Emerging phases and phase transitions in quantum...
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Emerging phases and phase transitions inquantum matter
Thomas VojtaDepartment of Physics, Missouri University of Science and Technology
• Condensed matter physics: complexity and emerging phenomena
• Phase transitions and quantum phase transitions
• Emerging phases close to quantum critical points
Missouri S&T Chemistry, Feb 10, 2020
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Acknowledgements
Collaboration
Jose Hoyos
(Sao Paulo)
Postdocs
Hatem Martin
Barghathi Puschmann
Collaboration: Experiment
Almut Schroeder Istvan Kezsmarki
(Kent State) (TU Budapest)
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What is condensed matter physics?
Condensed Matter Physics (Wikipedia):
deals with the macroscopic properties of matter; in particular ... the“condensed” phases that appear whenever the number of constituents ina system is large and their interactions ... are strong
Traditionally: Physics of solids and liquids
• What is the structure of crystals?
• How do solids melt or liquids evaporate?
• Why do some materials conduct an electric current and others do not?
Today: all systems consisting of a large number of interacting constituents
• biological systems: biomolecules, DNA, membranes, cells
• geological systems: earthquakes
• economical systems: fluctuations of stock markets, currencies
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Why condensed matter physics?
Applications: ”Helps you to make stuff.”
• semiconductors, transistors, microchips
• magnetic recording devices
• liquid crystal displays
• to come: qbits for quantum computers
Maglev train using levitation bysuperconducting magnets, cango faster than 350 mph
Read head, based on Giant Magnetoresistance effect(A. Fert + P. Grunberg, Physics Nobel Prize 2007)
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Why condensed matter physics II
Directions of fundamental physics research :
Astrophysics and cosmology:increasing length and time scales“physics of the very large”
Atomic, nuclear and elementaryparticle physics:decreasing length and time scales“physics of the very small”
Particle accelerator atFermilab
What fundamental direction does condensed matter research explore?“physics of the very complex”
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Emerging phenomena and the axis of complexity
“More is different!”
number of particles
100 1023103 - 105101 -102
atoms,molecules
chemical prop.(periodic table)optical prop.(e.g., color)
elementaryparticles
mesoscopicsystems
transportproperties(conductivity)
macroscopic solid or liquid
phases andphase transitions(ferromagnetism) ...biology, life
-1012
cluster
mechanicalproperties (rigidity, solid vs. liquid)
Emerging phenomena:
When large numbers of particles strongly interact, qualitatively new properties ofmatter emerge at every level of complexity
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New states of quantum matter and where to find them
Quantum matter:
• matter with emerging macroscopic properties that are intrinsically quantum(superconductors, superfluids, fractional quantum Hall states, spin liquids)
at low temperatures
F = E − TS
• thermal motion is suppressed• new types of order can form
at boundaries of existing phases
• two types of order compete, suppresseach other
• novel type of order may appear
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• Condensed matter physics: complexity and emerging phenomena
• Phase transitions and quantum phase transitions• Novel phases close to quantum critical points
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Phase diagram of water
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
Critical behavior at continuous transitions:
diverging correlation length ξ ∼ |T − Tc|−ν and time ξτ ∼ ξz ∼ |T − Tc|−νz
• 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 ≈ 36◦C: fluids are miscible
T < Tc: fluids separate into two phases
T → Tc: length scale ξ of fluctuations grows
When ξ reaches the scale of a fraction of amicron (wavelength of light):
strong light scatteringfluid appears milky
Pictures taken from http://www.physicsofmatter.com
46◦C
39◦C
18◦C
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How important is quantum mechanics close to a critical point?
Two types of fluctuations:
thermal fluctuations (thermal motion), energy scale kBTquantum fluctuations (quantum zero-point motion), energy scale ~ωc
Quantum effects unimportant if ~ωc ≪ kBT .
Critical slowing down:
ωc ∼ 1/ξτ ∼ |T − Tc|νz → 0 at the critical point
⇒ For any nonzero temperature, quantum fluctuations do not play a role closeto the critical point
⇒ Quantum fluctuations do play a role a zero temperature
Zero-temperature continuous phase transitions constitute a special class ofphase transitions, they are intrinsically quantum in nature
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Quantum phase transitions
occur at zero temperature as function of pressure, magnetic field, chemicalcomposition, ...
driven by quantum zero-point motion rather than thermal fluctuations
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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Toy model: transverse field Ising model
Quantum spins Si on a lattice: (c.f. LiHoF4)
H = −J∑i
SziSzi+1−h
∑i
Sxi
= −J∑i
SziSzi+1−
h
2
∑i
(S+i + S−
i )
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|)
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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 5p (kBar)
0
2
4
6
8
10
Tc
(K)
Paramagnet
AFM
0 2 4 6 8 10 12B (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-dimer interaction J
inter-dimer interaction J’
singlet =
ordered spin
)(1
2
Limiting cases:
|J | ≫ |J ′| spins on each dimer form singlet ⇒ no magnetic orderlow-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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Superconductor-metal QPT in ultrathin nanowires
• ultrathin MoGe wires (width ∼ 10 nm)
• produced by molecular templating usinga single carbon nanotube
• thicker wires are superconducting atlow temperatures
• thinner wires remain metallic
superconductor-metal QPT asfunction 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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Experiment: Ga thin films
Xing et al., Science 350, 542 (2015)
• three-monolayer Ga films
• superconductivity below Tc ≈ 3.62K,suppressed by magnetic field
• field-driven QPT well described by2D infinite-randomness critical point
• dynamical exponent diverges asz ∼ |B −Bc|−νψ with ν ≈ 1.2, ψ ≈ 0.5
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Mott transition in a Bose-Einstein condensate
Lattice
Beams
BEC
BEC
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• Condensed matter physics: complexity and emerging phenomena
• Phase transitions and quantum phase transitions
• Novel phases close to quantum critical points
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Exotic superconductivity in UGe2
Phase diagram:
• phase diagram of UGe2 has pocket of superconductivity close to ferromagneticquantum phase transition (electrical resistivity vanishes below about 1K)
• in this pocket, UGe2 is ferromagnetic and superconducting at the same time
• superconductivity appears only in superclean samples
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Character of superconductivity in UGe2
not compatible with conventional (BCS)superconductivity:
• in superconductor, electrons form (Cooper) pairsof spin-up and spin-down electrons
• ferromagnetism requires majority of spins to be inone direction
theoretical ideas:
• phase separation (layering or disorder): NO!
• partially paired FFLO state: NO!
• spin triplet pairs with odd spatial symmetry,magnetic fluctuations promote this type ofpairing
Magnetic quantum phase transition inducesspin-triplet superconductivity
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Metamagnetic transition in Sr3Ru2O7
H
T
H ,TC C
1st order
crossover
• Sr3Ru2O7 undergoes metamagnetictransition as function of field
• critical endpoint can be tuned to T =0 by tilting the field
⇒ metamagnetic quantum criticalpoint
• seen in temperature dependence ofresistivity ρ = ρ0 +ATα
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Electronic nematic phase in Sr3Ru2O7
• in pure samples at low temperatures,QCP preempted by novel phase
• resistivity highly anisotropic (C4 → C2)
⇒ new phase is electronic nematic(translational invariant but rotationalsymmetry spontaneously broken)
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Disorder and Griffiths phases
QPT in a disordered system:
• rare region can be locally in ordered phaseeven if bulk system is in disordered phase
• probability of rare region exponentially smallp(L) ∼ exp(−cLd):
• rare regions act as large superspins⇒ slow dynamics, large contribution to TD
0.1
1
10
100
1000
0 2 4 6 8 10 12 14 16
Tc
Tmax
Tcross
Tc (
K)
and o
ther
T-s
cale
s
x (%)
Ni1-x
Vx
PM
FM
GP
CG ?
0
0.2
0.4
0.6
0 5 10 15
- 5µB/V
µsa
t(µB)
x(%)
Can rare regions dominate thethermodynamic response?
⇒ quantum Griffiths phase
• example:diluted ferromagnet Ni1−xVx
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Quantum Griffiths phase in Ni1−xVx
0.1
1
10
100
1000
100 1000 104
Mm (
em
u/m
ol)
H (G)
Ni1-x
Vx
T=2K
(b)
0.0001
0.001
0.01
0.1
1
1 10 100
x=11.4%
11.6%
12.07%
12.25%
13.0%
15.0%!
"1+ "
dyn
"m (
em
u/m
ol)
T(K)
H=100G
(a)
0.1
1
10
100
1000
100 1000 104
Mm (
em
u/m
ol)
H (G)
Ni1-x
Vx
T=2K
(b)
0
0.2
0.4
0.6
0.8
1
11 12 13 14 15 16
!loT
!
1-"
!dyn!,
1- "
x (%)
(a)
Ni1-x
Vx
0.01
0.1
101
102
103
104
105
H=500GT=2KT=14K -- T=300K
Mm/H
0.5 (
em
u m
ol-1
G-0
.5)
H/T (G/K)
x=12.25%
(b)
at concentrations above xc:
• strongly enhanced magneticresponse
• susceptibility χ(T ) diverges asT → 0
• χ(T ) and m(H) follownonuniversal power lawsχ ∼ Tλ−1, m ∼ Hλ
(Griffiths singularities)
• Griffiths exponent λ = 1− γvaries systematically with x
• experiments agree withinfinite-randomness criticalpoint scenario
quantum Griffiths phase forx ≈ 11.5 to 15%.
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Rare regions and smeared phase transition in Sr1−xCaxRuO3
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Composition-tuned smeared phase transitions
1
2
3
0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.500.0
0.2
0.4
0.6
0.8
1.0
-150 0 150
0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52
-6
-5
-4
-3
-2
-1
0
1
2
xC
TC(x)
Theory
ln(T
C)
b
M
B [mT]
M [
B/R
u]
Composition, x
a T=2.9K
4.2K
10K
20K
30KI
0.1 B/Ru
0.52
0.49
0.48
0.47
0.46
0.45
0.44
M(x); T=2.9K
M(x); T=4.2K
M(B); T=4.2K
Theory
ln(M
)
Composition, x
Magnetization and Tc in tail:
M,Tc ∼ exp
[−C (x− x0c)
2−d/ϕ
x(1− x)
]for x→ 1:
M,Tc ∼ (1− x)Ldmin
T
x
Magneticphase
Tc
decreasesexponentially
0 1
power-law tail
Super-PMbehavior
xc
0
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Conclusions
• “More is different:” condensed matter physics explores emerging phenomenacaused by the interplay of many constituent particles
• new states of quantum matter can be found at low temperatures and atboundaries between existing phases
• quantum phase transitions occur at zero temperature as a function of aparameter like pressure, chemical composition, disorder, magnetic field
• quantum phase transitions are caused by quantum fluctuations (i.e,Heisenberg’s uncertainty principle) rather than thermal fluctuations
• quantum phase transitions can have fascinating consequences including thegenesis of new phases of matter
Quantum phase transitions provide a novel ordering principle incondensed matter physics
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Wonderland at low temperatures
273K (0C) water freezes
195K (-78C) carbon dioxide sublimates (dry ice)
133K (-140C) superconductivity in cuprate perovskites
77K (-196C) nitrogen (air) liquefies
66K (-207C) nitrogen (air) freezes
4.2K (-268.9C) helium liquefies
2.2K (-270.9C) helium becomes superfluid
170 nK Bose-Einstein condensation of rubidium
0K (-273.1C) absolute zero of temperature
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Magnetic phases in MnSi
Phase diagram: (Pfleiderer et al, 2004)
• magnetic transition at 30K atambient pressure
• transition tunable by hydrostaticpressure
• quantum phase transition at pc = 14kbar
Magnetic state:
• ordered state is helimagnet withq = 180A, pinned in (111) direction
• short-range order persists in para-magnetic phase, helical axis depinned
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Skyrmions and skyrmion lattices
• even more exotic magnetic states occur inmagnetic field B
• in “A” phase, magnetization vector formsknots, called skyrmions, by twisting in twodirections
• these skyrmions arrange themselves intoregular skyrmion lattice
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If the critical behavior is classical at any nonzero temperature, why arequantum 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 criticalbehavior
Path (b): temperature scaling ofquantum critical point
quantumdisordered
quantum critical
(b)
(a)
Bc
QCP
B0
ordered
kT ~ ��c
T
classicalcritical
thermallydisordered
T
Bc
QCP
kT -STc
quantum critical
quantumdisordered
(b)
(a)
B
0
order at T=0
thermallydisordered