Finite-size correction and bulk hole-excitations for special case of an open XXZ chain with nondiagonal boundary terms at roots of unity
aa r X i v : . [ h e p - t h ] M a y UMTG–253
Finite-size correction and bulk hole-excitations forspecial case of an open XXZ chain with nondiagonalboundary terms at roots of unity
Rajan MurganPhysics Department, P.O. Box 248046, University of MiamiCoral Gables, FL 33124 USA
Abstract
Using our solution for the open spin-1 / N spinsand two arbitrary boundary parameters at roots of unity, the central charge and theconformal dimensions for bulk hole excitations are derived from the 1 /N correction tothe energy (Casimir energy). Introduction
The integrable open spin-1 / q –Onsager algebra for general nondiagonal cases [14]. Ap-proaches based on boundary Temperley-Lieb algebra and its representations have also beenpresented recently, from which the spectral properties of the chain have been studied [15].Upon obtaining the desired solution, the next natural question that needed to be addresedis its practicality within various contexts. One important area where these solutions havefound creditable applications is in determining finite size corrections to the ground stateenergy. By relating to conformal invariance, these finite size corrections are shown to berelated directly to other crucial parameters like the critical indices, central charge and con-formal dimensions [17]–[20]. There are few methods and approaches to accomplish this task.De Vega and Woynarowich [21] derived integral equations for calculating leading finite-sizecorrections for models solvable by Bethe Ansatz approach [22]. This was then generalized tonested Bethe Ansatz models as well [23]. Another approach was introduced by Woynarowichand Eckel [24, 25], which utilizes Euler-Maclaurin formula and Wiener-Hopf integration tocompute these corrections for the closed XXZ chain. Others have also studied more generalintegrable spin chain models e.g., XXZ diagonal [2, 3], nondiagonal cases [28], quantum spin1 / / ,
1) whichcontains alternating spins of 1 / / , and extending the solution to account for odd number of sites as well, wecompute the correction of order 1 /N (Casimir energy) to the ground state energy togetherwith its low lying excited states (multi-hole states). We employ the method introducedby Woynarovich and Eckle [24] that makes use of Euler-Maclaurin formula [45] and Wiener-Hopf integration [46]. In particular, we compute the analytical expressions for central charge This solution, in contrast to [6]–[9] does not assume any constraint among the boundary parameters. N . InSection 3, we present the calculation of 1 /N correction to the ground state energy and henceour results for the central charge and conformal dimensions of low lying excited states. Wenotice that the lowest energy state for even N of this model has one hole. Hence, the trueground state (lowest energy state without holes) lies in the odd N sector. Similar behaviourare also found for the open chain with diagonal boundary terms, for certain values of bound-ary parameters [47]. It is known that (critical) XXZ model with nondiagonal boundaryterms corresponds to (conformally invariant) free Boson with Neumann boundary conditionwhereas the diagonal ones are related to the Dirichlet case [34, 35, 36, 42]. Although themodel we study here has nondiagonal boundary terms, we find that the conformal dimen-sions for this model resemble that of the Dirichlet boundary condition. Some numericalresults are presented in Section 4 to confirm and support the analytical results derived inSection 3. Here, we solve the model numerically for some large but finite N and furtheremploy an algorithm due to Vanden Broeck and Schwartz [39]–[40] to extrapolate the resultsfor N → ∞ limit. We conclude with a discussion of our results and some potential openproblems in Section 5. We begin this section by reviewing recently proposed Bethe Ansatz solution [11, 38] for thefollowing model [43, 44] H = H + 12 sinh η ( cosech α − σ x + cosech α + σ xN ) , (2.1)where the “bulk” Hamiltonian is given by H = 12 N − X n =1 (cid:0) σ xn σ xn +1 + σ yn σ yn +1 + cosh η σ zn σ zn +1 (cid:1) . (2.2)In the above expressions, σ x , σ y , σ z are the usual Pauli matrices, η is the bulk anisotropyparameter (taking values η = iπp +1 , with p odd), α ± are the boundary parameters, and N isthe number of spins/sites. Note that, this model has only two boundary parameters. Otherboundary parameters (as they appear in the original Hamiltonian in [43]) have been set to2ero. We restrict the values of α ± to be pure imaginary to ensure the Hermiticity of theHamiltonian. The Bethe Ansatz equations for both odd and even N are given by δ ( u (1) j ) h (2) ( u (1) j − η ) δ ( u (1) j − η ) h (1) ( u (1) j ) = − Q ( u (1) j − η ) Q ( u (1) j + η ) , j = 1 , , . . . , M ,h (1) ( u (2) j − η ) h (2) ( u (2) j ) = − Q ( u (2) j + η ) Q ( u (2) j − η ) , j = 1 , , . . . , M . (2.3)where δ ( u ) = 2 (sinh u sinh( u + 2 η )) N sinh 2 u sinh(2 u + 4 η )sinh(2 u + η ) sinh(2 u + 3 η ) sinh( u + η + α − )sinh( u + η − α − ) sinh( u + η + α + ) sinh( u + η − α + ) cosh ( u + η ) (2.4)and Q a ( u ) = M a Y j =1 sinh( u − u ( a ) j ) sinh( u + u ( a ) j + η ) , a = 1 , , (2.5) M and M are the number of Bethe roots, u (1) j and u (2) j (zeros of Q ( u ) and Q ( u ) respec-tively). However, h (1) ( u ) and h (2) ( u ) differ for odd and even values of N , as will be notedbelow. The energy eigenvalues in terms of the “shifted” Bethe roots ˜ u ( a ) j are given by E = 12 sinh η X a =1 M a X j =1 u ( a ) j − η ) sinh(˜ u ( a ) j + η ) + 12 ( N −
1) cosh η . (2.6)where ˜ u ( a ) j ≡ u ( a ) j + η . N We begin by recalling [38] the structure of roots distribution for this case. The Bethe roots˜ u ( a ) j for the lowest energy state have the form ( µλ ( a, j : j = 1 , , . . . , M ( a, µλ ( a, j + iπ , : j = 1 , , . . . , M ( a, , a = 1 , , (2.7)where λ ( a,b ) j are real. Here, M (1 , = M (2 , = N , and M (1 , = p +12 , M (2 , = p − . The µλ ( a, j are the zeros of Q a ( u ) that form real sea (“sea roots”) and µλ ( a, k are real parts of the “extraroots” (also zeros of Q a ( u )) which are not part of the “seas”. Hence, there are two “seas” ofreal roots. We employ notations similar to the one used in [28], e n ( λ ) = sinh (cid:0) µ ( λ + in ) (cid:1) sinh (cid:0) µ ( λ − in ) (cid:1) , g n ( λ ) = e n ( λ ± iπ µ ) = cosh (cid:0) µ ( λ + in ) (cid:1) cosh (cid:0) µ ( λ − in ) (cid:1) . (2.8)3ewriting bulk and boundary parameters [28], η = iµ , α ± = iµa ± , where µ = πp +1 andtaking h (1) ( u ) = 8 sinh N +1 ( u + 2 η ) cosh ( u + η ) cosh( u + 2 η )sinh(2 u + 3 η ) , h (2) ( u ) = h (1) ( − u − η ) , (2.9)the Bethe Ansatz equations (2.3) for the sea roots then take the following form [11, 38] e ( λ (1 , j ) N +1 h g ( λ (1 , j ) e a − ( λ (1 , j ) e − a − ( λ (1 , j ) e a + ( λ (1 , j ) e − a + ( λ (1 , j ) i − (2.10)= − N/ Y k =1 h e ( λ (1 , j − λ (2 , k ) e ( λ (1 , j + λ (2 , k ) i ( p − / Y k =1 h g ( λ (1 , j − λ (2 , k ) g ( λ (1 , j + λ (2 , k ) i , and e ( λ (2 , j ) N +1 g ( λ (2 , j ) − (2.11)= − N/ Y k =1 h e ( λ (2 , j − λ (1 , k ) e ( λ (2 , j + λ (1 , k ) i ( p +1) / Y k =1 h g ( λ (2 , j − λ (1 , k ) g ( λ (2 , j + λ (1 , k ) i , respectively, where j = 1 , . . . , N . The corresponding ground-state counting functions are h (1) ( λ ) = 12 π n (2 N + 1) q ( λ ) − r ( λ ) − q a − ( λ ) − q − a − ( λ ) − q a + ( λ ) − q − a + ( λ ) − N/ X k =1 h q ( λ − λ (2 , k ) + q ( λ + λ (2 , k ) i − ( p − / X k =1 h r ( λ − λ (2 , k ) + r ( λ + λ (2 , k ) i o , (2.12)and h (2) ( λ ) = 12 π n (2 N + 1) q ( λ ) − r ( λ ) − N/ X k =1 h q ( λ − λ (1 , k ) + q ( λ + λ (1 , k ) i − ( p +1) / X k =1 h r ( λ − λ (1 , k ) + r ( λ + λ (1 , k ) i o . (2.13)where q n ( λ ) and r n ( λ ) are odd functions defined by q n ( λ ) = π + i ln e n ( λ ) = 2 tan − (cot( nµ/
2) tanh( µλ )) ,r n ( λ ) = i ln g n ( λ ) . (2.14)These counting functions satisfy the following h ( l ) ( λ j ) = j , j = 1 , . . . , N l = 1 , Bethe Ansatz equations written in this and subsequent sections are true only for suitable values of a ± ,namely ν − < | a ± | < ν +12 , a + a − > ν = p + 1 .2 Odd N In this section, we present an extension of the previous results to include solutions forodd N values. The roots distribution is similar to the previous case, but now we have M (1 , = M (2 , = N +12 , and M (1 , = M (2 , = p − . Using the following in (2.3), h (1) ( u ) = sinh( u − α + + η ) sinh( u + α + + η ) sinh N +1 ( u + 2 η ) cosh ( u + η ) cosh( u + 2 η )sinh(2 u + 3 η ) ,h (2) ( u ) = h (1) ( − u − η ) , (2.16)we obtain the Bethe Ansatz equations e ( λ (1 , j ) N +1 h g ( λ (1 , j ) e a − ( λ (1 , j ) e − a − ( λ (1 , j ) i − (2.17)= − ( N +1) / Y k =1 h e ( λ (1 , j − λ (2 , k ) e ( λ (1 , j + λ (2 , k ) i ( p − / Y k =1 h g ( λ (1 , j − λ (2 , k ) g ( λ (1 , j + λ (2 , k ) i , and e ( λ (2 , j ) N +1 h g ( λ (2 , j ) e a + ( λ (2 , j ) e − a + ( λ (2 , j ) i − (2.18)= − ( N +1) / Y k =1 h e ( λ (2 , j − λ (1 , k ) e ( λ (2 , j + λ (1 , k ) i ( p − / Y k =1 h g ( λ (2 , j − λ (1 , k ) g ( λ (2 , j + λ (1 , k ) i , respectively, where j = 1 , . . . , N +12 . Note the presence of parameter-dependant terms inboth the equations above. One can also notice the number of extra roots changes from p +12 to p − for Q ( u ). The ground-state counting functions for this case read h (1) ( λ ) = 12 π n (2 N + 1) q ( λ ) − r ( λ ) − q a − ( λ ) − q − a − ( λ ) − ( N +1) / X k =1 h q ( λ − λ (2 , k ) + q ( λ + λ (2 , k ) i − ( p − / X k =1 h r ( λ − λ (2 , k ) + r ( λ + λ (2 , k ) i o , (2.19)and h (2) ( λ ) = 12 π n (2 N + 1) q ( λ ) − r ( λ ) − q a + ( λ ) − q − a + ( λ ) − ( N +1) / X k =1 h q ( λ − λ (1 , k ) + q ( λ + λ (1 , k ) i − ( p − / X k =1 h r ( λ − λ (1 , k ) + r ( λ + λ (1 , k ) i o , (2.20)As for even N , we again have the following h ( l ) ( λ j ) = j , j = 1 , . . . , N + 12 (2.21)5here l = 1 ,
2. Note that (2.15) and (2.21) can be written more compactly as h ( l ) ( λ j ) = j , j = 1 , . . . , ⌊ N + 12 ⌋ (2.22)where ⌊ . . . ⌋ denotes the integer part and µλ ⌊ N +12 ⌋ is the largest sea root for that “sea”.Subsequently, we shall denote largest sea roots as µ Λ l . /N In this section, we shall compute the finite-size correction for the ground state and low lyingexcited states. For these excited states, we restrict our analysis to excitations by holes whichare located to the right of the real sea roots. Applying (2.7) to (2.6), we get the lowest stateenergy eigenvalues for chain of finite length N , E = − π sin µµ n X a =1 ⌊ N +12 ⌋ X j = −⌊ N +12 ⌋ a ( λ ( a, j ) − a (0) + X a =1 M ( a, X j =1 b ( λ ( a, j ) o + 12 ( N −
1) cos µ . (3.1)where notations from [28] have again been adopted a n ( λ ) = 12 π ddλ q n ( λ ) = µπ sin( nµ )cosh(2 µλ ) − cos( nµ ) ,b n ( λ ) = 12 π ddλ r n ( λ ) = − µπ sin( nµ )cosh(2 µλ ) + cos( nµ ) . (3.2)Note that M ( a, in (3.1), refers to number of extra roots for Q a ( u ). The first and thirdterms in the curly bracket of (3.1) are summed over the number of sea roots and extraroots respectively. As one considers next lowest excited state, the number of sea roots andextra roots change. Hence, for these states of low lying excitations (with real sea), the verysame term in the first sum will again be summed over accordingly between approriate limitsdictated by the number of sea roots. As for the summation over extra roots, the functionsummed over depends on the imaginary part of these roots, especially in the presence of2-strings. However, as one shall see, for 1 /N correction (in the N → ∞ limit), only the sumover the sea roots contributes. The second sum in (3.1) contributes to order 1 correction(boundary energy) which we have considered elsewhere [38]. Equation [4.26] for the boundary energy in [38] holds both for even and odd values of N .1 Sum-rule and hole-excitations Now we present some results based on the solution of the model (2.1) for N = 2 , , . . . , N case. We find for even N , excited states contain odd number of holesfor each Q a ( u ). This can be seen from the following analysis on counting functions. Forthe lowest energy state the counting functions are given by (2.12) and (2.13). By using thefact that q n ( λ ) → sgn( n ) π − µn and r n ( λ ) → − µn as λ → ∞ and ρ ( l ) = N d h ( l ) dλ we have thefollowing sum rule Z ∞ Λ l dλ ρ ( l ) ( λ ) = 1 N ( h ( l ) ( ∞ ) − h ( l ) (Λ l ))= 1 N ( 12 + 1) (3.3) µ Λ l refers to the largest sea root. As before l = 1 ,
2. We make use of the fact that h ( l ) ( ∞ ) = N h ( l ) (Λ l ) = N Z ∞ Λ l dλ ρ ( l ) ( λ ) = 1 N ( h ( l ) ( ∞ ) − h ( l ) (Λ l ))= 1 N ( 12 + N H ) (3.5)where N H is the number of holes (odd) to the right of the corresponding largest sea root.To illustrate the results above, we consider the following low lying excited states with N − N − . The former case is found to have one hole with p − and p − extra roots in additionto a 2-string from each of the Q ( u ) and Q ( u ) respectively. From, h ( l ) ( ∞ ) = N h ( l ) (Λ l ) = N − N ( h ( l ) ( ∞ ) − h ( l ) (Λ l )) = 1 N ( 12 + 1) (3.7) The lowest energy state has N sea roots. As for the extra roots, there are p +12 and p − of them for Q ( u ) and Q ( u ) respectively N H = 1. The later case has three holes with p +12 and p − extra roots and a2-string from each of the Q a ( u ) with a = 1 ,
2. Similar analysis, h ( l ) ( ∞ ) = N h ( l ) (Λ l ) = N − N ( h ( l ) ( ∞ ) − h ( l ) (Λ l )) = 1 N ( 12 + 3) (3.9)giving N H = 3. The total number of roots are the same for all these states. There are alsoexcited states with equal number of sea and extra roots as for the state of lowest energy,but with position of the single hole nearer to the origin than that of the lowest energy state,suggesting the usual bulk hole-excitation scenario, E hole ( λ ( a ) ) increases as λ ( a ) → E hole ( λ ( a ) ) is the energy due to the presence of holes and λ ( a ) , with a = 1 , N case, we have the true ground state, namely state of lowest energywithout hole. From the counting functions, (2.19) and (2.20), we have Z ∞ Λ l dλ ρ ( l ) ( λ ) = 1 N ( h ( l ) ( ∞ ) − h ( l ) (Λ l ))= 12 N (3.10)As before l = 1 ,
2, and we make use of the fact that h ( l ) ( ∞ ) = N h ( l ) (Λ l ) = N + 12 (3.11)From (3.11), we see that this state of lowest energy for odd N has no hole, signifying thetrue ground state. Similar analysis for low lying excited states yields the following Z ∞ Λ l dλ ρ ( l ) ( λ ) = 1 N ( h ( l ) ( ∞ ) − h ( l ) (Λ l ))= 1 N ( 12 + N H ) (3.12)where N H is the number of holes (even) to the right of sea roots. Hence, for odd N case,there are even number of holes (for each Q a ( u )), with a = 1 ,
2, for the excited states, e.g.,8or the first excited state with N − sea roots, h ( l ) ( ∞ ) = N + 12 + 32 h ( l ) (Λ l ) = N −
12 (3.13)which signifies the presence of two holes.It is known for simpler models of spin chains e.g., closed XXZ chain that even numberof holes are present in chains with even number of spins and vice versa. Hence, the trueground state (lowest energy state with no holes) for these models is found to lie in even N sector. The reverse scenario (one hole in the lowest energy state for even N and groundstate in odd N sector) we find here for this model can be explained using some heuristicarguments based on spin and magnetic fields at the two boundaries, similar to the one givenin Section 3 of [38] . In footnote 2, we notice the signs of a + and a − must be the samefor boundary parameter region of interest. Hence, in Hamiltonian (2.1), the direction of themagnetic fields at the two boundaries are also the same (Both up or both down). This upsetsthe antiferromagnetic spin arrangement at the boundaries, favouring spin allignments alongthe same direction at the boundaries for chains with even N . This causes the following:presence of odd N behaviours in the even N chain, namely the lowest energy state for even N sector has one hole for each Q a ( u ). Spins at the boundaries for the odd N chain will notexperience such spin upset since the parallel magnetic fields favours the antiferromagneticarrangement of an odd N chain. Therefore, the lowest energy state for odd N chain has noholes. In other words, the true ground state exists in odd N sector. Further effects are thepresence of odd and even number of holes in chains with even and odd N respectively asshown in the analysis above.Now, the energy due to hole excitations can be presented. We consider first the lowestenergy state for even N case with one hole. Using1 N N X k = − N g ( λ − λ ( a, k ) ≈ Z ∞−∞ dλ ′ ρ ( l ) ( λ ′ ) g ( λ − λ ′ ) − N g ( λ − ˜ λ ( a ) ) (3.14)for some arbitrary function g ( λ ) and ρ ( l ) = 1 N d h ( l ) dλ (3.15)where l = 1 , µλ ( a, k ≡ sea roots, with a = 1 ,
2, and µ ˜ λ ( a ) ≡ position of the hole for eachof the Q a ( u ), one can write down the sum of the two densities ρ (1) ( λ ) + ρ (2) ( λ ) = 4 a ( λ ) − Z ∞−∞ dλ ′ ( ρ (1) ( λ ′ ) + ρ (2) ( λ ′ )) a ( λ − λ ′ ) Readers are urged to refer to Figures 2 and 3 in that Section N [ a ( λ − ˜ λ (1) ) + a ( λ − ˜ λ (2) )] + 1 N [2 a ( λ ) + 2 a ( λ ) − b ( λ ) − a a − ( λ ) − a − a − ( λ ) − a a + ( λ ) − a − a + ( λ ) − p − X k =1 ( b ( λ − λ (2 , k ) + b ( λ + λ (2 , k )) − p +12 X k =1 ( b ( λ − λ (1 , k ) + b ( λ + λ (1 , k ))] (3.16)Defining ρ total ( λ ) ≡ ρ (1) ( λ ) + ρ (2) ( λ ) and solving (3.16) using Fourier transform , we haveˆ ρ total ( ω ) = 4ˆ s ( ω ) + 1 N ˆ R ( ω )+ 1 N ˆ J ( ω )( e iω ˜ λ (1) + e iω ˜ λ (2) ) (3.17)where ˆ ρ total ( ω ) , ˆ a ( ω ) and ˆ s ( ω ) are the Fourier transforms of ρ total ( λ ) , a ( λ ) and a ( λ )1+ a ( λ ) respectively. Also ˆ J ( ω ) = ˆ a ( ω )1+ˆ a ( ω ) . ˆ R ( ω ) is the contribution from the second square bracketin (3.16), which will not enter the calculation for E hole (˜ λ ( a ) ) and will be omitted henceforth.The Fourier transform of hole density are the third and the fourth terms in (3.17), whichgives ρ hole ( λ ) = 1 N [ J ( λ − ˜ λ (1) ) + J ( λ − ˜ λ (2) )] (3.18)Using approximation (3.14) in (3.1), and making use of (3.18), one has E hole (˜ λ ( a ) ) = − N π sin µ µ Z ∞−∞ dλ a ( λ ) ρ hole ( λ )+ π sin µ µ X a =1 a (˜ λ ( a ) ) (3.19)which after some manipulation yields E hole (˜ λ ( a ) ) = π sin µ µ X a =1 π ˜ λ ( a ) (3.20) Our conventions areˆ f ( ω ) ≡ Z ∞−∞ e iωλ f ( λ ) dλ , f ( λ ) = 12 π Z ∞−∞ e − iωλ ˆ f ( ω ) dω . ˆ a n ( ω ) = sgn( n ) sinh (( ν − | n | ) ω/ νω/ , ≤ | n | < ν. α number of holes, one has ρ hole ( λ ) = 1 N X α X a =1 J ( λ − ˜ λ ( a ) α ) (3.21)and finally the following for the energy E hole (˜ λ ( a ) α ) = π sin µ µ X α X a =1 π ˜ λ ( a ) α (3.22)Note that E hole (˜ λ ( a ) α ) increases as ˜ λ ( a ) α → In this section, we give the derivation of 1 /N correction (Casimir energy) to the lowest energystate, for the even N case (with one hole). This result is then generalized to include odd N values as well as the low lying (multi-hole) excited states. We begin by presenting theexpression for the density difference between chain of finite length (with N spins), ρ (1) N ( λ ) + ρ (2) N ( λ ) and that of infinite length, ρ ∞ ( λ ) ρ (1) N ( λ ) + ρ (2) N ( λ ) − ρ ∞ ( λ ) = − Z ∞−∞ dγ a ( λ − γ )[ 1 N N X β = − N δ ( γ − λ (1 , β ) − ρ (1) N ( γ )] − Z ∞−∞ dγ a ( λ − γ )[ 1 N N X β = − N δ ( γ − λ (2 , β ) − ρ (2) N ( γ )] − Z ∞−∞ dγ a ( λ − γ )[ ρ (1) N ( γ ) + ρ (2) N ( γ ) − ρ ∞ ( γ )] (3.23)In (3.23) and henceforth, only terms that are crucial to the computation of 1 /N correctionare given. Other parameter dependant terms that contribute to order 1 correction have beenomitted here . Solving (3.23) yields ρ (1) N ( λ ) + ρ (2) N ( λ ) − ρ ∞ ( λ ) = − Z ∞−∞ dγ p ( λ − γ )[ 1 N N X β = − N δ ( γ − λ (1 , β ) − ρ (1) N ( γ )] − Z ∞−∞ dγ p ( λ − γ )[ 1 N N X β = − N δ ( γ − λ (2 , β ) − ρ (2) N ( γ )] (3.24) See [38] for details ρ ∞ ( λ ) = a ( λ )1+ a ( λ ) ≡ s ( λ ) and p ( λ ) = π R ∞−∞ dω e − iωλ ˆ a ( ω )1+ˆ a ( ω ) Similar equation ex-pressing the energy difference between finite and infinite system is also needed to computeCasimir energy. This is given by E N − E ∞ = − N π sin µ µ n Z ∞−∞ dλ a ( λ )[ 1 N N X β = − N δ ( λ − λ (1 , β ) − ρ (1) N ( λ )]+ Z ∞−∞ dλ a ( λ )[ 1 N N X β = − N δ ( λ − λ (2 , β ) − ρ (2) N ( λ )]+ Z ∞−∞ dλ a ( λ )[ ρ (1) N ( λ ) + ρ (2) N ( λ ) − ρ ∞ ( λ )] o (3.25)Using (3.24) and the fact that ˆ p ( ω )ˆ a ( ω ) = ˆ s ( ω )ˆ a ( ω ), we have E N − E ∞ = − N π sin µ µ n Z ∞−∞ dλ S (1) N ( λ ) ρ (1) ∞ ( λ ) + Z ∞−∞ dλ S (2) N ( λ ) ρ (2) ∞ ( λ ) o (3.26)where S ( l ) N ( λ ) ≡ N P N β = − N δ ( λ − λ ( l, β ) − ρ ( l ) N ( λ ) and ρ ( l ) ∞ ( λ ) = ρ ∞ ( λ ) ≡ s ( λ ) with l = 1 , E N − E ∞ = − N π sin µ µ n − Z ∞ Λ dλ ρ (1) ∞ ( λ ) ρ (1) N ( λ ) + 12 N ρ (1) ∞ (Λ ) + 112 N ρ (1) N (Λ ) ρ (1) ′ ∞ (Λ ) − Z ∞ Λ dλ ρ (2) ∞ ( λ ) ρ (2) N ( λ ) + 12 N ρ (2) ∞ (Λ ) + 112 N ρ (2) N (Λ ) ρ (2) ′ ∞ (Λ ) o (3.27)(3.24) can also be expressed in similar form ρ (1) N ( λ ) + ρ (2) N ( λ ) − ρ ∞ ( λ ) = Z ∞ Λ dγ p ( λ − γ ) ρ (1) N ( γ ) − N p ( λ − Λ ) − p ′ ( λ − Λ )12 N ρ (1) N (Λ )+ Z ∞ Λ dγ p ( λ − γ ) ρ (2) N ( γ ) − N p ( λ − Λ ) − p ′ ( λ − Λ )12 N ρ (2) N (Λ ) (3.28)As before, µ Λ and µ Λ are the largest sea roots from the two “seas” respectively. Fromthis point, the calculation very closely resembles the details found in Section 2 in [3]. Hence,we omit the details and give only the crucial steps. Note that (3.28) can be written in thestandard form of the Wiener-Hopf equation [46] after redefining the terms, χ (1) ( t ) + χ (2) ( t ) − Z ∞ ds p ( t − s ) χ (1) ( s ) − Z ∞ ds p ( t − s ) χ (2) ( s ) ≈ f (1) ( t ) − N p ( t ) + 112 N ρ (1) N (Λ ) p ′ ( t )+ f (2) ( t ) − N p ( t ) + 112 N ρ (2) N (Λ ) p ′ ( t ) (3.29)12here the following definitions have been adopted χ ( l ) ( λ ) = ρ ( l ) N ( λ + Λ l ) f ( l ) ( λ ) = ρ ( l ) ∞ ( λ + Λ l ) (3.30)and following change in variable is used : t = λ − Λ l with l = 1 , X ( l )+ ( ω ) which is the Fourier transfrom of χ ( l )+ ( t ) that is analytic in the upper half complex plane ,ˆ X ( l )+ ( ω ) = 12 N + iω N ρ ( l ) N (Λ l )+ G + ( ω ) h ig N ρ ( l ) N (Λ l ) − N − iω N ρ ( l ) N (Λ l )+ ππ − iω (cid:16) αN + 12 N − ig N ρ ( l ) N (Λ l ) (cid:17)i (3.31)where G + ( ω ) G + ( − ω ) = 1 + ˆ a ( ω ), g = i (2 + ν − νν − ) and α = G + (0) = ( ν ν − ) , with G + (0) = ν − ν .From (3.3), (3.30) and (3.31), one can then determine ρ (1) N (Λ ) and ρ (2) N (Λ ) explicitly from χ ( l )+ (0) ≡ ρ ( l ) N (Λ l ) = 12 π Z ∞−∞ dω ˆ X ( l )+ ( ω ) (3.32)by contour integration and some algebra. We give the result below ρ ( l ) N (Λ l ) = 14 N n π + 2 πα + ig + [ π + 2 ig π − g π α + 4 πα ( π + ig )] o (3.33)Finally, using ρ ( l ) ∞ ( λ ) ≈ e − πλ for λ → Λ l and (3.27), one arrives at the desired expressionfor 1 /N correction to the energy, E N − E ∞ = E Casimir = − π sin µ µN (1 − α ) (3.34)where the effective central charge is c eff = 1 − α = 1 − ν ( ν −
1) (3.35)We see that for this model, the central charge, c = 1 (Free boson). Also c eff is independentof boundary parameters, unlike for the Dirichlet case [3]. This is a feature expected for Again for complete details, refer to [3] − c eff ν ν −
1) (3.36)Note that the above results are derived for the lowest energy state for even N with one holefor each Q a ( u ). Reviewing the derivation above, one can notice that the results above canbe further generalized for any N and for low lying excited states with arbitrary number ofholes, provided these holes are located to the right of the largest sea root as mentioned inthe beginning of Section 3. For these excited states, the sum for S ( l ) N ( λ ) in (3.23) - (3.26) willinevitably have different limits since the number of sea roots vary. However, after applyingthe Euler-Maclaurin formula, one would recover (3.27) and (3.28). In addition to that, forstates with N H number of holes (all located to the right of the largest sea root), one usesthe more general result for the sum rule, namely (3.5) and (3.12) which eventually yields α = N H G + (0) (3.37)Thus, we have the following for the effective central charge and conformal dimensions forlow lying excited states c eff = 1 − ν ( ν − N H ∆ = ν ν − N H (3.38)Surprisingly, the results (3.36) and (3.38) appear to have more resemblance to spin chainswith diagonal boundary terms, as one could see from the νν − dependance [33]-[36], ratherthan ν − ν [42] which is the anticipated form for conformal dimensions for spin chains withnondiagonal boundary terms. Indeed the theory of a free Bosonic field ϕ compactified on acircle of radius r is invariant under ϕ ϕ + 2 πr , where r = β . β is the continuum bulkcoupling constant that is related to ν by β = 8 π ( ν − ν ). Further, the quantization of themomentum zero-mode Π , yields Π = nβ for Neumann boundary condition and Π = nβ forthe Dirichlet case, where n is an integer. Hence, the zero-mode contribution to the energy, E ,n ∼ Π implies E ,n ∼ ∆ ∼ ( ν − ν ) for Neumann and E ,n ∼ ∆ ∼ ( νν − ) for Dirichlet caserespectively. More complete discussion on this topic can be found in [35, 42]. Next, we willresort to numerical analysis to confirm our analytical results obtained in this section.14 Numerical results
We present here some numerical results for both odd and even N cases, to support ouranalytical derivations in Section 3.2. We first solve numerically the Bethe equations (2.3),(2.12), (2.13), (2.19) and (2.20) for some large number of spins. We use these solutions tocalculate Casimir energy numerically from the following E = E bulk + E boundary + E Casimir (4.1)In (4.1), E is given by (3.1). Thus, having determined the Bethe roots numerically, one usesknown expressions for E bulk [48] and E boundary [38] to determine E Casimir . Then using theexpression found above for E Casimir , namely (3.34), one can determine the effective centralcharge, c eff for that value of N , c eff = − µNπ sin µ ( E − E bulk − E boundary ) (4.2)Finally, we employ an algorithm due to Vanden Broeck and Schwartz [39]–[40] to extrapolatethese values for central charge at N → ∞ limit. Table 1 below shows the c eff values for somefinite even N , for the lowest energy state with one hole ( N H = 1). Equation (3.38) predicts c eff values of -11 and -7 for p = 1 and p = 3 respectively which are the extrapolated values(-11.000315 and -7.000410) we obtain from the Vanden Broeck and Schwartz method. N c eff , p = 1 , ν = 2 c eff , p = 3 , ν = 416 -9.365620 -2.85387224 -9.857713 -3.27127932 -10.122128 -3.55714840 -10.287160 -3.77088248 -10.399970 -3.93955456 -10.481956 -4.07765264 -10.544233 -4.193784... ... ... ∞ -11.000315 -7.000410Table 1: Central charge values, c eff for p = 1 ( a + = 0 . a − = 0 . p = 3( a + = 2 . a − = 1 . N = 16 ,24 ,. . . ,64 andextrapolated values at N → ∞ limit (Vanden Broeck and Schwartz algorithm).For odd N sector, since N H = 0, (3.38) predicts c eff = 1 (for the ground state) for anyodd p . We present similar numerical results for odd N in Table 2 below for p = 1 and ν = p + 1 = 3. We work out the c eff values numerically for N = 15 ,25 ,. . . ,65. Excellent agreementbetween the calculated and the extrapolated values of 1.000770 and 1.001851 again stronglysupports our analytical results. N c eff , p = 1 , ν = 2 c eff , p = 3 , ν = 415 0.898334 0.53150125 0.936128 0.63401235 0.953433 0.69275845 0.963360 0.73184155 0.969797 0.76014265 0.974311 0.781795... ... ... ∞ c eff for p = 1 ( a + = 0 . a − = 0 . p = 3( a + = 2 . a − = 1 . N = 15 ,25 ,. . . ,65 andextrapolated values at N → ∞ limit (Vanden Broeck and Schwartz algorithm). From the proposed Bethe ansatz equations for an open XXZ spin chain with special non-diagonal boundary terms at roots of unity, we computed finite size effect, namely the 1 /N correction (Casimir energy) to the lowest energy state for both even and odd N . We alsostudied the bulk excitations due to holes. We found some peculiar results for these excita-tions of this model. Firstly, the number of holes for excited states seem to be reversed: evennumber of holes for chains with odd number of spins and vice versa. However, one couldexplain this by resorting to heuristic arguments involving effects of magnetic fields on thespins at the boundary. We then computed the energy due to hole-excitations. We furthergeneralized the finite-size correction calculation to include multi-hole excited states, wherethese holes are situated to the right of the largest sea root. Having found the correction, weproceeded to compute the effective central charge, c eff and the conformal dimensions, ∆ forthe model. We found the central charge, c = 1. The effective central charge is independentof the boundary parameters, as expected for models with Neumann boundary condition.The result for ∆ however, turns out to be similar to models with diagonal boundary termsrather than the nondiagonal ones, to which the model studied here belongs to.As an independent check to our analytical results, we also solved the model numeri-cally for some large values of N . We used this solution to compute 1 /N correction for16hese large N values, then extrapolate them to the N → ∞ limit using Vanden Broeckand Schwartz algorithm. Our numerical results strongly support the analytical derivationspresented here. Spectral equivalences between diagonal-nondiagonal and diagonal-diagonal,nondiagonal-nondiagonal and diagonal-diagonal [15, 16, 50] open XXZ spin chains have beenshown to exist. Hence, one may attempt to explain the diagonal (Dirichlet) behaviour of themodel studied here by some such equivalence. However, to our knowledge, such equivalenceshave been found when the boundary parameters obey certain constraint [6]–[9], which is notthe case for the model we considered here, as already remarked in Footnote 1. Hence, thequestion about the “Dirichlet-like” behaviour remains for now. We hope to be able to resolvethis issue soon.There are many other open questions that one can explore and address further. Forexample, similar analysis involving boundary excitations can also be carried out. This canbe really challenging even for the diagonal (Dirichlet) case [34, 49]. Further, solution formore general XXZ model involving multiple Q ( u ) functions [12, 13], can also be utilized insimilar capacity to explore these effects. Last but not least, excitations due to other objectsthat we choose to ignore here, such as special roots/holes and so forth can also be exploredfor these models in order to make the study more complete. We look forward to addresssome of these issues in near future. Acknowledgments
I would like to thank R.I. Nepomechie for his invaluable advice, suggestions and commentsduring the course of completing this work. I also fully appreciate the financial supportreceived from the Department of Physics, University of Miami.
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