Download - Fermions with long-range interactions
Fermions with long-range interactionsMARCIN SZYNISZEWSKI*,1,2, SUPERVISOR: NEIL DRUMMOND1
1Physics Department, Lancaster University, Lancaster, LA1 4YB, UK2NoWNano DTC, University of Manchester, Manchester, M13 9PL, UK
*E-mail for corresponding author: [email protected]
The Hamiltonian of the generalised t-V model on a 1D chain is [1]:
π» = βπ‘
π=1
πΏ
ππβ ππ+1 + h.c. +
π=1
πΏ
π=1
π
ππππβ ππππ+π
β ππ+π
πΏ β system size (periodic), π β max interaction range, ππ β potentials
INTRODUCTION1
At the critical density, if the potential is not decreasing uniformly, wecan have the following phases in the system [2]: Charge-density wave (CDW) phases β insulating Luttinger liquid, despite the insulating density Bond-order phase β phase with long-range ordering
At the atomic limit π‘ = 0 ,one can determine what the CDW phases are for specific systems (e.g. π = 4, π = 1/3):
PHASE DIAGRAM2
β’ Luttinger liquidβ’ Highly degenerate GSβ’ Interacting βhard rodsβ, size π
Away from critical density
β’ Mott insulatorβ’ Simple ground stateβ’ βRodsβ filling the lattice or
equally overlapping
Critical density
ππΆ =1
π;π = 1,β¦ , π
Firstly, we transform the Hamiltonian to itsspin-half equivalent. The potential for π = 2:
π1
π=1
πΏ
βπββπ+1β + π2
π=1
πΏ
βπββπ+2β ,
where ββ = β β . We can construct theHamiltonian using the states of theautomaton [4] on the right.
The full Hamiltonian for any interactionrange can be constructed using the followingautomaton:
The one-site Hamiltonianin MPO representation hasdimensions π + 4 Γ π + 4 .
HAMILTONIAN AS MATRIX PRODUCT OPERATOR4
Although analysis of the atomic limit provides us with a completepicture of expected CDW phases, these phases may not be presentin real life situations (π‘ β 0).
We have decided to usematrix product statesapproach[3] in order to see1. Emergence of the
non-CDW phases.2. How long-range
correlations will affectthe bond dimensionneeded.
3. If any CDW phasecan never emerge.
MOTIVATION3
PRELIMINARY RESULTS5
References:[1] G. GΓ³mez-Santos, Phys. Rev. Lett. 70, 3780 (1993).[2] P. Schmitteckert and R. Werner, Phys. Rev. B 69, 195115 (2004);T. Mishra et al., Phys. Rev. B 84, 115135 (2011).[3] D. Perez-Garcia et al., Quantum Inf. Comput. 7, 401 (2007);F. Verstraete et al., Adv. Phys. 57, 143 (2008).[4] U. SchollwΓΆck, βDMRG: Ground States, Time Evolution, and SpectralFunctionsβ from βEmergent Phenomena in Correlated Matterβ.
OUTLOOK6
00
7
7
7
π1 π2
π3
π4 = 1
0
Ground state unit cell Energyβββ 1
3π3
ββββββ 1
6π1
ββββββ 1
6π2 + π4
βββββββββ 1
9π1 + 2π4
βββββββββ 1
92π2 + π4
ββββββββββββ 1
122π2 + π3
βββββββββββββββββββββ 1
212π1 + 3π2
Check the existence of bond-order
phases
Completethe results: more points near the liquid-insulator
transition
Bond dimension
vs. interaction
range
Determine which
CDW-phases will not
appear for π‘ β 0
0 2 4 6 8 10
10
8
6
4
2
0
π1
π2
Curvature:
Liquid
Insulator
Slice at π3 = 6
CDWββββββ
CDWβββββββββ
CDWββββββ
Liquid
Liquid
Phases(βββββββββ)
and (βββββββββββββββββββββ)
are absent.
Atomic limit