We study a model of strongly correlated fermions in one dimension with
extended N=2 supersymmetry. The model is related to the spin $S=1/2$ XXZ
Heisenberg chain at anisotropy $\Delta=-1/2$ with a real magnetic field on the
boundary. We exploit the combinatorial properties of the ground state to
determine its exact wave function on finite lattices with up to 30 sites. We
compute several correlation functions of the fermionic and spin fields. We
discuss the continuum limit by constructing lattice observables with well
defined finite size scaling behavior. For the fermionic model with periodic
boundary conditions we give the emptiness formation probability in closed form.Comment: 4 pages, 4 eps figure

B. A. LaMacchia

G. F. De Angelis

J. de Gier

K. Efetov

M. Beccaria

W. J. Le Veque

English

arXiv

Crossref

Exact Ground State and Finite-Size Scaling in a Supersymmetric Lattice Model

We study a one-dimensional lattice models of strongly correlated fermions with N=2 extended supersymmetry. By exploiting the combinatorial properties of the ground state, we determine its exact wave function on a finite lattice up to 30 sites

BECCARIA, Matteo

DE ANGELIS, Gian Fabrizio

Archivio Istituzionale della Ricerca- Università del Salento

PRL 94, 100401 (2005) P H Y S I C A L R E V I E W L E T T E R Sweek ending18 MARCH 2005Exact Ground State and Finite-Size Scaling in a Supersymmetric Lattice ModelM. Beccaria and G. F. De AngelisDipartimento di Fisica and INFN, Università di Lecce, Via Arnesano, 73100 Lecce, Italy(Received 30 July 2004; published 14 March 2005)0031-9007=We study a model of strongly correlated fermions in one dimension with extended N  2 supersym-metry. The model is related to the spin S  1=2 XXZ Heisenberg chain at anisotropy   1=2 with areal magnetic field on the boundary. We exploit the combinatorial properties of the ground state todetermine its exact wave function on finite lattices with up to 30 sites. We compute several correlationfunctions of the fermionic and spin fields. We discuss the continuum limit by constructing latticeobservables with well defined finite-size scaling behavior. For the fermionic model with periodicboundary conditions we give the emptiness formation probability in closed form.DOI: 10.1103/PhysRevLett.94.100401 PACS numbers: 05.50.+q, 05.30.–d, 11.30.PbSupersymmetry is well motivated in high energy physicswhere it offers a partial solution to fine-tuning problemsand improves gauge coupling unification. It also appears incondensed matter models although at a less fundamentallevel. Examples are disordered systems [1] and models ofstrongly correlated electrons like extended Hubbard [2] ort-J models [3] where supersymmetry relates fermionic andbosonic composite operators.A typical consequence of unbroken supersymmetry isthe prediction of the ground state energy. This is notsufficient to compute the ground state wave function, therelevant quantity for the calculation of vacuum expectationvalues. It is quite natural to ask whether supersymmetryand the knowledge of the ground state energy are useful forthis purpose.In this Letter, we analyze the problem in a recentlyproposed one dimensional model of itinerant fermions[4] with two supercharges obeying with the Hamiltonianan extended N  2 supersymmetry algebra. The knowl-edge of the ground state on large finite lattices is importantto study its continuum limit where the model is expected todescribe a minimal superconformal series.We are interested in boundary effects and thus considermainly free boundary conditions; see Ref. [4] for theperiodic case. The model can be mapped to the integrableopen XXZ Heisenberg spin 1=2 chain with anisotropy  1=2 and a suitable real surface magnetic field (seeRef. [5], Sec. 2.1). In principle, Bethe ansatz techniquescould be applied. The supersymmetry inherited from thefermionic model should allow us to compute the Baxterfunction whose zeros give the Bethe quantum numbers [6].However, the procedure is rather involved with openboundary conditions as discussed in Ref. [7].Here, we pursue a different approach starting with thefollowing remarks. The XXZ chain at   1=2 is inte-grable for a large class of boundary conditions. In somespecific cases [e.g., twisted or Uqsl2 symmetric ones]several remarkable conjectures have been claimed aboutthe combinatorial properties of the ground state wave05=94(10)=100401(4)$23.00 10040function [8,9]. They arise from the relation between theXXZ chain and Temperley-Lieb loop models [10].In this Letter, we first show that similar features arepresent in the fermionic model and in the related XXZchain with surface magnetic field. We then explain hownumber theoretical methods can be used to obtain exactexpressions for the ground state wave function of thefermionic model on long chains. Finally, we analyze insome detail the physical properties and finite-size scaling(FSS) behavior of the ground state. In particular, we dis-cuss the continuum limit and propose a way to extractscaling fields from the fermionic model and the associatedXXZ Heisenberg chain.The model [4] is defined on a one dimensional latticewith L sites and free boundary conditions. Let ci; cyi bespinless fermionic creation annihilation operators withalgebra fci; cyj g  ij, fci; cjg  fcyi ; cyj g  0. We denoteby Ni the set of nearest neighbors of site i. The projectorover states with a hard-core condition forbidding occu-pancy around site i is P i Qj2Ni1 nj, with nj cyj cj. Let Q	 be the supersymmetry charge Q	 Picyi P i and Q  Q	y. The operators Q are nilpotent.The HamiltonianH  fQ	; Qg XiXj2NiP icyi cjP j 	XiP i (1)is by construction Q symmetric. We restrict ourselves onthe subspace H L;F of states with F fermions and noadjacent occupied sites which is an invariant subspace ofthe full Fock space under the action of Q. We work in thebasis of simultaneous eigenstates of the number operatorsni. The structure of eigenstates of H follow from super-symmetry. The energy is non-negative and all energy ei-genstates with E> 0 are doublets connected by the actionof Q. The zero energy states are singlets annihilated byQ. They are supersymmetric ground states. A cohomo-logical analysis [4] shows that there is a unique zero energystate for Lmod3  0; 2 and none for Lmod3  1. In the1-1  2005 The American Physical Society0 5 10 15 20 25 30k-0.3-0.2-0.10.00.10.20.3< n k > - 1/3k mod 3 = 1k mod 3 = 2k mod 3 = 0FIG. 1. Z3 structure of the expectation hnki. The dashed linesconnect branches with the same value of kmod3.PRL 94, 100401 (2005) P H Y S I C A L R E V I E W L E T T E R Sweek ending18 MARCH 2005following we consider the case L  3n. The unique groundstate has then fermion number F  n. The dimension ofH L;L=3 increases rapidly as L ! 1. It should be clear thatthe determination of the unique ground state j0i is non-trivial on large lattices.At low L, the ground state can easily be obtained.Inspection of the explicit wave functions reveals a remark-able fact. Indeed, the ground state can always be written inthe very special formj0i j101010|{z}2F terms000|{z}F termsi	Xsxsjsi; (2)where xs 2 Z and s runs over all states (diagonal in theoccupation number basis) with the exception of the singlestate explicitly written. Lattice parity halves the number ofindependent coefficients. We used it as a consistencycheck. The integrality property of fxsg is definitely a non-trivial ansatz on the ground state wave function, and itwould be elusive in a Bethe ansatz approach. Similarresults already appeared in the literature for the XXZHeisenberg model at   1=2 [8,9] that is indeed closelyrelated to the present fermionic model.Let us assume the integrality property (2) as a workinghypothesis. The equation Hj0i  0 reduces to a linearproblem of the special kindAx  b; A 2 Zdd; b 2 Z1d; (3)where x is the vector of unknown coefficients xs and d dimH L;F  1. We know that this problem admits an in-teger solution x 2 Z1d, and it seems reasonable to be ableto find it exactly with modest effort. This is possible due toa well-known technique in cryptology. The problem can besolved by working in the finite field Zp of integers moduloa prime p. To solve Eq. (3), we choose a large prime p andfirst determine a solution modulo p by the Lanczos algo-rithm over finite fields [11]. We build the sequencesfbig0id and fcig1id where the initial values areb0  b; c1  Ab; b1  c1 c21b  c1b; (4)and the sequences are generated by iteratingci	1  Abi; (5)bi	1  ci	1 c2i	1bi  ci	1bi ci  ci	1ci  bi1bi1: (6)The solution is finally obtained asx Xd1i0bi  bbi  ci	1bi: (7)In Eqs. (4)–(7), all arithmetic operations, in particular,divisions, are done in Zp. The above algorithm exploitsthe sparsity of A and is not memory expensive since itrequires a storage Od. After a certain number of solution10040modulo (large) primes fpig have been found, they arecombined together to give the exact solution. This isdone by applying the Chinese remainder theorem [12]. Itis at this point that the assumed integer expansion (2) canbe checked. If it is actually true, the above process termi-nates after a finite number of steps. By the above proce-dure, we have determined the exact ground state forL  3n  30 ( dimH 30;10  352 716). The complete ex-pressions (i.e., the sequences fxsg) are available on request.We now illustrate the main physical properties of theresults. For any observable O we denote hOi h0jOj0i. Several remarkable combinatorial featuresappear immediately leading to conjectures in the spiritof Ref. [8]. We find that the dominant state in the groundstate is always ji  j010010010   i with coefficientmaxjxsj  xji  N82L=3	 2, where N82n Qn1k13k	 12k!6k!=4k!4k	 1! is the numberof cyclically symmetric transpose complement plane par-titions [9]. Also, the squared norm in our normalization is1	Psx2s  N82L=3	 2AV2L=3	 3, where AV2n	1  2nQnk16k 2!2k 1!=4k 1!4k 2! isthe number of vertically symmetric alternating sign matri-ces We checked these expressions for all L  30. Similarconjectures can be claimed on correlation functions, as in arecent work [13]. A simple example is the expectation ofthe potential energy hPiP ii. The operator 1=LPiP i isdiagonal on states jni and is minimum on ji, where itattains the value F=L  1=3. Our data are consistent withthe simple formula1L	XiP i135L	 214L	 15: (8)The asymptotic value is 5=12, slightly larger than 1=3, dueto the subdominant states jni  ji appearing in j0i.Another example is the fermion density on the boundaryhn1i. We find1-2PRL 94, 100401 (2005) P H Y S I C A L R E V I E W L E T T E R Sweek ending18 MARCH 2005hn1i L10L	 3324L	 94L	 15: (9)The asymptotic value 5=16 is smaller than 1=3, a plausiblefact since the dominant state ji has no fermions on thelattice boundary. We now examine the FSS behavior ofsimple correlation functions. In Fig. 1 we show the expec-tation value hnki  1=3 on the L  30 lattice. There is aclear Z3 substructure similar to that observed in the phasediagram of bosonic models with hard-core repulsion [14].The average number of fermions in the three branches canbe defined as Fk Pi;imod3khnii. We have F0  F1 andF0 	 F1 	 F2  F. Asymptotically, at large F  L=3, theupper branch has the asymptotic value F2 12 F	 1 	OF1. The three sublattices containing sites k with fixedkmod3 appear to reconstruct smooth curves for hnki as Lincreases. The Z3 structure can be exploited to constructscaling fields in the continuum limit.Let us define the effective length ~L  L=3	 1 and setk  L 1=2. Our data for hnki suggest to test thefollowing FSS laws:hnki  1=3  f	k k	= ~L ~L!; kmod3  2;hnk 	 nk	2i  2=3  fk k= ~L ~L!0 ;kmod3  1; (10)where ! and !0 are unknown exponents. The best collapseof data at different L is obtained with !  !0  0:332and is shown in Fig. 2. Notice that for kmod3  1, theseparate hnki and hnk	2i form a parity doublet. They shouldbe related in the continuum limit to the left and rightmoving parts of a single field. The value of !; !0 is closeto 1=3. In principle, this information is useful in the effortof identifying the proposed scaling fields with the operatorcontent of candidate superconformal field theories describ-ing the continuum limit of the model. The next simpleobservable built with local fermionic fields is the density-density (connected) correlation function Gi;j -2.0 -1.0 0.0 1.0 2.0(k-k+)/L~ k mod 3 = 20.400.450.500.550.600.65(< nk > - 1/3)   L~1/3L = 12L = 15L = 18L = 21L = 24L = 27L = 30-2.0 -1.0 0.0 1.0 2.0(k-k-)/L~  k mod 3 = 1-0.60-0.55-0.50-0.45-0.40(< nk + nk+2 >  - 2/3)  L~1/3FIG. 2. FSS of the occupation number hnki.10040hninji  hniihnji. On the left side of Fig. 3 we show theboundary correlation function G1;k on the L  30 lattice.Again the Z3 structure is evident. On the right side, we fitG1;k on lattices of various sizes with the simple form a0 	a1=k with good agreement. The term a0 is a small sizeeffect decreasing with L. A detailed scaling analysis shallbe reported elsewhere.The fermionic model can be mapped quasilocally to theXXZ spin chain at anisotropy   1=2 with a specificreal surface magnetic field [5]. The structure of Betheequations in the two models is closely related, but thefermionic and spin correlation functions can be quite dif-ferent. Following Ref. [5], we map a fermionic configura-tion fng to a well defined spin one f$g with the dichotomicvariable $k  1. The map is such that the spin chain haslength 2L=3	 1. The expectation value of the local spin isshown on the left side of Fig. 4.The structure is now Z3 ! Z2 and h$ki can be split intotwo independent channels with kmod2  0; 1.Equations (8) and (9) give the boundary spin h$1i 2hn1i  1 and the numbers of pairs of spins up or down,hN""i LL 334L	 9; hN##i LL	 634L	 15: (11)Again, it is possible to separate the two components inorder to build well defined scaling fields. We testh$ki  g	k ~L= ~L ~L&; kmod2  0; (12)h$ki  gk ~L= ~L ~L&0 ; kmod2  1: (13)The best values of the exponents are &  0:661, &0 0:801. The accuracy of the corresponding FSS laws isshown on the right side of Fig. 4. Notice that &;&0 are quiteclose to the simple rationals 2=3 and 4=5. This suggestsagain a possible simple identification of these scaling fieldswith superconformal operators.0 5 10 15 20 25 30k-0.2-0.10.00.10.20.3G1, kk mod 3 = 1k mod 3 = 2k mod 3 = 00 5 10 15 20 25 30k10-310-210-1G1, k  with  k mod L = 1L = 12L = 15L = 18L = 21L = 24L = 27L = 30a0 + a1/kFIG. 3. Boundary and bulk pair correlation function at L  30.Algebraic decay of the kmod3  1 component of the boundarycorrelation.1-30 5 10 15 20k-0.4-0.20.00.2< σk >k mod 2 = 1k mod 2 = 0-1.0 -0.5 0.0 0.5 1.0(k-L~)/L~  k mod 2 = 00.40.60.81.01.2<σ k> L~2/3L = 12L = 15L = 18L = 21L = 24L = 27L = 30-1.0 -0.5 0.0 0.5 1.0(k-L~)/L~  k mod 2 = 1-3.5-3.0-2.5-2.0-1.5-1.0-0.5<σ k> L~4/5FIG. 4. (left) Z2 structure; (right) FSS of h$ki.PRL 94, 100401 (2005) P H Y S I C A L R E V I E W L E T T E R Sweek ending18 MARCH 2005We stress at this point that the same methods can beapplied to any model with integrality properties likeEq. (2). In particular, a suitable version of Eq. (2) holdsin the model (1) with periodic boundary conditions. Themap to a model in the XXZ class is described in this case inSec. 3.2 of the second paper in Ref. [4]. To give an exampleof the efficacy of our number theoretical techniques in thiscontext, we have computed the unique supersymmetricground state for the model with periodic boundary con-ditions and L  3F	 1, up to L  28. A relevant non-trivial observable is the emptyness formation probabilityEk  hQki11 nii. The first values are fixed from trans-lation invariance and the constraint nini	1  0 which de-fines H L;F,E1 2F	 13F	 1; E2 F	 13F	 1: (14)For k > 2 we checked the validity of the relationEk  Ek1k 2!3k 5!2k 3!2k 4!Yk1!3kF	 !2F	 !: (15)We guessed the very specific form of Eq. (15) from similarconjectured relations proposed for the finite XXZ modelwith twisted boundary conditions [8]. The thermodynam-ical limit F ! 1 is"k  limF!1Ek 124Ak 12k3k	1; (16)where An is the number of alternating sign matricesAn Qn1k03k	 1!=n	 k! [9]. From Stirling’s ex-pansion, the asymptotic behavior of this expression is"k	1  2=3c3p=23k2k5=36; (17)where the constant c takes the form [8]10040c  expZ 105et36sinh5t=12 sinht=12sinh2t=2dtt: (18)To conclude, we have shown that the knowledge of theground state j0i on moderately large finite lattices allowsa precise FSS analysis of correlation functions guiding thedetailed identification of the continuum limit of the model(1). The exact j0i is also an effective heuristic tool toderive closed expressions for correlation functions.We acknowledge conversations with P. Fendley,J. De Boer, and V. Rittenberg.1-4[1] K. Efetov, Supersymmetry in Disorder and Chaos(Cambridge University Press, Cambridge, 1999).[2] F. H. L. Essler, V. E. Korepin, and K. Schoutens, Phys.Rev. Lett. 68, 2960 (1992); 70, 73 (1993); A. J. Bracken,M. D. Gould, J. R. Links, and Y-Zh. Zhang, Phys. Rev.Lett. 74, 2768 (1995); K. E. Hibberd, M. D. Gould, andJ. R. Links, Phys. Rev. B 54, 8430 (1996).[3] B. Sutherland, Phys. Rev. B 12, 3795 (1975); P.Schlottmann, Phys. Rev. B 36, 5177 (1987); P.-A. Baresand G. Blatter, Phys. Rev. Lett. 64, 2567 (1990); N.Kawakami and S.-K. Yang, Phys. Rev. Lett. 65, 2309(1990).[4] P. Fendley, K. Schoutens, and J. de Boer, Phys. Rev. Lett.90, 120402 (2003); P. Fendley, K. Schoutens, and B.Nienhuis, J. Phys. A 36, 12 399 (2003).[5] X. Yang and P. Fendley, cond-mat/0404682.[6] V. Fridkin, Yu. Stroganov, and D. Zagier, J. Phys. A 33,L121 (2000); Yu. G. Stroganov, J. Phys. A 34, L179(2001).[7] J. de Gier and P. Pyatov, J. Stat. Mech. Th. Exp. 03 (2004)002.[8] A. V. Razumov and Yu. G. Stroganov, J. Phys. A 34, 3185(2001); 34, 5335 (2001); J. de Gier, M. T. Batchelor, B.Nienhuis, and S. Mitra, J. Math. Phys. (N.Y.) 43, 4135(2002); J. de Gier, B. Nienhuis, P. A. Pearce, and V.Rittenberg, Phys. Rev. E 67, 016101 (2003); J. Stat.Phys. 114, 1 (2004).[9] M. T. Batchelor, J. de Gier, and B. Nienhuis, J. Phys. A 34,L265 (2001).[10] P. A. Pearce, V. Rittenberg, J. de Gier, and B. Nienhuis, J.Phys. A 35, L661 (2002).[11] B. A. LaMacchia and A. M. Odlyzko, in Advances inCryptology: Proceedings of Crypto ’90, edited by A.Menezes and S. Vanstone, Lect. Notes Comput. Sci.Vol. 537 (Springer-Verlag, New York, 1991), pp. 109–133.[12] W. J. Le Veque, Topics in Number Theory (Dover, NewYork, 2002), Vol. 1, 2nd printing.[13] S. Mitra, B. Nienhuis, J. de Gier, and M. T. Batchelor,cond-mat/0401245.[14] P. Fendley, K. Sengupta, and Subir Sachdev, Phys. Rev. B69, 075106 (2004).

Exact ground state and finite size scaling in a supersymmetric lattice model

Exact Ground State and Finite Size Scaling in a Supersymmetric Lattice
  Model

Exact Ground State and Finite Size Scaling in a Supersymmetric Lattice Model

Abstract

Similar works

Full text

Available Versions

Crossref

Archivio Istituzionale della Ricerca- Università del Salento