Thomas Vojta- Quantum phase transitions

8/3/2019 Thomas Vojta- Quantum phase transitions

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Quantum phase transitions

Thomas VojtaDepartment of Physics, Missouri University of Science and Technology

Phase transitions and critical points

Quantum phase transitions: How important is quantum mechanics?

Quantum phase transitions in metallic systems

Chennai, October 13, 2009


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Acknowledgements

at Missouri S&T

Chetan KotabageMan Young Lee

Adam FarquharJason Mast

former group members

Mark Dickison (Boston U.)Bernard Fendler (Florida State U.)

Jose Hoyos (Florida State U.)Shellie Huether (Missouri S&T)

Ryan Kinney (US Navy)Rastko Sknepnek (Iowa State U.)

external collaboration

Dietrich Belitz (U. of Oregon)Manuel Brando (MPI Dresden)

Adrian DelMaestro (UBC)Philipp Gegenwart (U. of Goettingen)

Ted Kirkpatrick (U. of Maryland)Wouter Montfrooij (U. of Missouri)Rajesh Narayanan (IIT Madras)Bernd Rosenow (MPI Stuttgart)Jorg Schmalian (Iowa State U.)Matthias Vojta (U. of Cologne)

Funding:

National Science FoundationResearch Corporation

University of Missouri Research Board


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Phase diagram of water

solid

liquid

gaseoustripel point

critical point

100 200 300 400 500 600 700

102

10 3

104

105

106

107

108

109

T(K)

Tc=647 K

Pc=2.2*10

8

Pa

Tt=273 KPt=6000 Pa

Phase transition:singularity in thermodynamic quantities as functions of external parameters


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Phase transitions: 1st order vs. continuous

1st order phase transition:phase coexistence, latent heat,short range spatial and time correlations

Continuous transition (critical point):

no phase coexistence, no latent heat,infinite range correlations of fluctuations

solid

liquid

gaseoustripel point

critical point

100 200 300 400 500 600 700

102

103

104

105

106

107

108

109

T(K)

Tc=647 K

Pc=2.2*108 Pa

Tt=273 K

Pt=6000 Pa

Critical behavior at continuous transitions:

diverging correlation length |T Tc| and time

z |T Tc|z

power-laws in thermodynamic observables: |T Tc|, |T Tc|

Manifestation: critical opalescence (Andrews 1869)

Universality: critical exponents are independent of microscopic details


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Critical opalescence

Binary liquid system:

e.g. hexane and methanol

T > Tc 36C: fluids are miscible

T < Tc: fluids separate into two phases

T Tc: length scale of fluctuations grows

When reaches the scale of a fraction of amicron (wavelength of light):

strong light scatteringfluid appears milky

Pictures taken from http://www.physicsofmatter.com

46C

39C

18C


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How important is quantum mechanics close to a critical point?

Two types of fluctuations:thermal fluctuations, energy scale kBTquantum fluctuations, energy scale c

Quantum effects are unimportant ifc kBT.

Critical slowing down:

c 1/ |T Tc|z 0 at the critical point

For any nonzero temperature, quantum fluctuations do not play a role close

to the critical point Quantum fluctuations do play a role a zero temperature

Thermal continuous phase transitions can be explained entirely in terms ofclassical physics


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Quantum phase transitions

occur at zero temperature as function of pressure, magnetic field, chemicalcomposition, ...

driven by quantum rather than thermal fluctuations

paramagnet

ferromagnet

0 20 40 60 80

transversal magnetic field (kOe)

0.0

0.5

1.0

1.5

2.0

Bc

LiHoF4

0

Doping level (holes per CuO2)

0.2

SCAFM

Fermiliquid

Pseudogap

Non-Fermiliquid

Phase diagrams of LiHoF4 and a typical high-Tc superconductor such as YBa2Cu3O6+x


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If the critical behavior is classical at any nonzero temperature, why are

quantum phase transitions more than an academic problem?


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Phase diagrams close to quantum phase transition

Quantum critical point controlsnonzero-temperature behaviorin its vicinity:

Path (a): crossover betweenclassical and quantum critical

behavior

Path (b): temperature scaling ofquantum critical point

quantumdisordered

quantum critical

(b)

(a)

Bc

QCP

B

0

ordered

kT~ $%cT

classicalcritical

thermally

disordered

T

Bc

QCP

kT-STc

quantum critical

quantumdisordered

(b)

(a)

B

0

order at T=0

thermallydisordered


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Quantum to classical mapping

Classical partition function: statics and dynamics decoupleZ =

dpdq eH(p,q) =

dp eT(p)

dq eU(q)

dq eU(q)

Quantum partition function: statics and dynamics are coupled

Z = TreH = limN(eT/NeU/N)N =

D[q()] eS[q()]

imaginary time acts as additional dimension, at zero temperature ( = ) theextension in this direction becomes infinite

Quantum phase transition in d dimensions is equivalent to classicaltransition in (d + 1) dimensions

Caveats: mapping holds for thermodynamics only resulting classical system is anisotropic if space and time enter asymmetrically if quantum action is not real, extra complications may arise, e.g., Berry phases


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Toy model: transverse field Ising model

Quantum spins Si

on a lattice: (c.f. LiHoF4

)

H = Ji

SziSzi+1h

i

Sxi = Ji

SziSzi+1

h

2

i

(S+i + Si )

J: exchange energy, favors parallel spins, i.e., ferromagnetic state

h: transverse magnetic field, induces quantum fluctuations between up and downstates, favors paramagnetic state

Limiting cases:

|J| |h| ferromagnetic ground state as in classical Ising magnet

|J| |h| paramagnetic ground state as for independent spins in a field

Quantum phase transition at |J| |h| (in 1D, transition is at |J| = |h|)

Quantum-to-classical mapping: QPT in transverse field Ising model in ddimensions maps onto classical Ising transition in d + 1 dimensions


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Magnetic quantum critical points of TlCuCl3

TlCuCl3 is magnetic insulator

planar Cu2Cl6 dimers forminfinite double chains

Cu2+ ions carry spin-1/2moment

antiferromagnetic ordercan be induced by

applying pressure

applying a magnetic field

0 1 2 3 4 5

p (kBar)

0

2

4

6

8

10

Tc

(K)

Paramagnet

AFM

0 2 4 6 8 10 12

B (T)

Paramagnet

canted AFM

QCP QCP

a) b)

Bc1 Bc2 B

T


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Pressure-driven quantum phase transition in TlCuCl3

quantum Heisenberg model

H =ij

Jij Si Sj h i

Si .

Jij = J intra-dimerJ

between dimers

pressure changes ratio J/J

intra-dimerinteraction J

inter-dimerinteractionJ

singlet=

orderedspin )(1

2

Limiting cases:

|J| |J| spins on each dimer form singlet no magnetic order

low-energy excitations are triplons (single dimers in the triplet state)|J| |J| long-range antiferromagnetic order (Neel order)

low-energy excitations are long-wavelength spin waves

quantum phase transition at some critical value of the ratio J/J


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Field-driven quantum phase transition in TlCuCl3

Single dimer in field:

field does not affect singlet groundstate but splits the triplet states

ground state: singlet for B < Bc and

(fully polarized) triplet for B > Bc

triplet

singlet

E

BBc0

-3 4J/

J/4

Full Hamiltonian:

singlet-triplet transition of isolated dimer splits into two transitions

at Bc1, triplon gap closes, system is driven into ordered state(uniform magnetization || to field and antiferromagnetic order to field)

canted antiferromagnet is Bose-Einstein condensate of triplons

at Bc2 system enters fully polarized state


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Ferromagnetic transitions: MnSi, UGe2, ZrZn2 pressure tuned

NixPd1x, URu2xRexSi2 composition tuned

Antiferromagnetic transitions: CeCu6xAux composition tuned

YbRh2Si2 magnetic field tuned

dozens of other rare earth compounds, tunedby composition, field or pressure

Other transitions:

metamagnetic transitions

superconducting transitions

Temperature-pressure phasediagram of MnSi (Pfleidereret al, 1997)


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Fermi liquids versus non-Fermi liquids

Fermi liquid concept (Landau 1950s):interacting Fermions behave like almost non-interacting quasi-particles withrenormalized parameters (e.g. effective mass)

universal predictions for low-temperature properties

specific heat C Tmagnetic susceptibility = constelectric resistivity = A + BT2

Fermi liquid theory is extremely successful, describes vast majority of metals.

In recent years:

systematic search for violations of the Fermi liquid paradigm high-TC superconductors in normal phase heavy Fermion materials (rare earth compounds)


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Non-Fermi liquid behavior in CeCu6xAux

specific heat coefficient C/T at quantum critical point diverges logarithmicallywith T 0

cause by interaction between quasiparticles and critical fluctuations

functional form presently not fully understood (orthodox theories do not work)

Phase diagram and specific heat ofCeCu6xAux (v. Lohneysen, 1996)

0.1 0.3 0.5

x

0.0

0.5

1.0

T

(K)

II

I

0.1 1.0

T (K)

0

1

2

3

4

C/T(

J/molK2)

x=0x=0.05x=0.1x=0.15x=0.2x=0.3

0.04 40

1

2

3

4


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Generic scale invariance

metallic systems at T = 0 contain many soft (gapless) excitations such asparticle-hole excitations (even away from any critical point)

soft modes lead to long-range spatial and temporal correlations

quasi-particle dispersion: (p) |ppF|3 log |ppF|

specific heat: cV(T) T3 log T

static spin susceptibility: (2)(q) = (2)(0)+ c3|q|2 log 1|q| + O(|q|

2)

in real space: (2)(r r) |r r|5

in general dimension: (2)(q) = (2)(0) + cd|q|d1 + O(|q|2)

Long-range correlations due to coupling to soft particle-hole excitations


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Generic scale invariance and the ferromagnetic transition

At critical point: critical modes and generic soft modes interact in a nontrivial way interplay greatly modifies critical behavior of the quantum phase transition

(Belitz, Kirkpatrick, Vojta, Rev. Mod. Phys., 2005)

Example:Ferromagnetic quantum phase transition

Long-range interaction due to generic scaleinvariance singular Landau theory

= tm2 vm4 log(1/m) + um4

Ferromagnetic quantum phase transitionturns 1st order

general mechanism, leads to singular Landau

theory for all zero wavenumber order parameters Phase diagram of MnSi.


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Quantum phase transitions and exotic superconductivity

Superconductivity in UGe2:

phase diagram of UGe2 has pocket of superconductivity close to quantum phasetransition

in this pocket, UGe2 is ferromagnetic and superconducting at the same time superconductivity appears only in superclean samples

Phase diagram and resistivity of UGe2 (Saxena et al, Nature, 2000)


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Character of superconductivity in UGe2

Conventional (BCS) superconductivity:

Cooper pairs form spin singletsnot compatible with ferromagnetism

Theoretical ideas:

phase separation (layering or disorder): NO!

partially paired FFLO state: NO!spin triplet pairing with odd spatial symmetry(p-wave)

magnetic fluctuations are attractive for

spin-triplet pairs

Ferromagnetic quantum phasetransition promotes magneticallymediated spin-triplet superconductivity


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Ferromagnetic QPT in a disordered metal: CePd1xRhx

ferromagnetic phase shows pronounced tail evidence for smeared transition

evidence for spin-glass like behavior in tail

above tail: nonuniversal power-laws characteristic of quantum Griffiths effects

Quantum phase transitions in disordered systems often lead to exoticcritical behavior


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Conclusions

quantum phase transitions occur at zero temperature as a function of aparameter like pressure, chemical composition, disorder, magnetic field

quantum phase transitions are driven by quantum fluctuations rather thanthermal fluctuations

quantum critical points control behavior in the quantum critical region at

nonzero temperatures

quantum phase transitions in metals have fascinating consequences: non-Fermiliquid behavior and exotic superconductivity

quantum phase transitions in disordered systems lead to unconventional critical

behavior

Quantum phase transitions provide a new ordering principle incondensed matter physics


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Superconductor-metal QPT in ultrathin nanowires

ultrathin MoGe wires (width 10 nm)

produced by molecular templating usinga single carbon nanotube[A. Bezryadin et al., Nature 404, 971 (2000)]

thicker wires are superconducting atlow temperatures

thinner wires remain metallic

superconductor-metal QPT as

function of wire thickness


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Pairbreaking mechanism

pair breaking by surface magnetic impurities random impurity positions quenched disorder

gapless excitations in metal phase Ohmic dissipation

weak field enhances superconductivitymagnetic field aligns the impuritiesand reduces magnetic scattering

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Thomas Vojta- Quantum phase transitions