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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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