Correlation Functions: Strange Metals and Spin Liquids - Michael Norman Materials Science Division - Argonne National Lab - Lorentz Center

Correlation Functions:
        Strange Metals and Spin Liquids

                Michael Norman

Materials Science Division – Argonne National Lab

                              Leiden – Jan. 7, 2019

Well, there have always been those resistant to change

A (very) brief introduction to AdS/CFT

The AdS/CFT conjecture asserts that certain strong
coupling gauge theories of dimension d in their large N
limit are dual to weakly coupled gravitational solutions
in a d+1 dimensional Anti-deSitter spacetime.

The extra dimension in “d+1” is an RG scale that allows
one to flow from the the boundary (i.e., the gauge theory
in its UV limit) to near the event horizon of a black hole
which sits at the center of this hyperbolic space. This
near region defines the IR limit of the gauge theory.

Advantage - strong coupling non-perturbative method

Disadvantage - the conjecture has only been made for
very special gauge theories, none of which seems to be
realized in Nature (i.e., QCD)

Hence, the interest of the AdS/CFT community
              in condensed matter physics

              (i.e., we have LOTS of theories,
                    and LOTS of materials,
            each of which has a different UV limit)

Hartnoll, Science (2008)           McGreevy, Adv. HEP (2010)

But, there are important “philosophical” differences …

It is natural to ask how surprised one should be that general
relativity can reproduce the basic properties of superconductors.
After all, Weinberg has shown that much of the phenomenology of
superconductivity follows just from the spontaneous breaking of
the U(1) symmetry. Once we have found the instability that leads
to charged scalar hair, doesn’t everything else follow?

Gary Horowitz - Introduction to Holographic Superconductors

arXiv:1002.1722

But their attention was focused on the details of the dynamics
rather than the symmetry breaking. This is not just a matter of
style. As BCS themselves made clear, their dynamical model was
based on an approximation, that a pair of electrons interact only
when the magnitude of their momenta is very close to a certain
value, known as the Fermi surface. This leaves a question: How
can you understand the exact properties of superconductors, like
exactly zero resistance and exact flux quantization, on the basis of
an approximate dynamical theory? It is only the argument from
exact symmetry principles that can fully explain the remarkable
exact properties of superconductors.

Steve Weinberg - From BCS to the LHC

(see also Prog. Theor. Phys. Suppl. 86, 43 (1986))

In this study, the researchers focused on two properties that
distinguish those cuprate strange metals from Fermi liquids. In
ordinary Fermi liquids, electrical resistivity and the rates of
electron scattering (deflection from their original course caused
by interactions with each other) are both proportional to the
temperature squared. However, in cuprates (and other
superconducting non-Fermi liquids), electron scattering and
resistivity are proportional to the temperature.

“There’s really no theory of how to explain that,” says Liu.

- MIT Press Release on Strange Metal Transport Realized by
Gauge/Gravity Duality (Science 329, 1043 (2010))

A Cautionary Tale
   Linear T Resistivity in the “free electron” metals Pt and Au

Fradin et al, PRB (1975)

Including fermions in AdS/CFT,
          there is a “Fermi surface” with gapless excitations.
  By tuning parameters, one can get a FL, a non FL, or a marginal FL.

Faulkner et al, Science (2010)

Add a scalar (superconducting condensate) and a
        spinor field (fermions), and then couple the two

                                       (G5 term mixes k and –k)

                         AdS2 x R2 → AdS4

Faulkner et al, JHEP (2010)

This change in geometry due to the scalar causes the “light cone”
       to open up (“timelike” bound states are quasiparticles)

 Coupling of the scalar to the spinor opens up a “Bogoliubov” gap

Faulkner et al, JHEP (2010)

And, voilà, out pops the peak/dip/hump in Im G (fermionic spectrum)

                                  peak – bound state outside of the
                                  IR light cone, gapped due to the
                                  G5 term (scalar-spinor coupling)

                                  dip/hump – measures the “speed of
                                  light” of the quantum critical CFT

Faulkner et al, JHEP (2010)

Experimental response functions (single and two-particle)

ARPES (photon in – electron out: single particle spectral function)

Optics (photon in – photon out: fermion bubble, phonons, etc.)
Raman (photon in – photon out: charge/spin response, phonons)
RIXS (photon in – photon out: charge/spin response, phonons)
EELS (electron in – electron out: charge response, phonons)
INS (neutron in – neutron out: spin response, phonons)

Photoemission spectrum above and below Tc
       at the “antinodal” momentum k=(p,0) for Bi2212
    (in 2D, ARPES measures the occupied part of Im G)

                                             peak
                                                    Incoherent normal state
           Intensity

                                                    Coherent superconductor

                                dip

                       0.12    0.08   0.04     0
                              Binding energy (eV)

Norman et al, PRL (1997)

What is the origin of the peak/dip/hump?

 1. Bilayer splitting?

 2. Scattering rate gap?

 3. Coupling to spin fluctuations?

 4. Coupling to current fluctuations?

 5. Coupling to phonons?

 6. Combination?

Electrons interacting with an Einstein mode

     S = ∫ G0 c, where S is the self-energy, G0 the bare Green’s
                    function, and c the boson spectral function
     G-1 = G0-1 - S, where G is the full Green’s function

Mode at W0 will cause a strong coupling feature (spectral dip) at D+W0
 Norman et al, PRB (1998)

Marginal Fermi Liquid from ARPES
(”nodal” region of the Brillouin zone)

      Valla et al, Science (1999)

Power law behavior in infrared conductivity
 (note sublinear power - w-2/3)

van der Marel et al, Nature (2003)

a2F

Norman & Chubukov, PRB (2006)

Norman & Chubukov, PRB (2006)

Phase Diagram of the Cuprates

Keimer et al, Nature (2015)

Linear T resistivity at low T in Bi2201 (left) and YbRh2Si2 (right)
     led to the Marginal Fermi Liquid conjecture of Varma et al
                                 (1989)

Martin et al, PRB (1990)                   Trovarelli et al, PRL (2000)

Raman Background in YBCO
           (left: spectral density; right: calculated resistivity)

Slakey et al, PRB (1991)

Similar backgrounds are seen in EELS (left) and RIXS (right)
 and have been attributed to the charge response function
            EELS is consistent with c ~ f(q)f(w)

Mitrano et al, PNAS (2018)      Suzuki et al, npj Quantum Matls (2018)

Similar Raman backgrounds are seen in candidate spin liquids
like herbertsmithite and have been attributed to the spin response

  Wulferding et al, PRB (2010)

With both fermionic (Majorana) and bosonic contributions
  to Raman speculated for RuCl3 (based on the “Kitaev” model)

Sandilands et al, PRL (2015)    Nasu et al, Nature Phys. (2016)

Introduction to Spin Liquids
     (Anderson – 1973)

Near neighbor Heisenberg model on a kagome lattice
       Certain rotations of spins cost no energy
leading to a suppression of long range magnetic order

                                Yildirim and Harris, PRB (2007)

Exact diagonalization studies of NN Heisenberg model
indicate a very different eigenvalue spectrum for the kagome case

      square                triangular                  Kagome

                                     Sindzingre & Lhuillier, EPL (2009)

Topology and Fractionalization in Spin Liquids
                   Spinons and Visons

Free spinon (yellow arrow)

                                      Vison pair at a and b

                                     Balents, Nature (2000)
  Ground state degeneracy            Misguich & Lhuillier, arXiv (2012)
    & topological sectors            He, Sheng, Chen, PRB (2014)

Herbertsmithite
       rare mineral first identified from a mine in Chile
(hydrothermal synthesis of single crystals by Nocera’s group)

                        Herbertsmithite
                (photo courtesy of Bruce Kelley)
                                            Shores et al, JACS (2005)
                                            Norman, RMP (2016)

Cu4 → ZnCu3
Canted AF → spin liquid

Shores et al, JACS (2005)
Lee et al, Nature Matls (2007)
Mendels & Bert, JPSJ (2010)

NMR indicates a decreasing static c below 50 K for
the kagome spins, despite the diverging behavior of
  the bulk static c due to impurity interlayer spins

                                  Mendels & Bert, JPSJ (2010)

INS data indicate a divergent response below 1 meV with
quantum critical scaling as also observed in heavy fermions
      (note w/T scaling with sublinear power – T-2/3)

                                       Helton et al., PRL (2010)

Is this due to novel quantum critical physics (left)
                                OR
       Is this a signature of random spin exchange (right)

                                  vs

Schroeder et al, Nature (2000)       Castro-Neto & Jones, PRL (1998)

INS data on single crystals indicate a continuum of
 spin excitations, with a dynamic susceptibility consistent
with near neighbor dimers and of the form c(q,w) = f(q) f(w)

                                          Han et al., Nature (2012)

Models with spinons & visons still show dispersive features
                  (as seen in 1D magnets)

Punk, Chowdhury, Sachdev, Nature Phys (2014)

Random spin Heisenberg model simulations
claim to give a better description of the INS data

                                Kawamura et al., JPSJ (2014)
                                Shimokawa et al., PRB (2015)

But INS data show that the impurities only affect the low w data
     (due to defect copper spins sitting on the Zn intersites)

                                        Han et al., PRB (2016)

INS data are a sum of impurity and kagome spin contributions
              (with the kagome spins having a spin gap)
Specific heat can be fit assuming a relaxational form for impurity c(w)

                                               Han et al., PRB (2016)

SUMMARY
• There is more than one way to skin a cat
• Finding the “right” way is a goal of condensed matter physics
• Linear T resistivity can be due to a variety of things – very low
  w phonons, broad boson continuum, quantum criticality, etc.
• Experimental two-particle correlation functions have a variety
  of contributions, both fermionic and bosonic – phonons,
  collective modes, spin response, charge response, etc.
• Raman background seems to be ubiquitous and not so well
  understood, though some progress has been made in RuCl3
• Role of disorder versus quantum criticality versus ”RVB” is an
  ongoing debate in spin liquids
• Is “locality”, i.e. c ~ f(q) f(w), also a consequence of disorder?