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keywords: Integrable systems , BCS model , Bethe Ansatz , Bethe root distributions Introduction The [MATH] -pairing Hamiltonian is an example of a Bardeen-Cooper-Schrieffer (BCS) model which admits an exact Bethe Ansatz solution. This result was initially established for the closed system which conserves particle numbe... |
For the open model there is no conservation of total particle number, due to interaction terms which accommodate particle exchange with the system’s environment. Consequently a [MATH] symmetry is broken, which generally renders the analysis of the system to be more complicated. Integrability of the open model was estab... |
through use of the Boundary Quantum Inverse Scattering Method. An alternative derivation, which is less technical, was later provided in |
. Topological properties of the open model, relating to zero-energy excitations, have been studied in Like its better known ancestor the Richardson model |
, which is associated with [MATH] -wave pairing, the existence of the exact solution for the closed [MATH] system provides a means to calculate the ground-state energy through a continuum limit approximation. Numerical studies suggest that, in the limit of infinite particle number, the ground-state roots become dense a... |
, and subsequently re-examined by Román et al. . One way to view this problem is to use the language of a two-dimensional electrostatic analogy. It was found in this manner that the solution for the ground-state energy coincides with the prediction coming from mean-field calculations. It is worthy of mention that an ex... |
Following this approach, the continuum limit approximation has also been adopted in for the closed [MATH] -pairing model. Despite having much more complicated patterns of Bethe root distribution compared to the Richardson model, the results were again found to give agreement with mean-field analysis. However, there are... |
The origins of the new approach that will be followed trace back to the work of Babelon and Talalaev who showed that, through a change of variables, the Bethe Ansatz equations for Richardson-Gaudin type systems could be recast into a set of coupled polynomial equations. The roots of these equations are related to the e... |
as a means of efficient numerical solution of the conserved operator spectrum, and in to compute wavefunction overlaps. In these instances the equations are quadratic. Extensions were given in |
to a setting suitable for the [MATH] Hamiltonian, for which the polynomial equations are also quadratic. Here it will be shown that in this form, the continuum limit approach can be formulated and solved in such a way that it does not require an Ansatz for the distribution of the roots of the equations. |
The second primary objective is to apply this methodology for the calculation of the ground-state energy in the case of the open |
[MATH] Hamiltonian. Here it will be shown how the alternative method developed to compute the ground-state energy in the closed case easily extends to the open case. It will also be shown that the result is again in complete agreement with mean-field calculations. |
The general form of the integrable Hamiltonian is introduced in Sect. . The closed model, for which the coupling constant of the environment interaction is set to zero, is then described in detail in Sect. . Two forms of Bethe Ansatz solution are presented, and the continuum limit approximation for calculating the grou... |
The Hamiltonian The annihilation and creation operators for two-dimensional fermions of mass [MATH] with momentum [MATH] are denoted by [MATH] [MATH] , satisfying |
[EQUATION] We consider the following Hamiltonian of the pairing model interacting with its environment [EQUATION] where [MATH] [MATH] are positive real constants. The sum of momenta is taken over an index set [MATH] with the properties (i) if [MATH] , then [MATH] ; (ii) for all [MATH] we have [MATH] , where [MATH] is c... |
[EQUATION] Let [MATH] , we then introduce the following notation: [EQUATION] These operators satisfy the [MATH] algebra commutation relations |
[EQUATION] From now on, we use integers to enumerate the pairs of momentum states [MATH] and [MATH] . Setting the mass to be [MATH] and [MATH] , we rewrite ( ) as |
[EQUATION] Defining the following operators [EQUATION] it can be shown that [EQUATION] and [MATH] is a set of mutually commuting conserved operators. These operators have been shown |
to satisfy the following quadratic identities [EQUATION] Hence the eigenvalues [MATH] corresponding to [MATH] give the energy expression |
[EQUATION] and due to ( [MATH] satisfy [EQUATION] The [MATH] model isolated from the environment The [MATH] Hamiltonian is isolated from the environment (closed model) when [MATH] . In this case, we adopt the letters [MATH] and [MATH] instead of [MATH] and [MATH] . The Hamiltonian reads |
[EQUATION] where [EQUATION] The quadratic identity ( ) in this case becomes [EQUATION] The eigenvalues [MATH] of [MATH] in ( ) then according to ( 10 ) satisfy |
[EQUATION] Also note that in this case we have [EQUATION] 3.1 First form of Bethe Ansatz solution The Bethe Ansatz solution for ( ) was obtained in |
. We state the solution and its connection to the [MATH] in ( 11 ): of the coupled Bethe Ansatz equations [EQUATION] where [MATH] is the quantum number of particle-pairs, for each solution [MATH] known as the Bethe roots, there exists a correspondence between [MATH] and [MATH] given by the following |
[EQUATION] The techniques involved in achieving this connection ( 13 ) first appeared in and are also adopted in later. The corresponding energy for ( ) is given by |
[EQUATION] Each eigenstate has the form [EQUATION] where [MATH] denotes the vacuum state and [EQUATION] 3.2 Second form of Bethe Ansatz solution |
Alternatively, a second form of Bethe Ansatz solution can be derived from the hole-pair perspective for the isolated case. Let [MATH] denote the quantum number of hole-pairs. For each solution [MATH] of the coupled equations |
[EQUATION] there exists a correspondence between [MATH] and [MATH] given by the following [EQUATION] The corresponding energy is given by |
[EQUATION] Each eigenstate has the form [EQUATION] where [MATH] denotes the completely filled state of [MATH] particle-pairs and |
[EQUATION] 3.3 Symmetries of Bethe Ansatz solutions Introduce the following parameters, [EQUATION] such that [MATH] takes values in [MATH] and [MATH] in [MATH] as shown in Fig. |
There exists a rotational symmetry around the point [MATH] between the two forms of Bethe Ansatz solutions. For instance, if we have a solution [MATH] to ( 12 ) with [MATH] [MATH] , then this solution corresponds to a solution [MATH] to ( 14 ) with [MATH] [MATH] and [MATH] being fixed. |
Apart from this correspondence, there exists another type of relation which we call inversion. Inversion establishes an invertible mapping, given by a skewed reflection against the line [MATH] , between solution sets in regions [MATH] and [MATH] , and between solution sets in regions [MATH] and [MATH] . Regions [MATH] ... |
[EQUATION] Note that under inversion, [MATH] is no longer preserved. In other words, knowing a solution [MATH] to ( 14 ) with a certain set of parameters [MATH] , we also obtain a solution [MATH] to ( 14 ) with transformed parameters [MATH] |
When we combine the rotational symmetry and inversion, we achieve a correspondence between solution sets to ( 12 ) and ( 14 ). For instance, a solution [MATH] to ( 12 ) with [MATH] [MATH] and parameters [MATH] is also a solution [MATH] to ( 14 ) with [MATH] [MATH] and the same parameters [MATH] . Then the inversion of ... |
3.4 Integral approximation for the first form of Bethe Ansatz solution Numerical solution for the ground-state Bethe roots [MATH] in ( 12 ) and its peculiar behaviour under certain choices of parameters are discussed in |
and . The distribution of the Bethe roots suggests that they lie on curves in the complex plane, allowing an integral approximation to be applied. The continuum limit approximation for ( 12 ) where [MATH] is large is studied in |
. In the limit, we require that [MATH] also becomes large while [MATH] is finite, and similarly [MATH] becomes small while [MATH] is finite. |
First we formally define the discrete density for each single particle energy level [MATH] . Since all the [MATH] are real and positive it is natural to relabel them as [MATH] with [MATH] whenever [MATH] such that [MATH] is the smallest and [MATH] is the largest. The discrete root density [MATH] is defined as |
[EQUATION] such that [EQUATION] In the continuum limit, we introduce [MATH] to be the continuum density for [MATH] with connected support being a subset of [MATH] and replace all [MATH] with a continuous variable [MATH] . the continuum approximation for summation over any given function [MATH] is undertaken by replacin... |
[EQUATION] Setting [MATH] in ( 16 ) gives the normalization condition for the density, [EQUATION] We also introduce a continuous curve [MATH] , which is invariant under complex conjugation, to approximate the distribution of the ground-state Bethe roots [MATH] in ( 12 ). Let [MATH] be the density for [MATH] in the cont... |
[EQUATION] gives rise to a singularity, we adopt the Cauchy principal value to approximate the summation [EQUATION] where [MATH] denotes the Cauchy principal value of an integral. The continuum limit approximation for the Bethe Ansatz solution [MATH] for ( 12 ) reads as |
[EQUATION] where equation ( 19 ) is the normalization condition for the density [MATH] . The ground-state energy is given by [EQUATION] |
The solution curve [MATH] and [MATH] have been solved in and are dependent on the choice of the parameters [MATH] [MATH] as shown in Fig. |
. The solution curve [MATH] consists of two parts, a complex part [MATH] depicted by a solid line and a real part [MATH] depicted by one or more dashed lines, see Fig. |
. In Fig. (a), the complex arc [MATH] intersects the real part [MATH] at [MATH] . In Fig. (b), the arc [MATH] closes with an endpoint [MATH] on the negative real line as [MATH] arrives at the Moore-Read line from right. In Fig. |
(c), a line segment [MATH] forms on the negative real line adding a component to [MATH] . Then the point [MATH] will approach [MATH] with the complex curve [MATH] shrinking until it vanishes when [MATH] and [MATH] arrives at the Read-Green line from right. In Fig. |
(d), as [MATH] departs from the Read-Green line to the left, [MATH] becomes negative and [MATH] consists of one real part [MATH] |
3.5 Approximation for the second form of Bethe Ansatz solution Following the approach as discussed in previous Sect. 3.4 , the continuum limit approximation for the ground-state Bethe Ansatz solution [MATH] of ( 14 ) reads as |
[EQUATION] where [MATH] is a continuous curve introduced to approximate the distribution of the Bethe roots [MATH] , and [MATH] is the density for [MATH] in the continuum limit satisfying |
[EQUATION] The ground-state energy is given by [EQUATION] Now we need to solve for [MATH] and density [MATH] for the ground state. The solution curve [MATH] is proposed to be classified under the four phases as of the approximation for the first form Bethe Ansatz solution ( 18 ), see Fig. |
Here we have modified shapes deduced from the rotational symmetry combined with inversion discussed in Sect. 3.3 . Fig. (a)(b)(c) are topological inversion of Fig. |
(a)(b)(c). In Fig. (c), the point [MATH] approaches zero as [MATH] approaches the Read-Green line from right. As [MATH] departs from the Read-Green line and continues to move left, a third real line segment [MATH] and a second complex loop intersecting the real line at [MATH] and [MATH] appear, see Fig. |
(d). Further calculation gives rise to the same energy expression as from the integral approximation ( 18 ) and mean-field analysis |
3.6 Limitations of the continuum limit approximation for the Bethe Ansatz solutions The Moore-Read line is an example of a ground-state phase boundary line associated with changes in the topology of the root distribution. In previous Sect. |
3.4 and 3.5 , as we send parameters [MATH] from the weak coupling BCS phase to the Moore-Read line, the complex part of the solution curve [MATH] evolves until it closes and forms a loop. However in the discrete case, all the Bethe roots condense at the origin |
when the parameters reach the Moore-Read line. This discrepancy between the integral approximation and the discrete case is not present in models such as the Richardson [MATH] -wave pairing model and the [MATH] -wave pairing Hamiltonian |
where a similar approach of integral approximation is adopted. In the case of the two-level Richardson [MATH] -wave pairing model |
, the solution curve of the integral approximation evolves until it closes and forms a loop as some governing parameters approach a ground-state phase boundary line, meanwhile the discrete Bethe roots do not condense and their distribution is predicted by the solution curve within small error in the limit. In the case ... |
, the solution curve of the integral approximation contracts to a point at the origin and the discrete Bethe roots also condense at the origin as the governing parameters approach a ground-state phase boundary line. |
Due to the aforementioned discrepancy in the closed model, we perform a closer inspection of the integral approximation ( 18 ) in the Moore-Read line case with solution curve [MATH] depicted in Fig. |
(b). The general form of the density [MATH] for this case is proposed in to be the following, [EQUATION] The arc [MATH] is obtained by solving the following integral equation |
[EQUATION] where [MATH] is any point of [MATH] We first consider the limiting case where [MATH] . In this case the Bethe roots [MATH] are all real and lie within the interval [MATH] where [MATH] is the upper bound for [MATH] and [MATH] . Hence in the integral approximation the solution curve [MATH] reduces to [MATH] an... |
[EQUATION] which determines the value of [MATH] . As [MATH] increases, the complex part [MATH] starts to form and [MATH] decreases away from [MATH] . Consequently the allowable bound for [MATH] is [MATH] |
Now we consider the special case of an inverse-square density [EQUATION] This is a limiting case for the density [MATH] as we let it vanish on [MATH] while sending [MATH] . When [MATH] , substituting the inverse-square density into ( 23 ) we have [MATH] . Hence the constraint for [MATH] is |
[EQUATION] The equation for [MATH] is then derived from ( 22 ), [EQUATION] Since [EQUATION] the arc [MATH] is determined by the following equation |
[EQUATION] Choosing [MATH] to be [MATH] [MATH] is given by [EQUATION] The remaining constraint [MATH] from ( 21 ) implies that [EQUATION] |
hence we have [EQUATION] Setting [MATH] , from ( 26 ) we numerically determine that [MATH] . From ( 25 ), the solution curve for [MATH] is plotted as in Fig. |
). However, with this choice of parameters and inverse-square density, numerical results show that the intersection [MATH] is outside the allowable bound [MATH] from ( 24 ). This inconsistency is verified when we study the solution curve in the integral approximation for the second form of Bethe Ansatz solution ( 20 ) ... |
Alternatively, if we choose a constant density [MATH] with support on [MATH] for ( 18 ), it can be shown that for the integral approximation to be valid [MATH] cannot exceed the numerically determined value [MATH] . These results suggest that the validity of the continuum limit approximation is dependent on the choice ... |
Conserved operator eigenvalue method The difficulties in the integral approximation for the Bethe roots [MATH] arise from the task to find a suitable solution curve [MATH] and solve for its density [MATH] in ( 18 ). The next step is to adopt an alternative approach that eliminates these requirements and accommodates to... |
. Defining [EQUATION] from ( 13 ) we have [EQUATION] Since each solution [MATH] to ( 12 ) consists of complex-conjugate pairs or real numbers, [MATH] and [MATH] are all real. Substituting into ( 11 ) we obtain the following quadratic equations |
[EQUATION] where [MATH] . A continuum limit approximation applied to ( 28 ) leads to the following integral equation, [EQUATION] |
The objective now is to derive a solution to ( 29 ). We will establish several useful results before the solution is stated. Let |
[EQUATION] Note that [MATH] are determined by [MATH] and [MATH] . We require them to be a complex-conjugate pair, or both negative real numbers, i.e. |
[EQUATION] As a result [MATH] and [MATH] . The function [MATH] is the elementary real-valued square-root with [MATH] . In the limit where [MATH] , we require [MATH] to vanish also. In this case ( 30 ) becomes |
[EQUATION] where [MATH] is real and determined by [MATH] . All the following results and proofs leading to the expression for the ground-state energy for the closed model assume [MATH] . In the special case where [MATH] , simply replace [MATH] with the finite value [MATH] in ( 32 ), the proof of which follows a similar... |
Lemma 1 For all [MATH] , the following identity holds [EQUATION] Proof: Let [EQUATION] [EQUATION] For [MATH] [EQUATION] the proof of which is straightforward and omitted here. It can be shown that |
[EQUATION] This representation of [MATH] has no singularities for all [MATH] . The same expression can be found for [MATH] . Therefore [MATH] continuously extends to vanish for all [MATH] |
\qed Corollary 2 Let [EQUATION] then [EQUATION] Proof: Exchanging the variables [MATH] and [MATH] in the integrand, and then performing a change of order of integration yields |
[EQUATION] where the second step is due to Lemma 1 . Now we add up two distinct expressions for [MATH] [EQUATION] Since [EQUATION] |
we conclude that [EQUATION] \qed Lemma 3 Let [EQUATION] then [EQUATION] Proof: Since [EQUATION] again by adding up two distinct expressions for [MATH] , we have |
[EQUATION] Next [EQUATION] Finally, following a similar calculation as that for [MATH] [EQUATION] \qed Now we prove the following: |
Proposition 4 The following function [EQUATION] is a solution to the integral equation ( 29 ) with [MATH] subject to ( 30 ) and ( 31 ). |
Proof: [EQUATION] and [EQUATION] hence [EQUATION] On the RHS of ( 29 ), we first use Lemma 1 and simplify the following term [EQUATION] |
Since [EQUATION] we have [EQUATION] Finally, [EQUATION] \qed The integral approximation for ( 27 ) corresponding to the solution [MATH] given in ( 34 ) is |
[EQUATION] As a direct consequence of Proposition , we have the following result, Corollary 5 The function [MATH] defined in ( 35 ) is a solution to the integral approximation of ( 11 ), |
[EQUATION] This particular solution [MATH] corresponds to the ground state since from ( ) in the continuum limit, [EQUATION] the last step is due to expression [MATH] in Lemma . The energy expression ( 37 ) coincides with the ground-state energy derived by mean-field analysis |
and integral approximation of both forms of Bethe Ansatz solutions discussed in Sect. 3.4 and 3.5 despite its limitations. The [MATH] model interacting with its environment |
Consider the open model ( ) with extra terms governed by parameter [MATH] [EQUATION] 5.1 Bethe Ansatz equations and numerics The Bethe Ansatz solution for ( ) was derived in |
: for each solution [MATH] of the coupled equations [EQUATION] where [MATH] [MATH] and [MATH] there is a correspondence between [MATH] in ( ) and [MATH] via a change of variables |
[EQUATION] This translates to the fact that ( ) and ( 38 ) are equivalent. It is worth mentioning that the difference between the closed and open model is that in the closed model, it is possible for the number of Bethe roots [MATH] to be less than [MATH] , while in the open model we must have exactly [MATH] Bethe root... |
. The energy is given by ( ), [EQUATION] The corresponding eigenstate reads as [EQUATION] We perform some numerical analysis to study the behaviour of the Bethe roots [MATH] . We consider a small-sized case where [MATH] . As we send [MATH] from [MATH] to a large number, numerical results (see Tab. ) show that one of [M... |
and 5.2 Conserved operator eigenvalue method for the open model The numerical analysis of the distribution of the Bethe roots gives no clear indications of a solution curve for a large particle number. However, since ( 38 ) is invariant under complex conjugation, again assuming the absence of degeneracy in the spectrum... |
[EQUATION] where [MATH] . Our task is to find a solution to ( 41 ) that corresponds to the ground state. We resort to mean-field analysis for suggestions of a possible solution. The mean-field analysis for the open model and its results are included in Appendix A. We consider the operators [MATH] defined in ( ). From (... |
[EQUATION] where [EQUATION] The integral approximation for ( 42 ) is [EQUATION] We will establish that ( 43 ) is in fact a solution to ( 41 ), i.e. |
[EQUATION] with the “gap” equation determining the value of [MATH] [EQUATION] In addition, this solution [MATH] corresponds to the ground state. We again establish some useful results to assist our proof. |
Lemma 6 Given equation ( 44 ), let [EQUATION] then [EQUATION] Proof: In ( 32 ), set [EQUATION] then we have [MATH] and [MATH] . Hence from ( 35 ) and by Corollary |
[MATH] satisfying ( 36 ), [EQUATION] Next [EQUATION] Finally, [EQUATION] \qed Now we show the following: Proposition 7 The following function |
[EQUATION] is a solution to the integral equation ( 41 ) with [MATH] subject to ( 44 ). Proof: We adopt the notation introduced in Lemma , rewriting |
[EQUATION] Since [EQUATION] and [EQUATION] then from ( 45 ) we are able to simplify equation ( 41 ) as [EQUATION] Since [EQUATION] |
[EQUATION] and we use the special case of [MATH] for expression [MATH] in Lemma , where [MATH] [MATH] and [MATH] are replaced by [MATH] [MATH] and [MATH] respectively, consequently we have |
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