Absence of high-temperature ballistic transport in the spin-$1/2$ $XXX$ chain within the grand-canonical ensemble

Whether in the thermodynamic limit, vanishing magnetic field $h\rightarrow 0$, and nonzero temperature the spin stiffness of the spin-$1/2$ $XXX$ Heisenberg chain is finite or vanishes within the grand-canonical ensemble remains an unsolved and controversial issue, as different approaches yield contradictory results. Here we provide an upper bound on the stiffness and show that within that ensemble it vanishes for $h\rightarrow 0$ in the thermodynamic limit of chain length $L\to\infty$, at high temperatures $T\rightarrow\infty$. Our approach uses a representation in terms of the $L$ physical spins $1/2$. For all configurations that generate the exact spin-$S$ energy and momentum eigenstates such a configuration involves a number $2S$ of unpaired spins $1/2$ in multiplet configurations and $L-2S$ spins $1/2$ that are paired within $M_{\rm sp}=L/2-S$ spin-singlet pairs. The Bethe-ansatz strings of length $n=1$ and $n>1$ describe a single unbound spin-singlet pair and a configuration within which $n$ pairs are bound, respectively. In the case of $n>1$ pairs this holds both for ideal and deformed strings associated with $n$ complex rapidities with the same real part. The use of such a spin $1/2$ representation provides useful physical information on the problem under investigation in contrast to often less controllable numerical studies. Our results provide strong evidence for the absence of ballistic transport in the spin-$1/2$ $XXX$ Heisenberg chain in the thermodynamic limit, for high temperatures $T\to\infty$, vanishing magnetic field $h\rightarrow 0$ and within the grand-canonical ensemble.


I. INTRODUCTION
The anisotropic spin-1/2 XXZ Heisenberg chain [1] with anisotropy parameter ∆ ≥ 0, exchange integral J, and Hamiltonian, J L j=1 (Ŝ x jŜ x j+1 +Ŝ y jŜ y j+1 + ∆Ŝ z jŜ z j+1 ), whereŜ x,y,z j are components of the spin-1/2 operators at site j = 1, ..., L, is a paradigmatic example of an integrable strongly correlated quantum many-body system. However, the isotropic point at ∆ = 1 (the spin-1/2 XXX Heisenberg chain [2,3]) is the most experimentally relevant [4][5][6]. It is also the case that poses the most challenging technical problems for theory. For instance, the problem of clarifying the possibility of ballistic spin transport at nonzero temperatures in the spin-1/2 XXX chain in a magnetic field h is one of the most intensely debated unsettled fundamental questions in the theory of strongly correlated systems. Its Hamiltonian with periodic boundary conditions reads, where h ∈ [−h c , h c ], µ B is the Bohr magneton and ±h c = ±J/µ B are the critical fields for fully polarized ferromagnetism. The model's spin stiffness D(T ), also called spin Drude weight, defined via the singularity in the real part of the spin conductivity, σ(ω, T ) = 2π D(T ) δ(ω) + σ reg (ω, T ) , can be interpreted as a quantitative measure of ballistic spin transport. In the thermodynamic limit (TL), L → ∞, the corresponding stiffness expressions given below in this paper involve the expectation values of the z-component spin current operator,Ĵ whereŜ ± j =Ŝ x j ± iŜ y j .
For instance, the schemes used in the studies of Refs. [11,13,15,21,22] lead to a finite value for the spin stiffness at nonzero temperature. In contrast, the investigations of Ref. [4] indicate that transport at finite temperatures is dominated by a diffusive contribution, the spin stiffness being very small or zero. Such studies exclude the large spin stiffness found in Ref. [15] by a phenomenological method that relies on a spinon and anti-spinon basis for the thermodynamic Bethe ansatz (TBA) [3]. The results obtained by a completely different and more direct use of the TBA in Refs. [8,9] as well as the more recent results of Ref. [20] that rely on the combination of several techniques find a vanishing spin stiffness for zero spin density.
The nature of the exotic spin transport properties at nonzero temperature of one-dimensional (1D) correlated lattice systems has been a problem of also experimental interest [5,6,[27][28][29][30][31]. The spin stiffness is directly related to the long-time asymptotic current-current correlation function as, (The angle brackets . denote here the thermal average.) In integrable models there is a lower bound for D(T ), which is encoded in an inequality due to Mazur [32], Here the sum runs over a complete set of linearly extensive orthogonal commuting conserved quantitiesQ j for which Q 2 j ∝ L, local and quasilocal [18,24,[33][34][35]. In the case of the spin-1/2 XXZ chain, the sum over strictly local conserved quantities responsible for integrability gives at nonzero temperatures (i) a finite value and thus ballistic spin transport for h = 0 and (ii) vanishes and is inconclusive at h = 0.
Two recent results provided some essential preliminary steps for the clarification of the problem studied in this paper. The first of these results is that the Mazur's inequality sum over quasilocal conservation laws associated with deformed symmetries gives for the spin-1/2 XXZ chain a stiffness lower bound at h = 0, D l (T ) ≤ D(T ), which for T → ∞ reads [24], It refers to a dense set of commensurate easy-plane anisotropies, ∆ = cos(πl/l ′ ), where l, l ′ ∈ Z + and l ≤ l ′ > 0 are such that 0 ≤ ∆ ≤ 1. Since this lower bound vanishes at the isotropic point, ∆ = 1, it does not discard the possibility that the spin stiffness of the spin-1/2 XXX chain is also vanishing as h → 0.
The second recent result presented in Ref. [26] is a upper bound for the spin stiffness of the spin-1/2 XXX chain, D u (T ) ≥ D(T ), valid within the canonical ensemble for spin densities m ∈ [0, 1] and the whole T > 0 range, in the TL. Its limiting behaviors are, That D u (T ) vanishes as m 2 L in the m → 0 limit ensures that within the canonical ensemble the stiffness vanishes as m → 0 yet leaves out, marginally, the grand canonical ensemble as h → 0 in which m 2 = O(1/L). A schematic phase diagram of temperature T versus spin density m of ballistic spin transport is shown in Fig. 1.
In this paper we provide new insights on the above unsolved problem concerning the spin stiffness for the spin 1/2 XXX chain in the TL. Specifically, we provide strong theoretical evidence that for high temperatures T → ∞ it also vanishes for h → 0, within the grand-canonical ensemble. While for a canonical ensemble one considers that the spin density m is kept constant, in the case of a grand-canonical ensemble it is the magnetic field h that is fixed. In general the canonical-ensemble and grand-canonical ensemble lead to the same results in the TL. This is generally true except near a phase transition or a critical point. Hence this issue deserves a careful analysis in the m → 0 and h → 0 limits, respectively. The use of effective spinon representations [36][37][38] provides a suitable description of the model low-energy physics and excitations of the S = 0 ground state. However, they do not apply to high-temperature problems at a magnetic field h ∈ [−h c , h c ] that involve all 2 L energy eigenstates, as that studied in this paper. Our approach then rather uses the representation of Ref. [26] in terms of the spin-1/2 XXX chain L physical spins 1/2. Within such a representation, all configurations that generate the exact energy and momentum eigenstates of spin S involve a number 2S of unpaired spins 1/2 in multiplet configurations and L − 2S spins 1/2 that are paired within M sp = L/2 − S spin-singlet pairs. Within the TBA, the imaginary part of the complex rapidities simplify in the TL, which corresponds to the ideal strings of length n > 1 [3]. For large L values there is in addition two types of deformed complex rapidities that deviate from such an ideal behavior [39][40][41].
Importantly, the general representation in terms of 2S unpaired physical spins 1/2 plus M sp = L/2 − S spin-singlet pairs of physical spins 1/2 used in the studies of this paper applies both to the TBA [3] and to BA schemes including three types of complex rapidities [39], respectively. On the one hand, both for an ideal string and a deformed string of length n > 1 the corresponding set of n complex rapidities with the same real part refer to an independent configuration with a number n of spin-singlet pairs bound within it. On the other hand, the real rapidities correspond to single unbound spin-singlet pairs.
Our derivation relies on the spin stiffness expression in terms of matrix elements of the z-component current operator, Eq. (3), and the operator algebra relating that operator to both the other two SU (2) symmetry operator components,Ĵ and the three generatorsŜ η = L j=1Ŝ η j , η = ±, z of that global symmetry. This includes the commutators, which follow directly from the SU (2) algebra for the operators under consideration. There is a general consensus that the use of ideal strings of TBA for energy and momentum eigenstates described by groups of real and complex rapidities [3] leads in the TL to exact results as long as either the temperature or the magnetic field are nonzero [39,42]. Our studies involve the spin stiffness at very hight temperature, T → ∞, so that concerning thermal effects they are not affected in the TL by the string deformations. On the one hand, concerning the case h = 0, we use a method other than the BA or TBA to compute the exact current operator expectation values of the corresponding S z = 0 energy and momentum eigenstates [26]. On the other hand, in what the contributions to the spin stiffness for the model at finite magnetic field from the square of current operator expectation values of finite-S z energy and momentum eigenstates is concerned, we rely on upper bounds. In contrast to those used in Ref. [26], the present upper bounds involve sums that run over a large, macroscopic number, of energy and momentum eigenstates. As justified below in Sec. VI, such upper bounds are in the TL insensitive to the use of ideal [3] or deformed [39] BA strings.
Our representation in terms of configurations of the L physical spins 1/2 provides useful physical information on the problem under investigation, in contrast to the often less controllable numerical studies on the occurrence or lack of ballistic spin transport in the spin-1/2 XXX chain as h → 0 in the TL.
The remainder of the paper is organized as follows. In Sec. II the finite-temperature spin stiffness and the representation in terms of configuration of the L physical spins 1/2 used in the studies of this paper are introduced. The general expressions of the spin stiffness at high temperature T → ∞ is the issue addressed in Sec. III. In Sec. IV a non-BA-related method used to compute the spin currents of the S z = 0 energy and momentum eigenstates for the strictly zero magnetic-field case is briefly reported and the physical consequences of the corresponding exact results are discussed. Useful and needed inequalities and corresponding current absolute values upper bounds are introduced in Sec. V. The effects of the string deformations on the spin currents in the TL at finite magnetic field is the issue addressed in Sec. VI. In Sec. VII the high-temperature stiffness upper bounds within the TL used in our study are derived. Finally, the concluding remarks are presented in Sec. VIII. Additional technical information useful for details of our analysis is provided in Appendices A and B.

II. THE FINITE-TEMPERATURE SPIN STIFFNESS AND L PHYSICAL SPINS 1/2 CONFIGURATIONS
We denote the energy eigenstate's spin and spin projection by S and S z = −(N ↑ − N ↓ )/2, respectively. Here N ↑ and N ↓ such that L = N ↑ + N ↓ are the numbers of spins 1/2 with up and down spin projection, respectively. For the so-called lowest-weight-states (LWSs) and highest-weight-states (HWSs) of the SU (2) algebra we have S = −S z and S = S z , respectively. The class of LWSs and the non-LWSs generated from those that are used in our analysis are energy and momentum eigenstates. They are as well eigenstates of (ˆ S) 2 andŜ z with eigenvalues S(S + 1) and S z , respectively. We thus label all 2 L energy, momentum (as well as spin and spin projection) eigenstates by |l r , S, S z . Here l r stands for all quantum numbers other than S and S z needed to specify an energy and momentum eigenstate, |l r , S, S z . This is independent of using the general BA or the TBA for these states, always holding that lr = N singlet (S) for the model in each fixed-S subspace. Here N singlet (S) = L L/2−S − L L/2−S−1 is that subspace number of independent spin-singlet configurations and thus N (S) = (2S + 1) N singlet (S) is its dimension. Since the LWSs and non-LWSs generated from them considered in this paper are energy and momentum eigenstates, these designations are often used for the latter states.
Within the canonical-ensemble description at fixed value of S z , the spin stiffness D(T ) expression involves the current operator expectation values, l r , S, S z |Ĵ|l r , S, S z , which in the TL and for nonzero temperatures are the current matrix elements that contribute to it [7,26,43]. As justified below in Sec. IV, for the non-LWSs, which are generated from the corresponding LWSs |l r , S, −S as |l r , S, S z = 1 √ C (Ŝ + ) ns |l r , S, −S where C = (n s !) ns j=1 ( 2S + 1 − j ) and n s ≡ S+S z = 1, ..., 2S, such current operator expectation values can be expressed in terms of that of the corresponding LWS by suitable use of the spin SU (2) operator algebra. From such considerations one finds that in the TL the spin stiffness reads D(T ) = 0 for S z = 0 and for |S z | ≥ 1/2 it can be written as [26], HereĴ z is the z component of the spin current operator, Eq. (3), p lr,S,S z are the Boltzmann weights, and Ĵ z (l r , S) ≡ l r , S, −S|Ĵ|l r , S, −S are the LWSs spin currents. In this and all following expressions for the spin stiffness, the sums over S always increase in steps of 1, whereas S z and S have to be integers (half-odd integers) for even (odd) L. For each S value there are N (S) = (2S + 1) N singlet (S) energy and momentum eigenstates. Our study accounts for all corresponding L 2S=0 (integers) N (S) = 2 L energy and momentum eigenstates. For S > 0 each such a state is populated by a set of 2S spins 1/2 that participate in its multiplet configuration, which is one of the 2S + 1 multiplet configurations, and a complementary set of even number L − 2S of spins 1/2 that form a tensor product of singlet states. Since all the N (S) states with the same S value have the sameˆ S 2 eigenvalue, the energy and momentum eigenstates are superpositions of such configuration terms. Each such terms is characterized by a different partition of L physical spins 1/2 into 2S such spins that participate in a 2S + 1 spin multiplet and a product of singlets involving the remaining even number L − 2S of spins 1/2.
As in Ref. [26], we call unpaired spins and paired spins the members of such sets of 2S and L−2S spins, respectively. In the TL this partition is common to the general BA solution and the TBA representation of its energy and momentum eigenstates. Both for large L and within the TL the L − 2S paired spins 1/2 are contained in a number, of spin-singlet pairs. Hence each fixed-S subspace is spanned by energy and momentum eigenstates with exactly the same number M sp = L/2 − S of such pairs. Moreover, M sp = L/2 − S also is the total number of BA rapidities that describe such states. And this is independent of such rapidities being all real or some being real and and other complex. Consistently, within the present representation each BA rapidity describes a spin-singlet pair. The derivation of the spin stiffness upper bound of Ref. [26], whose limiting behaviors are given in Eq. (7), used a large overestimate of the current absolute values | Ĵ z (l r , S) |. Specifically, for the whole set of energy and momentum eigenstates with the same S z value corresponding to the sums lr L/2 S=|S z | in Eq. (10) it used the largest magnitude of the current expectation value among these states. Since the probability distribution p lr,S,S z in each fixed-S z canonical ensemble is normalized as L/2 S=|S z | lr p lr,S,S z = 1, this then allowed performing exactly such sums for all nonzero temperatures, T > 0.
The large overestimate of the currents used in deriving that spin stiffness upper bound is behind its m → 0 behavior reported in Eq. (7) leaving out the grand canonical ensemble in which m 2 = O(1/L). Our main goal is to derive an alternative spin stiffness upper bound whose estimate of the current absolute values | Ĵ z (l r , S) | is closer to yet larger than those of the currents in Eq. (10). Here we perform such a program for high temperatures, T → ∞.
The The unbound and bound spin-singlet pairs of the L − 2S paired spins are indeed described by groups of real and complex solutions, respectively, of the model BA equation [2,3], Here the α = 1, ..., M p summation is over the subset of occupied q α quantum numbers out of the full set, The different occupancy configurations of the related quantum numbers I j (defined modulo L) such that j = 1, ..., M b generate different energy and momentum eigenstates. The latter are successive integers or half-odd integers according to the boundary conditions, The set of j = 1, ..., M b quantum numbers q j can only have occupancy zero and one, respectively. Within our representation, the α = 1, ..., M p occupied momentum values q α refer to the center of mass translation degrees of freedom of M p neutral composite pseudoparticles. The internal degrees of freedom of each of these M p pseudoparticles refer to one of the M p unbound spin-singlet pairs.
Our functional representation involves a q j distribution function M p (q j ) that reads 0 and 1 for the M h = 2S + 2(M B sp −M B st ) unoccupied and M p occupied q j values, respectively. Since the contribution to the momentum eigenvalues of the M p pseudoparticles reads π+ M b j=1 M p (q j ) q j , the set j = 1, ..., M b of quantum numbers q j such that q j+1 −q j = 2π/L may be associated with the discrete momentum values of a pseudoparticle spin band. Consistently with the 0 and 1 allowed occupancies of the spin-band momentum values, the LWS BA wave functions formally vanish when two rapidities Λ j and Λ j ′ in Eq. (12) become equal. If one considered all the rapidities to be real, this property could suggest that simply choosing α = 1, ..., M p distinct occupied momentum values q α among the set of j = 1, ..., M b allowed spin-band discrete momentum values q j , which gives a dimension M b M p = L/2+S L/2−S , would allow the reconstruction of all 2 L energy eigenstates that span the model Hilbert space.
However, only some of the solutions to the model general BA equation involve only a group of M sp = M p real rapidities Λ j . As mentioned above, there also exist solutions involving groups of real and complex rapidities [2,3]. There are M sp = M p + M B sp BA rapidities that describe the M sp spin-singlet pairs of a general energy and momentum eigenstate. Within our representation in terms of L − 2S paired physical spins 1/2, the M p real rapidities and M B sp complex rapidities describe their M p unbound spin-singlet pairs and their M B sp spin-singlet pairs bound within the state M B st independent configurations, respectively. The following general relations between the different numbers under consideration apply, Here gives the total number of both M p unbound spin-singlet pairs and corresponding spin-band pseudoparticles and M B st independent n-pair configurations with n > 1 spin-singlet pairs bound within them. The n complex rapidities with the same real part that describe each such a n-pair configuration is labelled by a quantum number l = 1, ..., n. It also labels each of the spin-singlet pairs bound within such a configuration. These l = 1, ..., n rapidities with the same real part have the general form [39], The roots of Eq. (12) are here partitioned in a configuration of strings. A n-string is a group of n roots also called rapidities. Within our representation such a string describes an independent n-pair configuration. The number n is often called the string length. The real part of the set of n rapidities, Λ n j , is called the string center [39]. Hence There is a one-to-one correspondence between an energy eigenstate M B st strings of length n > 1 and the M B st independent n-pair configurations with n > 1 spin-singlet pairs bound within them, respectively. The string length n > 1 is thus the number of spin-singlet pairs bound within the corresponding n-pair configuration. The present representation clarifies the physical meaning of the imaginary parts of the n > 1 complex rapidities with the same real part that refer to a string of length n, Eq. (16): Such imaginary parts are associated with the binding within the corresponding n-pair configuration of n > 1 spin-singlet pairs. Consistently and as mentioned above, for n = 1 the rapidity Λ 1,1 j is real and describes a single unbound pair. The maximum possible value of the number n of spin-singlet pairs bound within a n-pair configuration and corresponding string length is obviously given by the number of spin-singlet pairs, M sp = (L − 2S)/2, Eq. (11). The set of energy and momentum eigenstates that span each fixed-S subspace have all the same number M sp = (L − 2S)/2 of such pairs. Provided that (1 − m s ) is finite, that number is such that M sp → ∞ as L → ∞. Hence in general we consider in the TL that n has the range n = 1, ..., ∞.
For a given large L, the complex solutions of the spin-1/2 XXX chain BA equation, Eq. (12), are found to belong to three classes [39]. The first class refers to the ideal strings for which D n,l j = 0 in Eq. (16). The second class was first identified by Essler, Korepin and Schoutens (EKS) for n = 2 complex rapidities [40] yet also occurs for n > 2. The corresponding strings deviate from the ideal behavior and are known as EKS-strings [39]. The imaginary part of their complex rapidities are smaller than 1/2. It decreases upon increasing L, vanishing at some L value. The third class of solutions corresponds to another type of deformed strings usually called V-strings, which have been first found by Vladimirov (V) [41]. In the case of a system with a fixed large L, the number of energy eigenstates obtained by accounting for the three classes of BA equations groups of real and complex solutions is given by the correct Hilbert space dimension, 2 L [39].
In Sec. VI it is justified why concerning the model at finite magnetic field our final results are independent from the use in the TL of ideal or deformed strings of length n > 1 for the |S z | ≥ 1/2 energy and momentum states described by groups of real and complex rapidities. The unbinding of spin-singlet pairs by processes associated with the vanishing of the EKS-strings imaginary parts, usually called collapse of narrow pairs, is for a large system and finite magnetic field the aberration from the ideal strings that must be accounted for. The effects of the V-strings are unimportant in the TL for the physical quantities studied in this paper. For large finite systems they behave in a rather normal way, consistent with the predictions of the 1/L expansion methods [39].
The direct relation reported in the following of the TBA quantum numbers to our representation configurations of 2S unpaired spins 1/2, L − 2S paired spins 1/2, corresponding M sp = L/2 − S spin-singlet pairs, and M p and M B sp unbound and bound such pairs, respectively, is useful and needed for the studies of Secs. V and VII. Within the TBA, the l = 1, ..., n complex rapidities of a string, Eq. (16), simplify in the TL to their ideal form [3], Such rapidities are solutions of the TBA coupled integral equations given below. The number 2 L of energy eigenstates prevails under the use of the TBA in terms of only ideal strings, Eq. (17).
We call M n the number of n-pair configurations and corresponding strings of length n. Within our representation the st n-pair configurations involving for each spin-S energy and momentum eigenstate its M sp = L/2 − S spin-singlet pairs, Eq. (11). Consistently, the TBA quantum numbers obey the following sum rule [3], where m sp is the density of spin-singlet pairs and, Within the momentum-distribution functional notation used here and in Ref. [26], the TBA equations derived in Ref. [3] from the general BA equation, Eq. (12), by means of real and complex rapidities associated with ideal strings, Eq. (17), read, In this equation, and Θ n n ′ (x) is an odd function of x given by, Here n, n ′ = 1, ..., ∞ and δ n,n ′ is the usual Kronecker symbol. (The relation of the n = 1 rapidity momentum k 1 j = 2 arctan(Λ 1 j ), Eq. (21) for n = 1, to the rapidity momentum k j of Ref. [3], such that Λ 1 The function M n (q j ) in Eq. (20) is the n-band momentum distribution function associated with each energy and momentum eigenstate. It is such that M n (q j ) = 1 and M n (q j ) = 0 for occupied and non-occupied q j values, respectively. Such variables, are the momentum values of a n-band. It is associated with the set of M n n-pair configurations with the same n value. On the one hand, the TBA n = 1 band refers to the general BA spin band considered above. On the other hand, in the case of the TBA the n-pair configurations with n > 1 spin-singlet pairs bound within them are also associated with n-band sets of M b n real momentum values, Eq. (23). Here M b n = M n + M h n where the numbers {M n } of occupied momentum values in each such a n band obey the sum rule ∞ n=1 n M n = M sp , Eq. (18). The corresponding unoccupied values {M h n } are uniquely defined by the spin S and occupied values {M n } as follows [3,26], Moreover, the quantum numbers I n j on the right-hand side of Eq. (23) are successive integers or half-odd integers according to the boundary conditions, respectively. For each string of length n there is thus a BA branch momentum n-band whose successive set of momentum values q j , Eq. (23), have the usual separation, q j+1 − q j = 2π/L, and only occupancies zero and one. Often an index α = 1, ..., M n is used to label the subset of occupied quantum numbers I n α of an energy and momentum eigenstate [3,26].
In the case of the TBA, we associate a n-band pseudoparticle with each of the M n n-band occupied momentum values [26]. For n > 1 the n-band pseudoparticles are specific to the TBA. On the one hand, the M n occupied n-band momentum values q j refer to their translational degrees of freedom. They are associated with the center of mass motion of the M n n-band pseudoparticles of momentum q j . The corresponding M h n unoccupied momentum values q j left over are associated with M h n n-band holes. Within a corresponding real-space lattice representation, they interchange position with the n-band pseudoparticles under their center of mass motion. On the other hand, the internal degrees of freedom of a n-band pseudoparticle correspond to a single unbound spin-singlet pair for n = 1 and to a n-pair configuration with n spin-singlet pairs bound within it for n > 1.
The n-band momentum distribution function M n (q j ) obeys the sum rule This is consistent with the n-branch quantum numbers q j , Eq. (23), playing the role of n-band pseudoparticle momentum values. There are sum rules for the number of n-band pseudoparticles that populate the n = 1, ..., ∞ bands of a LWS or non-LWS. Such sum rules are related to those of spin-singlet pairs and density of spin-singlet pairs, Eqs. (11) and (18). Indeed, the latter sum rule implies that From the use of this relation in the number of pseudoparticles belonging to all n = 1, ..., ∞ bands, M ps ≡ ∞ n=1 M n , one confirms that the following exact sum rules for M ps and m ps = M ps /L are obeyed, where the density m h The numbers M B sp of bound spin-singlet pairs and M B ps of n > 1 band pseudoparticles within which they are bound and the corresponding densities m B sp = M B sp /L and m B ps = M B ps /L, respectively, appearing in Eq. (28) are given by, As in Ref. [26], m h,0 1 = M h,0 1 /L and m 0 1 = M 0 1 /L denote corresponding densities of energy and momentum eigenstates with spin S = 0. Those are given by m h,0 st ≤ M B ps that apply to energy and momentum eigenstates described by groups of real and complex rapidities within the general BA for a large system relative to those of the corresponding TBA states in the TL. The equalities in these relations are reached when the string deformations of the former states do not lead to the collapse of narrow pairs.
On the one hand, in the case of a LWS or non-LWS with M B st deformed strings of length n > 1 the corresponding independent n-pair configurations cannot be associated with n-band pseudoparticles carrying a real momentum q j . On the other hand, the M p real rapidities of a LWS or non-LWS are both within the general BA for a large system and the TBA in the TL associated with M p pseudoparticles whose internal degrees of freedom refer to a single unbound spin-singlet pair.

III. GENERAL EXPRESSIONS FOR THE SPIN STIFFNESS AT HIGH TEMPERATURE T → ∞
For |S z | ≥ 1/2, high temperature T → ∞, and L → ∞ the spin stiffness, Eq. (10), can in the TL be written as, where M sp = L/2−S and lr is the sum over the N singlet (S) = L Msp − L Msp−1 independent spin-singlet configurations of each fixed-S subspace. Those are associated with the N (S) = (2S +1) N singlet (S) energy and momentum eigenstates that span it.
The spin stiffness, Eq. (30), can alternatively be written as, where the summation {Mn}m S is over all n = 1, ..., ∞ band occupancies that refer to the same number M sp = L/2−S of spin-singlet pairs. Provided that one uses on the right-hand side of Eq. (31) the exact spin currents absolute values, | Ĵ z (l r , S) |, this spin stiffness expression is rigorous for |S z | ≥ 1/2, T → ∞, and L → ∞. It is approximation free because when written as N singlet (S) = {Mn}m S This is shown specifically in Appendix A of Ref. [3] for LWSs for which the number of unpaired spins 1/2 with down-spin projection reads 2S = −2S z . Due to symmetry, that proof applies as well to the non-LWSs in the fixed-S subspaces. The off-diagonal generators that transform a S > 0 LWS into its 2S tower states merely flip the spins of the 2S unpaired spins 1/2 without changing the LWS configurations of the M sp spin-singlet pairs involving that state L − 2S paired spins 1/2.
Within the general BA equations, Eq. (12), the spin current expectation values in Eq. (30) of energy and momentum eigenstates described only by groups of real rapidities read, both for large but finite chains and the TL. Here q α denotes the corresponding occupied values of the BA spin band and the elementary currents j S (q j ) are given by, The distribution 2πσ(k j ) in the j S (q j ) expression obeys the following equation that within the TL can be transformed into an integral equation, In this case the index l r in Eq. (30) labels the lr = N singlet (S) = L Msp − L Msp−1 independent spin-singlet configurations of the L − 2S paired spins 1/2 and corresponding M sp = L/2 − S spin-singlet pairs associated with the set of energy and momentum eigenstates that span each fixed-S subspace.
In the general case of energy and momentum eigenstates described by groups of both by real and complex rapidities, there appear new types of contributions to the current operator expectation value expression, Eq. (32). Such additional contributions emerge from the strings of length n > 1 associated with independent n-pair configurations with n > 1 spin-singlet pairs bound within them. They can be computed from the use in the general BA equation, Eq. (12), of the suitable sets of specific complex rapidities of general form given in Eq. (16).
Within the TBA, the spin currents Ĵ z (l r , S) in Eq. (31) of LWSs described by groups of real and complex rapidities can be written in the TL in terms of n-band pseudoparticle occupancies as follows [26], Here l r labels the Mn independent spin-singlet configurations of the L − 2S paired spins 1/2. They correspond to a well-defined set of numbers {M n } of n-pair configurations associated with the energy and momentum eigenstates that span each fixed-S subspace. The n-band elementary currents j n (q j ) in Eq. (35) read [26], where q b n = π m b n , the LWS rapidity functions k n (q j ) are obtainable from solution of the TBA equations, Eq. (22), and within the TL the distribution 2πσ n (k j ) is given by, Here q n (k) stands for the inverse function of the n-band rapidity momentum function k n (q).
In Appendix A 1 it is found that for LWSs for which m h 1 ≪ 1 and (1 − m h 1 ) ≪ 1 the elementary currents, Eq. (36), have the following exact limiting behaviors for the n = 1 band, For the n > 1 bands the corresponding exact limiting behaviors are, In addition, in that Appendix some of the exact behaviors useful for our studies of such elementary currents for a class of energy and momentum eigenstates whose currents absolute values reach largest values are reported.

IV. THE CASE OF STRICTLY ZERO MAGNETIC-FIELD
The general consensus is that the use of ideal strings for the energy and momentum eigenstates described by groups of real and complex rapidities of the spin-1/2 XXX Heisenberg chain leads in the TL to exact results as long as either the temperature or the magnetic field are nonzero [39]. Concerning the spin stiffness, our results refer to T → ∞, so that they are not affected in the TL by the finite-system string deformations.
A technical difference between the cases h = 0 and h = 0 is that for the former case of strictly zero magnetic-field there may occur deformations whose deviations D n,l j from the ideal string behavior may not occur in the strings themselves, Eq. (16). Hence at zero field the problem is more complex in terms of the BA solution than for h = 0 and the use of the ideal strings in the BA equations to compute current operator expectation values of the corresponding S z = 0 energy and momentum eigenstates is often considered questionable, even in the TL.
Fortunately, though, the current operator expectation values of these S z = 0 states, both with spin S = 0 and S > 0, can be computed by a method that does not rely on the BA and TBA. It is then found that such expectation values exactly vanish [26]. In the TL this applies both to energy and momentum eigenstates described by ideal and deformed strings of length n > 1.
In order to briefly revisit that problem, we consider a class of spin current operator expectation values l r , S, S z |Ĵ z |l r , S, S z for energy and momentum eigenstates with arbitrary S ≥ 1/2 and S z values for which the following relation is exact [26], where S z = −S + n s and n s = 1, ..., 2S. This relation is obtained by combining the systematic use of the commutators given in Eq. (9) with the state transformation lawsŜ − |l r , S, 0 = 0 andŜ + |l r , 0, 0 =Ŝ − |l r , 0, 0 = 0, which follow straight-forwardly from the corresponding spin SU (2) symmetry operator algebra. The calculations to reach Eq. (40) are relatively easy for non-LWSs whose generation from LWSs involves small n s = S − S z values. As discussed in Ref. [26], they become lengthy as the n s value increases, but they remain straightforward. The exact relation, Eq. (40), is behind the T > 0 spin stiffness expression given in Eq. (10). The form of the spin currents, Eq. (40), confirms that the S z = 0 expectation values l r , S, 0|Ĵ z |l r , S, 0 indeed all vanish exactly for S ≥ 1/2. The S = S z = 0 spin currents, l r , 0, 0|Ĵ z |l r , 0, 0 , are also found to vanish. They refer to energy and momentum eigenstates |l r , 0, 0 which are both LWSs and HWSs. It follows from Eq. (9) that the current operatorĴ z , Eq. (3), may be expressed in terms of the commutator,Ĵ z = 1 2 [Ĵ + ,Ŝ − ]. Thus the spin currents l r , 0, 0|Ĵ z |l r , 0, 0 can be written as, ( l r , 0, 0|Ĵ +Ŝ− |l r , 0, 0 − l r , 0, 0|Ŝ −Ĵ + |l r , 0, 0 )/2. That this expression vanishes is readily confirmed by applying the above state transformation laws. Hence all S z = 0 spin currents l r , S, 0|Ĵ z |l r , S, 0 vanish for S ≥ 0.
The Such a virtual current cancelation mechanism is encoded both in the general BA equation, Eq. (12), and in the n = 1, ..., ∞ TBA equations, Eq. (20), and corresponding general spin-current expressions. However, for increasingly larger numbers of spin/n = 1 band holes it is technically difficult to access from direct solution of these equations.
The problem can be explicitly solved in terms of such equations for the simplest case of the class of S = 0 energy eigenstates with two holes in the spin/n = 1 band. Such states thus have one n = 2-pair configuration described by one string of length two. (Within the TBA its two bound pairs refer to the internal degrees of freedom of one n = 2 composite pseudoparticle.) This simplest case has been studied within the BA solution, as in Ref. [44] for the present model, by use of the method of Ref. [45] for the related large-on-site-repulsion half-filled 1D Hubbard model. (In this paper the spin current operator, Eq. (3), and its expectation values are given in units of 1/2, which justifies that extra factor within the notation of Ref. [44].) One then explicitly finds that, independently of the momentum values q j and q j ′ of the two holes, their virtual spin currents exactly cancel each other.
As confirmed in the ensuing section, the virtual current mechanism also occurs for |S z | > 0 energy and momentum eigenstates. For such states it corresponds though to a partial cancellation [26].

V. USEFUL INEQUALITIES AND UPPER BOUNDS ON CURRENT ABSOLUTE VALUES
The inequalities and corresponding current absolute values upper bounds introduced in this section refer to the TBA. More general inequalities accounting for the effects of the string deformations on the spin currents at finite magnetic field are introduced below in Sec. VI.
The spin-1/2 XXX chain in a uniform vector potential Φ/L whose Hamiltonian is given in Eq. (A2) of Ref. [26] remains solvable by the BA. Within the TBA the LWSs momentum eigenvalues, P = P (Φ/L), have the general form, Here the Φ = 0 momentum eigenvalue P (0) is given in Eq. (26)  A second exact result is consistent with only the 2S unpaired physical spins 1/2 coupling to the vector potential also holding for non-LWSs. For simplicity, we consider that L is even yet within the TL the same results are reached for L odd. For a general LWS carrying a spin current Ĵ z LW S (l r , S) all 2S unpaired spins 1/2 have up-spin projection. Let S σ be the number of unpaired spins 1/2 with spin projection σ =↑, ↓ of a non-LWS such that σ=↑,↓ S σ = 2S. The exact relation, Eq. (40), can then be written simply as, where The exact relation, Eqs. (42) and (43), confirms that only the 2S = S ↑ + S ↓ unpaired spins 1/2 contribute to the spin currents. For each spin flip generated by application of the off-diagonal spin generatorŜ + (andŜ − ) onto an energy eigenstate with finite numbers S ↑ and S ↓ , the spin current exactly changes by a LWS current quantum 2j −1/2 (and 2j +1/2 .) Hence each unpaired spin 1/2 with spin projection ±1/2 carries an elementary current j ±1/2 , Eq. (43). For a LWS one has that S ↑ = 2S and S ↓ = 0, so that Ĵ z LW S (l r , S) = 2S × j +1/2 . That only the 2S = m S L unpaired physical spins 1/2 couple to the vector potential justifies the validity of the result of Ref. [26] that all spin currents exactly vanish as m S → 0. This exact result can be used to confirm that, as found in that reference, within the canonical-ensemble description at fixed value of S z , in the TL, and for nonzero temperatures the spin stiffness D(T ), Eq. (10), vanishes as m S → 0. The main goal of this paper is to extend that result to the grand-canonical-ensemble description for T → ∞.
Relying on the exact relation, Eq.
All remaining reduced subspaces of a LWS fixed-S subspace are called complex-rapidity reduced subspaces. Indeed those are spanned by complex-rapidity LWSs described by groups of both real and complex rapidities. Their m h 1 > m S values belong to the range m h 1 ∈ [m S , 1]. We denote by | Ĵ z LW S | T (mS ,m h 1 ) the largest current absolute value of each LWS reduced subspace of a given LWS fixed-S subspace. It is of the general form, The coefficient c T in this expression obeys the inequality c T ≤ π. It is a function of the densities m S and m h 1 with the following limiting behaviors, The LWSs spin currents result from processes that are simpler to be described in terms of local spins 1/2 occupancy configurations in the spin-1/2 XXX chain lattice. Within these processes, each 2n-site configuration of the M ps = ∞ n=1 M n pseudoparticles that populate a LWS interchanges position under its motion along the lattice with such a state single-site 2S unpaired physical spins 1/2. This justifies why the largest current absolute value of a LWS reduced subspace is proportional to 2S × M ps , as given in Eq. (45). Consistently, LWSs for which 2S = 0 and/or M ps = 0 carry no spin current.
The degrees of freedom of the 2S unpaired spins 1/2 are distributed over different quantum numbers of the exact BA solution. They are the physical spins 1/2 whose spin is flipped by the spin SU (2) symmetry algebra off-diagonal generators. The spin degrees of freedom of the S ↑ and S ↓ unpaired spins 1/2 with up and down spin projection, respectively, determine the spin S = (S ↑ + S ↓ )/2 and spin projection S z = −(S ↑ − S ↓ )/2 of all energy eigenstates. Their translational degrees of freedom are described in each n-band by its M h n = 2S + ∞ n ′ =n+1 2(n ′ − n) M n ′ holes. Hence in terms of the exact solution quantum numbers the above local processes that generate the spin currents refer to the relative occupancy configurations of the M n pseudoparticles and corresponding M h n holes in each n band for which M n > 0. Consistently, the LWSs spin currents Ĵ z LW S (l r , S) in the general spin current expression Ĵ z (l r , S ↑ , S ↓ ) = ([S ↑ − S ↓ ]/2S) Ĵ z LW S (l r , S) , Eq. (42), can alternatively be expressed in terms of pseudoparticles, as given in Eq. (35), or of n-band holes. Within the latter representation, they read Ĵ z (l r , S) = Here the sum The coefficient c A reads here c A = 1 for (1 − m h 1 ) ≪ 1 and otherwise obeys the inequality c A ≤ 1, being of the order of unity. The factor 1/ 2M ps that multiplies | Ĵ z LW S | L(mS ,m h 1 ) stems from the LWSs that span the reduced subspace being generated by all possible occupancy configurations of the M ps pseudoparticles.
In the case of the reduced subspace for which M ps reaches its maximum value at fixed S, the average current absolute value general form, Eq. (49), follows from the calculations of Appendix B. Its generalization to the remaining reduced subspaces involves in the TL lengthy yet straightforward calculations. The precise value of the coefficient c A remains though an involved open problem. Fortunately, the only related information needed for our studies is that c A is of the order of the unity.
At fixed spin S the number 2S of unpaired physical spins 1/2 that couple to a vector potential is fixed. Hence the current absolute values are largest for LWSs for which these 2S unpaired spins 1/2 have a larger number M ps of n-band pseudoparticles to interchange position with.
On the one hand, for a given LWS fixed-S subspace the average current absolute value is thus smallest for its M ps = 1 reduced subspace. For it the M sp = L/2 − S spin-singlet pairs are all bound within a single gigantic n = M sp = L/2 − S pair-configuration. The single pseudoparticle of the LWSs that span such a LWS reduced subspace has one of the j = 1, ..., 2S + 1 momentum values q j = 0, ± 2π L , ..., ± 2π L (S − 1), ± 2π L S. For such LWSs the M sp = L/2 − S spin-singlet pairs involving the L − 2S paired spins 1/2 reach the smallest dilution relative to the 2S unpaired spins 1/2. The spin current of these LWSs, Ĵ z LW S (l r , S) = Ĵ z LW S (q j , S) = −2J sin q j , results from the motion of the single gigantic pseudoparticle relative to a number 2S of n = L/2 − S band holes. Those describe the translational degrees of freedom of the 2S unpaired physical spins 1/2.
On the other hand, both the largest current absolute | Ĵ z LW S | T (mS ,m h 1 ) , Eq. (45), and the average current absolute value | Ĵ z LW S | A(mS ,m h 1 ) , Eq. (49), reach their maximum values for the real-rapidity reduced subspace for which M ps = M 1 = M sp = L/2 − S and thus M n = 0 for n > 1. Its average current absolute value, Eq. (48), can be written as, Here | Ĵ z LW S | A(mS ,mS) is the corresponding real-rapidity reduced subspace average current absolute value, Eq. (50). We call | Ĵ z LW S | A(mS) the average current absolute value of a LWS fixed-S subspace. It reads, That the inequalities, Eq. (51), are valid for all reduced subspaces of any LWS fixed-S subspace for which m h 1 > m S straightforwardly implies the validity of the following related inequality, Since that validity refers to all S > 0 values, it ensures the validity, within the TL, of the following important inequality used below in the analysis of Sec. VII, Before presenting such an analysis, a more general inequality accounting for the effects of the string deformations is introduced in the ensuing section.

VI. THE EFFECTS OF THE STRING DEFORMATIONS ON THE SPIN CURRENTS AT FINITE MAGNETIC FIELD
At finite magnetic field only the deviations D n,l j that occur in the strings themselves, Eq. (16), may have effects in the TL on the spin currents and other quantities. The set of these complex rapidities with the same real part of form Λ n,l j = Λ n j + i(n + 1 − 2l) + D n,l j remain being labelled by the quantum numbers n = 1, ..., ∞ and l = 1, ..., n that refer to the number of bound spin-singlet pairs and each of these pairs, respectively. Physically, this means that, as in the case of an ideal string, the distorted string associated with that set of complex rapidities also describes an independent configuration within which n = 1, ..., ∞ spin-singlet pairs are bound.
The two complex rapidities Λ n,l j and Λ n,l ′ j associated with two spin-singlet pairs labelled by the quantum numbers l and l ′ = n + 1 − l, respectively, being related as Λ n,l j = (Λ n,l ′ j ) * for l = 1, ..., n is actually a necessary condition for the binding of the l = 1, ..., n spin-singlet pairs within the n-pair configuration.
Importantly and due to self-conjugacy, the deviations D n,l j = R n,l j +iδ n,l j in Eq. (16) for the set of complex rapidities with the same real part associated with a distorted string are also such that D n,l j = (D n,n+1−l j ) * . This reveals that the symmetry Λ n,l j = (Λ n,n+1−l j ) * prevails under string deformations. This ensures that as for the ideal strings, the imaginary parts of the n real rapidities with the same real part associated with deformed strings also describe the binding within the corresponding n-pair configurations of l = 1, ..., n spin-singlet pairs.
The V-strings deformations [39] have in the TL and finite magnetic field no effects on the spin currents. At finite magnetic field the EKS-strings collapse of narrow pairs, described below within our representation in terms spinsinglet pair unbinding processes, is in the TL the only aberration from the ideal strings [39] that may have effects on the spin currents. This refers only to the currents of |S z | > 0 energy and momentum eigenstates described by groups of real and complex rapidities. Here we identify such effects and justify why in the TL they have no impact whatsoever in the high-temperature stiffness upper bounds introduced in the ensuing section.
The general consensus is that the use ideal strings for energy and momentum eigenstates described by groups of real and complex rapidities leads in the TL to exact results as long as either the temperature or the magnetic field are nonzero [39]. Consistently, although the collapse of narrow pairs is indeed found to enhance the spin currents absolute values of a few states, it does no change in the TL the stiffness upper bounds used in our study.
The string deviations from the TBA ideal strings do not change the value of the number of spin-singlet pairs. Hence their density is also exactly given by m sp = (1 − m S )/2 for the corresponding LWSs and non-LWSs. Narrow pairs refer to a string deformation originated by a deviation D n,l j that renders the separation between two rapidities Λ n,l j and Λ n,l+1 j in the imaginary direction less than i. Such a separation may become narrower and eventually merge and split back onto the horizontal axis [39]. Such a process is what is called the collapse of a narrow pair.
Within our representation in terms of the model physical spins 1/2, it then refers to an elementary process that leads to the unbinding of two spin-singlet pairs. On the one hand, for the set of n > 2 complex rapidities with the same real part associated with n bound pairs, it leads to the partition of the corresponding n-pair configuration into a n ′ -pair configuration where n ′ = n − 2. The latter is described by a smaller number n ′ = n − 2 of complex rapidities with the same real part in a string of smaller length n ′ = n − 2. The process also generates two unbound spin-singlet pairs described by real rapidities. On the other hand, for n = 2 complex rapidities with the same real part it leads in turn to the unbinding of the two spin-singlet pairs of the corresponding n = 2 pair configuration. This gives rise solely to the two unbound spin-singlet pairs described by real rapidities.
Hence the collapse of a narrow pair is a process that causes an increase in the value of the number of strings of all lengths, M st = M p + M B st , Eq. (15). It does not change though that of spin-singlet pairs, M sp = L/2 − S. Specifically, it always leads to a positive deviation δM p = 2 in the value of the number of spin-band pseudoparticles and corresponding unbound spin-singlet pairs. Moreover, it gives rise to a negative deviation δM B sp = −2 in the value of the number of bound spin-singlet pairs. There is as well either an additional negative deviation δM B st = −1 or no deviation δM B st = 0 in the number M B st of independent configurations with bound spin-singlet pairs within them. This depends on whether the deformed n-pair configuration that suffers the collapse of a narrow pair has n = 2 or n > 2 spin-singlet pairs bound within it, respectively.
We denote by | Ĵ z LW S | AD (mS ) the average current absolute value of the LWS fixed-S subspace spanned by energy eigenstates for which some of the complex strings are deformed. It is given by, The sum lr D in this expression runs over all L − 2S paired physical spins 1/2 occupancy configurations that generate the N singlet (S) LWSs with the same spin S and thus the same number M sp = L/2 − S of spin-singlet pairs. As given in Eq. (53), within the TBA the average of the current absolute values is largest in the fixed-S subspaces spanned by energy and momentum eigenstates described only by groups of real rapidities. Such an average is larger than that in the fixed-S subspaces spanned by all energy and momentum eigenstates of spin S. The main point is that a larger fraction of unbound spin-singlet pairs relative to bound spin-singlet pairs at the fixed number M sp = L/2 − S of such pairs tends to enhance the spin current absolute values.
A generalization of the inequality, Eq. (53), which accounts for the effects of the collapse of narrow pairs and thus of spin-singlet pair unbinding processes, involves the average current absolute value, Eq. (55), and reads, On the one hand, the validity of the inequality | Ĵ z LW S | A(mS ,mS) ≥ | Ĵ z LW S | AD (mS ) in this equation follows from the energy and momentum eigenstates described by real rapidities having no strings of length n > 1 and thus being string-deformation free. This is because all their M sp = L/2 − S spin-singlet pairs are unbound. The binding of spinsinglet pairs within n-pair configurations for which n > 2 in states with groups of real and complex rapidities lessens the current absolute values. The unbinding of spin-singlet pairs under string deformations only partially neutralizes this effect. Indeed, it does not refer to all spin-singlet pairs bound within n-pair configurations for which n > 2. In contrast, for the energy and momentum eigenstates described by real rapidities all M sp = L/2 − S spin-singlet pairs are unbound.
On the other hand, the inequality | Ĵ z LW S | AD (mS ) ≥ | Ĵ z LW S | A(mS ) in Eq. (56) is valid because the collapse of narrow pairs caused by complex rapidity string deformations may unbind some spin-singlet pairs. This effect tends to enhance the average of the current absolute values in the fixed-S subspaces whose strings of some states are deformed. This effect is though very small in the TL. Indeed most string deformations involve small variations in the string fine structure that do not lead to the collapse of narrow pairs and in the TL have no effects on the spin currents absolute values.
Since the inequalities in Eq. (56) are valid for all S > 0 values, the following important inequality, which is an extension of that given in Eq. (54), holds, . (57)

VII. HIGH-TEMPERATURE STIFFNESS UPPER BOUNDS WITHIN THE THERMODYNAMIC LIMIT
The high-temperature stiffness upper bounds introduced in this section rely on replacing averages of the spin current absolute values in the full LWS spin-S subspaces by those in the corresponding smaller LWS real-rapidity reduced subspaces. It follows from the inequalities, Eqs. (54) and (57), that our final results are independent from the use in the TL of ideal or deformed strings for the states described by groups of real and complex rapidities.
For simplicity, we use the number notation in Eq. (44), within which M p S (q j ) = M 1 (q j ), q j = (2π/L) The sums lS in this expression run over the real-rapidity LWSs whose number is M b S M p S that span each LWS realrapidity reduced subspace. The spin currents Ĵ z (l mS ) are given by, where M b S j=1 M p S (q j ) = M p S . The elementary current j S 1 (q j ) in this expression is that in Eq. (A20) of Appendix A 2. It reads j S 1 (q j ) = j 1 (q j ) for q j ∈ [−q b , q b ] and M 1 = M ps = M sp where j 1 (q j ) is the elementary current, Eq. (36) for n = 1.
For the present real-rapidity LWSs one has that m h 1 = m S . Hence the limits given in Eq. (38) apply. The elementary current j S 1 (q j ) changes thus from j S 1 (q j ) = − π 2 J sin q j for q j ∈ [−π/2, π/2] as m S → 0 to j S 1 (q j ) = −2J sin q j for q j ∈ [−π, π] as m S → 1. It can be written as j S 1 (q j ) = −j S 1 s S 1 (q j ) where |s S 1 (q j )| ≤ 1 for q j ∈ − π 2 (1 − m S ), π 2 (1 − m S ) . As justified in Appendix A 2, the elementary current coefficient j S 1 > 0 in that expression reaches its largest value j S 1 = 2J for the whole m S ∈ [0, 1] range for m S → 1. Moreover, in that Appendix it is found that the replacement in j S 1 (q j ) = −j S 1 s S 1 (q j ) of j S 1 and s S 1 (q j ) by 2J and sin q j , respectively, ensures that | | for all real-rapidity LWSs and the whole m S ∈ [0, 1] interval. This thus implies the validity of the following inequality, where J * (l mS ) = − respectively, where, Finally, we emphasize that our T → ∞ upper bound, Eq. (62), has been inherently constructed to the exact T → ∞ stiffness reading, for (1 − m) ≪ 1 and, for m ≪ 1. Here c is a m and T independent coefficient, c ≈ 1 such that c 2 < c u2 . The calculations of Appendix B that reached the expressions in Eqs.

VIII. CONCLUDING REMARKS
The upper bound on high-temperature spin stiffness derived in this paper, Eqs. (62)-(64), vanishes as m 2 in the m → 0 limit and is independent of the system size L. This ensures that the spin stiffness vanishes within the grandcanonical ensemble as h → 0 for high temperature T → ∞ in the TL. We believe that our result is exact in these limits. As discussed in the following, the possibility of the absence of ballistic spin transport for the whole finite-temperature range T > 0 within the grand-canonical ensemble in the limit of zero magnetic field remains though an interesting unsolved problem.
Concerning the relation of our results to previous results on the spin stiffness of the spin-1/2 XXX chain, the upper bound of Ref. [26] is valid for the whole temperature range T > 0 and vanishes as m 2 L in the m → 0 limit. This latter behavior reveals that within the canonical ensemble the model spin stiffness vanishes as m → 0 for finite temperature within the TL. However and as mentioned above, it leaves out, marginally, the grand canonical ensemble in which m 2 = O(1/L). The large overestimate of the current absolute values used in deriving the stiffness upper bound of that reference, whose limiting values are given in Eq. (7), leads for high temperature to an extra factor of the order O((1 − m)L) relative to our upper bound, Eq. (62). This refers to an overestimate of the method used in Ref. [26] that has ignored the factor 1/ 2M ps = 1/ (1 − m S )L in the corresponding spin current average value, first expression of Eq. (49) for m h 1 = m S where m S = m for LWSs. We note that our result on vanishing spin stiffness as h → 0 in the TL crucially depends on the existence of a global SU (2) symmetry where the current under comsideration is a part of the symmetry operator algebra. We thus expect that our result should be extendable to other integrable models with similar one or several global SU (2) symmetries, such as e.g. the fermionic 1D Hubbard model.
In conclusion, in this paper we addressed the important fundamental and highly debated question on the possibility of ballistic spin transport within the grand-canonical ensemble for h → 0 in what is arguably one of the simplest strongly correlated quantum many-body system, the spin-1/2 XXX chain. Our main result is the strong evidence of lack of such a ballistic transport within the grand-canonical ensemble as h → 0 in the TL at high temperature T → ∞.
Our results thus imply that the spin-1/2 XXX Heisenberg chain exhibits at infinite temperature anomalous subballistic spin transport. This is consistent with the studies of Ref. [19] that rely on a nonequilibrium open system approach.
Combination of the result of Ref. [26] that within the canonical ensemble the spin stiffness vanishes in the m → 0 limit at all nonzero temperatures with the absence of phase transitions in the spin-1/2 XXX chain at T > 0, could be an indication of the lack of ballistic spin transport for the whole nonzero temperature range, T > 0, also within the grand-canonical ensemble. This remains though an interesting open problem that deserves further studies.
Last but not least, our method uses a representation in terms of configurations of the L physical spins 1/2 that is more controllable than most numerical studies on the occurrence or lack of ballistic spin transport in the spin-1/2 XXX chain. Moreover, such a representation provides useful physical information on the microscopic processes involving the elementary currents carried by spin/n = 1 band holes and n-pair configurations with n > 1 spin-singlet pairs bound within them that control the very complex problem under investigation. That information may play a valuable role in future studies of the present problem for the whole nonzero temperature range, T > 0.

(A9)
The two methods used in Ref. [46] for the 1D Hubbard model and in Ref. [47] for the related t − J model to calculate the elementary spin current j n (q j ), Eq. (36), for reference LWSs with ground-state compact distributions by means of conservation laws and under twisting boundary conditions, respectively, apply as well to the present more general compact distributions, Eqs. (A1) and (A2). For the spin-1/2 XXX Heisenberg chain both such methods lead to exactly the same expression, ι a n ′ π ι=± (ι)f n n ′ (q j , q ι a n ′ ) , where a = p, h and, There are two limits in which the classes of LWSs considered here correspond to all existing such states: (i) (1 − m h 1 ) ≪ 1 when (m h 1 − m S ) ≪ 1 and (ii) m h 1 ≪ 1, respectively. In these two limiting cases the use of elementary current, Eq. (A10), gives for the n = 1 and n > 1 bands, respectively. The n-band group velocities, Eq. (A7), in these expressions have the following exact behaviors, By combining the relations, Eq. (A12), with the limiting group-velocity expressions provided in Eq. (A13) one arrives to the elementary current j n (q j ) expressions, Eqs. (38) and (39), which is one of the goals of this Appendix.

Elementary currents for LWSs described only by groups of real rapidities
The goal of this Appendix section is to justify the validity of the inequality, Eq. (60). It refers to the model in the LWS fixed-S real-rapidity reduced subspaces considered in Secs. V and VII.
For the class of LWSs described only by groups of real rapidities and generated from reference states with compact particle or hole n = 1 band distributions, Eqs. For such S > 0 ground states the corresponding n = 1 band elementary current reads, In the TL the relation j S 1 (q j ) = 2(ξ 1 ) 2 v 1 (q j ) is exact. The j S 1 (q j ) expression given here is exact both for m S ≪ 1 and (1 − m S ) ≪ 1. For intermediate m S ≈ 1/2 densities it has an absolute value |j S 1 (q j )| slightly larger than the corresponding exact value. Hence it is a very good approximation for the whole m S ∈ [0, 1] range.
Since the present symmetrical compact LWSs are ground states, one finds that the dressed phase-shift parameter ξ 1 , Eq. (A18), is directly related to the model zero-temperature spin stiffness, D = D(0). Indeed, the elementary current absolute value |j S 1 (πm 1 )| = 2(ξ 1 ) 2 v 1 (πm 1 ) at q j = πm 1 = π(1 − m S )/2 = π(1 − m)/2 fully controls such a zero-temperature stiffness for m = m S ∈ [0, 1] as follows [48], The dependence on m = m S of the zero-temperature spin stiffness, Eq. (A19), has been investigated in previous studies [48]. It is plotted in Fig. 2. The elementary currents j S 1 (q j ) of all fixed-S LWSs described only by groups of real rapidities can be written as, where j S 1 > 0 is the largest elementary current absolute value. As mentioned above, although the class of LWSs with symmetrical compact distributions, Eqs. (A1) and (A2), carry zero current, their elementary currents absolute values reach the largest values of each S-fixed subspace. The largest absolute value j S 1 = |j S 1 (q w )| of the symmetrical compact distribution ground-state elementary current, Eq. (A17), is reached at q j = q w ≈ ±π/2 and reads πD(0)v 1 (q w )/v 1 (πm 1 ) ≈ πD(0)/ cos (πm S /2). It is a continuous increasing function of m S that smoothly varies from its minimum value Jπ/2 for m S → 0 to its maximum value 2J as m S → 1. Moreover, for all fixed-S LWSs described only by groups of real rapidities the following two universal limiting behaviors hold, 1 of all the LWSs described only by groups of real rapidities is for m S < 1 smaller than 2J. Hence, The first inequality refers to the largest elementary current absolute value j S 1 = πD(0)v 1 (q w )/v 1 (πm 1 ) ≈ πD(0)/ cos π 2 m S reached for S > 0 ground states. It has been expressed in terms of the zero-temperature spin stiffness for m = m S . The second inequality in Eq. (A22) applies to the largest elementary current absolute value j S 1 of all LWSs with fixed spin S that are described only by groups of real rapidities.
The limiting behaviors, Eq. (A21), and inequalities, Eq. (A22), justify the largest elementary current absolute value j S 1 = 2J of the elementary current j(q j ) = −2J sin q j , Eq. (61), used in our T → ∞ spin stiffness upper bound scheme of Sec. VII. Next, we briefly describe the main mechanism that justifies the use of the function s S 1 (q j ) = − sin q j . First we discuss the suitable use of a odd function, s S 1 (q j ) = −s S 1 (−q j ), for that elementary current. We then justify the specific choice, s S 1 (q j ) = − sin q j . On the one hand, that we use a odd function for s S 1 (q j ) is all right for LWSs with symmetrical compact and symmetrical non-compact distributions such that M p S (q j ) = M p S (−q j ). On the other hand, analysis of the BA equation reveals that the exact function s S 1 (q j ) in Eq. (A20) such that |s S 1 (q j )| ≤ 1 is not a odd function of q j for general LWSs with asymmetrical compact and asymmetrical non-compact distributions such that M p S (q j ) = M p S (−q j ). Nonetheless, the use of a odd function s S 1 (q j ) for these states enhances in general their current absolute values, | Our following analysis applies to general LWSs described only by groups of real rapidities. Those do not necessarily have compact M p S (q j ) occupancies. Hence rather than the elementary current j S 1 (q j ) given in Eq. (A14), which is specific to such occupancies, here we use the more general elementary current j S 1 (q j ) = − 2J sin k 1 (qj ) 2πσ1(k 1 (qj )) . It is that given in Eq. (36) for n = 1 and LWSs described only by groups of real rapidities.
For all such LWSs the BA equation is of the same form, Eq. (12) and Eq. (20) for n = 1, for large finite L and the TBA, respectively. It can be written as, where j = 1, ..., M b S . If the momentum distribution is an even function, M p S (q j ′ ) = M p S (−q j ′ ), one finds that k 1 (0) = 0 at q j = 0. The elementary current, j S 1 (q j ) = − 2J sin k 1 (qj ) 2πσ1(k 1 (qj )) , is then a odd function. This follows from the distribution 2πσ 1 (k) turning out to be an even function in that case. The latter distribution can be written as 2πσ b 1 (k)M p S (k) and equivalently as 2πσ b (k)M p S (k). Here 2πσ b 1 (k) is the distribution, Eq. (37) for n = 1, and 2πσ b (k j ) is the solution of Eq. (34). For the present case of real rapidities they are the same distributions. Moreover,M p S (k j ) = M p S (q j ). In the general case of LWSs for which the momentum distribution M p S (q j ) is not an even function, M p S (q j ) = M p S (−q j ), the corresponding elementary current j S 1 (q j ) is not a odd function. Consistently, the n = 1 band momentum q j = 0 then corresponds to a finite momentum rapidity k 1 (0) given by, M p S (q j ) arctan tan(k 1 (q j )/2) + tan(k 1 (0)/2) 2 , such that k 1 (0) < π(1 − m S )/2. This implies that there is a positive or negative q j interval, where, M p S (q j ) arctan tan(k 1 (q j )/2) + (ι) tan(k 1 (0)/2) 2 , in which the elementary current, j S 1 (q j ) = − 2J sin k 1 (qj ) 2πσ1(k 1 (qj )) , has opposite signs for the two subintervals k 1 j ∈ [−k 1 (0), 0] and k 1 j ∈ [0, k 1 (0)], respectively. This refers to the corresponding momentum rapidity interval k 1 j ∈ [−k 1 (0), k 1 (0)]. Indeed the distribution 2πσ 1 (k 1 (q j )) = 2πσ 1 (k 1 j ) is for all LWSs such that 2πσ 1 (k 1 j ) ≥ 0. And this applies to its whole range k 1 j ∈ [−π, π] and thus corresponding q j range q j ∈ [−π(1 − m S )/2, π(1 − m S )/2]. In the q j interval q j ∈ [0, q 0 ] for q 0 > 0 and q j ∈ [q 0 , 0] for q 0 < 0 the band momentum q j has the same sign. However, the elementary current j S 1 (q j ) has opposite signs in two momentum q j subintervals of these intervals. For example, for q 0 > 0 such subintervals read q j ∈ [0, q(0)] and q j ∈ [q(0), q 0 ], respectively. Here, is the q j value at which the momentum rapidity vanishes, k 1 (q(0)) = 0. The function s S 1 (q j ) in Eq. (A20) such that |s S 1 (q j )| ≤ 1 has the same signs as k 1 j . It follows that the current contributions from occupancies in such q 0 > 0 subintervals, q j ∈ [0, q(0)] and q j ∈ [q(0), q 0 ], tend to cancel. This would not be so if s S 1 (q j ) was a odd function. Moreover, the cancelling momentum rapidity interval k 1 j ∈ [−k 1 (0), k 1 (0)] corresponds to q j alternative positive q j ∈ [0, q 0 ] and negative q j ∈ [q 0 , 0] intervals if the asymmetric distribution M p S (q j ) has integrated larger values for q j > 0 and q j < 0, respectively. Hence the use of a suitably chosen odd function s S 1 (q j ) enhances indeed the current absolute values | M b S j=1 M p S (q j ) j S 1 (q j )| of most LWSs. Finally, we justify the choice of the specific odd function, s S 1 (q j ) = − sin q j . As follows from Eq. (38) for m h 1 = m S , one finds for all LWSs described only by groups of real rapidities that in the limits m S ≪ 1 and (1 − m S ) ≪ 1 their elementary currents j S 1 (q j ) are exactly given by, respectively. The simplest odd function s S 1 (q j ) = −s S 1 (−q j ) that in these two limits reaches the exact behavior of the elementary currents carried by such LWSs is indeed s S 1 (q j ) = − sin q j . Additionally, we have confirmed that this choice enhances the current absolute values | and b j ≡ M (q j ) are binary occupation numbers, which we sum over. The δ-constrain can be analytically treated by means of a counting field parameter λ. This is done by defining, We then find immediately that, Indeed, due to δ-constraint one has that e λM p S = M b S k=1 e λb k . We have been using the property that 0 = ( b k sin q k ) 2 = k sin 2 q k + k =l sin q k sin q l . From it we find I(M p s ), Furthermore, we can explicitly calculate the sum over sin 2 q k . This gives, From the combination of such procedures, we arrive at the following final compact upper bound valid for m = −2S z /L → 0 in the TL, (B11) This is the expression given in Eq. (63) for m ≪ 1.
Note that the lower limit of the sum in Eq. (B8) can in the TL be pulled up to the S = |S z | for any |S z | ≤ L/3. This is so that the sum still starts before the maximum of the binomial symbol, which in the TL can be approximated with a gaussian This yields the same asymptotic inequality, Eq. (54).
Finally, we evaluate the behavior of the stiffness upper bound, Eq. (62), in the regime m → 1, i.e., −S z = L/2 − δ, where δ ≪ L. This is a simple task fulfilled by using the leading order asymptotic in δ/L = 1 − m, which gives, This is the behavior reported in Eq. (63) for (1 − m) ≪ 1.