Download - Thomas Vojta- Quantum phase transitions
-
8/3/2019 Thomas Vojta- Quantum phase transitions
1/27
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
-
8/3/2019 Thomas Vojta- Quantum phase transitions
2/27
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
-
8/3/2019 Thomas Vojta- Quantum phase transitions
3/27
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
-
8/3/2019 Thomas Vojta- Quantum phase transitions
4/27
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
-
8/3/2019 Thomas Vojta- Quantum phase transitions
5/27
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
-
8/3/2019 Thomas Vojta- Quantum phase transitions
6/27
Phase transitions and critical points
Quantum phase transitions: How important is quantum mechanics?
Quantum phase transitions in metallic systems
-
8/3/2019 Thomas Vojta- Quantum phase transitions
7/27
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
-
8/3/2019 Thomas Vojta- Quantum phase transitions
8/27
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
-
8/3/2019 Thomas Vojta- Quantum phase transitions
9/27
If the critical behavior is classical at any nonzero temperature, why are
quantum phase transitions more than an academic problem?
-
8/3/2019 Thomas Vojta- Quantum phase transitions
10/27
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
-
8/3/2019 Thomas Vojta- Quantum phase transitions
11/27
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
-
8/3/2019 Thomas Vojta- Quantum phase transitions
12/27
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
-
8/3/2019 Thomas Vojta- Quantum phase transitions
13/27
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
-
8/3/2019 Thomas Vojta- Quantum phase transitions
14/27
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
-
8/3/2019 Thomas Vojta- Quantum phase transitions
15/27
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
-
8/3/2019 Thomas Vojta- Quantum phase transitions
16/27
Phase transitions and critical points
Quantum phase transitions: How important is quantum mechanics?
Quantum phase transitions in metallic systems
-
8/3/2019 Thomas Vojta- Quantum phase transitions
17/27
Quantum phase transitions in metallic systems
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)
-
8/3/2019 Thomas Vojta- Quantum phase transitions
18/27
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)
-
8/3/2019 Thomas Vojta- Quantum phase transitions
19/27
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
-
8/3/2019 Thomas Vojta- Quantum phase transitions
20/27
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
-
8/3/2019 Thomas Vojta- Quantum phase transitions
21/27
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.
-
8/3/2019 Thomas Vojta- Quantum phase transitions
22/27
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)
-
8/3/2019 Thomas Vojta- Quantum phase transitions
23/27
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
-
8/3/2019 Thomas Vojta- Quantum phase transitions
24/27
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
-
8/3/2019 Thomas Vojta- Quantum phase transitions
25/27
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
-
8/3/2019 Thomas Vojta- Quantum phase transitions
26/27
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
-
8/3/2019 Thomas Vojta- Quantum phase transitions
27/27
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