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[Pages:70]Quantum phase transitions
? ? cond-mat/0109419
Quantum Phase Transitions
Cambridge University Press
What is a quantum phase transition ?
Non-analyticity in ground state properties as a function of some control parameter g
E
E
g
True level crossing: Usually a first-order transition
g
Avoided level crossing which becomes sharp in the infinite
volume limit: second-order transition
Why study quantum phase transitions ?
T
Quantum-critical
gc
g
? Theory for a quantum system with strong correlations:
describe phases on either side of gc by expanding in deviation from the quantum critical point.
? Critical point is a novel state of matter without quasiparticle excitations
? Critical excitations control dynamics in the wide quantum-critical region at non-zero temperatures.
Important property of ground state at g=gc : temporal and spatial scale invariance;
characteristic energy scale at other values of g: ~ g - gc z
Outline
I. Quantum Ising Chain
II. Landau-Ginzburg-Wilson theory Mean field theory and the evolution of the excitation spectrum.
III. Superfluid-insulator transition Boson Hubbard model at integer filling.
IV. Tilting the Mott insulator Density wave order at an Ising transition.
V. Bosons at fractional filling Beyond the Landau-Ginzburg-Wilson paradigm.
I. Quantum Ising Chain
I. Quantum Ising Chain
Degrees of freedom: j = 1... N qubits, N "large"
,
j
j
( ) ( ) or = 1 + , = 1 -
j
2
j
j
j
2
j
j
Hamiltonian of decoupled qubits:
H0 = -Jg
x j
j
j
2Jg
j
Coupling between qubits:
H1 = -J
z z j j+1
j
( )( ) + +
j
j
j +1
j +1
Prefers neighboring qubits
are either or
j
j +1
j
j +1
Full Hamiltonian
(not entangled)
( ) H = H0 + H1 = -J
g
x j
+
zj
z j +1
j
leads to entangled states at g of order unity
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