Okay. So  thanks  to  the  organizers for  inviting  me  to  this  very  nice  meeting. So  my  talk  will  be  maybe  a little  bit  different  from what  you  heard  earlier  today. This  is.  You  can  you  hear  me.  Yeah. So  a  bunch  of different  things  that  I'll try  to  get  through. Maybe  it's,  you  know, quite  a  few  things  going  on  here. So  superconductivity,  we'll  talk about  a  new  way  to  get  superconductors. Anions  will  make  an  appearance. But  really,  the  way  we  connect to  conformal  field  theory  is  through this  critical  point  between a  pair  of  topological  phases. And  I'm  hoping  that the  connection  to  the  program through  conformal  field  theory, I  can  maybe  not  solve some  of  the  problems  that  have  come  up, but  I  can  give  you reasons  maybe  to  look  at,  you  know, certain  subset  of  problems  that are  of  great  interest  to  us  in  Kenz  matter. Okay.  So  by  the  way, feel  free  to  ask  questions  anytime. I  realize  there's  a  major  language  barrier in  terms  of  science  terms  and  so  on. So  feel  free  anytime  to interrupt  me,  any  questions. Okay,  so  a  big  question in  Kinetics  matter  ofphsics  is  can we  get  superconductivity  through a  non  BCS  mechanism,  okay? That's  this  is  roughly  BCS. You  start  at  the  metallic  state, weak  attraction  between  the  electrons, perhaps  from  pons, you  end  up  getting  your  superconductor. Okay?  So  what  I'll  talk  to  you talk  about  today  is  way, way  different  scenario  where we  believe  we  get  superconductivity  as  well. There  will  be  no  metal.  In  fact, we'll  have  a  pair  of  insulators  that undergo  a  quantum  phase  transition. And  when  we  add  charge, so  this  is  chemical  potential, we  will  end  up  getting  a  superconductor, so  we  call  this  doping,  adding charge  near  the  critical  point. That's  where  the  CFT  emerges. But  over  some  range, so  you  don't  have  to  be  exactly  here. If  you're  somewhere  here,  if  you raise  the  chemical  potential  enough, you  can  end  up  in  the  superconducting  state. Okay,  so  that's  kind  of  the  motivation. We'd  like  to  have  more  tools  to such  study  conformal  field  theories  with the  additional  feature  that they  also  have  a  conserved  UN  charge. So  if  I  have  a  charge,  I  can  add a  finite  density  of  particles,  and  I  can  ask, what  is  the  fate  of  the  system, this  finite  density  of  particles when  I  perturb  away  from  the  CFT. And  in  some  cases, we  believe  this  is  a  superconductor  and it's  a  good  way  to  get  a  superconductor. Okay,  so  of  course, we  don't  want  to  begin.  Yeah. CFT  is  Lorenzen  variant in  some  number  of  dimensions. In  this  particular  case,  it will  be  Lorenzen  variant. At  least  we  have  a  reasonable argument  to  expect  that. Um,  and  of  course, always  there  is  a  question whether  we  write  on these  theories,  lagranians  look  nice, Laurentian  variant,  et  cetera, h.  We  don't  quite  know if  the  fluctuations  will  drive that  first  order,  so  we  have  to  work. We  can  give  some  numerical  evidence, but  that's  always  going  to  be a  question  hanging  in  the  background. And  doing  things  like  Epsilon  expansion large  and  helps  to provide  some  evidence  that, you  know,  maybe  it  is  continuous,  um,  Okay. So  yeah,  so  I'll  have a  few  so  this is  my  slide  with  the  sphere  on  it. Make  sure  to  have  one  so  my  time  doesn't suddenly  end  and  Yeah, so  how  does  this  relate  to  things  that people  are  interested  in  in this  sphere  workshop? So  we're  going  to be  talking  about  superfluids, and  actually,  thanks  to Max  for  pointing  out  this  connection. So  people  study,  the  operator spectrum  of  CFTs  at  large  global  charge. And  by  the  operator  state  mapping, this  corresponds  to  taking  a  sphere. We're  going  to  do  everything  in two  plus  one  dimensions. On  the  sphere,  you  have a  finite  density  of  U  one  charge. And  in  fact,  people  have  made,  you  know, statements  about,  uh, the  scaling  dimensions,  how  they  scale, at  least  some  of  the  terms  in  these, um,  um,  uh,  in  the  primary  operator, uh,  dimensions  that  relate  to the  Goldstone  modes  of a  superfluid  if  it  emerged  on  the  sphere. Okay,  so  there  is  this  connection  between the  ground  state  of  finite  charge  density  or, you  know,  perturbing  or  CFT  on the  sphere  and  the  operator spectrum  at  large  global  charge. Okay,  so  maybe  there  are  other ways  to  get  at,  you  know, maybe  you  don't  really  form  a  superfluid, and  you  use  these  tools  to  figure out  what  actually  happens,  um. So  that's  kind  of  the  connection,  um,  Okay, but  let  me  give  you  a  few  slides  of overview  before  I  actually show  you  the  Finally, we're  going  to  talk about  a  microscopic  model. It  turns  out  in  this  business, it's  kind  of  important  to  work with  a  microscopic  model, but  I'll  give  you  some  more  general  framework which  may  be  more  familiar  to people  in  this  audience. So  we'll  be  considering a  phase  transition  upon phase  transition,  one  critical  point, which  will  turn  out  to be  Lorenz  invariant  CFT, in  the  best  case  scenario, between  two  gap  phases, we'll  call  them  phase  one  and  phase  two. There  is  some  distinction  between  these, and  for  our  purpose,  we want  them  both  to  be  gapped. So  for  example,  we're  now  going to  talk  about  the  uh, the  mo  superfluid  transition, the  usual  complex scalar  condensation  transition. One  of  those  states  is,  uh,  gapless. So  instead,  we'll  be talking  about  two  gap  phases. There's  going  to  be  a  UN  to symmetry  in  our  CFTs  in  our  theories. Um,  But  of  course, there's  a  gap  to  these  charge excitations  in  the  phases. So  really,  the  phases  are  insulators. As  far  as  the  charge,  degrees of  freedom  are  concerned,  they're  all  gap. And  we're  going  to  probe  these  things  with the  external  gauge  field that  couples  to  this  globally  one  charge. Okay,  now,  despite  the  fact that  you  have  two  different  insulators, you  have a  phase  transition  occurring  between  them, you'll  see  that  there  is  a topological  distinction  between  these  phases. That  is  shows  that  shows up  in  the  charge  conductance. There's  a  thermal  there's electrical  hole  conductance  difference between  these  two  insulators, which  will,  you  know,  so  this  has  to change  from  one  hole  response  to  another. And  that  requires  that  in  between, you  have  a  phase  transition. And  if  the  phase  transition  is  continuous, you're  going  to  have  gapless charge  degrees  of  freedom between  these  two  insulators. So  they're  going  insulator  to insulator  in  between,  it  becomes  gapless. And  we'd  like  to understand  what  is  the  nature  of the  gapless  charge  degree of  freedom  at  this  transition. In  many  cases,  you  know,  for  example, in  the  integer  quantum  hole you  go  from  g  max, one  integer  to  the  next, the  gapless  charge  is  just  the  electron. You  just  get  electrons  at low  energies  that  change  the  response. In  our  example,  it'll  turn  out although  you  have  an  electronic  model, it'll  automatically  automatically  turn out  to  be  Cooper  pairs. So  in  a  way,  it's a  natural  way  to get  pairing  between  electrons. So  in  order  to  make  this  transition, the  system  is  forced  to bring  the  Kuper  pairs  down  in energy  and  then  change  the  charge  response. And  then  you  have  an  opportunity  to get  a  supercduct. If  you  add  charge,  it  can  go  into the  Kuper  pairs  and  give  you  a  superinductor. Okay.  So  a  little  more  pictorally, this  is  sort  of  the  phases  that we  are  going  to  be  interested  in. On  one  side, the  gap  phase  is  just  going  to  be a  simple  integer  quantum  hall  phase. There  are  two  spins  down. So  you  know,  it's going  to  have  a  hall  conductance, one  for  each  spin. On  the  other  side,  we're  going  to  have a  phase  which  has  a  spin  liquid, so  the  charged  degrees  of  freedom  are  frozen. All  you  have  is  the  spin  degree of  freedom  that's  active, but  it's  going  to  be  a  particular  kind  of spin  liquid  called  the  Cal  spin  liquid. Yeah,  and  in  between  this  phase  transition, we  can  write  down  a  theory  for a  potential  transition  between  these  two. And  this  is  what  the  theory  looks  like. Okay,  so  it's  a  complex  scalar, Sos  is  a  complex  scalar,  but  it's  gauged. There's  a  little  A  that  couples  to  it as  a  fluctuating  gauge  field. There's  some  potential  for  the  psi. And  then  there's  a  churn  Salmons  term, which  includes  the  external  probe  field. And  the  churn  Salmon's  term  is  at  level  two. Um,  And  the  way  you  get  this  two  phases. So  this  is  within  one  set  of  variables, we  can  get  both  of  these  phases depending  on  whether  psi  condenses  or  not. So  if  si  is  condensed, it's  easy  to  see  that  this  little  is  hexed. So  you  can  just  set  a  little  at  zero, and  all  you  get  is the  response  of  the  probe  gauge  field. If  in  order  to  read  this,  this  is  really a  hall  conductance  of  two,  okay? So  this  is  the  inter  hall  effect. On  the  other  hand  of  size  gap, so  this  is  also,  you  could  call  it an  invertible  topological  phase. There  are  no  anions,  but  there  are non  trivial  red  states. Again,  this  is  non  trivial  hall  conductance. Now,  on  the  other  hand  of  size  gap, then  you  can  integrate  out, you  know,  then  this,  you  know,  size  gone, you  can  forget  about  it,  and you're  left  with  this  particular  theory, you  can  absorb  the  capital  A  into the  little  A,  just  a  redefinition. This  is  the  effective  theory  that  you  have. So  this  is  a  theory which  has  no  charge  response. It  doesn't  matter  how  you kind  of  manipulate  your  capital  A, can  simply  be  absorbed  by a  fluctuating  gauge  field. So  it's  an  insulator which  has  no  hall  conductances  either, but  it  does  have  topological  order. It's  a  Charsans  theory  at a  non  trivial  Level.  Yeah. I'm  sorry,  what  is varied  across  the  transition? So  you  vary  the  potential.  So  as  usual, there  is  a  quadratic  term,  size  squared, coefficient  of  that  change  of  sign. Yeah.  But  sor normally,  it's  going  to  be  different. So  how  do  you  get  rid  of  that  how  do  you  get a  transition  between  states  with  different? You're  talking  about  the  edge  states? Yeah. Yes.  So  this  is. So  this  theory  does  that  for  you. Yeah,  it's  kind  of  weird  because  usually the  missing  plurality  disappear  because it  purelity  can  usually only  anlate  with  anticlo. Yes.  So  it  percolates  through  the  system. Just  like  in  the  dignomol  transition, the  system  becomes  gapless and  one  of  the  edge  modes  percolates  through. And  that  is  really  the  fact  that  the  child becomes  gapless  at  the  transition.  Okay. All  right.  Yeah.  Okay.  So  it's still  a  critical  thing  here. Lorenz.  The  transition  has the  It  could,  yeah. Yeah.  So  this  theory  has  Lorenz  variance, captures  both  phases. And  yeah,  so  I drop  this  gravitational  on  Simmons  which keeps  track  of  the  edged eight  degrees  of  freedom. But  clearly  this  one will  have  on  number  of  two, current  central  charge  of  two, and  this  is  current  central  charge  one. This,  uh,  the  compact  boson. You  can  have  a  critical  state where  it's  essentially. Yes,  it  turns  out  that,  that  is  the  side. So  somehow  there  is, of  course,  this  geometrical  picture  of the  edge  modes  going  in  and  all  that. But  somewhat  amazingly,  it  can  be  captured streak  The  edges  are  the  edge  edge  modes. Yes,  but  it  is  written  just  in  terms  of a  very  conventional  andnsburg I  never  knew  that conventional  land  of Ginsburg  could  actually  have the  cleral  aspect  of clearly  not  conventional. Well,  it's  really  the  Tn  Simon's  term combined  with  the  andagensburg. So  you  just  do  this  exercise, you  get  these  two  phases,  right? There  is  no  topological  order  on  this  side, and  there  is  the  semi on  topological  order  on  the  side. Yeah,  I  have  a  question and  what  about  the  nature of  the  CFT  at  critical  point. The  CFT  mean  the  edge  CFT. Because  the  edges, it's  going  to  be  a  bulk.  Yes,  yes. The  question  was,  is the  bulk  theory  invariant, two  plus  one  dimensional  theory? Yes,  that  is  the  claim. Yes.  And  that  is  just  this  thing set  at  the  master of  squared  set  equal  to  zero. But  then  it's  got this  additional  feature  chiral. It  apparently  it  does,  yeah. But  it's  all  contained  in  the  lagrangen. Yeah.  So  this  side  of  the  theory is  actually  has  an  on. So  it's  a  topologically  ordered  phase, a  scalar  QED  theory,  right? It's  basically  scalar  QED. Plus,  yeah. But  Scalar  acuity  with  John  Simon  Star. Yeah.  And  scalar  acuity  three? Yes.  Acuity.  Yeah.  Single  component. Yes.  We  have  a  critical  theory of  close  critical  theory  of. Yeah.  Somehow,  that  this  is  capturing  that. And  what  we're  going  to  try  to  do is  we're  going  to  add  another  axis  to  this. We'll  never  be  happy  with  this, we're  going  to  add another  axis  chemical  potential. You  need a  finite  chemical  potential  to  end  up adding  charge  to  these  gapped  phases. And  what  we'll  see  is  that  when  you  do  that, you  end  up  we'll  make an  argument  why  you'll end  up  in  a  superconductor. Yeah.  That's  where  we're  going. The  external  gauge  field  is  A. Yes.  The  product  field  is  capital  H.  Yeah. So  I'm  going  to  turn  on  chemical  potential, I'm  going  to  turn  on  a  zero and  critical  for  example, usually  these  things  are kind  of  first  order  fair  transitions. Yes,  it  could  be.  I'll  show  you some  numericals  example  of  something called  Gap  which  is  interesting. Obscure  but  they're  known  to  be  a  kind  of da  state  but  it's usually  people  speculates  two  theories. So  what  guarantees  that  you're  going  to  get? There  no  guarantees  will  actually. There  are  no  guarantees  in  this. So  it  may  not  be  SPFT.  It  may  not  be.  Yeah. Exactly.  So  the  best  we  can  do is  is  not  a  vehicle,  it  may  not  be. Yeah.  So  the  best  we  can  do  is  write  down a  candidate  theory  Then we  can  direct  through  numerical  calculations, see  if  there's a  diverging  correlation  length. We  can  do  those  kind  of  things. But  the  level  of  difficulty of  these  problems  is  there's  no  certainty, unlike  one  D  or  something  where  you  can. Is  there  a  kind  of other  kind  of  known  examples of  of  simplest  one  stuff  with  chiral. Well,  you  know,  you  could  say the  integer  got  them. But,  you  know,  a  very  simple  example  is just  the  integer  quomal effect  in  the  clean  limit. You  have,  let's  say,  SG max  one  to  zero  transition. The  edge  mode  of  the  Sig  max  one, the  integer  quomal  has  to  penetrate the  bulk  and  annalate in  order  to  get  the  churn  zero  state. Yeah,  right?  We  have  a  theory. Yeah,  just  the  free  drag  theory. What?  Free  drag. So  you  change  the  sine  of  mass  on free  drag  theory  that will  take  you  between churn  zero  and  churn  one. At  the  critical  point,  you have  a  relativistic  theory, which  is  just  non  interacting  fermions with  direct  dispersion. Yeah. So  this  is the  more  complicated  analog  of  that. You  have  a  normally  have  a  single. Yeah.  Yeah.  So  here you  have  a  chirality  built  in. Yeah. So  here  you  can  rewrite  this  if  you  like  in terms  of  drug  fermions covered  gauge  fields  as  well. There  is  a  bot  for  me.  You  can have  QED  with  transon  terms. Those  are  known  to  be  CFDs.  That  too. Yeah.  Yeah.  Yeah.  I  see  you  need an  odd  number  of.  Yeah.  Yeah. Right.  Yeah.  So  chiral.  Right.  Yeah.  Yeah. So  I  think  you  can  run  the  odd  number  of drug  fermions  and some  other  level  of  this  gss. Just  good  decomposite  fermions  from  this. Okay,  so  let  me  move  on. The  third  the  third  point of  view  I  wanted  to  put  in  is that  we  have  one  side  where  we  have  anions, and  imagine  adding electric  charge  into  that  state. So  you  get  a  finite  density  of  anions  there. And  there's  an  old  idea,  due  to Loughlin  and  also,  you  know, folks  here  at  that  time, that  this  is  another  way  to get  superconductivity. Per  density  of  electrically  charged  anions, under  certain  circumstances  can give  you  a  very, you  know,  a  different  way  to  get superductivity. And  this  is  one  of  the  few  examples, I'd  say  it's  fairly controlled  compared  to,  you  know, like  top  other  kinds  of  spilled  liquids, where,  you  know,  you  can  try  to realize  this  anion  superductivity. Okay,  so  we'll  have  that  flavor  as well  because  we're  going  to  have  anions. The  reason  this  is  of  interest these  days,  of  course, we've  always  had  for  many  years, we've  had  anions  in the  fractional  quantum  Hall  effect. But  there  if  you  add  electric  charge  in the  form  of  anions, they  don't  really  condense. Because  it's  a  strong  magnetic  field, they're  going  to  just sit  where  you  put  them  in. Recently,  we  have  these  mod  systems, we  have  potentials  that modulate  the  quantum  Hall  effect  if  you  like, and  give  dispersion  to these  extations  we  are  in  the  regime, really  in  the  regime  where  one  can  talk  about finite  density  of  anions which  have  kinetic  energy, and  we  can  see  if  they  can  make  a  superuted. So  that's  the  third  kind  of  motivation. So  now  let  me  kind  of  get  microscopic, talk  about  an  actual  lattice  model  that, you  know,  we  could  try  to  see  how that  effective  field  theory  comes  out  of. We're  going  to  start  with  the  lattice  model that  shows  these  two  phases, and  then  we'll  write  down  a  theory  that, you  know,  we  can  see the  effective  theory  coming  out  of. So  this  is  an  extremely  simple  model. It's  called  the  Hubbard  model, but  with  one  twist. So  it's  some  hopping  of  electrons, fermions  on  a  lattice, in  this  particular  case,  triangular  lattice. So  at  each  side  of  the  triangle  latice, you  can  have  an  electron  side  up  or  down. You  can  have  no  electron and  you  can  have  two  of  them. That's  the  That's  the  Albert  space. And  they  hop  from  side  to side  with  this  hopping  TIJ. And  then  there  is  a  repulsion. The  only  repulsion  we put  in  is  on  the  same  side, you  like  to  have  one  electron. You  don't  like  to  have  zero. Um,  the  other  thing we're  going  to  do  is  we're  going  to change  the  hopping  from  regular  hopping to  one  that  includes  flux,  okay? So  I'm  going  around  this  pet over  here  and  you  get  a  flux, which  is  a  phase  factor  R.  Okay, so  it's  sort  of  like  you  took the  triangle  As  Hubbard  model  and  print a  strong  megalic  field to  give  you  these  hoppings. And  this,  of  course, has  spin  rotation  symmetry, and  it  has  the  conservation of  electric  charge  as  well. One  very  nice  thing  about  this  model, it  actually  has  more  symmetry. In  fact,  it  has  what  we call  the  pseudopin  symmetry that  the  charge  you  want is  part  of  a  larger  SU  two  group. You  can  get  it  by  arranging the  fermions  in  this  way. Rotation  on  the  left  gives  you  spin  rotation, rotation  on  the  right  gives you  the  pseudospin  symmetry. So  it's  some  kind  of  particle  hole symmetry  is  present  in  this  model. Okay,  but  we'll  just  refer to  these  symmetries  by  and  large. You  can  explore  this  model in  two  opposite  limits. One  limit  is  when  the interactions  are  extremely  weak. It's  essentially  three  fermions. You  can  solve the  band  structure  very  trivially, and  you  get  two  bands. So  because  of  the  flux,  there's a  doubling  the  unit  cell, two  sides  in  the  unit  cell, translates  to  two  bands  in  momentum  space. These  bands  have  churn  number. Okay,  so  you  completely  fill up  one  of  these  bands, one  electron  per  site, you  end  up  in  a  churn  insulator, which  has  a  whole  conductance  of  two  units, two  units  because  you have  spin  up  and  spin  up. There's  also  something  called  a spin  hall  effect  that  you  can  define, the  quanti  spin  off  effect, which  is  different  from the  quantum  spin  off  effect, but  this  is  purely  a  spin  response. This  is  also  two  because  we  have  two of  these  edge  modes for  spin  up  and  spin  down, each  of  which  can  transport  spin  at  the  edge. This  is  an  integer  quantum  hall  insulator, a  very  simple  kind  of  state. But  what  is  interesting  is  I  imagine cranking  up  the  repulsive  interaction. And  as  the  interaction gets  stronger  and  stronger, you'd  like  to,  you  know, have  exactly  one  electron  per  site. You  want  to  go  towards that  limit,  and  one,  um, in  a  possible  state  that  you  can  get, you'll  see  in  numerics whether  this  works  out, is  what  is  called  a  chiral  spin  liquid. In  the  chiral  spin  liquid, the  charge  response  is extinguished  because  of the  stronger  propulsion  the spin  response  persists. The  same  spin  response, but  to  have  this  kind  of  distinction means  something  funny  has  happened. You've  got  a  state  which  has  got an  on  stop  lot  order  of this  UN  level  two  variety, whereas  this  is  some  in  vertical  phase. Okay,  so  that  is  the  schematic  phase  diagram, just  looking  at  this  model and  different  limits  of  this  model. The  Sigma  spin  XY? How  should  I  think  of  this? Yeah.  So  think  of  it  as, you  know,  there's  an  acetosinsmmetry. Let  me  break  it  down  to  U  one. So  now  I  can  put  in an  external  probe  field for  that  U  one  spin  symmetry. It  couples  in  opposite  way, so  spin  up  and  spin  down. So  it's  different  from  the  capital A  that  was  the  U  one  charge. Now,  you  know,  it's  a  gap  phase. These  are  both  gap  phases  that  can integrate  out  all  the  gap  excitations. And  look,  what  is  the  probe  field, you  know,  response,  give  me. I  got  a  Churn  Sam's  term  for both  the  spin  gauge  field and  for  the  charged  gauge  field  in  this  case, but  only  for the  spin  gauge  field  in  this  case. Yeah.  So  for  free  fins easy  to  do,  both  of  them  kind  of  add. You  don't  care  about  the  sign,  whether  this is  coupling  with  the  plasma minus  sign  to  the  two  spins. Yeah,  so  this  is  just  a  way  to  keep  track of  the  topology  of  the  states of  these  systems. Yeah.  I  just  want  to  make  sure. So  there's  one  way  to get  to  the  same  topologygical  order. That  is,  say  we  have  pairs  of ferme  from  the  boss and  then  boss  from  one  half. Yes,  that's  different,  right? I'm  going  to  explain  that  in  the  next  slide. Actually,  I  think  I  learned  this  from  you, so  I'm  going  to  repeat this  in  the  next  slide. Actually,  maybe  two  slides.  Yeah,  for  now. So  this  is  some earlier  work  that  people  looked  at this  model  numerically  to  see whether  this  expectation  was  correct  or  not. You  know,  of  course, on  the  Bac  contraction  side, they  saw  the  ing  quantumho  effect. This  was  done  DMRG what  you  can  do  is  you  can do  an  entanglement  cut. Effectively,  it's  a  probe  of  the  edge  states. And  then  as  a  function  of momentum,  the  entanglement  spectrum, actually  introduced  by  Duncan and  several  years  back, you  can  look  at  the  counting  of states  in  this  entanglement  spectrum, and  that  is  effectively  a  proxy  for  the  edge. And  these  numbers, the  degencies  that  you  get is  characteristic  of  the  edge  mode, so  the  topological  order  that  you  have. And  in  this  case,  you  can also  do  this  pumping  experiment. You  put  in  flux  through the  cylinder  and  see how  much  charge  gets  pumped. It's  a  probe  of  San  Sun's  term for  the  external  gauge  fields, and  you  can  isolate both  a  charge  pump  and  a  spin  pump. And  this  tells  you  that  SigmaxY  charge is  two  and  Sigma  max  spin  is  also. Just  some  normalization  makes  them  different. But  the  more  interesting  thing is  the  strong  interaction  side. So  once  this  U  over T  ratio  goes  above  ten  or  12, you  get  into  a  different  phase according  to  these  numerics, and  it  seems  completely consistent  with  this  car  spin  liquid. You  get  the  counting  for the  edge  of  the  current  spin  liquid. And  when  you  do  this  pumping  experiment, you  get  the  spin  response, but  not  the  charge. Okay,  so  it's  consistent with  that  distinction. So  there  is  numerical  evidence on  small  cylinders. Circumference  is,  like, always  smaller  than  six,  but, you  know, there's  some  non  trivial  evidence  for  this, um,  in  these  numerics. Okay,  now  this  critical  point,  um,  you  know, what  I  flashed  before  actually  has an  emergent  SU  two  symmetry. It  was  pointed  out  in  a  paper  by  Natty, and  then  we  also  wrote a  paper  on  it  with  chen. Um,  so  there  are many  interesting  things about  that  critical  point, which  I  didn't  really  talk  about. It  has  an  SU  two  symmetry. That's  the  pseudospin  SU  two,  actually, which  relates  the  operator that  inserts  cop  pairs to  a  conserved  current. So  it  actually  fixes the  scaling  dimension of  that  particular  operator. But  I  won't  talk  too  much  about that  maybe  if  there  are  questions  at  the  end. So  there's  evidence  for  this  critical  point, and  for  these  two  phases, to  the  best  that people  can  do  in  the  nematic. What  CFT  do  you  think is  describing  that  critical  point? That's  the  one  I  wrote  in  the  beginning. That's  one,  you  know, version  of  it,  scale acuity  with  the  Tsims  term. It's  not  apparent  at  all  in  that  CFT, that  in  that  presentation  that  it  has an  SU  two  symmetric  that  relates the  monopole  insertion  operator to  some  conserved  current. But  there  are  other  ways  of writing  it  where  it's  more  apparent. Okay,  so  let  us  recover  that  theory. And  one  way  we  can  do  that  is using  the  so  called  parton  construction. You  break  up  the  electron  into  two  partons, sort  of  like  breaking  up  the  hadron  or a  proton  or  whatever  into  quarks. And  there  is  one  fermionic  object  spin on  and  a  bosonic  object, the  charge  on,  we'll  call  it. We'll  write  it  like  a  phase  field. And,  of  course,  there  is some  ambiguity  in  the  separation. So  there's  an  emergent gauge  field  that  you  need  to include  that  couples  to these  objects  tells  you that  they  are  partons, not  really  local  objects. So  then  the  way  you  proceed  is  that this  F  fermion  actually ends  up  in  a  churn  insulator  phase. You  write  on  a  me  field  theory  in  terms  of these  fermions  and  charge  on, and  this  is  a  mature  insulator  phase, so  you  can  integrate  out  the  fermions. But  because  it's  coupled  to a  emergen  gauge  field, it  gives  you  a  churns  in  start. Okay?  So  that's  the  so the  entire  fate  of this  problem  depends  on what  happens  to  the  charge. So  there  are  two  things  this charge  on  can  do  naively. One  is  it  can  condense,  right? There's  a  finite  density of  them.  It  can  condense. And  when  it  does,  essentially, if  you  look  at  this  equation,  this  is like  a  C  number  now,  it's  condensed. There's  no  real  distinction  between the  F  and  the  C.  Okay, that's  the  Higs  phase. So  you  can  say  that  the  F  being  in the  chlator  is  like the  C  being  in  the  chlator. This  is just  the  regular  integer  quantum  phase. So  that's  the  condensate  of  this  charge  on. But  the  other  option  is that  you  continue  with  this, you  know,  answers  for  F, but  you  gap  out  the  charge  on,  okay? So  that  is  the  symmetric  phase. And  then  when  you,  you  know, integrate  out  these  fermions,  you  get  this, you  know,  the  U  on level  two  turn  Silence  term for  a  fluctuating  gauge  field. Okay,  so  that  is  the  way  in  which  you  can derive  the  effective  theories for  these  two  phases. And  of  course,  the  transition  is whether  or  not  the  spin  condenses. So  if  you  add  it  as  a  soft  spin, then  this  is kind  of  the  effective  theory of  the  transition. It's  what  I  wrote  before, except  I've  shifted  around the  iteration  of  this  capital A.  I  think  then  I  had  it  there, but  I  can  always  absorb  it and  it  will  reappear  over  here. Okay,  so  that's  roughly the  justification,  if  you  like. How  from  a  microscopic  theory, you  can  implement,  you  know, this  procedure,  and  you  can intert  this  particular  effective  theory. Okay.  Okay,  so  now let's  look  at  some aspects  of  this  transition, which  are  slightly  surprising, I  think,  which  is  the  following. So  let's  imagine  we're going  across  this  transition, and  we  keep  track  of all  the  different  degrees  of  freedom, especially  based  on  the  symmetric  charges. So  one  thing  you  can argue  is  that  the  spin  gap  remains  open. So  any  particle  that  carries spin  is  always  gapped  across  this  transition. And  that  has  to  do  with  the  fact  you keep  track  of  this  a  spin, it  simply  doesn't  enter into  the  low  energy  theory, except  as  some  background  to  Simonst. So  there's  a  spin  gap, but  the  charge  gap  has  to  close  because the  Sigma  X  has  to  change  2-0. Okay.  So  it's  a  closing  and reopening  of  the  charge  gap. Um,  and  so  the  only  thing  that,  you  know, so  you  can  ask  what  has gone  gapless  over  here, and  the  first  suspect  would  be  the  electron. Electron  came  down  low  energies, but  that's  an  object  that  carries both  electric  charge  and  spin. Okay,  so  that's  not  allowed. It's  spin  gapped. So,  what  actually  turns  out  to  be gapless  is  some  even  numbers  of  electrons. So  the  simplest  would  be  a  cooper  pair. So  this  transition  is  really  a  cooper  pair, becomes  gapless  at  the  transition. Okay, so  that's  sort  of  one  way  to  argue  this. You  can  also  see  it from  this  effective  theory. If  you  look  at  this  effective  theory, I  also  put  in  the  spin  gauge  field  to  show you  that  everything  is spin  gap  at  low  energy, just  some  background to  salmon's  contribution. If  you  look  at how  the  dynamical  gauge  field  couples to  the  external  potential  A, it  tells  you  that really  the  magnetic  monopole of  little  A  is  the  pair. Okay,  so  you  have  a  flux  of two  Pi  for  little  A, that'll  give  you  something with  electric  charge  too. Okay,  so  the  way  it  couples  to the  fluctuating  gauge  field tells  you  that  it's  really the  per  pair  that's  coming down  to  low  energy  at  the  transition. Okay,  there's  another  way  which, I  think  I  learned  it  from  Chen, but  which  is, you  know,  you  can  do  the  following. You  can  take  this  model  and  stick  onto it,  a  invertible  phase. Okay.  So  because  these  are  invertible  phases, they  don't  really  affect  the  physics. So  just  add  two  layers of  integer  quantum  hall  effect, again,  with  the  spin  up  and  spin  down. So  Squal  to  minus  two  on  both  sides of  this  just  to  the  system across  this  phase  diagram. So  what  this  does  is  it's  going  to modify  the  responses  that  you  see  over  here. The  hall  conductance  on the  left  is  now  going  to  be  zero. It's  been  designed  that  way. I  pick  something  that  will  be  zero. And  on  the  right,  the  hall  conductance becomes  minus  two  now.  Okay. And  at  the  same  time,  the  spin  conductances has  been  zeroed  out  by  putting this  inverse  phase. Okay,  so  this  is  a  transition  between something  that  is  no  change in  the  spin  response, and  the  hall  conductance goes  from  zero  to  minus  two. And  actually,  you  can  think  of this  as  a  plateau  transition, what's  called  a  plateau  transition. You're  going  from  a  fractional quantum  hole  state  a  trivial  one, trivial  insulator  to  a  Loughlin  state of  these  a  peds. So  you  have  a  nuclear  one  half Loughlin  state  of  copper  pairs. And  if  you  work  out  the  whole  conductors of  that,  it's  exactly  two. So  there's  a  two  E  squared  in  the  charge, but  there's  a  one  half that  gives  you  the  two. Okay.  So  that's  another  way  of  saying  it. It's  really  a  plateau transition  of  couple  peds. Going  from  trivial  insulator to  the  Loughlin  state. So  of  course,  at  the  transition, you  have  gapless  coup  peds. That's  our  way  to  see  intuitively  why the  couple  pair  should  be gapless  in  the  transition. Okay.  So  how  am  I  doing  on  time, actually?  5  minutes  left. A  little  more  10  minutes. Okay.  All  right. I  have  another  sphere  at  the  end  of  my  talk, so  Okay. So  now  let's  talk  about  finite  doping, putting  on  this  chemical  potential. There's  one  side  of  the  phase  diagram where  it's  very  easy  to  argue. So  because  of  this  trajectory  of  the  pair, the  gap  to  the  Cooper  pair, if  you  were  to  dope an  electric  charge  on  the  side, the  cheapest  charge  excitation is  the  charge  to  Cooper  pair. It  pays  for  you  to  just  put in  pairs  to  begin  with. At  least  for  infinite  decimal  doping, you  will  just  get  a  dilute  set of  a  pairs  when  you  add  charge  to  the  system. Okay,  simply  because  that's the  cheapest  thing  that  you  can do  as  you  approach  the  transition. Again.  So  for  this, you  do  not  even  need  to  be  very  close  to the  transition  or  even have  a  continuous  transition. All  you  need  is  some  softening of  this  Couper  pay  gap. Okay, so  this  mechanism  will  give  that  to  you. And  you  can  make  an  analysis  of  that  limit. So  really,  this  is  the  picture. You  have  some  density  delta, and  you  want  to  argue  that the  limit  of  small  delta, you  have  a  superfluid, right,  a  superfluid  of  these  super  fls. Now,  the  thing  that  you  have  to  worry about  is  you  could  get  crystallization. You  add  these  charges  in. Instead  of  becoming  a  superfluid, they  could  form  a  crystal. Okay. But  in  order  to  form  a  crystal, they  need  to  have  long  range  interactions. They  need  really  the interactions  between  the  particles, at  least  in  our  model. The  interactions  are  at  best, the  length,  the  correlation  length. Okay, so  the  critical point  that  you're  approaching, there's a  correlation  length  that  is  diverging. You  make  a  perturbation  on  your  system, it  affects  stuff  out  to the  correlation  length  at  most. Right?  So  the  correlation  length is  she  and  so this  gives  you  a  small  parameter. If  you're  sufficiently  dilute, you  can  make  this  small,  and  then you're  in  the  dilute  limit  of  the  Bs  skins. Okay,  I  think  it's  still  not  solvable. What  you  can  say  is  that this  is  really  can  be modeled  by  just  two  body  interactions, some  scattering  length. But  in  fact,  if  that  scattering length  also  satisfies  this, then  you  have analytic  control  over  this  problem, and  you  can  guarantee  that  it's  a  superfluid. Okay,  it  seems  that  if you're  at  small  density away  from  the  critical  point, you  can  more  or  less  guante this  thing  is  going  to  become  a  superfluid. Okay.  And  you  can  figure  out what  kind  of  superfluid  it  becomes. It's  a  conventional  in  most  ways, except  that  it  has  ed  states. It  inherits  the  ed  states from  the  digit  one  mono. So  there  are  a  current  center  charge  of  two, what  we  call  a  D  plus  ID, at  least  topologically  D  plus  ID  superfluid. Okay,  so,  of  course, the  open  question  is,  what  if you're  really  close  to  the  critical  point? And  I  think  that's  where,  you  know, having  technology  like understanding  what  happens  when  you perturb  your  CFT  with finite  charge  density  becomes  important. Can  you  distinguish  the  case  where, if  you're  close  to  the  critical  point, the  density  and  this  correlation  length, they  both  have  the  same  scales that  you  have  a  diverging  correlation  length, they  could  in  principle,  talk  to  each  other. It  doesn't  necessarily  mean they  want  to  form  a  crystal, but  how  do  you  rule  it  out? And  if  they  did  form  a  crystal, what  does  it  mean  for  all  of these  predictions  that  people  make  for, um,  you  know,  scaling dimensions  of  operators? It's  a  fully  gap  so. Yeah,  so  that  is,  you  know, maybe  a  connection  that would  be  interesting  to  figure  out. So  the  last  point  I  want to  talk  about  is  what  if  you  think about  doping  the  system  from the  current  spin  liquid  site. Okay?  So  now  your  excitations, the  lowest  charge  extation turns  out  to  be  a  charge  E  object, which  is  non  local  in  terms  of  the  electrons. It's  called  a  semi  onion. It's  a  anion  which  has  got topological  spin  of  Pi  over  two.  Okay. So  you  can  think  of  this  if  you  like. If  you're  familiar  with  Kern  spin  liquid, it's  got  spin a  half  extations  that  are  seions. You  can  bind  that  to an  electron  to  make  a  sinlet. So  a  spinless  particle  which carries  charge  that's  the  object that  is  essentially  being  doped  when  you  add charge  from  the  s.  So  the  question  is, if  you  have  a  finite  density  of those  charge  objects, what  is  the  fate  of  this  system? What  is  this  system  doing? Okay,  so  if  these  were  bosons, maybe  you  would  say  that  it's  a  superfluid. If  they  are  fermions,  you  would  say  it  forms a  fermiy  liquid  probably  if the  attractions  are  weak. What  happens  when  they're semion  somewhere  in  between? Okay,  so  there's  an  old  work  by  Laughlin, basically  a  very  nice  argument, intuitive  argument. He  argued  this  is  superconductor. And  essentially  the  idea  is  that having  finite  density  of  anions, you  know,  it's  very  difficult  for  them  to form  any  state  other  than  superfluid. You  know,  they  have  these  non  trivial phase  factors  when  they  go  around  each  other. That's  unique  to  anions, not  true  of  bosons  or  fermions, ends  up,  you  know, giving  you  the  state,  which is  a  superconductor. Again,  it's  an  argument  that's  not, um,  you  know, completely  rigorous,  but  we'll  try  to  see how  this  works  out  in  our  picture. Okay,  so  you  are  going  to  use  the  part  on approach  to  make  sense  of this  anion  superconductivity. What  happens  when  you  have  a  finite  density of  these  Chi  semions? Okay,  so  we  already  talked about  how  we  got  this  phase, condensate  of  charge  on. Second  phase  gapped  charge  on, the  symmetric  phase  for  charge  ons. So  it  seems  like  we  have exhausted  all  possibilities. The  two  options  we  add  either condense  or  gap  it  out, seem  to  have  exhausted  it. How  do  we  get  this  third  phase when  you  have  finite  doping? Okay,  but  you  have  one  other  option. You  can  keep  it  symmetric, but  you  can  have  several  different  states that  have  the  same  symmetry, but  differ  at  the  level  of  topology. You  can  have  symmetry protected  topological  phases. And  that  actually  turns  out  what happens  if  you  want  to  get  a  superfluid. So  imagine  that  these  charge ons  form  not  just  the  trivial  insulator, but  a  bosonic  integer  quantum  horn  state. So  again,  this  expectation  value  is  zero, but  it  has  a  non  trivial  response. So  you  can  integrate  this  out. I'll  explain  this  a little  more  in  the  next  slide. And  you  get  a  term  in  addition to  the  Kern  spin  liquid  term  we  had  before, L  two,  U  one,  U  one,  level  two. You  get  this  term,  the  charge response  of  this  bosonic integer  quantum  hole  state. And  this  exactly  cancels the  topological  order  of the  kernel  spin  liquid. You're  left  with  this.  So  you can  forget  about  this  last  term, it's  higher  order  derivatives. You  have  this  term,  and  this  is  really the  effective  action  of a  superconductor  superfluid. Because  you  can  integrate  out  the  little  a, it  enforces  dA  to  be  zero. That's  just  the  Meissner. Okay.  So  if  you can  put  this  in  a  particular  state, the  effective  thing  that  you  get  is  the. You  could  have  a  maxwell  term  for  little  A, but  it's  higher  order  in  derivative. So  when  you  integrate  a  little  A, you  could  have  some  stuff  of derivatives  so  you  can  work  it  out  carefully. Yeah,  but  it's  like  giving  you a  finite  landscape  for  the  penetration  depth. Plus  two plus  penetration  in  three  dimensions. Yeah.  So  this  is  flatland  superconductivity. All  the  gauge  fields  live  inside  the flat  the  emergent  gauge  field is  two  plus  one. Yeah.  No  three  D  superconductivity. Yeah. Okay,  so  you  can  see  this in  the  part  on  picture. Let  me  just  say  one  or two  things  before  ending. The  key  thing  is  that  you  add  charge, you're  supposed  to  add  both  the  charge  ons and  the  spin  ons  F.  Okay, but  these  Fs  are  gapped. So  how  do  you  add  density of  them  to  the  state? What  you  do  is  you  actually  adjust  the  flux, the  background  flux  in such  a  way  that  by  then  Son's  term, you  adjust  the  density  of  F.  Okay. So  that's  an  option  you  have  when  this  is  in a  topologically  non  trivial  bank. Just  adjust  the  flux. But  now  the  charge  ons  begin  to  see  flux. Yeah.  And  when  they  see  flux, you  can  figure  out  what the  effective  filling  fraction  is. The  filling  fraction  is  exactly  minus  two. This  is  perfect  to  get  this bosonic  integer  coal  state, and  you  can  get that  response  that  will  cancel  off the  Carl  spin  liquid topological  order  and  give  you  the  superdut. Okay. One  of  the  surprises,  the  superconductor you  get  is  exactly  the  same  as  before. So  it's  not  an  exotic  superconductor. It's  deepest  idea. You  can  work  out  from  this  theory in  terms  of  red  states, but  otherwise  it's  conventional. So  it's  actually  different from  taking  pairs  of semions  binding them  together  and  condensing  them. You  could  work  out  the  theory  of that  superconductor  as  well. That  turns  out  to be  an  exotic  superconductor. It  has  some  other  topological  order  as  well. So  what  is  the  maximum  term? I  mean,  to  get  flat  land  gauge  field, you  need  the  maximum  term  flatland. Yeah.  I  need. So  actually,  maybe  I  should  be  a  bit  more careful  about  my  terminology. So  we  don't  ever  gauge  the  capital  A. The  external  thing  is  only  used  as  a  probe. So  what  we're  getting  is actually  a  superfluid. So  it's  capitals. It  has  a  Goldstone  mode. Yes. Yeah.  Okay.  That's  what  he  meant.  Yeah,  yeah. It  has  a  Goldstone  mode.  Yeah.  Yeah.  Right. Yeah. So  the  Hubbard  model  is  just  a  pair  of electrons  and there's  no  issue  of  penetration. That's  right.  Yeah. If  you  were  to gauge  the  capital  A  and  make  it a  If  you  have  to  invent a  maximum  term  in  two  D, you  can  find  your  lab. Yes,  that. Yeah.  Yeah.  Right.  It's  not gap.  It's  gapless. It's  capitalss  because  the  Goldstone  mode. Yes.  Yeah.  Yeah,  but the  fermions  are  all  gapped, so  there's  no  room. Yeah.  Yeah.  Right. Okay.  So  this  is  actually,  if  you  like, a  regularization  of  this  old  theory of  any  on  superdtivity, and  the  flux  is  important. So  Loflin  proposed  it for  the  triangle  lattice. Without  flux,  you  would  get a  different  theory  if you  actually  went  through  that. Okay,  so  very  quickly,  the  numerics, if  you  look  at  the  numerics,  in  fact, you  do  see  signs  of  pairing. You  can  just  compare  the  energy  of charge  two  versus  charge one  and  see  if  they  differ. If  this  is  lower  energy than  two  of  the  charged  ones, and  that's  the  plot  over  here, you  get  a  binding  of the  pairs  of  electrons across  the  phase  diagram. So  a  wide  range  of  Us  over  here, and  the  binding  strength  is  about  0.1 P  So  for  us,  this  is  a  big  gap. Hopping  is  very  large  in  electronic  systems, so  this  could  support  a  fairly robust  superconductor  that's  from a  different  technique  called exact  aglization  Also  sees  this  binding. And  let  me  conclude  over  there. So  we  found  superconductor  in  the  new  regime, which  is  maximally  unfavorable  in  fact. It's  very  strong  propulsion and  strong  time  reversal  rate. The  new  mechanism  is  the  softening of  the  Cooper  Pair  gap. Superconductor  is  deepest  idea, otherwise  it's  conventional. And  there  are  ways you  could  think  of  trying  to realize  this  using  all  the  more  stuff, but  maybe  that's  for  a  different  dog. And  the  future  directions connected  to  perhaps  this  workshop, I  think  I  already mentioned  a  few  of  these  already. Um,  but  there  is  a  large  N, there's  a  generalization  of  this  with  N  N, and  there's  a  very  interesting  difference depending  on  whether  N  is  even  or  odd. So  you  get  fermions  or  bosons, you  get  upper  pairs  or  triions depending  on  whether  N  is  even  or  odd. And  the  hope  is  that  some  of the  tools  in  CFTs  can  give  us a  non  perturbative  handle  on  this  thing  of doping  and  seeing  whether  you  can  get a  superconductor  across  some of  these  phase  transitions. Okay  let  me  end  over  there. Yeah.  We  have  some  estimates  of the  critical  exponents  of the  AT  that  you're  trying  with  the  mono. Is  there  any  hope  that  there will  be  something  to  compare  to  in the  future  with  a  critical  exponent? Um  Can  you  look  at  the  experiments or  maybe  for  your  methods  I  mean, numerics  is  your  best  bet, I  think,  if  you  want  to  compare. So  for  this  equilar  two, there  is  a  prediction  for the  lowest  monopole  operator. We  know  what  the  scaling  dimension  is. Yes.  Because  you  can  relate  it  using the  SU  two  symmetry  to  conserve  charge. So  it  has  to  have  the  same  scaling  dimension. So  two,  basically. Where  we  start.  You're  talking  about  turns  on this  level  two  scalar  city three  with  two  scalars? Just  one  scalar. Oh,  just  one  scalar  adjustment.  Enhanced? Yes.  Well,  in  our  model actually  is  microscopic, but  when  written  like  this, it  looks  like  an  enhanced  symetry. Yes,  this  has  an  ST  assymmetry  when  energy would  it  to  that  is not  apparent  that  rotates, the  monopool  into  some  conserved. So  that's something  you  can  certainly  compare  to. But  yeah,  see  this  operator, it's  a  current,  so  it's. Yeah.  Yeah.  Yeah.  So  interesting to  the  a  single  direct  point  gives  you a  critical  theory  when you  change  the  line  one. Are  there  any  other  possibilities to  the  points  or  is  that the  only  known  critical  theory  which  is parallel  should  say  you can  only  change  the  one? Well,  there  are  case  if you  allow  for  most  symmetries, you  can  have  a  quadratic  band touching  that  gives  you  a  change  of two  It's  only  known. Yes.  And  it's  dual  form. So  this  is  some  dual  form  of  that.  Yeah. Yeah. Questions.  Diction  top  quotation of  the  superfid.  Could  you  chart  that? Yeah,  that's  a  good  point. I  mean,  there  is  like  a  background  field. Let  me  see.  Did  I  have  it  here? Yeah,  I  added  here.  Yeah. Like  you  have  this  over  here. Is  that  what  you  mean?  If  you go  to  the  delete  ta, the  level  dictate your  disposity  of  the  flats. It's  like  a  West  for the  su  data  values  level. I  think  the  sofa  fluid  is  conventional. You  want  to  add  a  breaking. It's  pay  breaking,  yes. Yes. Okay.  It  sounds  I'd  like  to  hear  more.  Yeah. We  can  postpone  further  questions to  the  break  and  thank  A.