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]:
𝐻 = −𝑡
𝑖=1
𝐿
𝜙𝑖†𝜙𝑖+1 + h.c. +
𝑖=1
𝐿
𝑚=1
𝑝
𝑈𝑚𝜙𝑖†𝜙𝑖𝜙𝑖+𝑚
† 𝜙𝑖+𝑚
𝐿 – system size (periodic), 𝑝 – max interaction range, 𝑈𝑚 – 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 – insulating Luttinger liquid, despite the insulating density Bond-order phase – phase with long-range ordering
At the atomic limit 𝑡 = 0 ,one can determine what the CDW phases are for specific systems (e.g. 𝑝 = 4, 𝑄 = 1/3):
PHASE DIAGRAM2
• Luttinger liquid• Highly degenerate GS• Interacting “hard rods”, size 𝑝
Away from critical density
• Mott insulator• Simple ground state• “Rods” filling the lattice or
equally overlapping
Critical density
𝑄𝐶 =1
𝑚;𝑚 = 1,… , 𝑝
Firstly, we transform the Hamiltonian to itsspin-half equivalent. The potential for 𝑝 = 2:
𝑈1
𝑖=1
𝐿
ℙ𝑖↑ℙ𝑖+1↑ + 𝑈2
𝑖=1
𝐿
ℙ𝑖↑ℙ𝑖+2↑ ,
where ℙ↑ = ↑ ↑ . 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 𝑝 + 4 × 𝑝 + 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 (𝑡 ≠ 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. Gómez-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. Schollwöck, “DMRG: Ground States, Time Evolution, and SpectralFunctions” from “Emergent Phenomena in Correlated Matter”.
OUTLOOK6
00
7
7
7
𝑈1 𝑈2
𝑈3
𝑈4 = 1
0
Ground state unit cell Energy●○○ 1
3𝑈3
●●○○○○ 1
6𝑈1
●○●○○○ 1
6𝑈2 + 𝑈4
●●○○○●○○○ 1
9𝑈1 + 2𝑈4
●○●○●○○○○ 1
92𝑈2 + 𝑈4
●○●○○●○●○○○○ 1
122𝑈2 + 𝑈3
●●●○○○○●○●○○○○●○●○○○○ 1
212𝑈1 + 3𝑈2
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 𝑡 ≠ 0
0 2 4 6 8 10
10
8
6
4
2
0
𝑈1
𝑈2
Curvature:
Liquid
Insulator
Slice at 𝑈3 = 6
CDW●●○○○○
CDW●●○○○●○○○
CDW●○●○○○
Liquid
Liquid
Phases(●○●○●○○○○)
and (●●●○○○○●○●○○○○●○●○○○○)
are absent.
Atomic limit