The Mott Transition: a CDMFT study

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The Mott Transition a CDMFT study

• G. Kotliar
• Physics Department and Center for Materials
  Theory
• Rutgers

Sherbrook July 2005
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References

• Model for kappa organics. O. Parcollet, G.
  Biroli and G. Kotliar PRL, 92, 226402. (2004))
  
• Model for cuprates O. Parcollet (Saclay), M.
  Capone (U. Rome) M. Civelli (Rutgers) V.
  Kancharla (Sherbrooke) GK(2005).
• Cluster Dynamical Mean Field Theories a Strong
  Coupling Perspective. T. Stanescu and G. Kotliar
  (in preparation 2005)
• Talk by B. Kyung et. al. Tomorrow.
  cond-mat/0502565 Short-Range Correlation Induced
  Pseudogap in Doped Mott InsulatorsTalk by V.
  Kancharla Sarma (this morning)

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Outline

• Motivation and Objectives.Schematic Phase
  Diagram(s) of the Mott Transition.
• Finite temperature study of very frustrated
  anisotropic model. O. Parcollet
• Low temperature study of the normal state of the
  isotropic Hubbard model. M. Civelli, T.
  Stanescu See also B. Kyungs talk
• Superconducting state near the Mott transition.
  M. Capone. V. Kancharla Sarma
• Conclusions.

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RVB phase diagram of the Cuprate Superconductors

• P.W. Anderson. Connection between high Tc and
  Mott physics. Science 235, 1196 (1987)
• Connection between the anomalous normal state of
  a doped Mott insulator and high Tc.
• Slave boson approach. ltbgt
  coherence order parameter. k, D singlet formation
  order parameters.

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RVB phase diagram of the Cuprate Superconductors.
Superexchange.
G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988)

• The approach to the Mott insulator renormalizes
  the kinetic energy Trvb increases.
• The proximity to the Mott insulator reduce the
  charge stiffness , TBE goes to zero.
• Superconducting dome. Pseudogap evolves
  continously into the superconducting state.

Related approach using wave functionsT. M. Rice
group. Zhang et. al. Supercond Scie Tech 1, 36
(1998, Gross Joynt and Rice (1986) M. Randeria
N. Trivedi , A. Paramenkanti PRL 87, 217002
(2001)
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Problems with the approach.

• Neel order
• Stability of the pseudogap state at finite
  temperature. Ubbens and Lee
• Missing incoherent spectra . fluctuations of
  slave bosons
• Dynamical Mean Field Methods are ideal to remove
  address these difficulties.

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COHERENCE INCOHERENCE CROSSOVER
T/W
Phase diagram of a Hubbard model with partial
frustration at integer filling.  M. Rozenberg
et.al., Phys. Rev. Lett. 75, 105-108 (1995). .
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Focus of this work

• Generalize and extend these early mean field
  approaches to systems near the Mott transition.
• Obtain the solution of the 2X 2 plaquette and
  gain physical understanding of the different
  CDMFT states.
• Even if the results are changed by going
• to larger clusters, the short range physics is
  general and will teach us important lessons.
  Follow states as a function of parameters.
  Adiabatic continuity. Furthermore the results can
  be stabilized by adding further interactions.

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Finite T Mott tranisiton in CDMFT Parcollet
Biroli and GK PRL, 92, 226402. (2004))
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Evolution of the spectral function at low
frequency.
If the k dependence of the self energy is weak,
we expect to see contour lines corresponding to
t(k) const and a height increasing as we
approach the Fermi surface.
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Evolution of the k resolved Spectral Function at
zero frequency. (QMC study Parcollet Biroli and
GK PRL, 92, 226402. (2004)) )
U/D2.25
U/D2
Uc2.35-.05, Tc/D1/44. Tmott.01 W
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Momentum Space Differentiation the high
temperature story T/W1/88
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Physical Interpretation

• Momentum space differentiation. The Fermi liquid
  Bad Metal, and the Bad Insulator - Mott
  Insulator regime are realized in two different
  regions of momentum space.
• Cluster of impurities can have different
  characteristic temperatures. Coherence along the
  diagonal incoherence along x and y directions.
• Connection with slave
• Boson theory divergence of
• Sigma13 . Connections with
• RVB (Schmalian and Trivedi)

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Cuprate superconductors and the Hubbard Model .
PW Anderson 1987
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CDMFT study of cuprates

• Allows the investigation of the normal state
  underlying the superconducting state, by forcing
  a symmetric Weiss function, we can follow the
  normal state near the Mott transition.
• Earlier studies (Katsnelson and Lichtenstein, M.
  Jarrell, M Hettler et. al. Phys. Rev. B 58, 7475
  (1998). T. Maier et. al. Phys. Rev. Lett 85,
  1524 (2000) . ) used QMC as an impurity solver
  and DCA as cluster scheme.
• We use exact diag ( Krauth Caffarel 1995 with
  effective temperature 32/t124/D ) as a solver
  and Cellular DMFT as the mean field scheme.
  Connect the solution of the 2X2 plaquette to
  simpler mean field theories.

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Follow the normal state with doping. Evolution
of the spectral function at low frequency.
If the k dependence of the self energy is weak,
we expect to see contour lines corresponding to
Ek const and a height increasing as we approach
the Fermi surface.
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Hole doped case t-.3t, U16 t n.71
.93 .97
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K.M . Shen et. al. Science (2005). For a review
Damascelli et. al. RMP (2003)
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Approaching the Mott transition CDMFT Picture

• Qualitative effect, momentum space
  differentiation. Formation of hot cold regions
  is an unavoidable consequence of the approach to
  the Mott insulating state!
• D wave gapping of the single particle spectra as
  the Mott transition is approached.
• Similar scenario was encountered in previous
  study of the kappa organics. O Parcollet G.
  Biroli and G. Kotliar PRL, 92, 226402. (2004)
  and Senechal and tremblay for cuprates with VCPT.

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Results of many numerical studies of electron
hole asymmetry in t-t Hubbard models Tohyama
Maekawa Phys. Rev. B 67, 092509 (2003) Senechal
and Tremblay. PRL 92 126401 (2004) Kusko et. al.
Phys. Rev 66, 140513 (2002)
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Experiments. Armitage et. al. PRL
(2001).Momentum dependence of the low-energy
Photoemission spectra of NCCO
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Approaching the Mott transition CDMFT picture.

• Qualitative effect, momentum space
  differentiation. Formation of hot cold regions
  is an unavoidable consequence of the approach to
  the Mott insulating state!
• General phenomena, BUT the location of the cold
  regions depends on parameters.
• Quasiparticles are now generated from the Mott
  insulator at (p, 0).
• Results of many l studies of electron hole
  asymmetry in t-t Hubbard models Tohyama Maekawa
  Phys. Rev. B 67, 092509 (2003) Senechal and
  Tremblay. PRL 92 126401 (2004) Kusko et. al.
  Phys. Rev 66, 140513 (2002). Kusunose and Rice
  PRL 91, 186407 (2003).

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Comparison with Experiments in Cuprates
Spectral Function A(k,??0) -1/p G(k, ? ?0) vs k
hole doped
electron doped
P. Armitage et.al. 2001
K.M. Shen et.al. 2004
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To test if the formation of the hot and cold
regions is the result of the proximity to
Antiferromagnetism, we studied various values of
t/t, U16.
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Introduce much larger frustration t.9t
U16tn.69 .92 .96
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Approaching the Mott transition

• Qualitative effect, momentum space
  differentiation. Formation of hot cold regions
  is an unavoidable consequence of the approach to
  the Mott insulating state!
• General phenomena, but the location of the cold
  regions depends on parameters.
• With the present resolution, t .9 and .3 are
  similar. However it is perfectly possible that
  at lower energies further refinements and
  differentiation will result from the proximity to
  different ordered states.

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Understanding the result in terms of cluster self
energies (eigenvalues)
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Cluster Eigenvalues
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Evolution of the real part of the self energies.
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Fermi Surface Shape Renormalization (
teff)ijtij Re(Sij(0))
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Fermi Surface Shape Renormalization

• Photoemission measured the low energy
  renormalized Fermi surface.
• If the high energy (bare ) parameters are doping
  independent, then the low energy hopping
  parameters are doping dependent. Another failure
  of the rigid band picture.
• Electron doped case, the Fermi surface
  renormalizes TOWARDS nesting, the hole doped case
  the Fermi surface renormalizes AWAY from nesting.
  Enhanced magnetism in the electron doped side.

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Understanding the location of the hot and cold
regions. Interplay of lifetime and fermi surface.
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Comparison of periodization methods for A(w0,k)
S M
S M
S M
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Pseudogap insights from cumulant periodization.

• Qualitative Difference between the periodization
  methods. The cumulant periodization, is a non
  linear interpolation of the self energies (linear
  interpolation of the cumulants). When the off
  diagonal elements of the self energy get large,
  it gives rise to lines of poles of the self
  energy, in addition to the Fermi lines.
• Quasi-one d, Essler and Tsvelik.

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• Subtle topological phase transition at
  intermediate doping ?
• Is there a quantum critical point related to the
  change of topology of the Fermi surface ?
• Is there a quantum critical point associated to
  the emergence of lines of zeros.
• This is NOT a Fermi surface instability,
  invisible to weak coupling analysis.
• We checked with PCMDFT that the results survive,
  unlike the corresponding quasi-1d case (Arrigoni
  et. al. Giamarchi Georges and Biermann).

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Large Doping
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Small Doping
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How is the Mott insulatorapproached from the
superconducting state ?
Work in collaboration with M. Capone, see also V.
Kancharlas talk.
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Superconductivity in the Hubbard model role of
the Mott transition and influence of the
super-exchange. (M. Capone, V. Kancharla.
CDMFTED, 4 8 sites t0) .
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• The superconductivity scales
• with J, as in the RVB approach.
• Qualitative difference between large and small U.
  The superconductivity goes to zero at half
  filling ONLY above the Mott transition.

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Order Parameter and Superconducting Gap .
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• In BCS theory the order parameter is tied to the
  superconducting gap. This is seen at U4t, but
  not at large U.
• How is superconductivity destroyed as one
• approaches half filling ?

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Evolution of the low energy tunneling density of
state with doping. Decrease of spectral weight as
the insulator is approached. Low energy particle
hole symmetry.
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• Superconductivity is destroyed at half filling
  due to a reduction of the one electron weight.
  Just like in the slave boson.
• High energy ph asymmetry.
• Low energy ph symmetry.

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Alternative view
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Conclusions

• DMFT is a useful mean field tool to study
  correlated electrons. Provide a zeroth order
  picture of a physical phenomena.
• Provide a link between a simple system (mean
  field reference frame) and the physical system
  of interest. Sites, Links, and Plaquettes
• Formulate the problem in terms of local
  quantities (which we can usually compute better).
• Allows to perform quantitative studies and
  predictions . Focus on the discrepancies between
  experiments and mean field predictions.
• Generate useful language and concepts. Follow
  mean field states as a function of parameters.
• K dependence gets strong as we approach the Mott
  transition. Fermi surfaces and lines of zeros of
  G.

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Conclusions

• Qualitative effect, momentum space
  differentiation. Formation of hot cold regions
  is an unavoidable consequence of the approach to
  the Mott insulating state!
• General phenomena, but the location of the cold
  regions depends on parameters. Study the normal
  state of the Hubbard model is useful.
• Character of the superconductivity is different
  for small and large U.

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the Hubbard model does not capture the trend of
supra with t. Need augmentation.(Venky Kancharla
Sarma).Same trend observed in DCA Maier and
Jarrell.
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D wave Superconductivity and Antiferromagnetism
t0 M. Capone V. Kancharla (see also VCPT
Senechal and Tremblay ).
Antiferromagnetic (left) and d wave
superconductor (right) Order Parameters
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Estimates of upper bound for Tc exact diag. M.
Capone. U16t, t0, ( t.35 ev, Tc 140 K.005W)
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Site? Cell. Cellular DMFT. C-DMFT. G..
Kotliar,S. Savrasov, G. Palsson and G. Biroli,
Phys. Rev. Lett. 87, 186401 (2001)
tˆ(K) hopping expressed in the superlattice
notations.

• Other cluster extensions (DCA Jarrell
  Krishnamurthy, M Hettler et. al. Phys. Rev. B 58,
  7475 (1998)Katsnelson and Lichtenstein periodized
  scheme, Nested Cluster Schemes , causality
  issues, O. Parcollet, G. Biroli and GK Phys.
  Rev. B 69, 205108 (2004)

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Understanding in terms of cluster self-energies.
Civelli et. al.
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k-(ET)2X are across Mott transition
ET
X- Ground State U/t t/t
Cu2(CN)3 Mott insulator 8.2 1.06
CuN(CN)2Cl Mott insulator 7.5 0.75
CuN(CN)2Br SC 7.2 0.68
Cu(NCS)2 SC 6.8 0.84
Cu(CN)N(CN)2 SC 6.8 0.68
Ag(CN)2 H2O SC 6.6 0.60
I3 SC 6.5 0.58
Insulating anion layer
X-1
conducting ET layer
(ET)21
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Electron doped case t.9t U16tn.69 .92 .96
Color scale x.9,.32,.22
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Two paths for calculation of electronic
structure of strongly correlated materials
Crystal structure Atomic positions
Model Hamiltonian
Correlation Functions Total Energies etc.
DMFT ideas can be used in both cases.
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Outline

• _________________________________________________
  ___________
• INTUITIVE NOTIONS OF DMFT AND WEISS FIELD CAVITY
  CONSTRUCTION.
• Mapping of lattice onto a cluster in a medium.
  With
• a prescription for building the medium from the
  computation of
• the cluster quantities.
• Prescription for reconstructing lattice
  quantities.
• Weiss field describe the medium.
• Cavity Construction is highly desireable. Delta
  is non zero
• on the boundary.
• -------------------------------------------------
  ------
• -------------------------------------------------
  -----
• EFFECTIVE ACTION CONSTRUCTION.

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• a) Baym Kadanoff Functional
• b) Self energy functional.
• c) DFT.
• Want to generate good approximations.
• and their hybrids.
• ------------------------------------------------
• ------------------------------------------------
• WEIS FIELD.
• I NEED TO GET THIS RIGHT HOW I DO THE
  SEPARATION,
• INTO PIECES WHAT IS THE EXACT AND WHA IS TEH

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• Nested Cluster Schemes.
• Explicit Cavity constructions.
• -------------------------------------------------
  
• -------------------------------------------------
  -
• CAUSALITY PROBLEMS.
• Parcollet Biorli KOtliar clausing.
• ----------------------------------------
• -------------------------------------------
• CONVERGENCE. 1/L vs e-L.
• DCA converges as 1/L2 lt put reference gt
• Classical limits. See what olivier ahs done
• on w-cdmft.

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Medium of free electrons impurity
model. Solve for the medium using Self Consistency

G.. Kotliar,S. Savrasov, G. Palsson and G.
Biroli, Phys. Rev. Lett. 87, 186401 (2001)
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Mott transition in layered organic conductors
S Lefebvre et al. cond-mat/0004455, Phys. Rev.
Lett. 85, 5420 (2000)
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k-(ET)2X are across Mott transition
ET
Insulating anion layer
X-1
conducting ET layer
(ET)21
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References
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CDMFT one electron spectra n.96 t/t.-.3 U16
t

• i

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Other cluster extensions (DCA Jarrell
Krishnamurthy, M Hettler et. al. Phys. Rev. B
58, 7475 (1998)Katsnelson and Lichtenstein
periodized scheme. Causality issues O. Parcollet,
G. Biroli and GK Phys. Rev. B 69, 205108 (2004)
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Medium of free electrons impurity
model. Solve for the medium using Self Consistency
G.. Kotliar,S. Savrasov, G. Palsson and G.
Biroli, Phys. Rev. Lett. 87, 186401 (2001)
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Difference in Periodizations
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Advantages of the Weiss field functional. Simpler
analytic structure near the Mott transition.
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How to determine the parameters of the bath ?

• Extremize Potthoffs self energy functional. It
  is hard to find saddles using conjugate
  gradients.
• Extremize the Weiss field functional.Analytic for
  saddle point equations are available
• Minimize some distance.

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Convergence of Cluster Schemes as a function of
cluster size.

• Aryamanpour et. al. DCA observables converge as
  1/L2.
• cond-mat/0205460 abs, ps, pdf, other
• Title Two Quantum Cluster ApproximationsAuthors
  Th. A. Maier (1), O. Gonzalez (1 and 2), M.
  Jarrell (1), Th. Schulthess (2) ((1) University
  of Cincinnati, Cincinnati, USA, (2) Oak Ridge
  National Laboratory, Oak Ridge, USA)
• Aryamampour et. al. The Weiss field in CDMFT
  converges as 1/L. Title The Dynamical Cluster
  Approximation (DCA) versus the Cellular Dynamical
  Mean Field Theory (CDMFT) in strongly correlated
  electrons systemsAuthors K. Aryanpour, Th. A.
  Maier, M. JarrellComments Comment on Phys. Rev.
  B 65, 155112 (2002). 3 pages, 2
  figuresSubj-class Strongly Correlated
  ElectronsJournal-ref Phys. Rev. B 71, 037101
  (2005)
• Biroli and Kotliar. Phys. Rev. B 71, 037102
  (2005)cond-mat/0404537. Local observables (i.e.
  observables contained in the cluster ) converge
  EXPONENTIALLY at finite temperatures away from
  critical points.

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Hole doped case t-.3t, U16 t n.71
.93 .97
Color scale x .37 .15 .13
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• Qualitative Difference between the hole doped and
  the electron doped phase diagram is due to the
  underlying normal state. In the hole doped, it
  has nodal quasiparticles near (p/2,p/2) which
  are ready to become the superconducting
  quasiparticles. Therefore the superconducing
  state can evolve continuously to the normal
  state. The superconductivity can appear at very
  small doping.
• Electron doped case, has in the underlying normal
  state quasiparticles leave in the (p, 0) region,
  there is no direct road to the superconducting
  state (or at least the road is tortuous) since
  the latter has QP at (p/2, p/2).

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ED and QMC
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Systematic Evolution