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: mszynisz@gmail.com

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