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Physical Review B

Vanishing spin stiffness in the spin- 12 Heisenberg chain for any nonzero temperature

J. M. P. Carmelo, Tomaž Prosen, David K Campbell

Published Oct 30, 2015• 29 citations
Publisher

Abstract

Whether at the zero spin density $m=0$ and finite temperatures $T>0$ the spin stiffness of the spin-$1/2\phantom{\rule{4pt}{0ex}}XXX$ chain is finite or vanishes remains an unsolved and controversial issue, as different approaches yield contradictory results. Here we explicitly compute the stiffness at $m=0$ and find strong evidence that it vanishes. In particular, we derive an upper bound on the stiffness within a canonical ensemble at any fixed value of spin density $m$ that is proportional to ${m}^{2}L$ in the thermodynamic limit of chain length $L\ensuremath{\rightarrow}\ensuremath{\infty}$, for any finite, nonzero temperature, which implies the absence of ballistic transport for $T>0$ for $m=0$. Although our method relies in part on the thermodynamic Bethe ansatz (TBA), it does not evaluate the stiffness through the second derivative of the TBA energy eigenvalues relative to a uniform vector potential. Moreover, we provide strong evidence that in the thermodynamic limit the upper bounds on the spin current and stiffness used in our derivation remain valid under string deviations. Our results also provide strong evidence that in the thermodynamic limit the TBA method used by X. Zotos [Phys. Rev. Lett. 82, 1764 (1999)] leads to the exact stiffness values at finite temperature $T>0$ for models whose stiffness is finite at $T=0$, similar to the spin stiffness of the spin-$1/2$ Heisenberg chain but unlike the charge stiffness of the half-filled 1D Hubbard model.

DOI
10.1103/physrevb.92.165133

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Whether at the zero spin density m=0m=0m=0 and finite temperatures T>0T>0T>0 the spin stiffness of the spin-1/2XXX1/2\phantom{\rule{4pt}{0ex}}XXX1/2XXX chain is finite or vanishes remains an unsolved and controversial issue, as different approaches yield contradictory results. Here we explicitly compute the stiffness at m=0m=0m=0 and find strong evidence that it vanishes. In particular, we derive an upper bound on the stiffness within a canonical ensemble at any fixed value of spin density mmm that is proportional to m2L{m}^{2}Lm2L in the thermodynamic limit of chain length L\ensuremath→\ensuremath∞L\ensuremath{\rightarrow}\ensuremath{\infty}L\ensuremath→\ensuremath∞, for any finite, nonzero temperature, which implies the absence of ballistic transport for T>0T>0T>0 for m=0m=0m=0. Although our method relies in part on the thermodynamic Bethe ansatz (TBA), it does not evaluate the stiffness through the second derivative of the TBA energy eigenvalues relative to a uniform vector potential. Moreover, we provide strong evidence that in the thermodynamic limit the upper bounds on the spin current and stiffness used in our derivation remain valid under string deviations. Our results also provide strong evidence that in the thermodynamic limit the TBA method used by X. Zotos [Phys. Rev. Lett. 82, 1764 (1999)] leads to the exact stiffness values at finite temperature T>0T>0T>0 for models whose stiffness is finite at T=0T=0T=0, similar to the spin stiffness of the spin-1/21/21/2 Heisenberg chain but unlike the charge stiffness of the half-filled 1D Hubbard model.
Whether at zero spin density m=0m=0m=0 and finite temperatures T>0T>0T>0 the spin stiffness of the spin-1/21/21/2 XXXXXXXXX chain is finite or vanishes remains an unsolved and controversial issue, as different approaches yield contradictory results. Here we provide an exact upper bound on the stiffness within a canonical ensemble at any fixed value of spin density mmm and show that it is proportional to m2Lm^2 Lm2L in the thermodynamic limit of chain length L→∞L\to\inftyL→∞, for any finite, nonzero temperature. Moreover, we explicitly compute the stiffness at m=0m=0m=0 and confirm that it vanishes. This allows us to exactly exclude the possibility of ballistic transport within the canonical ensemble for T>0T>0T>0.
Whether in the thermodynamic limit, vanishing magnetic field h→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→0 in the thermodynamic limit of chain length L→∞, at high temperatures T→∞. 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 Msp=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→∞, vanishing magnetic field h→0 and within the grand-canonical ensemble.
Finite-temperature T > 0 transport properties of integrable and nonintegrable one-dimensional (1D) many-particle quantum systems are rather different, showing ballistic and diffusive behavior, respectively. The repulsive 1D Hubbard model is a prominent example of an integrable correlated system. For electronic densities n ≠ 1 (and spin densities m ≠ 0) it is an ideal charge (and spin) conductor, with ballistic charge (and spin) transport for T ⩾ 0. In spite of the fact that it is solvable by the Bethe ansatz, at n = 1 (and m = 0) its T > 0 charge (and spin) transport properties are an issue that remains poorly understood. Here we combine this solution with symmetry and the explicit calculation of current-operator matrix elements between energy eigenstates to show that for on-site repulsion U > 0 and at n = 1 the charge stiffness D η ( T ) vanishes for T > 0 in the thermodynamic limit. A similar behavior is found by such methods for the spin stiffness D s ( T ) for U > 0 and T > 0, which vanishes at m = 0. This absence of finite temperature n = 1 ballistic charge transport and m = 0 ballistic spin transport are exact results that clarify long-standing open problems.
The studies of this paper on the spin- 1 / 2 XXZ chain at finite temperatures T > 0 have two complementary goals. The first is to identify the spin carriers of all its S q > 0 energy eigenstates and to show that their spin elementary currents fully control the spin-transport quantities. Here, S q is the q -spin of the continuous S U q ( 2 ) symmetry of the model for anisotropy Δ > 1 . To achieve this goal, our studies rely on a suitable exact valid in throughout the Hilbert space and associated with normal spin operators, rather than fractionalized particles such as spinons. Both the spin stiffness and the zero-field spin-diffusion constant are expressed in terms of thermal expectation values of the square of the elementary currents carried by the spin carriers. Our second goal is to confirm that the zero-field and finite-temperature spin transport is normal diffusive for Δ > 1 . We use two complementary methods that rely on an inequality for the T > 0 spin stiffness and the above thermal expectation values, respectively, to show that the contributions to ballistic spin transport vanish. Complementarily, for T > 0 and Δ > 1 , the spin-diffusion constant is found to be finite and enhanced upon lowering T , reaching its largest yet finite values at low temperatures. Evidence suggests that it diverges in the Δ → 1 limit for T > 0 , consistent with T > 0 anomalous superdiffusive spin transport at Δ = 1 and zero field.
Whether in the thermodynamic limit of lattice length L→∞, hole concentration mηz=−2Sηz/L=1−ne→0, nonzero temperature T>0, and U/t>0 the charge stiffness of the 1D Hubbard model with first neighbor transfer integral t and on-site repulsion U is finite or vanishes and thus whether there is or there is no ballistic charge transport, respectively, remains an unsolved and controversial issue, as different approaches yield contradictory results. (Here Sηz=−(L−Ne)/2 is the η-spin projection and ne=Ne/L the electronic density.) In this paper we provide an upper bound on the charge stiffness and show that (similarly as at zero temperature), for T>0 and U/t>0 it vanishes for mηz→0 within the canonical ensemble in the thermodynamic limit L→∞. Moreover, we show that at high temperature T→∞ the charge stiffness vanishes as well within the grand-canonical ensemble for L→∞ and chemical potential μ→μu where (μ−μu)≥0 and 2μu is the Mott–Hubbard gap. The lack of charge ballistic transport indicates that charge transport at finite temperatures is dominated by a diffusive contribution. Our scheme uses a suitable exact representation of the electrons in terms of rotated electrons for which the numbers of singly occupied and doubly occupied lattice sites are good quantum numbers for U/t>0. In contrast to often less controllable numerical studies, the use of such a representation reveals the carriers that couple to the charge probes and provides useful physical information on the microscopic processes behind the exotic charge transport properties of the 1D electronic correlated system under study.
We present evidence suggesting that spin transport in the gapless phase of the S=1/2XXZS=1/2 \mathrm{XXZ}S=1/2XXZ model is ballistic rather than diffusive. We map the model onto a spinless fermion model whose charge stiffness determines the spin transport of the original model. By means of exact numerical diagonalization and finite size scaling we study both the stiffness and the level statistics. We show that the stiffness is nonzero at any temperature so that the transport is ballistic. Our results suggest that the nonzero stiffness arises because even in the presence of umklapp scattering a nonzero fraction of states remains degenerate in the thermodynamic limit.