FERMIONIC LADDERS IN MAGNETIC FIELDcoqusy06/SLIDES/narozhny.pdf · • produces circulating...
Transcript of FERMIONIC LADDERS IN MAGNETIC FIELDcoqusy06/SLIDES/narozhny.pdf · • produces circulating...
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FERMIONIC LADDERS IN MAGNETIC FIELD
BORIS NAROZHNY
SAM CARR, ALEXANDER NERSESYAN
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spinless fermions on a two-leg ladder
hamiltonian physical quantities
bond current
bond density
i=1
i=2n n+1n-1
t0 eiϕ
t⊥ V⊥
V7
t0 e-iϕ
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outline
comment on bosonization
n=1/4 : charge fractionalization
n=1/2 : field-induced phase transitions
physics beyond bosonization –persistent current
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bosonization in ladders
single-chain single-particle spectrum
ladder spectrum
ladder spectrum in the presence of the magnetic field
interaction terms
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bosonization approach to quarter-filled ladder
single-particle spectrum
single band partially occupied
effective low-energy hamiltonian
interaction parameters
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strong-coupling cartoon
repulsive interaction• strong V┴ - no rung doubly occupied• hopping delocalizes electrons on links• for V║ > 0 – avoid neighboring sites
attractive in-chain interaction• phase separation
role of the magnetic field• delocalizes electrons around plaquettes• produces circulating currents
the external field is uniform!
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bosonization approach to quarter-filled ladder
possible states with long-range order (K < 1/2)
• g2 > 0 - bond density wave
• g2 < 0 - staggered flux phase (orbital anti-ferromagnet)
charge quantization
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fractional quantum numbers in spin chains
anti-ferromagnetic Heisenberg model
doubly degenerate ground state
elementary excitation – spin flip (S=1)
spinons – S=1/2 excitations
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fractionalization in polyacetylene
hamiltonian
the Schrieffer counting argument• local neutrality : 1 σ-electron per H; 2 core, 3 σ, 1 π-electron per C• soliton: charge: +e, spin: 0 (since all electrons are paired)• remaining non-bonding π-orbital on central C: if singly occupied, the soliton
is neutral with spin ½, if doubly occupied, the soliton is spinless, charge -e
electron content• two core (1s) electrons per C• two electrons in a bonding σ-orbital
(sp2 hybrid) per • two π-electrons (out-of-plane 2p
orbital of C) per
Su, Schrieffer, Heeger (1979) Brazovskii (1978) ; Rice (1979)
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conclusions for n=1/4
we have considered electrons on the two-leg ladder at arbitrary values of the external field, inter-chain hopping and interaction strength
we have found a new ordered phase in the model – the orbital anti-ferromagnet – that exists only when the field is applied
this new ground state is doubly degenerate, so the elementary excitations carry charge ½
we showed that fractionally charged excitations that exist in the absence of the field are stable with respect to the external magnetic field
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bosonization approach to half-filled ladder
single-particle spectrum
both bands partially occupied
effective low-energy hamiltonian
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half-filled ladder – phase diagram
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half-filled ladder – ordered phases
bond current (OAF)
density (CDW)
relative density (Rel. CDW)
bond density (BDW)
bond density (Rel. BDW)
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half-filled ladder – phase boundaries
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bosonization approach to half-filled ladder
example 1: states with long-range order (K+ < 1, K- > 1)• g4 < 0 and g5 < 0 - charge density wave (CDW)
• g4 > 0 and g5 < 0 - staggered flux phase (OAF)
example 2: states without long-range order• K+ < 1 and K- < 1 - Mott insulator (only charge sector is gapped)
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conclusions for n=1/2
we have considered electrons on the two-leg ladder at arbitrary values of the external field, inter-chain hopping and interaction strength
at half filling the model exhibits several ordered phases as well as phases without long-range order
we have found field-induced (sometimes re-entrant) quantum phase transitions between phases with different types of long-range order and between ordered and gapless phases
gapless phases are characterized by the algebraic decay of dominant correlations
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persistent current
current operator
ground state value – relative current
small flux, small inter-chain tunneling
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persistent current
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conclusions for persistent current
persistent current is an example of a non-universal quantity contributed to by all electrons – not only those in the vicinity of the Fermi points
not an infra-red quantity – non zero even in the insulating phase.
can not be addressed in terms of any Lorentz-invariant effective low-energy field theory
gapless phases are characterized by the algebraic decay of dominant correlations
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SUMMARY
fermionic ladders exhibit interesting physics charge fractionalization a quarter-fillingfield-induced quantum phase transitions at half-filling
there exist physical quantities that cannot be described by means of low energy effective theory
such as persistent current
possible generalizations: multiple-leg ladders, spinful fermions, …