Exploring universality of many-body quantum phases and phase transitions with cold atoms, molecules, and ions    Eugene Demler Harvard University

Collaborators:  Mehrtash Babadi (Harvard-> Caltech), Michael Knap (Harvard), Fabian Grusdt (Harvard/Kaiserslautern), Brian Skinner (University of Minnesota), Mikhail Fogler (UCSD), Mikhail Lukin (Harvard), Immanuel Bloch (MPQ), Thierry Giamarchi (Geneva), Adrian Kantian (Geneva)

Funded  by  NSF,  Harvard-­‐MIT  CUA,    DARPA  OLE,  MURI  polar  molecules,  MURI  quantum  simulaGons,  MURI  atomtronics  

Outline

Liquid to crystal quantum phase transitions in 2D Surprising Universality (Lindemann and Hansen-Verlet criteria) Variational approach to the equation of state and collective modes. Experimental tests with polar molecules

Exploring dynamical response functions in spin models using many-body Ramsey interference

Universality of liquid to crystal quantum phase transitions in 2D

M. Babadi, B. Skinner, M. Fogler, E. Demler, arXiv:1212.1493

Universality in physics Spontaneous symmetry breaking and order

Phase transitions and universality Universality of phase diagrams, phase transitions

Universality of collective excitations

Goldstone modes: spin waves, Bogoliubov modes

Amplitude Mode: Anderson-Higgs excitation

Lindemann criterion of crystal melting Classical transitions

a u

Maximum vibration amplitude is a fixed fraction of the distance to the nearest neighbor

Hansen-Verlet criterion of freezing Classical transitions

Liquid freezes when the height of the principal peak in the static structure factor reaches

Density waves in a liquid spontaneously lock and transition into a crystal takes place

Sf(km)=2.8  

Hansen,  Verlet,    Phys  Rev  (1969)  

TheoreGcal  model  with    Lennard-­‐Jones  potenGal,  Comparison  to  Ar  data  

Quantum liquid to crystal transitions in 2d

Quantum Wigner crystals in 2d Originally  proposed  for  electrons  with  Coulomb                                                    interacGons    

Nearly free Fermi gas

Correlated Landau Fermi

Liquid

Wigner crystal (WC)

High density Low density

kx   ky  

Quantum Wigner crystals in 2d

Free Fermi gas

Strongly correlated Fermi

Liquid (FL)

Wigner crystal (WC)

Fermions  with  Coulomb  1/r  interacGon:  Wigner,  Ceperley  et  al.,  Drummond  et  al.,…  Fermions  with  dipolar  1/r3  interacGons:  Matveeva  and  Giorgini  Fermions  with  hard  core  interacGons;  Drummond  et  al.,    Bosons  with  dipolar  interacGon:    Buchler  et  al.,  Polini  et  al.,  Bosons  with  Yukawa  interacGons:  Ceperley  et  al.,  

Free Fermi gas

Strongly correlated Fermi

Liquid (FL)

Wigner crystal (WC)

Universality of quantum liquid to crystal transitions in 2d

Lindemann   Hansen-­‐Verlet  

Assuming Bijl-Feynman single mode approximation

•  Almost identical structure factor at the transition:

•  Universal features: –  “roton” gap, “roton” mass –  Density wave dispersion in the���

roton regime

•  Non-universal features: –  Long-wavelength collective modes –  2DEG: (plasmon) –  Dipolar and hard-core: (ZS/phonon) –  Short-range physics (PDF at small r)

•  This suggests that the physics of strongly-correlated 2D fermions with repulsive interactions is universal in the regime, independent of their interaction law

–  Short-range details are masked by the localized exchange-correlation hole –  Long-range tail of the interaction only affects the long-wavelength modes –  WC transition: is dominated by softened rotons

Universality of quantum liquid to crystal transitions in 2d

Pushing universality further …

–  Use the ground state wavefunctions of A as a variational set for the ground state energy estimation of B

Pair distribution function is known from QMC –  Kinetic energy can be separated from the interaction energy using

the Virial theorem

Utilizing universaliy of strongly correlated systems in 2d

Conjecture: the ground-state energy, to a large part, is dominated by softened density-waves. Strongly-correlated ground-states of one model will be well suited as variational trial states for another model

Use Coulomb 1/r QMC to calculate properties of dipolar 1/r3 systems with variational conjecture. Compare to existing QMC results for dipolar interactions

VariaGonal  using  2DEG  

FN-­‐DMC  [N.  Matveeva  et  al.,  PRL  109,  (2012)]    

More than 95% of the correlation energy is captured in the strongly correlated regime.

WC transition: Hansen-Verlet + 2DEG wave functions: The FN-DMC result is:

A test for the universality conjecture

Application to experiments with dipolar fermions

•  Finite transverse confinement modifies the dipolar interaction at short distances:

•  Use the ground states of the pure dipolar fermions as ���variational trial states for the quasi-2D dipolar gas

Hartree-­‐Fock  

VariaGonal  using  2d  dipolar  

VariaGonal  using  2d  Coulomb  

Phase  Diagram  The  energy  of  quasi-­‐2D  dipolar  gas  

Collective modes in a trap as a probe of equation of state

CollecGve  modes  in  the  BCS/BEC  regime  allowed  to  probe  equaGon  of  state  with  sub  1-­‐percent  accuracy  

Radial compression mode

BCS QMC

 [A.  Altmeyer  et  al.,  PRL  (2007)]  

Monopole (Breathing) mode of quasi 2D Dipolar fermions

Mon

opol

e (B

reat

hing

)

Typical routes to universality:

long wavelength properties protected by RG (collective modes, universality of transitions)

Surprise of liquid to crystal quantum transition in 2d: universality at intermediate scales.

Lindemann and Hansen-Verlet criteria for quantum liduid to crystal transtion. New variational approach to calculating equation of state and analyzing collective modes  

Exploring dynamical response functions in spin models using many-body Ramsey interference

M. Knap, A. Kantian, T. Giamarchi, I. Bloch, M. Lukin, E. Demler

Cold atoms Trapped ions Polar molecules

n Heisenberg model of XXZ type

n super-exchange

n e.g. 87Rb mixtures of and

n LR transverse field Ising model

n  interactions mediated by phonons

n e.g. 171Yb

n LR XX model n dipolar interactions

n e.g. KRb

MPQ  group   JQI  group   JILA  group  

Probing spin dynamics in synthetic matter

n  condensed  mader    →  common  framework  to  understand  diverse  probes  n  neutron/X-­‐ray  scadering  n  opGcal  response  n  STM  n  ...    

n  retarded  Green's  funcGons:  

n  informaGon  about  excitaGon  spectra  and  quantum  phase  transiGon  (e.g.  scaling)  

Dynamic probes of many-body systems

n  SyntheGc  many-­‐body  systems  (atoms,  molecules,  ions):  →  typically  dynamics  explored  through  quench  experiments  

n  no  direct  informaGon  about  excitaGons  →  excepGons:  RF-­‐spectroscopy  

Propose  to  use  many-­‐body  Ramsey  interferometry  to  measure  dynamic  spin-­‐correla9on  func9ons  

Quench  EvoluGon  

Measurement  

Dynamic probes of many-body systems

p/2  pulse  

 EvoluGon  

Tools  of  atomic  physics:  Ramsey  interference    

Used  for    atomic  clocks,  gravitometers,    accelerometers,  magneGc  field    measurements  

p/2  pulse  +  measurement  ot  Sz  gives  relaGve    phase  accumulated  by  the  two  spin  components  

EvoluGon    EvoluGon  

Spin  rotaGons  

p/2 pulse:

p/2 pulse:

Many-­‐body  spin  Ramsey  protocol  

ggggggggggggg  

n  for many relevant cases terms with odd number of spin-x/spin-y operators vanish

n  additional degree of freedom: → phases of the laser field

Many-­‐body  spin  Ramsey  protocol  

n  global  symmetry    n  U(1)  symmetry  around  z  axis  

Heisenberg  model  

n  Problem  of  shot  to  shot  fluctuaGons  of  magneGc  field:  p/2  pulse  makes  a  superposiGon  of  

       states  with  different  Sz  

n   Need  to  implement  spin  echo.            Add  p  pulse  at  t/2  

n  Heisenberg  is  invariant  under  this  transformaGon  n  Zeeman  term  is  cancelled  

p/2 pulse:

Spin  echo  for  Heisenberg  model  

AnGferromagneGc  Heisenberg  model  On-site correlations Nearest neighbor correlations

Momentum (p,p) correlations (frequency)

Momentum (p,p) correlations (time)

Long-ranged transverse field Ising model with ion chains

n  interactions decay as power-law

n  Ramsey: n  global symmetry n  odd terms vanish through special choice of phases

Theory: Porras and Cirac, PRL (2004) Experiments: Monroe et al., Science (2013)

Phase diagram

n  unstable for n  quantum critical line separates FM from PM

Phase diagram

n  Ginzburg-Landau���

Irrelevant for MF exponents

A.  Duda  et  al,    Phys.  Rev.  B  64,  184106  (2001).  

Phase diagram

n  Ginzburg-Landau���

n  lower, critical

n  below which���MF exact

Phase diagram

n  BKT transition at n  extends to finite field���

Can we see change of critical behavior���with currently available systems?

Local dynamic correlation function

Power  law  decay  at  the  criGcal  line  Extracted  criGcal  exponent  

Summary

Surprising universality of liquid to crystal quantum phase transitions in 2D can be explored with polar molecules

Exploring dynamical spin response functions with Many-body Ramsey interference

Funded  by  NSF,  Harvard-­‐MIT  CUA,    DARPA  OLE,  MURI  polar  molecules,  MURI  quantum  simulaGons,  MURI  atomtronics