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arXiv:2610.16915

Mutual information across the commensurate--incommensurate crossover and near the Kosterlitz--Thouless point of the spin-$\tfrac12$ $J_1$--$J_2$ Heisenberg chain

Wuttichai Pankeaw, Pruet Kalasuwan

cond-mat.str-elquant-phSubmitted Oct 12, 2026• Announced Oct 13, 2026• 0 citations
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Abstract

What quantum mutual information adds to conventional correlation functions is often unclear. We study the spin-1/2 $J_1$-$J_2$ Heisenberg chain, using DMRG on infinite chains and periodic rings and an SU(2) decomposition of the mutual information into correlation channels. For singlet ground states the two-site mutual information is an exact function of the spin correlator, so any additional content must come from blocks. In the dimerized phase, blocks resolve bond-energy correlations in the SU(2)-scalar channel without an independent asymptotic length: singlet and triplet correlation lengths agree to about 1%. The scalar channel dominates block mutual information at large distances in the commensurate regime up to $J_2/J_1=0.40$, but falls to a few percent beyond the disorder point, which coincides numerically with the Majumdar-Ghosh point $J_2/J_1=1/2$; the structure-factor peak splits only at the Lifshitz point $J_2/J_1=0.520622(2)$. On the gapless side we compare block mutual information on rings extrapolated to infinite size with the two-interval prediction of SU(2)$_1$ conformal field theory across the Kosterlitz-Thouless point $J_2/J_1=0.241167$. There, for block lengths $l\le6$, the correction decays as $l^{-p}$ with $p=1.96$-$2.07$, and any logarithmic term is at least 60 times smaller than in the Heisenberg chain. Expanding in the marginal coupling (from level spectroscopy) the first-order coefficient is consistent with $l^{-2}$ scaling and the second-order coefficient is positive for $x\le1/4$, suggesting that the mutual information relative to the field-theory value becomes stationary at the KT point for large blocks. The scalar-channel share approaches 1/4 at small cross ratio $x$, as expected from SO(4) symmetry; under two stated assumptions its first-order response to the coupling is $(3/8)\ln x-5/8$ at small $x$, matched by the $l=4$ data within 1-3% at $x\le1/9$.

Comments
57 pages, 19 figures, 16 tables

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Mutual information across the commensurate--incommensurate crossover and near the Kosterlitz--Thouless point of the spin-12\tfrac1221​ J1J_1J1​--J2J_2J2​ Heisenberg chain
What quantum mutual information adds to conventional correlation functions is often unclear. We study the spin-1/2 J1J_1J1​-J2J_2J2​ Heisenberg chain, using DMRG on infinite chains and periodic rings and an SU(2) decomposition of the mutual information into correlation channels. For singlet ground states the two-site mutual information is an exact function of the spin correlator, so any additional content must come from blocks. In the dimerized phase, blocks resolve bond-energy correlations in the SU(2)-scalar channel without an independent asymptotic length: singlet and triplet correlation lengths agree to about 1%. The scalar channel dominates block mutual information at large distances in the commensurate regime up to J2/J1=0.40J_2/J_1=0.40J2​/J1​=0.40, but falls to a few percent beyond the disorder point, which coincides numerically with the Majumdar-Ghosh point J2/J1=1/2J_2/J_1=1/2J2​/J1​=1/2; the structure-factor peak splits only at the Lifshitz point J2/J1=0.520622(2)J_2/J_1=0.520622(2)J2​/J1​=0.520622(2). On the gapless side we compare block mutual information on rings extrapolated to infinite size with the two-interval prediction of SU(2)1_11​ conformal field theory across the Kosterlitz-Thouless point J2/J1=0.241167J_2/J_1=0.241167J2​/J1​=0.241167. There, for block lengths l≤6l\le6l≤6, the correction decays as l−pl^{-p}l−p with p=1.96p=1.96p=1.96-2.072.072.07, and any logarithmic term is at least 60 times smaller than in the Heisenberg chain. Expanding in the marginal coupling (from level spectroscopy) the first-order coefficient is consistent with l−2l^{-2}l−2 scaling and the second-order coefficient is positive for x≤1/4x\le1/4x≤1/4, suggesting that the mutual information relative to the field-theory value becomes stationary at the KT point for large blocks. The scalar-channel share approaches 1/4 at small cross ratio xxx, as expected from SO(4) symmetry; under two stated assumptions its first-order response to the coupling is (3/8)ln⁡x−5/8(3/8)\ln x-5/8(3/8)lnx−5/8 at small xxx, matched by the l=4l=4l=4 data within 1-3% at x≤1/9x\le1/9x≤1/9.
For the antiferromagnetic J1{J}_{1}J1​-J2{J}_{2}J2​ quantum spin chain with an even number of sites, the point J2d=J1/2{J}_{2}^{d}={J}_{1}/2J2d​=J1​/2 is a disorder point. It marks the onset of incommensurate real space correlations for J2>J2d{J}_{2}>{J}_{2}^{d}J2​>J2d​. At a distinct larger value of J2L=0.52036(6)J1{J}_{2}^{L}=0.520\phantom{\rule{0.16em}{0ex}}36(6){J}_{1}J2L​=0.52036(6)J1​, the Lifshitz point, the peak in the static structure factor begins to move away from $k=\ensuremath{\pi}$. Here, we focus on chains with an odd number of sites. In this case, the disorder point is also at J2d=J1/2{J}_{2}^{d}={J}_{1}/2J2d​=J1​/2 but the behavior close to the Lifshitz point, ${J}_{2}^{L}\ensuremath{\simeq}0.538{J}_{1}$, is quite different: starting at J2L{J}_{2}^{L}J2L​, the ground state goes through a sequence of level crossings as its momentum changes away from $k=\ensuremath{\pi}/2$. An even length chain, on the other hand, is gapped for any J2>0.24J1{J}_{2}>0.24{J}_{1}J2​>0.24J1​ and has the ground-state momentum k=0k=0k=0. This gradual change in the ground-state wave function for chains with an odd number of sites is reflected in a dramatic manner directly in the ground-state on-site magnetization as well as in the bipartite von Neumann entanglement entropy. Our results are based on DMRG calculations and variational calculations performed in a restricted Hilbert space defined in the valence bond picture. In the vicinity of the point J2=J1/2{J}_{2}={J}_{1}/2J2​=J1​/2, we expect the variational results to be very precise.
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The zero-temperature phase diagram of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}\phantom{\rule{4pt}{0ex}}\mathrm{SU}(N)$ antiferromagnetic Heisenberg spin chain is investigated by means of complementary field theory and numerical approaches for general NNN. A fully gapped SU(N)\mathrm{SU}(N)SU(N) valence bond solid made of NNN sites is formed above a critical value of J2/J1{J}_{2}/{J}_{1}J2​/J1​ for all NNN. We find that the extension of this NNN-merized phase for larger values of J2{J}_{2}J2​ strongly depends on the parity of NNN. For even NNN, the phase smoothly interpolates to the large J2{J}_{2}J2​ regime where the model can be viewed as a zigzag SU(N)\mathrm{SU}(N)SU(N) two-leg spin ladder. The phase exhibits both a NNN-merized ground state and incommensurate spin-spin correlations. In stark contrast to the even case, we show that the NNN-merized phase with odd NNN has only a finite extent with no incommensuration. A gapless phase in the SU(N)1\mathrm{SU}{(N)}_{1}SU(N)1​ universality class is stabilized for larger J2{J}_{2}J2​ that stems from the existence of a massless renormalization group flow from SU(N)2\mathrm{SU}{(N)}_{2}SU(N)2​ to SU(N)1\mathrm{SU}{(N)}_{1}SU(N)1​ conformal field theories when NNN is odd.
We explore quantum and classical correlations along with coherence in the ground states of spin-1 Heisenberg chains, namely the one-dimensional XXZ model and the one-dimensional bilinear biquadratic model, with the techniques of density matrix renormalization group theory. Exploiting the tools of quantum information theory, that is, by studying quantum discord, quantum mutual information, and three recently introduced coherence measures in the reduced density matrix of two nearest neighbor spins in the bulk, we investigate the quantum phase transitions and special symmetry points in these models. We point out the relative strengths and weaknesses of correlation and coherence measures as figures of merit to witness the quantum phase transitions and symmetry points in the considered spin-1 Heisenberg chains. In particular, we demonstrate that, as none of the studied measures can detect the infinite-order Kosterlitz-Thouless transition in the XXZ model, they appear to be able to signal the existence of the same type of transition in the biliear biquadratic model. However, we argue that what is actually detected by the measures here is the SU(3) symmetry point of the model rather than the infinite-order quantum phase transition. Moreover, we show in the XXZ model that examining even single site coherence can be sufficient to spotlight the second-order phase transition and the SU(2) symmetry point.
We report on a detailed investigation of the spin-$1 {J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}$ Heisenberg model, a frustrated model with nearest-neighbor coupling J1{J}_{1}J1​, next-nearest neighbor coupling J2{J}_{2}J2​, and a three-site interaction ${J}_{3}\left[({\mathbf{S}}_{i\ensuremath{-}1}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i})({\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i+1})+\mathrm{H}.\mathrm{c}.\right]$ previously studied in [Phys. Rev. B 93, 241108(R) (2016)]. Using density matrix renormalization group (DMRG) and exact diagonalizations, we show that the phase boundaries between the Haldane phase, the next-nearest neighbor Haldane phase, and the dimerized phase can be very accurately determined by combining the information deduced from the dimerization, the ground-state energy, the entanglement spectrum and the Berry phase. By a careful investigation of the finite-size spectrum, we also show that the transition between the next-nearest neighbor Haldane phase and the dimerized phase is in the Ising universality class all along the critical line. Furthermore, we justify the conformal embedding of the SU(2)2\mathrm{SU}{(2)}_{2}SU(2)2​ Wess-Zumino-Witten conformal field theory in terms of a boson and an Ising field, and we explicitly derive a number of consequences of this embedding for the spectrum along the SU(2)2\mathrm{SU}{(2)}_{2}SU(2)2​ transition line between the Haldane phase and the dimerized phase. We also show that the solitons along the first-order transition line between the Haldane phase and the dimerized phase carry a spin-1/2, while the domain walls between different dimerization domains inside the dimerized phase carry a spin 1. Finally, we show that short-range correlations change character in the Haldane and dimerized phases through disorder and Lifshitz lines, as well as through the development of short-range dimer correlations in the Haldane phase, leading to a remarkably rich phase diagram.
We study slightly generalized quantum fidelity susceptibilities where the differential change in the fidelity is measured with respect to a different term than the one used for driving the system toward a quantum phase transition. As a model system we use the spin-1/21/21/2 J1{J}_{1}J1​-J2{J}_{2}J2​ antiferromagnetic Heisenberg chain. For this model, we study three fidelity susceptibilities, ${\ensuremath{\chi}}_{\ensuremath{\rho}}$, ${\ensuremath{\chi}}_{D}$, and ${\ensuremath{\chi}}_{\mathrm{AF}}$, which are related to the spin stiffness, the dimer order, and antiferromagnetic order, respectively. All these ground-state fidelity susceptibilities are sensitive to the phase diagram of the J1{J}_{1}J1​-J2{J}_{2}J2​ model. We show that they all can accurately identify a quantum critical point in this model occurring at ${J}_{2}^{c}\ensuremath{\sim}0.241{J}_{1}$ between a gapless Heisenberg phase for J2<J2c{J}_{2}<{J}_{2}^{c}J2​<J2c​ and a dimerized phase for J2>J2c{J}_{2}>{J}_{2}^{c}J2​>J2c​. This phase transition, in the Berezinskii-Kosterlitz-Thouless universality class, is controlled by a marginal operator and is therefore particularly difficult to observe.