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Phases and phase transitions of quantum materials. Subir Sachdev Yale University. Talk online: http://pantheon.yale.edu/~subir or Search for Sachdev on . Phase changes in nature James Bay. Summer. Winter. Water. Ice. At high temperatures, maximize entropy.

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Phases and phase transitions of quantum materials

Subir Sachdev

Yale University

Talk online:

http://pantheon.yale.edu/~subir

or

Search for Sachdev on


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Phase changes in nature

James Bay

Summer

Winter

Water

Ice

At high temperatures,

maximize entropy

At low temperatures,

minimize energy


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Quantum physics: By Heisenberg’s uncertainty principle, the precise specification of the particle positions implies that their velocities are uncertain, with a magnitude determined by Planck’s constant . The kinetic energy of this -induced motion adds to the energy cost of the classically predicted phase of matter.

Tune : If we are able to vary the “effective” value of , then we can change the balance between the interaction and kinetic energies, and so change the preferred phase:

matter undergoes a quantum phase transition

Classical physics: In equilibrium, at the absolute zero of temperature ( T = 0 ), all particles will reside at rest at positions which minimize their total interaction energy. This defines a (usually) unique phase of matter e.g. ice.


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Outline

Varying “Planck’s constant” in the laboratory

  • The quantum superposition principle – a qubit

  • Interacting qubits in the laboratory - LiHoF4

  • Breaking up the Bose-Einstein condensate Bose-Einstein condensates and superfluids The Mott insulator

  • The cuprate superconductors

  • Conclusions


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1. The Quantum Superposition Principle

The simplest quantum degree of freedom –

a qubit

These states represent e.g. the orientation of the electron spin on a Ho ion in LiHoF4

Ho ions in a crystal of LiHoF4


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An electron with its “up-down” spin orientation uncertain has a definite “left-right” spin


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2. Interacting qubits in the laboratory uncertain has a definite “left-right” spin

In its natural state, the potential energy of the qubits in LiHoF4 is minimized by

or

A Ferromagnet


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Enhance quantum effects by applying an external “transverse” magnetic field which prefers that each qubit point “right”

For a large enough field, each qubit will be in the state


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Quantum phase transition “transverse” magnetic field which prefers that each qubit point “right”

Phase diagram

Absolute zero of temperature

g = strength of transverse magnetic field


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Quantum phase transition “transverse” magnetic field which prefers that each qubit point “right”

Phase diagram

g = strength of transverse magnetic field


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3. Breaking up the Bose-Einstein condensate “transverse” magnetic field which prefers that each qubit point “right”

Certain atoms, called bosons (each such atom has an even total number of electrons+protons+neutrons), condense at low temperatures into the same single atom state. This state of matter is a Bose-Einstein condensate.

A. Einstein and S.N. Bose (1925)


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The Bose-Einstein condensate in a periodic potential “transverse” magnetic field which prefers that each qubit point “right”

“Eggs in an egg carton”


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The Bose-Einstein condensate in a periodic potential “transverse” magnetic field which prefers that each qubit point “right”

“Eggs in an egg carton”


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The Bose-Einstein condensate in a periodic potential “transverse” magnetic field which prefers that each qubit point “right”

“Eggs in an egg carton”


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The Bose-Einstein condensate in a periodic potential “transverse” magnetic field which prefers that each qubit point “right”

“Eggs in an egg carton”


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The Bose-Einstein condensate in a periodic potential “transverse” magnetic field which prefers that each qubit point “right”

“Eggs in an egg carton”

Lowest energy state of a single particle minimizes kinetic energy by maximizing the position uncertainty of the particle


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The Bose-Einstein condensate in a periodic potential “transverse” magnetic field which prefers that each qubit point “right”

Lowest energy state for many atoms

Large fluctuations in number of atoms in each potential well – superfluidity (atoms can “flow” without dissipation)


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The Bose-Einstein condensate in a periodic potential “transverse” magnetic field which prefers that each qubit point “right”

Lowest energy state for many atoms

Large fluctuations in number of atoms in each potential well – superfluidity (atoms can “flow” without dissipation)


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The Bose-Einstein condensate in a periodic potential “transverse” magnetic field which prefers that each qubit point “right”

Lowest energy state for many atoms

Large fluctuations in number of atoms in each potential well – superfluidity (atoms can “flow” without dissipation)


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The Bose-Einstein condensate in a periodic potential “transverse” magnetic field which prefers that each qubit point “right”

Lowest energy state for many atoms

Large fluctuations in number of atoms in each potential well – superfluidity (atoms can “flow” without dissipation)


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The Bose-Einstein condensate in a periodic potential “transverse” magnetic field which prefers that each qubit point “right”

Lowest energy state for many atoms

Large fluctuations in number of atoms in each potential well – superfluidity (atoms can “flow” without dissipation)


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The Bose-Einstein condensate in a periodic potential “transverse” magnetic field which prefers that each qubit point “right”

Lowest energy state for many atoms

Large fluctuations in number of atoms in each potential well – superfluidity (atoms can “flow” without dissipation)


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The Bose-Einstein condensate in a periodic potential “transverse” magnetic field which prefers that each qubit point “right”

Lowest energy state for many atoms

Large fluctuations in number of atoms in each potential well – superfluidity (atoms can “flow” without dissipation)


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3. Breaking up the Bose-Einstein condensate “transverse” magnetic field which prefers that each qubit point “right”

By tuning repulsive interactions between the atoms, states with multiple atoms in a potential well can be suppressed. The lowest energy state is then a Mott insulator – it has negligible number fluctuations, and atoms cannot “flow”


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3. Breaking up the Bose-Einstein condensate “transverse” magnetic field which prefers that each qubit point “right”

By tuning repulsive interactions between the atoms, states with multiple atoms in a potential well can be suppressed. The lowest energy state is then a Mott insulator – it has negligible number fluctuations, and atoms cannot “flow”


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3. Breaking up the Bose-Einstein condensate “transverse” magnetic field which prefers that each qubit point “right”

By tuning repulsive interactions between the atoms, states with multiple atoms in a potential well can be suppressed. The lowest energy state is then a Mott insulator – it has negligible number fluctuations, and atoms cannot “flow”


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3. Breaking up the Bose-Einstein condensate “transverse” magnetic field which prefers that each qubit point “right”

By tuning repulsive interactions between the atoms, states with multiple atoms in a potential well can be suppressed. The lowest energy state is then a Mott insulator – it has negligible number fluctuations, and atoms cannot “flow”


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Quantum phase transition “transverse” magnetic field which prefers that each qubit point “right”

Phase diagram

Bose-Einstein

Condensate


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4. The cuprate superconductors “transverse” magnetic field which prefers that each qubit point “right”

A superconductor conducts electricity without resistance below a critical temperature Tc


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Cu “transverse” magnetic field which prefers that each qubit point “right”

La2CuO4 ---- insulator

La2-xSrxCuO4 ---- superconductor for 0.05 < x < 0.25

Quantum phase transitions as a function of Sr concentration x

O

La,Sr


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La “transverse” magnetic field which prefers that each qubit point “right” 2CuO4 --- an insulating antiferromagnet

with a spin density wave

La2-xSrxCuO4 ----

a superconductor


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Zero temperature phases of the cuprate superconductors as a function of hole density

Insulator with a spin density wave

Applied magnetic field

Theory for a system with strong interactions: describe superconductor and superconductor+spin density wave phases by expanding in the deviation from the quantum critical point between them.

Superconductor

Superconductor with a

spin density wave

~0.12

~0.05

x


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Accessing quantum phases and phase transitions by varying “Planck’s constant” in the laboratory

  • Immanuel Bloch: Superfluid-to-insulator transition in trapped atomic gases

  • Gabriel Aeppli: Seeing the spins (‘qubits’) in quantum materials by neutron scattering

  • Aharon Kapitulnik: Superconductor and insulators in artificially grown materials

  • Matthew Fisher: Exotic phases of quantum matter


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