Supersymmetry in strongly correlated fermion models

Transcription

1 Supersymmetry in strongly correlated fermion models Dimitrios Galanakis () Stefanos Papanikolaou (2) Chris Henley (3) () Nanyang Technological University (Singapore) (2) Yale University (3) Cornell University

2 Hard-core spinless fermions on a lattice Nearest neighbor exclusion Square Lattice Hubbard model 4 N Spinless nn excluded fermions 2 N/2 2

3 Supersymmetric Lattice Hamiltonian Creation operator of hard-core nn excluded fermions where: d i c i P i,excl P i,excl ( c j c j) <j,i> If we define: Q i d i a natural form of Hamiltonian is: H {Q,Q} <ij> d i d j + i P i,excl [H, Q] H, Q Fendley and Schoutens 5 Q 2 (Q ) 2 Supersymmetry ψ> and Q ψ> are degenerate 3

4 Zero energy states re may be eigenstates: Q ψ > Q ψ > zero energy states allow for a very large degeneracy (even exponential) of groundstate. Triangular lattice super-frustration. This may be root to exotic phases. 4

5 Computational methods Lanczos diagonalization (via ARPACK) Periodic clusters of arbitrary shape (3-45 sites) Momentum resolution 5

6 Range of zero energy states Jonsson () conjectured that cross-cycles are zero energy states (Electr. J. Comb., 6(2):#R5, 29) For triangular lattice /5 > f > /7 Interesting physics arises at boundaries of this interval 6

7 Energy vs filling.7 45 Ground state energy per site appears like phase coexistence Filling 7

8 Entropy vs filling.25!"#$ N3...4!"#$%%%&$ N4...45!"&'%%%&(.2!"#.,/23 Entropy.5!"%$.!"%.5!"!$!.!"%.2!"%#.4!"%&.6!"%'.8!"%(.2!"#.22!"##.25!"#$ )*++*,- Filling 8

9 Clusters with a zero energy state at f>/5 (9,3)x(,4)33 (8,4)x(3,7)44 (7,4)x(2,6)34 (,)x(,4)43 (3,4)x(2,4)44 (,)x(,4)44 (2,2)x(2,4)44 (9,2)x(,4)34 (,2)x(,4)38 9

10 reference fermion ( origin); zero occupancy being ne origin. Black is max. occupancy, dark and light gray are 2nd 3rd highest occ.; I didn t tryand distingush lightest sites from figurei was sent Isotropic cluster (8,4)x(3,7), f9/44 Density - Density correlation 5 usual transverse hops Abacus order layer 6 layer 5 2 inter column hops from a pair to a new pair order destroying hop: pair new pair (after this hop)

11 8% which an approximate state containing only twosuggest low energy components: form for ground state containing only Triangular lattice, K, 3-4 sites two low energy components: (n) (2 column) (n ) Triangular lattice, K, 3-4 sites Ψ s,.9 a ψa Ψa (n) sa ψa(2 column) Ψ(n ), a,2 Ψ a Stripes in (2n,)x(,5) ladders, f/5 density-density where sa is For coefficient of Schmidt decomposi n2,3,4,5 and filling2n (n ) where sa is coefficient of Schmidt decomposition. It turns out that Ψ feature a nearly iden,2 correlation Unique zero energy state (n ) a,2 tion. It turns that feature a nearly idenψ titical entanglement which allows for,2this prostripes inout d-d correlation spectrum, cedure to be carried out entanglement iteratively for spectrum, remaining of allows for this protitical which.4 Factorizable wave function system, which cedure gives a to leading contribution form be carried out iteratively for remaining of.3 system, which gives a leading contribution form n n 2 (i) (i).2 (n) (i) (i) Φ + λ, Ψ Φ + λφ3 Φ3 n + n 2 2 (n) (i) (i) (i) (i). i i + λ Φ3 +, Ψ Φ Φ2 + λ Φ i i Filling where.35 i,..., n is index of a double column and Filling where double column and,..., n is index 2 of a i(i) te energy of KCM momen Φ2 S (i) lar lattice. Because of superfrustra S : ground state energy of KCM momen2 Φ(i) (i) triangular lattice is varies little Φ S Φ S 2 tor for triangular lattice. Because of superfrustra d size for a given filling. We observe (i) ground state of triangular lattice is varies little S for.4 < v <.22. Φ e cluster shape and size for a given filling. We observe (i) ergy ground states for.4 < v <.22. Φ S 2 (i) (i) ty-density corellation profile,nn () n (x), nsity-density corellation profile, () n (x), S S 3S S Φ3 Φ to unique zero energy state for 5 unique zero energy state for 5 (6,)x(,5) (i) boundary conditions for /5 andand profile, n () n (x), Φ S Figure 2:conditions density-density corellation S cdic boundary forfilling fillingofof /5 3 In this graphical representation that corresponds to unique zero energy state for 5 this graphical representation nearest nearest h site are 4 nearest sites toger with ladder with periodic boundary conditions for filling of /5 and te are 4 nearest sites toger with (i) where S is a symmetrization operator which performs (8,)x(,5) lower corellation left diagonal neighbors. numbers nsity profile, n ()graphical n (x), symmetrizes S operator Φ3 nearest along column where S is S asymmetrization which performs for Kdiagonal In this representation CM.neighbors. wer left numbers a circular perumutation of indices along v (, 5). coloring schemestate is such site zero is unique zero energy for that 5 indices along v (, 5). neighbors of each site are 4 nearest sites toger with a circular perumutation of endary coloring scheme is such that site zero is parameter λ ( λ / for is a variational d or sites are where S 6 5) conditions for colored filling ofaccoring /5 andto is a operator which perfo upper right and lower left diagonal numbers parameter / forsymmetrization is a variational or sites are colored accoring to neighbors. parameter. (x), such that darker indicates higher values. For 6λ (5λladder overlap of6 5) this wave graphical representation nearest circular perumutation are site coloring scheme is such thatwith site ground zero nearest neighbor exclusion, all with neighbors function statea6is about 88.9%., such thatlabels. darker indicates higher values. parameter. Foris 5 ladder overlap of of this indices wave along v ( 4 nearest sites toger parameter λ (88.9%. λ / for 6 5) is a variatio where S accoring is a symmetrization operator which performs pty. However rest of system forms colored white and or are colored to ft diagonal neighbors. numbers rest neighbor exclusion, all sites neighbors overlap of this approximate function with function with ground state iswave about.5

12 Conclusions Strong evidence for systematic violation of bound set by Johnson on triangular lattice, reporting zero energy and supersymmetric states for f>.5 We report novel patterns for high filling zero energy states f.5 and we propose idealized wavefunctions in order to understand novel quantum order. Glimpses of a phase transition at f~.5, possibly discontinuous 2