Frustrated Antiferromagnetism Fluctuation induced first order versus deconfined quantum criticality

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Frustrated AntiferromagnetismFluctuation induced
first order versus deconfined quantum criticality
Europhys. Lett. 74, 896 (2006)

• ? Frank Krüger1 Stefan Scheidl2
• 1Instituut Lorentz for theoretical physics,
  Leiden
• 2Institut für theoretische Physik, Köln

Workshop on Quantum Criticality Leiden 2006
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Plan of talk

• Introduction
• J1-J2 Heisenberg AF on a square lattice
• Effective non-linear sigma models for frustrated
  quantum AFs
• Lattice symmetry breaking due to Berry phase
  effects ?VBS order
• Deconfined quantum criticality ? Direct 2nd
  order transition Néel/VBS ? Fractional
  excitations at the QCP
• What can we learn from numerics?
• Novel renormalization approach
• underlying lattice geometry
• frustration mechanism

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Introduction
Frustrated quantum AF on a square lattice
1) Stability of Neel order against quantum
fluctuations 1/S and frustration a ?2) Nature
of transitions out of Neel phase ?
S1/2
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CNLs-model for QAFs

• represent spins as unit vectors Ni (coherent
  states)

• Trotter Formula Action in Imaginary time t

• Coarse graining

• Effective long-wavelength theory

M. Berry
J1-J2 model
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Momentum-shell RG
(Polyakov 75 Chakravarty,Halperin,Nelson,89)

• Decompose into slow and fast fluctuating fields

• Sucessively eliminate modes of highest energy

2nd order transition

• preserves Lorentz invariance, no renormalization
  of c and a
• misses renormalization effects from small scales
• no instability towards 1st order

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Crucial role of Berry phases

• Smooth configurations of the Néel vector
  field n(r,t) admit skyrmions

Total skyrmion number
is conserved, .

• We come from a lattice!

Monopole tunneling event
Haldane
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Breaking of lattice symmetries due to Berry phase
effects
Berry phases of proliferating instantons break Z4
lattice rotation symmetry
(Read, Sachdev 89)
VBS order parameter
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VBS phases and Violation of LGW
(Sushkov et al. 01)

• J1-J2 model for S1/2

• Results of numerical simulations (quantum XY
  with ring exchange) joggle fundamental
  concepts of statistical physics direct 2nd
  order transition Neel / VBS

(Sandvik et al. 02)
Landau
Breakdown of the Landau-Ginzburg-Wilson paradigm
!!!
Ginzburg
Wilson
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LGW theory of multiple order parameters
Write down an effective action for the AF order
parameter and the VBS order parameter
by expanding in powers of , and
their spatial and temporal derivatives, while
preserving all symmetries of the microscopic
Hamiltonian.
No direct 2nd order transition without fine
tuning to a multicritical point
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Deconfined QC
Fractionalized spinor fields
Coupled to compact U(1) gauge field

• skyrmion number conserved

• Proliferation of monopoles

Monopole tunneling event

• Confinement due to instanton proliferation

• Confinement by Higgs mechanism

Deconfinement, fractionalized excitations
Two divergent length scales in paramagnet
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Motivation

• Careful numerical finite-size analysis of the
  Neel / VBS transition (Kuklov, Prokofev,
  Svistunov 05)

Evidence for extremely weak first-order
transition that in small systems can be confused
with continuous or high symmerty points.

• What is the way out??? Q What can we do
  better ?
  

A We have to go back to the lattice !

• No naive coarse graining
• Fully account for the underlying lattice
  geometry
• consistently treat fluctuations all over the
  magnetic BZ in the framework of a
  renormalization approach

Scenario of fluctuation induced 1st order
transition appears in a natural way
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Discrete spin-coherent state path integral

• Trotter formula ? imaginary time action
  (discretized in intervals Dt)

• Novel path-integral parametrization
• Expansion action in terms of stereographic
  coordinates a (sublattice A) and b (sublattice B)

Umklapp processes can be captured on an equal
footing with spin-wave interactions
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Diagonalization of bilinear action

• Dispersion

• Correlators

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Renormalization approach
Successively eliminate fraction of modes of
highest energy
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Néel phase

• RBZ become circular with decreasing radius e-l

• Asymptotically flow of CNLs model is recovered

• Order out of disorder

Stabilization of Néel order above a1/2
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Paramagnetic phase and instability towards first
order

• Paramagnetic phase

• At some scale Néel order is destroyed
  by quantum fluctuations

• Correlation length

• Fluctuation induced first order

• Unstable modes are not lifted

• No information about order in adjacent phase
  (discontinuous transition)

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Phase diagram
New phenomenon Fluctuation induced first order
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Conclusion discussion
Lattice model
Fluctuation induced 1st order
Coarse graining
Instability towards 1st order
Effective long-wavelength theory
Deconfined quantum criticality