Introduction to Quantum Phase Transition

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Introduction to Quantum Phase Transition
Condensed Matter dpt SCES group
Emanuele Dalla Torre

• Student Seminar June 12th 2006

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Introduction

• What happens at zero Temperature?
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• Outline
• From Classical
• to Quantum
• My Master Thesis

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Classical - example

• Look for stable state minimizes free energy
  (G) for given intrinsic parameters

vapor ?? ice (sublimation)
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Classical - order

• Two different phases
• Ordered phase ? crystal (ice)
• Disordered phase ? gas (vapor)

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Classical - order

• Order parameter long range order
• (f.e. direction of maximal concentration)
• Crystal ? finite (vector)
• Gas ? zero
• Spontaneous Symmetry breaking (crystal)
• Environment ? rotation symmetry O(3)
• Stable state ? no rotation symmetry

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Classical - fluctuations

• Crystal rise temperature (intrinsic param.)
• ? phase transition (order param.)
• Thermal fluctuations increase ? mean fluctuations
  (dx) reach the order of lattice constant (d) ?
  crystal melts
• Lindemann criterion

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Classical pressure

• Another intrinsic parameter
• pressure ? concentration ? inter-atomic distance
• Sublimation temperature is determined by the
  pressure
• ? two independent parameters ? we can draw a
  phase diagram

high pressure high concentration
low pressure low concentration
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Classical phase diagram

• T Temperature (x-axis), P Pressure (y-axis)
• ? which is the stable state?

• When we study the phase diagram we can find
  other unexpected phases liquid (water)

LL, Statistical Mechanics
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Classical - summary

• Stable state
• Ordered phase / Disorder phase
• Order parameter
• Thermal fluctuations
• Pressure
• Phase diagram

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Quantum - example

• Example Spin-1 chain (one dimensional)
• We look for the ground state minimizes the
  energy (GE-ST ? this is the stable state for
  zero T)

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Quantum - order

• Ordered phase ? Anti-Ferromagnetic (AF)
• Disordered phase ? Mott insulator

A. Auerbach, Interacting Electrons and Quantum
Magnetism (1994)
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Quantum - order

• AF Order parameter
• AF ? constant (complex number)
• MOTT? zero
• Spontaneous Symmetry Breaking SU(2)
• (direction of the first spin)

In reality power low
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Quantum - fluctuations

• If we choose Sz direction, we cannot exactly
  define Sx
• ? fluctuations (hopping term)
• Temperature (parameterize fluctuations)

S.Sachdev, Quantum Phase Transition (1999)
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Quantum phase diagram

• Another parameter
• ? concentration of non zero spin
• Spin-1 Phase diagram
• (Neel Crystal,
• Mott Gas,
• Haldane Liquid)

Mott
AF
1/Thop
H. Tasaki, PRL,66(6) (1991)
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Quantum phase diagram

• Haldane phase hidden order
• 1 Spin up / zero(es) / 1 spin down / zero(es) /
  .
• String Order Parameter like AF order, but skips
  zeros

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Master Thesis

• From spin-1 to bosons
• Methods
• Analytical Results (Mean Field, RG, Symmetries)
• Direct calculations (Density Matrix
  Renormalization Group)
• Experiments with ultra cold atoms

T.D. Kuhner, S.R. White, H. Monien PRB 61(18)
(2000)
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Master Thesis

• ultra-cold atoms in periodic lattice
• Interaction atoms light (intensity)
• Superposition of opposite laser beam ? periodic
  potential
• Tunneling hopping
• On site interaction

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Thank you!
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• God
• My wife, my family
• dr. Ehud Altman and Erez Berg, prof. Moti Heiblum