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[EQUATION] and the right hand side [EQUATION] where the integral over [MATH] was performed as in Eq. ( 2.28 ). Taking the Fourier transform of this equation yields |
[EQUATION] and so the Fourier transformed density is [EQUATION] To obtain the density, one need only Fourier transform Eq. ( 2.33 ). First note that on the real line it is equal to the analytic function given by simply removing the absolute values. With a small perturbation which can later be removed, this function shr... |
[EQUATION] where the residues are [MATH] . Therefore the density is [EQUATION] Thus [EQUATION] This equation is the starting point for studies of the thermodynamics of this model. |
We will need explicit expressions for the various functions of [MATH] . To find these, we must solve Eq. ( 2.36 ). Multiplying through by [MATH] cosh [MATH] and integrating one obtains |
[EQUATION] which is easily inverted to obtain [EQUATION] Substituting this into Eqs. ( 2.14 ) and ( 2.15 ) gives the needed results |
[EQUATION] Similarly one finds [EQUATION] Note that the function [MATH] given in Eq. ( 2.38 ) is an exact solution of the continuum equation ( 2.22 ) but not of the exact discrete equation ( 2.18 ). At large [MATH] with [MATH] constant, the [MATH] th equation in Eq. ( 2.18 ) is violated by [MATH] where [MATH] is indepe... |
In fact in Subsec. we will see that the [MATH] contribution to the variance of the anchored [MATH] is the difference between two terms which differ by about 1%. In principle, it is possible that such a small difference is an artefact of the continuum approximation and would vanish if we solved the original discrete sys... |
The Anchor Recall that the coordinate Bethe Ansatz expresses the matrix elements in the form [EQUATION] where the phase [EQUATION] |
depends on the permutation [MATH] . At large [MATH] , the sum ( 3.1 ) becomes an integral ( 1.1 ) with measure given by the density [MATH] . Eq. ( 1.1 ) states that the matrix elements are given by the Fourier transform of [MATH] . This function is shown in Fig. in the case of the matrix element [MATH] at [MATH] and [M... |
The role of the anchor [MATH] is to shift [EQUATION] so as to cancel out the substructure. The anchor [MATH] will be an integral multiple of [MATH] and so the shift will not affect [MATH] . Therefore the substitution of [MATH] with [MATH] leaves the matrix elements invariant. To see how the anchor works, and to motivat... |
3.1 Type I Cyclic Permutations One can define a free action of the cyclic group [MATH] on the permutation group [MATH] as follows. Let the generator [MATH] act on [MATH] by |
[EQUATION] Restrict our attention to matrix elements with the classical ground state [MATH] , which corresponds to [MATH] . In this case, and only in this case, we will now show that [MATH] is an exact symmetry of the phases |
[EQUATION] Indeed, [MATH] is easily calculated [EQUATION] Now, fixing [MATH] we find [EQUATION] Using the antisymmetry of [MATH] this simplifies to |
[EQUATION] [MATH] is symmetrically distributed about [MATH] and so the first term on the right hand side is just [MATH] . Bethe’s equation ( 2.8 ) on the other hand gives the sum of the second and third terms to be [MATH] . Putting this all together we obtain |
[EQUATION] This is an integer multiple of [MATH] . Thus we have shown that these cyclic permutations leave each summand in the matrix elements invariant, and yet they affect the arguments [MATH] and so complicate the distribution [MATH] . Clearly, to calculate this distribution, it would be desirable to remove these sp... |
Consider a second action of the generator of the cyclic group. Now [EQUATION] and so [EQUATION] In general the element [MATH] of the cyclic group shifts the arguments by |
[EQUATION] How can we modify [MATH] to prevent these spurious shifts? Recall that [MATH] is a bijection and so it is invertible and the inverse transforms under the cyclic action by |
[EQUATION] Choose any integer [MATH] and define [EQUATION] How does this transform? [EQUATION] and so the difference is [EQUATION] |
As [MATH] and [MATH] transform identically under the cyclic permutations, their difference [MATH] is invariant. Thus [MATH] so defined is an anchor which fixes these cyclic permutations. However it is not the only such anchor. One may add to it any other integral multiple of [MATH] which is invariant under these cyclic... |
3.2 Type II Cyclic Permutations The symmetric group [MATH] admits another free [MATH] action, whose generator acts by [EQUATION] |
This action does not leave the phases invariant. But, for [MATH] not too close to the boundaries, it leaves the phases [MATH] reasonably invariant while dramatically shifting the arguments [MATH] . Repeating the calculation as above, with this action, one obtains |
[EQUATION] This time the calculation is more difficult. Again consider the classical ground state [MATH] . Now if [MATH] is a cyclic permutation |
[EQUATION] then [EQUATION] and so type II cyclic permutations are, in this case, identical to type I cyclic permutations. Therefore as before |
[EQUATION] The trivial rewriting in the last step will allow this result to approximately generalize to other permutations [MATH] as we will now explain. |
Any two permutations [MATH] in the symmetric group [MATH] are related by a series of basic permutations in which pairs of adjacent numbers are permuted. In particular, any [MATH] is related to a cyclic permutation, for which ( 3.21 ) holds, by some series of basic permutations. We have checked numerically, for [MATH] ,... |
Again it is not difficult to construct an anchor which reproduces this transformation law for an arbitrary type II cyclic permutation |
[EQUATION] where [MATH] is arbitrary. However this anchor does not leave the type I cyclic permutations invariant, and as can be seen in Fig. it causes a reduction in the scatter of [MATH] which is comparable to that of [MATH] |
3.3 A Universal Anchor We propose that the argument [MATH] in the Bethe Ansatz ( 3.1 ) and ( 3.2 ) be replaced by the anchored argument |
[EQUATION] where the anchor [MATH] is defined by [EQUATION] where the Heaviside step function is [EQUATION] The trivial permutation [MATH] gives [MATH] . More generally, this counts the number of pairs of sites whose order is flipped by [MATH] |
We will see that the anchored argument [MATH] has a number of nice properties, not shared by [MATH] . In this subsection we will see that it is invariant under type I permutations. [MATH] jumps by [MATH] under type II permutations, and so while [MATH] is not invariant under these permutations, its shift is relatively m... |
How does the anchor work? For example, begin with the identity permutation [MATH] . Now consider the type I cyclic permutation, it yields |
[EQUATION] In this case [MATH] and so a single entry has moved from the right to the left of all other [MATH] entries, all of which were smaller. Thus the sum gains contributions from all elements with [MATH] |
[EQUATION] And so we see that [MATH] transforms just like [MATH] under this cyclic permutation of type I. In fact, the transformation ( 3.26 ) is not only the generator of type I cyclic permutations, but also type II cyclic permutations, which coincide in this example because [MATH] is just a shift. Now |
[EQUATION] and so the anchor compensates for the type II permutation as well, as it must since this is also a type I permutation. |
What about general elements of [MATH] ? Beginning with an arbitrary element [MATH] , a type I permutation yields [EQUATION] and so our anchor transforms to |
[EQUATION] yielding a difference of [EQUATION] which equals [MATH] calculated in Eq. ( 3.9 ). Therefore [MATH] is invariant under type I permutations. |
What about type II permutations? Now [EQUATION] and so [EQUATION] The difference is then [EQUATION] which agrees with the approximation to the shift in [MATH] found in Subsec. 3.2 . Therefore [MATH] is approximately invariant under both kinds of cyclic permutations. |
We need more. We need [MATH] to be free of substructure, so that its moments yield a well-behaved expansion about a Gaussian. In the case of the matrix element of the classical and quantum ground states [MATH] at [MATH] , so that [MATH] , these properties are demonstrated numerically in Fig. . One sees that the full wi... |
This is our first main result. With the anchor ( 3.24 ) the distribution of phases [MATH] in the CBA becomes approximately a Gaussian and so the calculation of the matrix elements in Eq. ( 1.1 ) requires only that one determine its moments. In the rest of this note, we will describe a method for the calculation of thes... |
3.4 Other matrix elements Of course we are not only interested in the matrix element [MATH] . Our anchor was motivated by the fact that the type I shift symmetry leaves [MATH] invariant in the case of the classical ground state [MATH] . This is not true for other states. So how well does the anchor perform when [MATH] ... |
First let us consider a small change, leaving all [MATH] except for [MATH] . Let us call this state [MATH] . In Fig. we see that this shift in [MATH] leads to a shift in [MATH] , but the shape and variance are not noticeably affected. What about matrix elements with states that are further from the classical vacuum? Co... |
[EQUATION] at [MATH] and [MATH] respectively. In Fig. we compare the distribution of [MATH] in the case of [MATH] with that of [MATH] . The values of [MATH] in the state [MATH] were chosen at random, so that it may represent a generic state. One sees that for this state the shape of [MATH] is still quite similar to the... |
Binning Exact calculations of matrix elements have been a major industry for decades. However as we are interested in the continuum field theory, our goal is somewhat different. It is more difficult, because we will need a method which calculates matrix elements for states which differ at arbitrarily many lattice sites... |
, which is reviewed in Appendix A 4.1 The Binning This motivates the following approach. Let [MATH] be an integer. We will divide the interval [MATH] into [MATH] bins |
[EQUATION] Recall that an element [MATH] is completely characterized by a bijection [MATH] . Let [EQUATION] where [MATH] is the cardinality of the set [MATH] . In other words, [MATH] is the number of entries of [MATH] which [MATH] maps into [MATH] . Clearly [MATH] contains only some of the information in [MATH] , while... |
The Binning Postulate: For the calculation of a given quantity [MATH] to any precision [MATH] , there exists a sufficiently high [MATH] such that, if [MATH] is calculated replacing all [MATH] with the same [MATH] by the same [MATH] then the introduced error in [MATH] will be bounded by [MATH] |
It may be that the binning postulate is false, or that it is true only at some leading orders in [MATH] . Certainly it is false for many quantities [MATH] . It is our hope that the binning postulate is true, however, for all [MATH] accessible in the continuum field theory. This requires that, in the continuum limit, th... |
At least at the small values of [MATH] accessible to brute force numerical calculations, there is no evidence that the binning postulate holds for [MATH] itself. As shown in Fig. 10 the intrabin and interbin variances of [MATH] at [MATH] are comparable. Whether it holds at large [MATH] may depend on the relation betwee... |
With these strong conjectures in hand our strategy is clear. We will recast our problem in terms of [MATH] , assuming that with a suitable choice of [MATH] the intrabin contributions to various quantities vanish in the [MATH] limit. |
We have checked this in some cases as follows. The expressions below often contain nested sums over bins with inequalities, such as [MATH] . The summand in which two bins are equal, such as [MATH] , is not clearly defined by our procedure. For example, in terms involving [MATH] or [MATH] it depends on the permutations ... |
Now our binning approximation is [EQUATION] Here and from now on, we drop the prime on the anchored argument [MATH] as we will no longer need the unanchored [MATH] . These expressions are the definitions of our binned [MATH] , and so no large [MATH] or [MATH] limit needs to be taken. However, even in the case of quanti... |
Our strategy will be as follows. The matrix elements of interest can be expressed in terms of moments of [MATH] where [MATH] is a function of [MATH] . Therefore the moments are averages over the group [MATH] . The binning approximation lets us replace [MATH] with [MATH] . The moments of [MATH] are averages over the spa... |
4.2 Simplifications at First Order This can be somewhat simplified. First note that each of the [MATH] elements of [MATH] is mapped to some [MATH] by [MATH] . This yields the sum rule |
[EQUATION] Similarly all [MATH] elements of [MATH] are in some [MATH] yielding the second sum rule [EQUATION] These sum rules hold individually for every [MATH] |
Let us define the expectation value of [MATH] by [EQUATION] Higher correlators are defined similarly. It is quite clear that [MATH] is independent of [MATH] and [MATH] . Therefore the expectation value of either sum rule yields |
[EQUATION] This quantity will appear so often that we will name it [EQUATION] Many quantities are more simply expressed in terms of the reduced |
[EQUATION] From the corresponding properties of [MATH] one finds [EQUATION] These sum rules hold exactly for any value of [MATH] and [MATH] , so long as [MATH] is an integer. |
We can now use ( 4.3 ) to express the Bethe phases in terms of [MATH] . The first is [EQUATION] Let us fix our reference state to be the classical ground state [MATH] and so [MATH] . Then this becomes |
[EQUATION] where we used the fact that [MATH] is symmetric about [MATH] . The [MATH] correction to the first term is an artefact of our treatment of interbin effects, and could be changed if we changed our prescription for these by, for example, adding terms [MATH] to consider cases in which [MATH] but nonetheless a gi... |
Next we will treat [MATH] [EQUATION] Note that the term with no [MATH] vanishes because [EQUATION] This expression is exact only at [MATH] and also in the large [MATH] limit for any [MATH] . The deviation from zero at subleading orders in the [MATH] expansion is an artefact of the binning approximation, which should no... |
It may appear that the term linear in [MATH] in Eq. ( 4.18 ) vanishes as a result of the sum rule, but it does not as [MATH] and so it is not summed over all bins. However [MATH] and [MATH] are summed over all bins, and so we can apply the binned version of the Bethe equation ( 2.8 ), which in the case of the ground st... |
[EQUATION] Now we are ready to evaluate the terms linear in [MATH] . It turns out that they are equal, so we will show the evaluation of the [MATH] term |
[EQUATION] which can be cleaned using the sum rule [EQUATION] As the [MATH] term is equal to the [MATH] term, we have found [EQUATION] |
Here we see our first major cancellation. The second term of [MATH] exactly cancels the second term in [MATH] as written in Eq. ( 4.2 ). Thus the function [MATH] disappears from the phase factor [MATH] , and only a constant remains of [MATH] |
Finally we turn to [MATH] [EQUATION] The term with no [MATH] is easily evaluated [EQUATION] This cancels half of the remaining constant term in [MATH] in Eq. ( 4.2 ). These constant terms then yield |
[EQUATION] In the case [MATH] , corresponding to no binning , the expectation values for [MATH] and for this full anchored combination are visible in Fig. and one indeed sees that the later is a bit more than half of the former. Why a bit more? Should not [MATH] be an artifact? When [MATH] [MATH] should either tend to ... |
The Bethe phase [MATH] can be simplified yet further. We have seen that it contains terms which are constant, linear and quadratic in [MATH] . The constant terms where summed in Eq. ( 4.21 ). The two linear terms are equal, and so to evaluate their sum we will simply multiply the [MATH] term by [MATH] |
[EQUATION] This is equal to the first term in [MATH] as written in Eq. ( 4.18 ), leading to our second major cancelation. Putting all remaining terms together we have found our master formula for the anchored phase |
[EQUATION] We will soon see that [MATH] at leading order in [MATH] and so we may already try to estimate the fluctuations of the anchored phase in the large [MATH] and [MATH] limit. The first term is a constant and so does not contribute. The second two have [MATH] . The [MATH] cancels with the sums, up to a factor of ... |
[MATH] . On the other hand the four point function of [MATH] in the Gaussian approximation would give [MATH] , and so we find a variance of [MATH] and so a standard deviation of [MATH] |
The canceled term in Eq. ( 4.22 ) has a larger variance. Consider the square of this term. The term contains [MATH] and so its square contains [MATH] , yielding [MATH] as above. Again, as above, the [MATH] in the [MATH] is canceled by eight sums over bins. The difference is that this term only contains a single power o... |
4.3 Bin Statistics from Partitions: One Point Finally we are ready to calculate correlation functions of [MATH] . These are averages of products of [MATH] over the symmetric group [MATH] . To calculate them, one must count how many members [MATH] give each value for a given polynomial in [MATH] . Let us warm up by cons... |
Let us call this number [EQUATION] As the symmetric group [MATH] has [MATH] elements, the probability that a given [MATH] satisfies [MATH] is then |
[EQUATION] Recall that [MATH] must map each integer in [MATH] to a distinct integer in [MATH] . If [MATH] elements of [MATH] are to map to [MATH] , one needs to choose which [MATH] elements of [MATH] are in [MATH] . Recalling that each bin has [MATH] elements, the number of choices is [MATH] . One must also choose the ... |
[EQUATION] These later factors are independent of [MATH] and so will not be important in future calculations, as they only contribute to the overall normalization which is fixed by the fact that |
[EQUATION] So let us separate all of the [MATH] -independent terms into a constant [MATH] [EQUATION] where we have defined the falling and rising factorials |
[EQUATION] Curiously, [MATH] is the [MATH] th term in the Gauss series for the hypergeometric function [MATH] So far these expressions are exact for all [MATH] and [MATH] . We will be interested in the limit where [MATH] while [MATH] , which is of order [MATH] , will be finite or slowly tend to [MATH] . In this limit t... |
To find a suitable approximation for the ratios of factorials in this limit, we combine the expansion [EQUATION] with Stirling’s approximation |
[EQUATION] and the binomial expansion to obtain our main tool [EQUATION] With this tool in hand, we can approximate [MATH] . If we let [MATH] and expand to order [MATH] , for example, we find |
[EQUATION] Note that the leading term is a Poisson distribution times [MATH] . Therefore the expectation value of any function of [MATH] can be given in terms of Poisson correlators |
[EQUATION] In particular, by setting the expectation value of [MATH] to be equal to [MATH] , we can fix [MATH] at any desired order. In this case the relation between Eq. ( 4.33 ) and the Poisson distribution yields |
[EQUATION] Then inserting the Poisson expectation values from Eq. ( 4.34 ) one finds [EQUATION] and so obtains [MATH] at [MATH] [EQUATION] |
Any other correlator can be found similarly, using ( 4.33 ) to relate the desired correlator to a combination of Poisson correlators. For example, |
[EQUATION] This spectacular order by order cancellation is in fact required by the sum rule, as was argued above, and so provides a consistency check of our approximations. |
Higher orders in [MATH] have useful information for correlators of distinct [MATH] . However, for our purposes in this Subsection, for correlators at a single [MATH] it suffices to use the leading term, given by the Poisson distribution. At this order |
[EQUATION] We can then find arbitrary correlators of [MATH] at the same point. For example [EQUATION] This is reasonable. It means that so long as [MATH] [MATH] will stay away from its minimal value of [MATH] , where [MATH] vanishes, and so is reasonably well approximated by a Gaussian. As [MATH] is quadratic in [MATH]... |
[EQUATION] The first term is usual disconnected contribution to the four point function, in which the [MATH] s are paired into 3 possible pairs of pairs and their two point correlations are used. These give a result of order [MATH] which, combined with the [MATH] in [MATH] in Eq. ( 4.23 ) yields [MATH] and so a varianc... |
4.4 Bin Statistics from Partitions: Multiple Points In general we will need correlators of [MATH] with different indices. There are two ways to generalize the above calculation to multiple indices. The first is to use the sum rule to extrapolate new correlators from old correlators. This is sufficient to derive all of ... |
Let us begin with the sum rule approach. Once we know that in the large [MATH] and [MATH] limit, with [MATH] unconstrained [EQUATION] |
the sum rule ( 4.10 ) implies that [EQUATION] for all [MATH] and [MATH] . In the last expression we have used the large [MATH] limit. A repeated application of the same sum rule yields |
[EQUATION] for [MATH] and [MATH] We will denote these correlations using the following diagrams [EQUATION] Here the rows are the [MATH] indices which are contracted with [MATH] in our master formula ( 4.23 ), while the columns are the [MATH] indices which are ordered. Recall that [MATH] is represented as a map [MATH] a... |
The [MATH] approximation to the four point functions then follow from simply summing together the three pairs of products of two point functions. For example, if [MATH] and [MATH] then at leading order |
[EQUATION] while [EQUATION] corresponding to the diagrams [EQUATION] There are contributions from other combinations of pairings of the points, but these are subdominant in [MATH] |
In general to calculate correlators at distinct points, the sum rules are not sufficient. However the above partition argument can be generalized. For concreteness, let us consider a correlator corresponding to a diagram with 2 rows and 2 columns. This means that we will be interested in two domain bins [MATH] and two ... |
[EQUATION] The joint probability [MATH] is just the number of elements [MATH] satisfying ( 4.60 ) divided by [MATH] It can be calculated as in the [MATH] case treated above. First, one needs to choose [MATH] elements of [MATH] to be in [MATH] . There are [MATH] such choices. Similarly there are [MATH] choices for the i... |
[EQUATION] Again this expression is exact for all [MATH] and [MATH] . One sees that the terms with isolated [MATH] ’s cancel, only those with entire rows [MATH] or columns [MATH] remain. |
The first four ratios enforce the correlations caused by the sum rules corresponding to each of the two rows and each of the two columns, while the last enforces the sum rule on the entire matrix. This may be expanded using our main tool ( 4.3 ) and any correlation function may then be calculated as a sum of the corres... |
The generalization to [MATH] domain bins (columns) and [MATH] image bins (rows) is clear. There are [MATH] choices of maps and so a factor of [MATH] in the denominator. The numerator consists of [MATH] descending factorials, each [MATH] with an argument equal to the sum of the [MATH] ’s in the corresponding row or colu... |
Testing the Anchor In the large [MATH] limit, what is the variance of [MATH] 5.1 The Variance of [MATH] at [MATH] Let us warm up with [MATH] as given in Eq. ( 4.2 ). There are two terms. First, a constant term, which doesn’t contribute. We will drop it. Next is |
[EQUATION] As [MATH] [MATH] and so the variance is [EQUATION] This is the sum of four terms depending on whether [MATH] and whether [MATH] , each summand corresponding to a diagram. |
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