label) 

ax[0].spines['right'].set_visible(False)
ax[0].spines['top'].set_visible(False)
ax[0].set_ylabel(r"$E[t]$",fontsize=26)
ax[0].set_xlabel(r"$K_z$",fontsize=26)


k=[[0,0,0],[0,0,0.5]] # Punto por los cuales que quiero que pase. Son los punto de al simetria
                                 #unidades en unidades de V de red 1=2pi/a

(k_vec,k_dist,k_node)=PWSM.k_path(k,1001,report=False)
Ek,U=PWSM.solve_all(k_vec,eig_vectors=True)
ax[1].plot(k_dist,Ek[1,:],zorder=1,lw=1.5)
ax[1].plot(k_dist,Ek[2,:],zorder=1,lw=1.5)
 # puntos de alta simetria, valor que correspomde en k_path
a=0.01
ax[1].set_yticks([-a,0.,a])
ax[1].set_ylim([-a,a])
ax[1].set_xlim([0.1,0.4])
# xsticklabels(k_label) 

ax[1].spines['right'].set_visible(False)
ax[1].spines['top'].set_visible(False)
ax[1].set_ylabel(r"$E[t]$",fontsize=26)
ax[1].set_xlabel(r"$K_z$",fontsize=26)

savefig("WSMKoshinoBulkGap.pdf",bbox_inches="tight")

---------------------------------------------------------------------------
FileNotFoundError                         Traceback (most recent call last)
File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/texmanager.py:255, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
    254 try:
--> 255     report = subprocess.check_output(
    256         command, cwd=cwd if cwd is not None else cls.texcache,
    257         stderr=subprocess.STDOUT)
    258 except FileNotFoundError as exc:

File ~/psi4conda/envs/jbook/lib/python3.11/subprocess.py:466, in check_output(timeout, *popenargs, **kwargs)
    464     kwargs['input'] = empty
--> 466 return run(*popenargs, stdout=PIPE, timeout=timeout, check=True,
    467            **kwargs).stdout

File ~/psi4conda/envs/jbook/lib/python3.11/subprocess.py:548, in run(input, capture_output, timeout, check, *popenargs, **kwargs)
    546     kwargs['stderr'] = PIPE
--> 548 with Popen(*popenargs, **kwargs) as process:
    549     try:

File ~/psi4conda/envs/jbook/lib/python3.11/subprocess.py:1024, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask, pipesize, process_group)
   1021             self.stderr = io.TextIOWrapper(self.stderr,
   1022                     encoding=encoding, errors=errors)
-> 1024     self._execute_child(args, executable, preexec_fn, close_fds,
   1025                         pass_fds, cwd, env,
   1026                         startupinfo, creationflags, shell,
   1027                         p2cread, p2cwrite,
   1028                         c2pread, c2pwrite,
   1029                         errread, errwrite,
   1030                         restore_signals,
   1031                         gid, gids, uid, umask,
   1032                         start_new_session, process_group)
   1033 except:
   1034     # Cleanup if the child failed starting.

File ~/psi4conda/envs/jbook/lib/python3.11/subprocess.py:1917, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session, process_group)
   1916         err_msg = os.strerror(errno_num)
-> 1917     raise child_exception_type(errno_num, err_msg, err_filename)
   1918 raise child_exception_type(err_msg)

FileNotFoundError: [Errno 2] No such file or directory: 'latex'

The above exception was the direct cause of the following exception:

RuntimeError                              Traceback (most recent call last)
Cell In[3], line 86
     83 ax[1].set_ylabel(r"$E[t]$",fontsize=26)
     84 ax[1].set_xlabel(r"$K_z$",fontsize=26)
---> 86 savefig("WSMKoshinoBulkGap.pdf",bbox_inches="tight")

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/pyplot.py:1023, in savefig(*args, **kwargs)
   1020 @_copy_docstring_and_deprecators(Figure.savefig)
   1021 def savefig(*args, **kwargs):
   1022     fig = gcf()
-> 1023     res = fig.savefig(*args, **kwargs)
   1024     fig.canvas.draw_idle()  # Need this if 'transparent=True', to reset colors.
   1025     return res

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/figure.py:3343, in Figure.savefig(self, fname, transparent, **kwargs)
   3339     for ax in self.axes:
   3340         stack.enter_context(
   3341             ax.patch._cm_set(facecolor='none', edgecolor='none'))
-> 3343 self.canvas.print_figure(fname, **kwargs)

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/backend_bases.py:2342, in FigureCanvasBase.print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, pad_inches, bbox_extra_artists, backend, **kwargs)
   2336     renderer = _get_renderer(
   2337         self.figure,
   2338         functools.partial(
   2339             print_method, orientation=orientation)
   2340     )
   2341     with getattr(renderer, "_draw_disabled", nullcontext)():
-> 2342         self.figure.draw(renderer)
   2344 if bbox_inches:
   2345     if bbox_inches == "tight":

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
     93 @wraps(draw)
     94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95     result = draw(artist, renderer, *args, **kwargs)
     96     if renderer._rasterizing:
     97         renderer.stop_rasterizing()

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
     69     if artist.get_agg_filter() is not None:
     70         renderer.start_filter()
---> 72     return draw(artist, renderer)
     73 finally:
     74     if artist.get_agg_filter() is not None:

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/figure.py:3140, in Figure.draw(self, renderer)
   3137         # ValueError can occur when resizing a window.
   3139 self.patch.draw(renderer)
-> 3140 mimage._draw_list_compositing_images(
   3141     renderer, self, artists, self.suppressComposite)
   3143 for sfig in self.subfigs:
   3144     sfig.draw(renderer)

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/image.py:131, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
    129 if not_composite or not has_images:
    130     for a in artists:
--> 131         a.draw(renderer)
    132 else:
    133     # Composite any adjacent images together
    134     image_group = []

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
     69     if artist.get_agg_filter() is not None:
     70         renderer.start_filter()
---> 72     return draw(artist, renderer)
     73 finally:
     74     if artist.get_agg_filter() is not None:

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/axes/_base.py:3064, in _AxesBase.draw(self, renderer)
   3061 if artists_rasterized:
   3062     _draw_rasterized(self.figure, artists_rasterized, renderer)
-> 3064 mimage._draw_list_compositing_images(
   3065     renderer, self, artists, self.figure.suppressComposite)
   3067 renderer.close_group('axes')
   3068 self.stale = False

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/image.py:131, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
    129 if not_composite or not has_images:
    130     for a in artists:
--> 131         a.draw(renderer)
    132 else:
    133     # Composite any adjacent images together
    134     image_group = []

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
     69     if artist.get_agg_filter() is not None:
     70         renderer.start_filter()
---> 72     return draw(artist, renderer)
     73 finally:
     74     if artist.get_agg_filter() is not None:

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/axis.py:1377, in Axis.draw(self, renderer, *args, **kwargs)
   1374 renderer.open_group(__name__, gid=self.get_gid())
   1376 ticks_to_draw = self._update_ticks()
-> 1377 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
   1379 for tick in ticks_to_draw:
   1380     tick.draw(renderer)

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/axis.py:1304, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
   1302 if renderer is None:
   1303     renderer = self.figure._get_renderer()
-> 1304 return ([tick.label1.get_window_extent(renderer)
   1305          for tick in ticks if tick.label1.get_visible()],
   1306         [tick.label2.get_window_extent(renderer)
   1307          for tick in ticks if tick.label2.get_visible()])

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/axis.py:1304, in <listcomp>(.0)
   1302 if renderer is None:
   1303     renderer = self.figure._get_renderer()
-> 1304 return ([tick.label1.get_window_extent(renderer)
   1305          for tick in ticks if tick.label1.get_visible()],
   1306         [tick.label2.get_window_extent(renderer)
   1307          for tick in ticks if tick.label2.get_visible()])

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/text.py:959, in Text.get_window_extent(self, renderer, dpi)
    954     raise RuntimeError(
    955         "Cannot get window extent of text w/o renderer. You likely "
    956         "want to call 'figure.draw_without_rendering()' first.")
    958 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 959     bbox, info, descent = self._get_layout(self._renderer)
    960     x, y = self.get_unitless_position()
    961     x, y = self.get_transform().transform((x, y))

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/text.py:378, in Text._get_layout(self, renderer)
    375 ys = []
    377 # Full vertical extent of font, including ascenders and descenders:
--> 378 _, lp_h, lp_d = _get_text_metrics_with_cache(
    379     renderer, "lp", self._fontproperties,
    380     ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
    381 min_dy = (lp_h - lp_d) * self._linespacing
    383 for i, line in enumerate(lines):

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/text.py:97, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
     94 """Call ``renderer.get_text_width_height_descent``, caching the results."""
     95 # Cached based on a copy of fontprop so that later in-place mutations of
     96 # the passed-in argument do not mess up the cache.
---> 97 return _get_text_metrics_with_cache_impl(
     98     weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/text.py:105, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
    101 @functools.lru_cache(4096)
    102 def _get_text_metrics_with_cache_impl(
    103         renderer_ref, text, fontprop, ismath, dpi):
    104     # dpi is unused, but participates in cache invalidation (via the renderer).
--> 105     return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/backends/_backend_pdf_ps.py:107, in RendererPDFPSBase.get_text_width_height_descent(self, s, prop, ismath)
    104 def get_text_width_height_descent(self, s, prop, ismath):
    105     # docstring inherited
    106     if ismath == "TeX":
--> 107         return super().get_text_width_height_descent(s, prop, ismath)
    108     elif ismath:
    109         parse = self._text2path.mathtext_parser.parse(s, 72, prop)

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/backend_bases.py:645, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
    641 fontsize = prop.get_size_in_points()
    643 if ismath == 'TeX':
    644     # todo: handle properties
--> 645     return self.get_texmanager().get_text_width_height_descent(
    646         s, fontsize, renderer=self)
    648 dpi = self.points_to_pixels(72)
    649 if ismath:

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/texmanager.py:368, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
    366 if tex.strip() == '':
    367     return 0, 0, 0
--> 368 dvifile = cls.make_dvi(tex, fontsize)
    369 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
    370 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/texmanager.py:300, in TexManager.make_dvi(cls, tex, fontsize)
    298     with TemporaryDirectory(dir=cwd) as tmpdir:
    299         tmppath = Path(tmpdir)
--> 300         cls._run_checked_subprocess(
    301             ["latex", "-interaction=nonstopmode", "--halt-on-error",
    302              f"--output-directory={tmppath.name}",
    303              f"{texfile.name}"], tex, cwd=cwd)
    304         (tmppath / Path(dvifile).name).replace(dvifile)
    305 return dvifile

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/texmanager.py:259, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
    255     report = subprocess.check_output(
    256         command, cwd=cwd if cwd is not None else cls.texcache,
    257         stderr=subprocess.STDOUT)
    258 except FileNotFoundError as exc:
--> 259     raise RuntimeError(
    260         'Failed to process string with tex because {} could not be '
    261         'found'.format(command[0])) from exc
    262 except subprocess.CalledProcessError as exc:
    263     raise RuntimeError(
    264         '{prog} was not able to process the following string:\n'
    265         '{tex!r}\n\n'
   (...)
    272             exc=exc.output.decode('utf-8', 'backslashreplace'))
    273         ) from None

RuntimeError: Failed to process string with tex because latex could not be found

Error in callback <function _draw_all_if_interactive at 0x7fc9d8df5940> (for post_execute):

---------------------------------------------------------------------------
FileNotFoundError                         Traceback (most recent call last)
File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/texmanager.py:255, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
    254 try:
--> 255     report = subprocess.check_output(
    256         command, cwd=cwd if cwd is not None else cls.texcache,
    257         stderr=subprocess.STDOUT)
    258 except FileNotFoundError as exc:

File ~/psi4conda/envs/jbook/lib/python3.11/subprocess.py:466, in check_output(timeout, *popenargs, **kwargs)
    464     kwargs['input'] = empty
--> 466 return run(*popenargs, stdout=PIPE, timeout=timeout, check=True,
    467            **kwargs).stdout

File ~/psi4conda/envs/jbook/lib/python3.11/subprocess.py:548, in run(input, capture_output, timeout, check, *popenargs, **kwargs)
    546     kwargs['stderr'] = PIPE
--> 548 with Popen(*popenargs, **kwargs) as process:
    549     try:

File ~/psi4conda/envs/jbook/lib/python3.11/subprocess.py:1024, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask, pipesize, process_group)
   1021             self.stderr = io.TextIOWrapper(self.stderr,
   1022                     encoding=encoding, errors=errors)
-> 1024     self._execute_child(args, executable, preexec_fn, close_fds,
   1025                         pass_fds, cwd, env,
   1026                         startupinfo, creationflags, shell,
   1027                         p2cread, p2cwrite,
   1028                         c2pread, c2pwrite,
   1029                         errread, errwrite,
   1030                         restore_signals,
   1031                         gid, gids, uid, umask,
   1032                         start_new_session, process_group)
   1033 except:
   1034     # Cleanup if the child failed starting.

File ~/psi4conda/envs/jbook/lib/python3.11/subprocess.py:1917, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session, process_group)
   1916         err_msg = os.strerror(errno_num)
-> 1917     raise child_exception_type(errno_num, err_msg, err_filename)
   1918 raise child_exception_type(err_msg)

FileNotFoundError: [Errno 2] No such file or directory: 'latex'

The above exception was the direct cause of the following exception:

RuntimeError                              Traceback (most recent call last)
File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/pyplot.py:120, in _draw_all_if_interactive()
    118 def _draw_all_if_interactive():
    119     if matplotlib.is_interactive():
--> 120         draw_all()

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/_pylab_helpers.py:132, in Gcf.draw_all(cls, force)
    130 for manager in cls.get_all_fig_managers():
    131     if force or manager.canvas.figure.stale:
--> 132         manager.canvas.draw_idle()

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/backend_bases.py:2082, in FigureCanvasBase.draw_idle(self, *args, **kwargs)
   2080 if not self._is_idle_drawing:
   2081     with self._idle_draw_cntx():
-> 2082         self.draw(*args, **kwargs)

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/backends/backend_agg.py:400, in FigureCanvasAgg.draw(self)
    396 # Acquire a lock on the shared font cache.
    397 with RendererAgg.lock, \
    398      (self.toolbar._wait_cursor_for_draw_cm() if self.toolbar
    399       else nullcontext()):
--> 400     self.figure.draw(self.renderer)
    401     # A GUI class may be need to update a window using this draw, so
    402     # don't forget to call the superclass.
    403     super().draw()

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
     93 @wraps(draw)
     94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95     result = draw(artist, renderer, *args, **kwargs)
     96     if renderer._rasterizing:
     97         renderer.stop_rasterizing()

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
     69     if artist.get_agg_filter() is not None:
     70         renderer.start_filter()
---> 72     return draw(artist, renderer)
     73 finally:
     74     if artist.get_agg_filter() is not None:

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/figure.py:3140, in Figure.draw(self, renderer)
   3137         # ValueError can occur when resizing a window.
   3139 self.patch.draw(renderer)
-> 3140 mimage._draw_list_compositing_images(
   3141     renderer, self, artists, self.suppressComposite)
   3143 for sfig in self.subfigs:
   3144     sfig.draw(renderer)

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/image.py:131, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
    129 if not_composite or not has_images:
    130     for a in artists:
--> 131         a.draw(renderer)
    132 else:
    133     # Composite any adjacent images together
    134     image_group = []

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
     69     if artist.get_agg_filter() is not None:
     70         renderer.start_filter()
---> 72     return draw(artist, renderer)
     73 finally:
     74     if artist.get_agg_filter() is not None:

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/axes/_base.py:3064, in _AxesBase.draw(self, renderer)
   3061 if artists_rasterized:
   3062     _draw_rasterized(self.figure, artists_rasterized, renderer)
-> 3064 mimage._draw_list_compositing_images(
   3065     renderer, self, artists, self.figure.suppressComposite)
   3067 renderer.close_group('axes')
   3068 self.stale = False

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/image.py:131, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
    129 if not_composite or not has_images:
    130     for a in artists:
--> 131         a.draw(renderer)
    132 else:
    133     # Composite any adjacent images together
    134     image_group = []

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
     69     if artist.get_agg_filter() is not None:
     70         renderer.start_filter()
---> 72     return draw(artist, renderer)
     73 finally:
     74     if artist.get_agg_filter() is not None:

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/axis.py:1377, in Axis.draw(self, renderer, *args, **kwargs)
   1374 renderer.open_group(__name__, gid=self.get_gid())
   1376 ticks_to_draw = self._update_ticks()
-> 1377 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
   1379 for tick in ticks_to_draw:
   1380     tick.draw(renderer)

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/axis.py:1304, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
   1302 if renderer is None:
   1303     renderer = self.figure._get_renderer()
-> 1304 return ([tick.label1.get_window_extent(renderer)
   1305          for tick in ticks if tick.label1.get_visible()],
   1306         [tick.label2.get_window_extent(renderer)
   1307          for tick in ticks if tick.label2.get_visible()])

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/axis.py:1304, in <listcomp>(.0)
   1302 if renderer is None:
   1303     renderer = self.figure._get_renderer()
-> 1304 return ([tick.label1.get_window_extent(renderer)
   1305          for tick in ticks if tick.label1.get_visible()],
   1306         [tick.label2.get_window_extent(renderer)
   1307          for tick in ticks if tick.label2.get_visible()])

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/text.py:959, in Text.get_window_extent(self, renderer, dpi)
    954     raise RuntimeError(
    955         "Cannot get window extent of text w/o renderer. You likely "
    956         "want to call 'figure.draw_without_rendering()' first.")
    958 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 959     bbox, info, descent = self._get_layout(self._renderer)
    960     x, y = self.get_unitless_position()
    961     x, y = self.get_transform().transform((x, y))

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/text.py:378, in Text._get_layout(self, renderer)
    375 ys = []
    377 # Full vertical extent of font, including ascenders and descenders:
--> 378 _, lp_h, lp_d = _get_text_metrics_with_cache(
    379     renderer, "lp", self._fontproperties,
    380     ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
    381 min_dy = (lp_h - lp_d) * self._linespacing
    383 for i, line in enumerate(lines):

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/text.py:97, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
     94 """Call ``renderer.get_text_width_height_descent``, caching the results."""
     95 # Cached based on a copy of fontprop so that later in-place mutations of
     96 # the passed-in argument do not mess up the cache.
---> 97 return _get_text_metrics_with_cache_impl(
     98     weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/text.py:105, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
    101 @functools.lru_cache(4096)
    102 def _get_text_metrics_with_cache_impl(
    103         renderer_ref, text, fontprop, ismath, dpi):
    104     # dpi is unused, but participates in cache invalidation (via the renderer).
--> 105     return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/backends/backend_agg.py:226, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
    224 _api.check_in_list(["TeX", True, False], ismath=ismath)
    225 if ismath == "TeX":
--> 226     return super().get_text_width_height_descent(s, prop, ismath)
    228 if ismath:
    229     ox, oy, width, height, descent, font_image = \
    230         self.mathtext_parser.parse(s, self.dpi, prop)

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/backend_bases.py:645, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
    641 fontsize = prop.get_size_in_points()
    643 if ismath == 'TeX':
    644     # todo: handle properties
--> 645     return self.get_texmanager().get_text_width_height_descent(
    646         s, fontsize, renderer=self)
    648 dpi = self.points_to_pixels(72)
    649 if ismath:

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/texmanager.py:368, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
    366 if tex.strip() == '':
    367     return 0, 0, 0
--> 368 dvifile = cls.make_dvi(tex, fontsize)
    369 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
    370 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/texmanager.py:300, in TexManager.make_dvi(cls, tex, fontsize)
    298     with TemporaryDirectory(dir=cwd) as tmpdir:
    299         tmppath = Path(tmpdir)
--> 300         cls._run_checked_subprocess(
    301             ["latex", "-interaction=nonstopmode", "--halt-on-error",
    302              f"--output-directory={tmppath.name}",
    303              f"{texfile.name}"], tex, cwd=cwd)
    304         (tmppath / Path(dvifile).name).replace(dvifile)
    305 return dvifile

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/texmanager.py:259, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
    255     report = subprocess.check_output(
    256         command, cwd=cwd if cwd is not None else cls.texcache,
    257         stderr=subprocess.STDOUT)
    258 except FileNotFoundError as exc:
--> 259     raise RuntimeError(
    260         'Failed to process string with tex because {} could not be '
    261         'found'.format(command[0])) from exc
    262 except subprocess.CalledProcessError as exc:
    263     raise RuntimeError(
    264         '{prog} was not able to process the following string:\n'
    265         '{tex!r}\n\n'
   (...)
    272             exc=exc.output.decode('utf-8', 'backslashreplace'))
    273         ) from None

RuntimeError: Failed to process string with tex because latex could not be found

---------------------------------------------------------------------------
FileNotFoundError                         Traceback (most recent call last)
File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/texmanager.py:255, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
    254 try:
--> 255     report = subprocess.check_output(
    256         command, cwd=cwd if cwd is not None else cls.texcache,
    257         stderr=subprocess.STDOUT)
    258 except FileNotFoundError as exc:

File ~/psi4conda/envs/jbook/lib/python3.11/subprocess.py:466, in check_output(timeout, *popenargs, **kwargs)
    464     kwargs['input'] = empty
--> 466 return run(*popenargs, stdout=PIPE, timeout=timeout, check=True,
    467            **kwargs).stdout

File ~/psi4conda/envs/jbook/lib/python3.11/subprocess.py:548, in run(input, capture_output, timeout, check, *popenargs, **kwargs)
    546     kwargs['stderr'] = PIPE
--> 548 with Popen(*popenargs, **kwargs) as process:
    549     try:

File ~/psi4conda/envs/jbook/lib/python3.11/subprocess.py:1024, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask, pipesize, process_group)
   1021             self.stderr = io.TextIOWrapper(self.stderr,
   1022                     encoding=encoding, errors=errors)
-> 1024     self._execute_child(args, executable, preexec_fn, close_fds,
   1025                         pass_fds, cwd, env,
   1026                         startupinfo, creationflags, shell,
   1027                         p2cread, p2cwrite,
   1028                         c2pread, c2pwrite,
   1029                         errread, errwrite,
   1030                         restore_signals,
   1031                         gid, gids, uid, umask,
   1032                         start_new_session, process_group)
   1033 except:
   1034     # Cleanup if the child failed starting.

File ~/psi4conda/envs/jbook/lib/python3.11/subprocess.py:1917, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session, process_group)
   1916         err_msg = os.strerror(errno_num)
-> 1917     raise child_exception_type(errno_num, err_msg, err_filename)
   1918 raise child_exception_type(err_msg)

FileNotFoundError: [Errno 2] No such file or directory: 'latex'

The above exception was the direct cause of the following exception:

RuntimeError                              Traceback (most recent call last)
File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/IPython/core/formatters.py:340, in BaseFormatter.__call__(self, obj)
    338     pass
    339 else:
--> 340     return printer(obj)
    341 # Finally look for special method names
    342 method = get_real_method(obj, self.print_method)

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/IPython/core/pylabtools.py:152, in print_figure(fig, fmt, bbox_inches, base64, **kwargs)
    149     from matplotlib.backend_bases import FigureCanvasBase
    150     FigureCanvasBase(fig)
--> 152 fig.canvas.print_figure(bytes_io, **kw)
    153 data = bytes_io.getvalue()
    154 if fmt == 'svg':

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/backend_bases.py:2342, in FigureCanvasBase.print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, pad_inches, bbox_extra_artists, backend, **kwargs)
   2336     renderer = _get_renderer(
   2337         self.figure,
   2338         functools.partial(
   2339             print_method, orientation=orientation)
   2340     )
   2341     with getattr(renderer, "_draw_disabled", nullcontext)():
-> 2342         self.figure.draw(renderer)
   2344 if bbox_inches:
   2345     if bbox_inches == "tight":

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
     93 @wraps(draw)
     94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95     result = draw(artist, renderer, *args, **kwargs)
     96     if renderer._rasterizing:
     97         renderer.stop_rasterizing()

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
     69     if artist.get_agg_filter() is not None:
     70         renderer.start_filter()
---> 72     return draw(artist, renderer)
     73 finally:
     74     if artist.get_agg_filter() is not None:

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/figure.py:3140, in Figure.draw(self, renderer)
   3137         # ValueError can occur when resizing a window.
   3139 self.patch.draw(renderer)
-> 3140 mimage._draw_list_compositing_images(
   3141     renderer, self, artists, self.suppressComposite)
   3143 for sfig in self.subfigs:
   3144     sfig.draw(renderer)

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/image.py:131, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
    129 if not_composite or not has_images:
    130     for a in artists:
--> 131         a.draw(renderer)
    132 else:
    133     # Composite any adjacent images together
    134     image_group = []

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
     69     if artist.get_agg_filter() is not None:
     70         renderer.start_filter()
---> 72     return draw(artist, renderer)
     73 finally:
     74     if artist.get_agg_filter() is not None:

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/axes/_base.py:3064, in _AxesBase.draw(self, renderer)
   3061 if artists_rasterized:
   3062     _draw_rasterized(self.figure, artists_rasterized, renderer)
-> 3064 mimage._draw_list_compositing_images(
   3065     renderer, self, artists, self.figure.suppressComposite)
   3067 renderer.close_group('axes')
   3068 self.stale = False

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/image.py:131, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
    129 if not_composite or not has_images:
    130     for a in artists:
--> 131         a.draw(renderer)
    132 else:
    133     # Composite any adjacent images together
    134     image_group = []

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
     69     if artist.get_agg_filter() is not None:
     70         renderer.start_filter()
---> 72     return draw(artist, renderer)
     73 finally:
     74     if artist.get_agg_filter() is not None:

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/axis.py:1377, in Axis.draw(self, renderer, *args, **kwargs)
   1374 renderer.open_group(__name__, gid=self.get_gid())
   1376 ticks_to_draw = self._update_ticks()
-> 1377 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
   1379 for tick in ticks_to_draw:
   1380     tick.draw(renderer)

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/axis.py:1304, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
   1302 if renderer is None:
   1303     renderer = self.figure._get_renderer()
-> 1304 return ([tick.label1.get_window_extent(renderer)
   1305          for tick in ticks if tick.label1.get_visible()],
   1306         [tick.label2.get_window_extent(renderer)
   1307          for tick in ticks if tick.label2.get_visible()])

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/axis.py:1304, in <listcomp>(.0)
   1302 if renderer is None:
   1303     renderer = self.figure._get_renderer()
-> 1304 return ([tick.label1.get_window_extent(renderer)
   1305          for tick in ticks if tick.label1.get_visible()],
   1306         [tick.label2.get_window_extent(renderer)
   1307          for tick in ticks if tick.label2.get_visible()])

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/text.py:959, in Text.get_window_extent(self, renderer, dpi)
    954     raise RuntimeError(
    955         "Cannot get window extent of text w/o renderer. You likely "
    956         "want to call 'figure.draw_without_rendering()' first.")
    958 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 959     bbox, info, descent = self._get_layout(self._renderer)
    960     x, y = self.get_unitless_position()
    961     x, y = self.get_transform().transform((x, y))

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/text.py:378, in Text._get_layout(self, renderer)
    375 ys = []
    377 # Full vertical extent of font, including ascenders and descenders:
--> 378 _, lp_h, lp_d = _get_text_metrics_with_cache(
    379     renderer, "lp", self._fontproperties,
    380     ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
    381 min_dy = (lp_h - lp_d) * self._linespacing
    383 for i, line in enumerate(lines):

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/text.py:97, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
     94 """Call ``renderer.get_text_width_height_descent``, caching the results."""
     95 # Cached based on a copy of fontprop so that later in-place mutations of
     96 # the passed-in argument do not mess up the cache.
---> 97 return _get_text_metrics_with_cache_impl(
     98     weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/text.py:105, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
    101 @functools.lru_cache(4096)
    102 def _get_text_metrics_with_cache_impl(
    103         renderer_ref, text, fontprop, ismath, dpi):
    104     # dpi is unused, but participates in cache invalidation (via the renderer).
--> 105     return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/backends/backend_agg.py:226, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
    224 _api.check_in_list(["TeX", True, False], ismath=ismath)
    225 if ismath == "TeX":
--> 226     return super().get_text_width_height_descent(s, prop, ismath)
    228 if ismath:
    229     ox, oy, width, height, descent, font_image = \
    230         self.mathtext_parser.parse(s, self.dpi, prop)

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/backend_bases.py:645, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
    641 fontsize = prop.get_size_in_points()
    643 if ismath == 'TeX':
    644     # todo: handle properties
--> 645     return self.get_texmanager().get_text_width_height_descent(
    646         s, fontsize, renderer=self)
    648 dpi = self.points_to_pixels(72)
    649 if ismath:

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/texmanager.py:368, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
    366 if tex.strip() == '':
    367     return 0, 0, 0
--> 368 dvifile = cls.make_dvi(tex, fontsize)
    369 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
    370 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/texmanager.py:300, in TexManager.make_dvi(cls, tex, fontsize)
    298     with TemporaryDirectory(dir=cwd) as tmpdir:
    299         tmppath = Path(tmpdir)
--> 300         cls._run_checked_subprocess(
    301             ["latex", "-interaction=nonstopmode", "--halt-on-error",
    302              f"--output-directory={tmppath.name}",
    303              f"{texfile.name}"], tex, cwd=cwd)
    304         (tmppath / Path(dvifile).name).replace(dvifile)
    305 return dvifile

File ~/psi4conda/envs/jbook/lib/python3.11/site-packages/matplotlib/texmanager.py:259, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
    255     report = subprocess.check_output(
    256         command, cwd=cwd if cwd is not None else cls.texcache,
    257         stderr=subprocess.STDOUT)
    258 except FileNotFoundError as exc:
--> 259     raise RuntimeError(
    260         'Failed to process string with tex because {} could not be '
    261         'found'.format(command[0])) from exc
    262 except subprocess.CalledProcessError as exc:
    263     raise RuntimeError(
    264         '{prog} was not able to process the following string:\n'
    265         '{tex!r}\n\n'
   (...)
    272             exc=exc.output.decode('utf-8', 'backslashreplace'))
    273         ) from None

RuntimeError: Failed to process string with tex because latex could not be found

<Figure size 1200x600 with 2 Axes>

En este Hamiltoniano se observa una relacion de dispersion propia d eun semimetal de Weyl tipo I. Adicionalmente, se observa una brecha energetica alrededor de \(E[t]=0\)

Ahora se analizará el comportamiento de la estructura de bandas en un sistema finito en una dirección e infinito en dos direcciones

Exploración de las bandas de semimetal de Weyl#

Para estudiar cómo se comportan las bandas en distintos planos del material, es necesario pensr que nuestro muestra es un sistema cubico infinito, es decir tiene simetria de traslación en la dirección \(X, Y,Z\). Ahora, si se desea estudiar al sistema en \(X\), se debe cortar en esa dirección, es decir, se debe romper la simetria de traslación en \(X\), pero amnteniendola en \(Y\) y en \(Z\). Esto mismo aplica cuandos e quieres estudiar en \(Y\) y \(Z\), se deben romper las simetrías respectivas.

Sistemas finitos en una dirección

Hide code cell source 
NY_WSM=PWSM.cut_piece(100,1,glue_edgs=False) 
NX_WSM=PWSM.cut_piece(100,0,glue_edgs=False)
NZ_WSM=PWSM.cut_piece(100,2,glue_edgs=False)
Hide code cell source 
k=[[0,0.0],[0.0,0.25],[0,.5]]                                
k_label1=[r"$-Z$",r"$\Gamma$",r"$Z$"hide-input]
(k_vec,k_dist,k_node)=NX_WSM.k_path(k,101, report=False)
EkfinX=NX_WSM.solve_all(k_vec)
k=[[0,0.0],[0.25,0],[.5,0]]  
k_label2=[r"$-X$",r"$\Gamma$",r"$X$"]
(k_vec,k_dist,k_node)=NY_WSM.k_path(k,101, report=False)
EkfinY=NY_WSM.solve_all(k_vec)
k_label3=[r"$-Y$",r"$\Gamma$",r"$Y$"]
(k_vec,k_dist,k_node)=NY_WSM.k_path(k,101, report=False)
EkfinZ=NZ_WSM.solve_all(k_vec)
Hide code cell source 
fig,ax  = plt.subplots(ncols=3,nrows=1,figsize=(20,6),
                       gridspec_kw = {'wspace':0.2, 'hspace':0, 'width_ratios': [1, 1,1]})
for n in range(0, len(EkfinX)):
    
    ax[0].plot(k_dist,EkfinX[n,:], c='purple', alpha=0.6) #la long de Ek fin tiene todo
    ax[1].plot(k_dist,EkfinY[n,:], c='purple', alpha=0.6) #la long de Ek fin tiene todo
    ax[2].plot(k_dist,EkfinZ[n,:], c='purple', alpha=0.6) #la long de Ek fin tiene todo

for i in range(0,3):
    ax[i].set_xticks(k_node) # puntos de alta simetria, valor que correspomde en k_path
    
    ax[i].set_yticks([-2,0.,2])
    # xsticklabels(k_label) 

    ax[i].spines['right'].set_visible(False)
    ax[i].spines['top'].set_visible(False)
ax[0].set_xticklabels(k_label1 )
ax[1].set_xticklabels(k_label2 )
ax[2].set_xticklabels(k_label3 )
ax[0].set_ylabel(r"$E[t]$",fontsize=26)
ax[0].set_xlabel(r"Finite System in $X|_{K_Y=0}$",fontsize=26)
ax[1].set_xlabel(r"Finite System in $Y|_{K_X=0}$",fontsize=26)
ax[2].set_xlabel(r"Finite System in $Z|_{K_X=0}$",fontsize=26)

# puntos de alta simetria, valor que correspomde en k_path

savefig("Finite_systems_Koshino.pdf",bbox_inches='tight')
_images/5afbadd2d1e4582fb46e38c12fd03a00087a990fbe825204f0c52e9c422d8591.png

Se denota que los conos de Dirac sólo aparecen cuando el sistema es finito en la dirección \(X\) y con un recorrido en \(Z\). Sin embargo, no hay estados de superficie.