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arXiv:2302.14090 · Physical review. B./Physical review. B

Even-odd effects in the J1−J2 SU(N) Heisenberg spin chain

Loïc Herviou, Sylvain Capponi, P. Lecheminant

Published May 18, 2023• 5 citations
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Abstract

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 $N$. A fully gapped $\mathrm{SU}(N)$ valence bond solid made of $N$ sites is formed above a critical value of ${J}_{2}/{J}_{1}$ for all $N$. We find that the extension of this $N$-merized phase for larger values of ${J}_{2}$ strongly depends on the parity of $N$. For even $N$, the phase smoothly interpolates to the large ${J}_{2}$ regime where the model can be viewed as a zigzag $\mathrm{SU}(N)$ two-leg spin ladder. The phase exhibits both a $N$-merized ground state and incommensurate spin-spin correlations. In stark contrast to the even case, we show that the $N$-merized phase with odd $N$ has only a finite extent with no incommensuration. A gapless phase in the $\mathrm{SU}{(N)}_{1}$ universality class is stabilized for larger ${J}_{2}$ that stems from the existence of a massless renormalization group flow from $\mathrm{SU}{(N)}_{2}$ to $\mathrm{SU}{(N)}_{1}$ conformal field theories when $N$ is odd.

DOI
10.1103/physrevb.107.205135

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The zero-temperature phase diagram of the J1\ensuremath-J2SU(N){J}_{1}\text{\ensuremath{-}}{J}_{2}\phantom{\rule{4pt}{0ex}}\mathrm{SU}(N)J1​\ensuremath-J2​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.
According to field theory computations, the zero-temperature phase diagram of the J1-J2 SU(N) spin chains depends on the parity of N. While a transition from critical to a gapped SU(N) valence bond solid is always observed at small J2s, we find that for odd N a gapless phase reopen in the large J2 limit. This phase is in the SU(N)1 universality class due to the existence of a massless renormalization group flow from SU(N)2 to SU(N)1 conformal field theories.In this talk, after presenting our RG analysis, I will discuss the strengths and limits of (i)DMRG applied to SU(N) spin chains, which allowed us to confirm (some of) the field theory predictions, and also revealed the existence of a commensurate to incommensurate transition within the valence bond phase for even N.
Motivated by near-term experiments with ultracold alkaline-earth atoms confined to optical lattices, we establish numerically and analytically the phase diagram of two-leg SU(N)\mathrm{SU}(N)SU(N) spin ladders. Two-leg ladders provide a rich and highly nontrivial extension of the single chain case on the way towards the relatively little explored two-dimensional situation. Focusing on the experimentally relevant limit of one fermion per site, antiferromagnetic exchange interactions, and $2\ensuremath{\le}N\ensuremath{\le}6$, we show that the phase diagrams as a function of the interchain (rung) to intrachain (leg) coupling ratio, ${J}_{\ensuremath{\perp}}/{J}_{\ensuremath{\parallel}}$, strongly differ for even versus odd NNN. For even N=4N=4N=4 and 6, we demonstrate that the phase diagram consists of a single valence bond crystal (VBC) with a spatial period of N/2N/2N/2 rungs. For odd N=3N=3N=3 and 5, we find surprisingly rich phase diagrams exhibiting three distinct phases. For weak rung coupling, we obtain a VBC with a spatial period of NNN rungs, whereas for strong coupling we obtain a critical phase related to the case of a single chain. In addition, we encounter intermediate phases for odd NNN, albeit of a different nature for N=3N=3N=3 as compared to N=5N=5N=5. For N=3N=3N=3, we find a novel gapless intermediate phase with ${J}_{\ensuremath{\perp}}$-dependent incommensurate spatial fluctuations in a sizeable region of the phase diagram. For N=5N=5N=5, there are strong indications for a narrow potentially gapped intermediate phase, whose nature is not entirely clear. Our results are based on (i) field theoretical techniques, (ii) qualitative symmetry considerations, and (iii) large-scale density matrix renormalization group (DMRG) simulations keeping beyond a million of states by fully exploiting and thus preserving the SU(N)\mathrm{SU}(N)SU(N) symmetry.
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.
We study the ground-state properties of antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ chains with half-integer spins ranging from S=32S=\frac{3}{2}S=23​ to S=112S=\frac{11}{2}S=211​ using the density-matrix renormalization group method. We map out the ground-state phase diagrams as a function of J2J1\frac{{J}_{2}}{{J}_{1}}J1​J2​​ containing topological phases with alternating $\frac{2S\ensuremath{-}1}{2}$ and 2S+12\frac{2S+1}{2}22S+1​ valence bonds. We identify these topological phases and their boundaries by calculating the string order parameter, the dimer order parameter, and the spin gap for those $\mathrm{high}\text{\ensuremath{-}}S$ systems in the thermodynamic limit (finite-size scaling). We find that these topological regions narrow down inversely with SSS and converge to a single point at J2J1=14\frac{{J}_{2}}{{J}_{1}}=\frac{1}{4}J1​J2​​=41​ in the classical limit---a critical threshold between commensurate and incommensurate orders. In addition, we extend the discussion of the Majumder-Ghosh state, previously noted only for S=12S=\frac{1}{2}S=21​, and speculate about its possible presence as a ground state in half-integer high-spin systems over a substantial range of J2J1\frac{{J}_{2}}{{J}_{1}}J1​J2​​ values.
The antiferromagnetic ${J}_{1}\ensuremath{-}{J}_{2}$ model is a spin-1/2 chain with isotropic exchange J1>0{J}_{1}>0J1​>0 between first neighbors and ${J}_{2}=\ensuremath{\alpha}{J}_{1}$ between second neighbors. The model supports both gapless quantum phases with nondegenerate ground states and gapped phases with $\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})>0$ and doubly degenerate ground states. Exact thermodynamics is limited to $\ensuremath{\alpha}=0$, the linear Heisenberg antiferromagnet (HAF). Exact diagonalization of small systems at frustration $\ensuremath{\alpha}$ followed by density matrix renormalization group calculations returns the entropy density $S(T,\ensuremath{\alpha},N)$ and magnetic susceptibility $\ensuremath{\chi}(T,\ensuremath{\alpha},N)$ of progressively larger systems up to N=96N=96N=96 or 152 spins. Convergence to the thermodynamic limit, $S(T,\ensuremath{\alpha})$ or $\ensuremath{\chi}(T,\ensuremath{\alpha})$, is demonstrated down to $T/J\ensuremath{\sim}0.01$ in the sectors $\ensuremath{\alpha}<1$ and $\ensuremath{\alpha}>1$. $S(T,\ensuremath{\alpha})$ yields the critical points between gapless phases with ${S}^{\ensuremath{'}}(0,\ensuremath{\alpha})>0$ and gapped phases with ${S}^{\ensuremath{'}}(0,\ensuremath{\alpha})=0$. The ${S}^{\ensuremath{'}}(T,\ensuremath{\alpha})$ maximum at ${T}^{*}(\ensuremath{\alpha})$ is obtained directly in chains with large $\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})$ and by extrapolation for small gaps. A phenomenological approximation for $S(T,\ensuremath{\alpha})$ down to T=0T=0T=0 indicates power-law deviations ${T}^{\ensuremath{-}\ensuremath{\gamma}(\ensuremath{\alpha})}$ from $exp[\ensuremath{-}\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})/T]$ with exponent $\ensuremath{\gamma}(\ensuremath{\alpha})$ that increases with $\ensuremath{\alpha}$. The $\ensuremath{\chi}(T,\ensuremath{\alpha})$ analysis also yields power-law deviations, but with exponent $\ensuremath{\eta}(\ensuremath{\alpha})$ that decreases with $\ensuremath{\alpha}$. Spin correlation functions account for $S(T,\ensuremath{\alpha})$ differences between frustration $\ensuremath{\alpha}<1$ within a chain and $\ensuremath{\alpha}>1$ between HAFs on sublattices. $S(T,\ensuremath{\alpha})$ and the spin density $\ensuremath{\rho}(T,\ensuremath{\alpha})=4T\ensuremath{\chi}(T,\ensuremath{\alpha})$ probe the thermal and magnetic fluctuations, respectively, of strongly correlated spin states. Gapless chains have constant $S(T,\ensuremath{\alpha})/\ensuremath{\rho}(T,\ensuremath{\alpha})$ for T<0.10T<0.10T<0.10. Remarkably, the ratio decreases (increases) with TTT in chains with large (small) $\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})$.
We study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic (J1FJ_1^FJ1F​) and antiferromagnetic (J1AJ_1^AJ1A​) nearest-neighbor interactions and ferromagnetic next-nearest-neighbor interaction (J2FJ_2^FJ2F​). In this model frustration is induced by the non-zero J2FJ_2^FJ2F​ and the effect of dimerization is realized by changing the strength of J1AJ_1^AJ1A​ for a fixed value of J1FJ_1^FJ1F​. The model with site spin sss behaves like a Haldane spin chain with site spin 2sss in the limit of vanishing J2FJ_2^FJ2F​ and large J1F/J1AJ_1^F/J_1^AJ1F​/J1A​. We show that the exact ground state of the model can be found along a line in the parameter space. The phase diagram in the space of J1A−J2FJ_1^A-J_2^FJ1A​−J2F​ is determined using numerical techniques complemented by some analytical calculations. A number of quantities, including the ground state spin, energy gap, structure factor, entanglement entropy, and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and application-wise important feature of this model is that it exhibits a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian.