当前位置: 首页 > >

Exact valence bond entanglement entropy and probability distribution in the XXX spin chain

发布时间:

Exact valence bond entanglement entropy and probability distribution in the XXX spin chain and the Potts model
J.L. Jacobsen1,2 , H. Saleur2,3
LPTMS, Universit? Paris-Sud, B?timent 100, 91405 Orsay, France e a 2 Service de Physique Th?orique, CEN Saclay, 91191 Gif Sur Yvette, France and e 3 Department of Physics, University of Southern California, Los Angeles, CA 90089-0484 (Dated: February 2, 2008) By relating the ground state of Temperley-Lieb hamiltonians to partition functions of 2D statistical mechanics systems on a half plane, and using a boundary Coulomb gas formalism, we obtain in closed form the valence bond entanglement entropy as well as the valence bond probability distribution in these ground states. We ?nd in particular that for the XXX spin chain, the number Nc of valence bonds connecting a subsystem of size L to the outside goes, in the thermodynamic 4 limit, as Nc (?) = π2 ln L, disproving a recent conjecture that this should be related with the von 1 Neumann entropy, and thus equal to 3 ln 2 ln L. Our results generalize to the Q-state Potts model.
PACS numbers: 03.67.-a 05.50+q
1

arXiv:0711.3391v1 [cond-mat.stat-mech] 21 Nov 2007

Introduction. Entanglement is a central concept in quantum information processing, as well as in the study of quantum phase transitions. One of the widely used entanglement measures is the von Neumann entanglement entropy SvN , which quanti?es entanglement of a pure quantum state in a bipartite system. To de?ne SvN precisely, let ρ = |Ψ Ψ| be the density matrix of the system, where |Ψ is a pure quantum state. Given a complete set X of commuting observables, let X = A ∪ B be a bipartition thereof. Then SvN is de?ned as SvN (A) = ?TrA ρA ln ρA (1)

where ρA = TrB ρ is the reduced density matrix with respect to A. One readily establishes that SvN (A) = SvN (B). In most applications, the subset A corresponds to a commuting set of observables characterizing only a part of the whole system, and so one may think of A denoting simply that subsystem. Critical ground states in 1D are known to have entanglement entropy that diverges logarithmically in the subsystem size with a universal coe?cient proportional to the central charge c of the associated conformal ?eld theory. Let L = |A| be the size of the subsystem, and N the size of the whole system, both measured in units of the lattice spacing, with 1 ? L ? N (as we shall invariably assume in what follows). Then [1, 2]
a SvN (A) = (c/3) ln L .

entanglement entropy. In the case of systems admitting in?nite randomness ?xed points, in one [5, 6, 7] as well as in higher [8, 9] dimensions, the ground state |? can be represented as a single valence bond state, and SvN coincides with the number of singlets that cross the boundary of A times the logarithm of the number of states per site. In an interesting recent paper [10] (see also [11] for a related, independent work) it was suggested that, even when |? is not a single valence bond state but a superposition of such states, the average number of singlets Nc (?) crossing the boundary of the subsystem (multiplied e.g. by ln 2 for spins 1/2) could still be used as a measure of the entanglement entropy with all the required qualitative properties. Moreover, it was observed numerically in [10] that, up to statistical errors, this valence bond entanglement entropy SVB had the same asymptotic behavior as SvN for the XXX quantum spin chain, namely (2) with c = 1. In other words, the observation of [10] was that Nc (?) ≈
a

1 ln L ? 0.481 ln L . 3 log 2

(3)

(2)

a where by = we denote asymptotic behaviour. Away from the critical point, the entanglement entropy saturates to a ?nite value, which is related to the correlation length. The von Neumann entanglement entropy is not an easy quantity to calculate, analytically or numerically; nor is it easy to grasp intuitively. The plethora of algebraic and geometric reformulations of quantum hamiltonians in one and higher dimensions [3, 4] suggests that there should be a more convenient, geometric way to de?ne an

Apart from this, the valence bond basis has been actively studied recently, in particular from rigorous [12, 13] and probabilistic [14] points of view. Studying entanglement entropy from a geometrical point of view seems particularly appealing in view of the quantum dimer model of Rokhsar and Kivelson [15] and its many recent generalizations. Indeed, Henley [16] has shown that for any classical statistical mechanics model equipped with a discrete state space and a dynamics satisfying detailed balance, there is a corresponding quantum Hamiltonian whose ground state |? is precisely the classical partition function. Of particular interest are then statistical mechanics models in which the microscopic degrees of freedom directly de?ne the valence bonds. This is the case for a certain class of lattice models of loops, to be studied below.

2 √ Uq sl(2) singlet ( Q = q + q ?1 ) so a valence bond can be drawn between the two corresponding sites. (Note that the generator ei is nothing but the operator that projects sites i and i + 1 onto the singlet.) For the dots, the state must be chosen such that the application of the projection operator onto the Uq sl(2) singlet for any two dots that are adjacent (when parentheses are ignored) annihilates the state. Thus those sites are “non-contractible”. With these de?nitions, it is clear that the TL algebra does not mix states with di?erent numbers of non-contractible sites. For Q generic, the set of basis states with ?xed number 2j of such sites provides an irreducible representation of TL, of well-known dimension dj = N N/2 + j ? N N/2 + j + 1 (6)

FIG. 1: Loop representation of the periodic TL algebra (with N = 8) on the in?nite half cylinder C? . The basis state corresponding to the upper rim is ?(())()?.

We show in this Letter that the probability distribution of the number of singlets crossing the boundary can be exactly determined for the XXX spin chain as well as for the related Q-state Potts model hamiltonians. We ?nd that (3) is not quite correct: the exact leading asymptotic a 4 behaviour is in fact Nc (?) = π2 ln L ? 0.405 ln L. All other cumulants have similar closed form expressions. Entanglement and the TL algebra. The 2D classical Q-state Potts model can be de?ned for Q non integer through an algebraic reformulation where Q enters only as a parameter. For this, recall that the transfer matrix in the anisotropic limit gives rise to the hamiltonian [22] H =?
N ?1 i=1

ei

(4)

where N/2 + j must be an integer. This dimension coincides with the number of representations of spin j appearing in the decomposition of the product of N spins 1/2. This is no accident: it is well-known [17] that the uncrossed diagrams are linearly independent and form a basis of the spin j sector in the sl(2) case; the results extends trivially to the Uq sl(2) case with q generic. √ When Q ≥ 0 the ground state is found in the sector with j = 0 for N even (and j = 1/2 for N odd). Note that the valence-bond basis is not orthonormal. The simplest way to proceed is thus not to calculate matrix elements of the hamiltonian H in this basis wi |H|wj but rather to de?ne a non-symmetric matrix hij by expressing the action of H on any state as a linear combination of states H|wi =
j

Here the ei are elements of an associative unital algebra called the Temperley-Lieb (TL) algebra, de?ned by e2 = i Qei (5)

hij |wj

(7)

[ei , ej ] = 0 for |i ? j| ≥ 2 ei ei+1 ei = ei

The ground state of H depends on the representation of the algebra. For our purpose, it is natural to use the loop model representation, where the generators act on the following non-orthogonal but linearly independent basis states. Each basis state corresponds to a pattern of N parentheses and dots, such as () ? (())?. The parentheses must obey the typographical rules for nesting, and the dots must not be inside any of the parentheses. These rules imply that the () pairs consist of one even and one odd site, and that dots are alternately on even and odd sites. We start by convention with an odd site on the left. Note that we have used an open chain for convenience, but that a periodic chain can be considered as well. This requires the introduction of an additional generator eN coupling the N th and ?rst site, and in the graphical representation, parentheses can now be paired cyclically so ) ? (()) ? ( is now a valid pattern. This is shown in Fig. 1. One can interpret these states in terms of spin 1/2 by associating with each pair of nearest parentheses () a

The matrix hij is unique due to the linear independence of the states. The eigenvalues and right eigenvectors of h give those of H. Since all entries hij are (strictly) positive, the PerronFrobenius theorem implies that the ground state |? expands on the basis words with positive coe?cients [23] |? = λw |w , λw > 0 (8)

We de?ne the number of valence bonds Nc connecting the subsystem to the outside as the number of unpaired parentheses in the subsystem. We are here interested in its mean value Nc (?) =
w

λw Nc (w) w λw

(9)

and more generally in the probability distribution p(Nc ) =
w:Nc (w)=Nc w

λw

λw

(10)

Below we establish the leading asymptotic behaviour of Nc (and the higher cumulants) in the scaling limit 1 ?

3 L ? N . Note that the TL formulation shows relationship √ π between the Potts hamiltonian when Q = 2 cos k+2 , with k integer, and the interacting anyons (coming in k + 1 species) hamiltonian in [18]. The valence bond entanglement entropy can be de?ned for these models as well, and, in the sector of vanishing topological charge, coincides with the one we are studying. Mapping onto a boundary problem. The wave function in the ground state of a certain hamiltonian with periodic (free) boundary conditions [24] can be obtained as the path integral of the equivalent Euclidian theory on a in?nite half cylinder (annulus), denoted C? (or A? ). To translate this in statistical mechanics terms, note that if we consider the square lattice with axial (diagonal) direction of propagation [cf. Fig. 1], the hamiltonian belongs to a family of commuting transfer matrices describing the Q-state Potts model with various degrees of anisotropy. The ground state of all these transfer matrices is given by |? . Let us chose for instance the particular case where the Potts √ model is isotropic, with coupling constant eK = 1 + Q. Now the ground state |? can be obtained by applying a large number of times the transfer matrix on an arbitrary initial state (corresponding to boundary conditions at the far end of C? or A? .) Clearly, by the mere de?nition of the transfer matrix, this means that the coe?cients of the ground state |? on the words |w are (up to a common proportionality factor) equal to the partition function of the 2D statistical system on C? or A? with boundary conditions speci?ed by |w . We must now study such partition functions. We move immediately to the limit N → ∞. We then have a system in the half plane, which, in the geometrical description, √ corresponds to a gas of loops with fugacity Q in the bulk, with half loops ending up with open extremities on the boundary. To go to the continuum limit it is convenient to transform this loop model into a solid-on-solid √ model [19]. For this parametrize Q = 2 cos πe0 with 0 ≤ e0 < 1 [25] Give to all the loops an orientation, and introduce complex weights exp ± iπe0 for the left 4 and right turns. Since on the square lattice the number of left (nL ) minus the number of right (nR ) turns √ equals ±4, this gives closed loops the correct weight Q. Meanwhile, loops ending on the boundary will get, with √ this construction, the weight Qb = 2 cos πe0 since for 2 them nL ? nR = ±2 [20]. Although no such boundary weight appeared in the initial lattice model and partition function, we note that for the fully packed loop model we are interested in it does not matter: the number of open loops touching the boundary is just N/2 ? j, a constant. Introducing this boundary loop weight allows complete mapping to the SOS model (or free six-vertex model). Now it is known that in the continuum limit, the dynamics of the SOS height variables turns into the one of a free bosonic ?eld [19]. In a renormalization scheme where loops carry a constant height step ?Φ = ±π, the propagator of the ?eld evaluated at two points x, x′ on the boundary reads, in the in?nite size limit [20] Φ(x)Φ(x′ )
N

1 2 = ? ln |x ? x′ | g

(11)

where g = 1 ? e0 . Here the subscript N indicates Neumann boundary conditions, corresponding to the presence of loop extremities on the boundary. Let us now single out a segment of length L on this boundary and attempt to count the number of loops connecting this segment to the rest of the boundary. To do this we insert a pair of vertex operators, one at each a extremity of the segment, V = exp [i(±e1 + e0 /2)Φ]. These operators do not a?ect the loops encircling the whole interval L since they modify the weight of such loops from e±iπe0 /2 to√e±iπe0 /2 e?iπe0 = e?iπe0 /2 , thus giving the same sum Qb . But for loops connecting the inside to the outside, the weight is now w = 2 cos πe1 . The boundary dimension of the ?elds V is, using the propagator h= 4e2 ? e2 1 0 4g (12)

so their two-point function decays as L?2h . We can then ?nd the average number of loops separating two given points by taking appropriate derivatives, and setting e1 = e0 /2 in the end. This leads to our main result
a Nc (?) =

a 4 For the XXX chain (e0 = 0) this reads Nc = π2 ln L ≈ 0.405 ln L, while for bond percolation (Q = 1 or e0 = 1/3) √ a we have Nc = π3 ln L ≈√ 0.551 ln L. The slope becomes 1 exactly as e0 → 1, or Q → ?2. We note that the result for the XXX case is close but de?nitely di?erent from the one proposed in [10]. It is amusing to observe that one can exactly interpret the valence bond as singlet contractions for an ordinary supergroup in the case Q = 1, by taking a lattice model where the fundamental three-dimensional representation of SU (2/1) and its conjugate alternate. The hamiltonian is again (4), but this time the ei are projectors onto the singlet in 3 ? ? The e?ective central charge for the this 3. 9 spin chain is ce? = 1 + π2 [arccosh(3/2)]2 ? 1.845, and extending the argument suggested in [10] for the XXX eff case gives a slope of 3cln 3 ? 0.559, even closer to the exact result (13). Of course, by taking higher derivatives of the two-point function of the vertex operators one can access the higher moments of (10). In fact, the two-point function itself is nothing but the characteristic function of p(Nc ), although carrying out the Fourier transform in general is somewhat cumbersome. We will content ourselves here by giving the ?rst few cumulants Ck = (ck /π k ) ln L, with, in the XXX case (top) and the Q = 1 case (bottom):

2 cos(πe0 /2) e0 ln L π(1 ? e0 ) sin(πe0 /2)

(13)

c1 =

4 √ c = 3π 2

8/3 √ c = 2(2π 3 ? 9) 3

16/15 √ 8(5π 3 ? 27) (14)

4
1

be seen what happens for other models, and whether in particular a c-theorem of sorts is obeyed for SVB . Acknowledgments. We thank F. Alet for helpful exchanges, and for pointing out Ref. [18]. This work was supported by the European Community Network ENRAGE (grant MRTN-CT-2004-005616) and by the Agence Nationale de la Recherche (grant ANR-06BLAN-0124-03).

0.8

0.6

0.4

1st cumulant 2nd cumulant

0.2

0

0.2

0.4

0.6

0.8

1

FIG. 2: Comparison between exact and numerically determined values of the slopes c1 and c2 , shown as functions of the parameter e0 .

√ together with the observation that, as Q → ?2, the probability distribution becomes Poissonian:
√ lim P (Nc ) Q→?2

= e? ln L

(ln L)Nc Nc !

(15)

Numerical calculations. We have computed the distribution (10) numerically by exactly diagonalizing the transfer matrix, for periodic chains of size up to Nmax = 32. The cumulants Ck ∝ ln L of p(Nc ) obey a very simple ?nite size scaling (FSS) form, where ln L has to be replaced by N ln sin Lπ ; this follows from standard forπ N mulas for two-point functions of our vertex operators V . Precise values of the slopes ck can then be extracted from a careful analysis of the residual FSS e?ects. As shown in Fig. 2 they agree well with our analytical results, except for Q → 4, where we expect logarithmic FSS corrections. For Q = 1, the combinatorial nature of |? implies that all λw in (8) are integers. This allows to obtain p(Nc ) exactly for ?nite L and N ≤ Nmax . Using this, we can in some cases conjecture p(Nc ) for any value of N [21]. In particular we have established that
N/2?1

Nc = (N ? L )pk (N )

2

2

2

n=0

(N 2 ? (2n + 1)2 )n?N/2

(16) 1 where pk is a polynomial of degree k = 8 (N + 4)(N ? 2) in N 2 . This exact FSS form allows to obtain for the slope c1 = 0.5517 ± 0.0003, in very precise agreement with the value 0.551329 from (13). Conclusions. As already argued in [10], SVB is a measure of entanglement which seems as good qualitatively as SvN , and easier to obtain numerically. We have shown in this Letter that it is also possible to tackle it analytically in the 1D case. The results are less easily expressed than for SvN (which is proportional to c) . On the other hand, they ?t considerably more naturally within the transfer matrix and Coulomb gas formalism. It remains to

[1] C. Holzhey, F. Larsen and F. Wilczek, Nucl. Phys. B 424, 44 (1994). [2] P. Calabrese and J. Cardy, J. Stat. Mech. P06002 (2004). [3] P. Martin, Potts models and related problems in statistical mechanics, World Scienti?c, Singapore (1991). [4] P. Martin and H. Saleur, Comm. Math. Phys. 158, 155 (1993). [5] G. Refael and J.E. Moore, Phys. Rev. Lett. 93 260602 (2004). [6] G. Refael and J.E. Moore, cond-mat/0703038. [7] R. Santachiara, J. Stat. Mech. L06002 (2006). [8] Y.C. Lin, F. Igloi and H. Rieger, Phys. Rev. Lett. 99, 147202 (2007). [9] R. Yu, H. Saleur and S. Haas, cond-mat/0709.3840. [10] F. Alet, S. Capponi, N. La?orencie and M. Mambrini, Phys. Rev. Lett. 99, 117204 (2007). [11] R.W. Chhajlany, P. Tomczak and A. Wojcik, Phys. Rev. Lett. 99, 167204 (2007). [12] M. Mambrini, cond-mat/0706.2508. [13] K.S.D. Beach and A.W. Sandvik, Nucl. Phys. B 750, 142 (2006). [14] A. W. Sandvik, Phys. Rev. Lett. 95, 207203 (2005). [15] D.S. Rokhsar and S.A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988). [16] C.L. Henley, J. Phys.: Cond. Matt. 16, S891 (2004). [17] K. Chang, I. A?eck, G. Hayden and Z. Soos, J. Phys. C 1, 153 (1989). [18] A. Feiguin, S. Trebst, A.W.W. Ludwig, M. Troyer, A. Kitaev, Z. Wang and M.H. Freedman, Phys. Rev. Lett. 98, 160409 (2007). [19] B. Nienhuis, in C. Domb and J.L Lebowitz (eds.), Phase Transitions and Critical Phenomena, vol. 11 (Academic, London, 1987). [20] I.K. Kostov, B. Ponsot and D. Serban, Nucl. Phys. B 683, 309 (2004). [21] J.L. Jacobsen and H. Saleur, to be published. [22] The scale of (4) a?ects the sound velocity and is important when studying the scaling of gaps. But it does not matter when dealing with entanglement issues. [23] Note that the scalar product of a state |w with itself √ N/2?j is equal to Q for all w, so there is no need to consider “normalized” basis words. [24] The boundary conditions at large N are not expected to a?ect the leading behaviour of SVB . √ 1 [25] When Q < 0 (i.e. 2 < e0 < 1) the true ground state has spin j > 0. Nevertheless, we continue to let |? denote the j = 0 ground state.√ Numerical studies then indicate that (8) holds even for Q < 0, if N is large enough.




友情链接: