Frustration and multicriticality in the antiferromagnetic spin-1 chain

The antiferromagnetic spin-1 chain has a venerable history and has been thought to be well understood. Here, we show that inclusion of both next-nearest-neighbor $\left(\alpha \right)$ and biquadratic $\left(\beta \right)$ interactions results in a rich phase diagram with a multicritical point that has not been observed before. We study the problem using a combination of the density matrix renormalization group (DMRG), an analytic variational matrix product state wave function, and conformal field theory. For negative $\beta <{\beta }^{*}$, we establish the existence of a spontaneously dimerized phase, separated from the Haldane phase by the critical line ${\alpha }_c\left(\beta \right)$ of second-order phase transitions. In the opposite regime, $\beta >{\beta }^{*}$, the transition from the Haldane phase becomes first order into the next-nearest-neighbor (NNN) AKLT phase. Based on the field theoretical arguments and DMRG calculations, we find that these two regimes are separated by a multicritical point $({\beta }^{*},{\alpha }^{*})$ of a different universality class, described by the level-4 SU(2) Wess-Zumino-Witten conformal theory. From the DMRG calculations, we estimate this multicritical point to lie in the range $-0.2<{\beta }^{*}<-0.15$ and $0.47<{\alpha }^{*}<0.53$. We further find that the dimerized and NNN-AKLT phases are separated from each other by a line of first-order phase transitions that terminates at the multicritical point. We establish that transitions out of the Haldane phase into the dimer or NNN-AKLT phases are topological in nature and occur either with or without closing of the bulk gap, respectively. We also study short-range incommensurate-to-commensurate transitions in the resulting phase diagram. Inside the Haldane phase, we show the existence of two incommensurate crossovers: the Lifshitz transition and the disorder transition of the first kind, marking incommensurate correlations in momentum and real space, respectively. Notably, these crossover lines stretch across the entire $(\beta ,\alpha )$ phase diagram, merging into a single incommensurate-to-commensurate transition line for negative $\beta \lesssim {\beta }^{*}$ inside the dimer phase. This behavior is qualitatively similar to that seen in classical frustrated two-dimensional spin models, by way of the quantum (1+1)D to classical 2D correspondence.

Figure 1

Schematic rendering of spin-1 chain with nearest (brown dashed), next-nearest (yellow dashed), and biquadratic (red dashed) interactions. Three “typical” ground states (GS) illustrated corresponding to the AKLT state, which is the exact GS at $\beta =1/3,\alpha =0$, the “next-nearest-neighbor” AKLT (NNN-AKLT), which takes over for large $\alpha$, and the dimerized states characteristic of $\beta <-1$. We complement numerical studies of the model with an analytic variational wave function represented as a matrix product state which interpolates the three states described above.

Figure 2

Schematic phase diagram obtained in this work. Symbols marking the transitions between different phases are the DMRG data, solid lines are guide to the eye, and dashed lines are obtained from the analytical matrix-product state ansatz (see Sec. 8c). The critical line (thick black line) ${\alpha }_c\left(\beta \right)$ starts at the TB point $\beta =-1$ (solid circle), and is terminated at a multicritical point $\Omega$, beyond which the transition becomes first order (thin black line ${\alpha }_T)$. The critical line ${\alpha }_c$ separates the dimer and Haldane phases, whereas first-order-transition ${\alpha }_T$ separates the Haldane and NNN-AKLT phases. The transition between the dimer and NNN-AKLT phases is of first order, marked by ${\alpha }_{\delta }$ line (crosses). We have not studied the effect of $\alpha$ on the ULS point (solid diamond) and the region $\beta \ge 1$.

Figure 3

DMRG phase diagram indicating short-range order within the phases illustrated earlier in Fig. 2. Various commensurate (C) and incommensurate (IC) phases are described in the text. Note that the onset of real-space incommensuration (IC-R) across the ${\alpha }_d$ line (diamonds) is distinct from the Lifshitz line ${\alpha }_L$ (circles) inside the Haldane phase. The two lines merge into a single C-IC transition upon entering the long-range dimer phase at $\beta \lesssim -0.15$.

Figure 4

Schematic RG flow diagram in the vicinity of the Takhtajan-Babudjian (TB) point. The signs of conformal theory parameters $m$ and $\lambda$ in Eq. (5) determine the nature of the phases (Dimer, Haldane, or ferromagnet). The blue and red arrows indicate the flow of the relevant parameter $m$ which opens up a spectral gap upon entering the Haldane or dimer phase, respectively. The solid black line is critical, and is governed by the conformal WZW theory at the TB point. The black arrows indicate the marginally irrelevant flow of $\lambda$ towards the TB point (if $J_1>0$ is antiferromagnetic), whereas the green arrows indicate marginally relevant flow (if $J_1<0$, not discussed in this work). We identify a new mutlicritical point $\Omega$, characterized by an unstable RG flow (purple arrows), marking the first-order phase transition everywhere along the dashed line (see text for discussion).

Figure 5

(a) The bulk gap $\Delta (\alpha ,\beta =-0.2)$ in the thermodynamic limit for different number $M$ of kept DMRG states as a function of $\alpha$. (Inset) At the critical point ${\alpha }_c=0.480\left(5\right)$, we find a small gap $\Delta =0.018\left(3\right)$ for the largest $L$ and $M$ values studied. However, extrapolating to the infinite system size, we find an excellent fit to a gapless form $\Delta \left(L\right)=a_{\Delta }/L+b_{\Delta }/L^2$ for a fixed number of states $M=200$. (b) The correlation length for various system sizes $L$ with $M=200$ [see equation (2) for a definition of $\xi ]$, as we move through the critical line, with a disorder point at the minimum of $\xi$ at ${\alpha }_d(\beta =-0.2)=0.600\left(5\right)$.

Figure 6

(a) The bulk gap $\Delta (\alpha ,\beta =0.2)$ in the thermodynamic limit for different number $M$ of kept DMRG states as a function of $\alpha$. We find that the gap goes through a minimum in the vicinity of ${\alpha }_T=0.855\left(5\right)$, with a minimum value of ${\Delta }_0=0.11\left(2\right)$. (Inset) At the transition point, we find an excellent fit to a gapped form $\Delta \left(L\right)={\Delta }_0+a_{\Delta }/L+b_{\Delta }/L^2$ for a fixed number of states $M=200$. (b) Correlation length for various system sizes $L$ with $M=200$ [see Eq. (2) for a definition of $\xi ]$, as we move through the critical line and a disorder point at ${\alpha }_d(\beta =0.2)=0.100\left(5\right)$.

Figure 7

Schematic representation of two coupled $S=1$ spin chains. (a) The spin coupling within each chain is given by the interaction $J_2=\alpha J_1$ in the original model Eq. (1). (b) The dimerized ground state, with the circled rungs corresponding to the spin singlet (quintuplet) of two spins, for antiferromagnetic (ferromagnetic) interchain coupling $J_{\perp }=J_1$. (c) The NNN-AKLT ground state, with the blue links representing singlet bonds between composite spin-$1/2$ objects that form the AKLT ground state between second neighbors.

Figure 8

(a) Central charge $c$ for different values of $\beta$ along the critical line in Fig. 4, extracted by fitting the DMRG entanglement entropy to Eq. (7) for $M=200$ kept states. The red circles mark the values of $c$ determined reliably, and the blue squares denote the data points where conformal scaling does not apply, as illustrated in (b): the error of fitting DMRG data to Eq. (7) for $\beta =-0.2$ (reliable) and $\beta =0$ (scaling fails).

Figure 9

The gap to edge excitations ${\Delta }_{\mathrm{edge}}$ at a fixed system size $L$ and fixed number of states $M=200$, opening up as $\alpha$ crosses the critical line from the AKLT phase (a) at ${\alpha }_c(\beta =-0.2)=0.480\left(5\right),L=80$, and (b) ${\alpha }_T(\beta =0.2)=0.855\left(5\right),L=100$. We clearly see the edge excitations become gapped out. (c) The absolute value of magnetization along the chain in the $S_{\mathrm{tot}}=0$ symmetry sector approaching the critical line from the AKLT phase for $\beta =0.2$. We see the edge excitations begin to bleed into the bulk of the chain as the correlation length increases and then vanish as we cross the transition line.

Figure 10

The bulk gap $\Delta$ between symmetry sectors $S_{\mathrm{tot}}=0$ and 2 for fixed $\beta =-1$ tuning away from the TB point extrapolated in $L$ with $M=200$ kept states (a). We see the gap opens continuously for a finite value of $\alpha$. (Inset) Extrapolating the gap at the TB point $(\beta =-1.0$ and $\alpha =0)$ to the thermodynamic limit, the fit is in excellent agreement with a gapless point namely, $\Delta \left(L\right)=a_{\Delta }/L+b_{\Delta }/L^2$. When fit to a functional form for a finite gap [see equation (10)] we find a small gap at the TB equal to $0.017\left(2\right)$. (b) Dimer order parameter, defined in Eq. (8) as a function of the system size $L$ and extrapolated to the thermodynamic limit for $M=200$ states, clearly displaying the system enters the dimer phase immediately upon tuning $\alpha$ away from the TB point. The finite value of $D(\alpha =0,\beta =-1)$ at the TB point is attributed to not reaching large enough system sizes at the critical point.

Figure 11

Bulk gap $\Delta$ in the vicinity of the TB point for a fixed NNN coupling $\alpha =0.2$, as a function of $\beta$ extrapolated in system size $L$ with $M=200$ kept states. From the behavior of the gap, it is clear the critical line will move towards a smaller value of $\beta$, as we increase $\alpha$ away from the TB point, verifying the slope of ${\alpha }_c\left(\beta \right)$ shown in the schematic phase diagram 2.

Figure 12

(a) The dimer order parameter $D(\alpha ,\beta )$ as a function of $\alpha$ for $\beta =-0.2$ for various system sizes $L$ and $M=200$ states. In the dimer phase, we find the order parameter is independent of system size implying $D(\alpha ,\beta )$ is finite in the thermodynamic limit, whereas deep inside the NNN-AKLT phase $\left(\alpha \gtrsim 1.2\right)$, we find that $D(\alpha ,\beta )$ does not saturate in $L$ and likely becomes vanishingly small in the thermodynamic limit. (b) The ground-state energy as a function of $\alpha$ for $\beta =-0.2$ obtained within DMRG extrapolated in $L$ and $M$ (circles) compared to the AKLT, NNN-AKLT, dimer, and variational wave functions. Note, the naive estimate of the ground-state energy of the dimer and NNN-AKLT wave functions agrees with the variational result and is therefore not shown. We have clearly marked the critical line ${\alpha }_c$ as well as the first-order transition line ${\alpha }_{\delta }$ from the dimer to the NNN-AKLT phase (see Fig. 2). (Inset) The numerical second derivative of the ground-state energy as a function of $\alpha$, we find a discontinuity in $d^2{E}_{gs}/d{\alpha }^2$ at the critical point, which suggests the transition into the dimer phase is second order.

Figure 13

(a) The dimer order parameter $D(\alpha ,\beta )$ as a function of $\alpha$ for $\beta =0.2$ for various system sizes $L$ and $M=200$ states. The dimer order parameter is identically zero in the Haldane phase, whereas inside the NNN-AKLT phase, we find that $D(\alpha ,\beta )$ does not saturate in $L$ and may become vanishingly small in the thermodynamic limit. (b) Comparison of the ground-state energy for $\beta =0.2$ as a function of $\alpha$ between the DMRG results extrapolated in $L$ and $M$ (circles), the AKLT ansatz (dashed line) and the variational wave function (continuous line). For large values of $\alpha$ the ground state approaches the NNN-AKLT ansatz. The first-order transition shows up clearly in both the AKLT and variational solutions, but the first numerical derivative of the ground-state energy obtained within DMRG shows no sign of a discontinuity at ${\alpha }_T$, consistent with Refs. [13, 14].

Figure 14

Schematic figures displaying a disorder point in the correlation length $\xi$ (a) and the wave number $q$ (b). C and IC denote commensurate and incommensurate correlations in real space. Generic phase diagram of breaking a discrete symmetry via tuning an effective energy scale $T_{\mathrm{eff}}$ in two dimensions in the presence of incommensurate correlations $\lambda$ (c).

Figure 15

The real-space correlation function $C\left(x\right)$, plotted in terms of $K\left(x\right)\equiv C\left(x\right){(-1)}^x\sqrt{x}$ to extract the correlation length $\xi (\alpha ,\beta )$ for $\beta =0.1$ on either side of the disorder point at ${\alpha }_d(\beta =0.1)=0.180\left(2\right)$ with (a) $\alpha =0.1<{\alpha }_d\left(\beta \right)$ and (b) $\alpha =0.2>{\alpha }_d\left(\beta \right)$. Note the presence of incommensurate real-space correlations in (b) in the form of peaks. Extracting the wave number $q(\alpha ,\beta )$ by plotting $C\left(x\right)$ in terms of $P\left(x\right)\equiv K\left(x\right)exp(x/\xi )$ for $\beta =0.1$ and $\alpha =0.2$, the numerical data are circles and the solid line is a fit to the data (c).

Figure 16

(a) The correlation length $\xi (\alpha ,\beta )$ for various values of $\alpha$ and $\beta$. We identify the minimum in $\xi (\alpha ,\beta )$ as a disorder point. Note that the value of the correlation length at the disorder point is increasing as $\beta$ decreases. (b) The wave number $q(\alpha ,\beta )$ for fixed $\beta$ as a function of $\alpha$. We find $q(\alpha ,\beta )$ rises continuously from $\pi$ at ${\alpha }_d\left(\beta \right)$ to an incommensurate value.

Figure 17

Comparison of the ground-state energy for $\beta =0.2$ as a function of $\alpha$ between the DMRG results (points), the AKLT ansatz (dashed line) and the variational wave function (continuous line). Zoomed in region around the disorder and Lifshitz points (dashed lines), upon crossing the disorder point the ground state is very close to the AKLT ansatz.

Figure 18

The momentum-space correlation function $S\left(q\right)$ for $\beta =0.3$ and various values of $\alpha$. We find a Lifshitz point at ${\alpha }_L(\beta =0.3)=0.095\left(2\right)$.

Figure 19

The real-space correlation function $C\left(x\right)$, plotted in terms of $K\left(x\right)\equiv C\left(x\right){(-1)}^x\sqrt{x}$ for $\beta =-0.2$ with (a) $\alpha =0.50$ in the dimer phase with $q(\alpha ,\beta )=\pi$ and (b) $\alpha =0.80$ in the incommensurate dimer phase, plotted for even (black) and odd (blue) values of $x$ to show clearly finite dimerization on top of an OZ form with $q(\alpha ,\beta )>\pi$. The dimerization $\delta (\alpha ,\beta )$ is plotted as a function of $\alpha$ for (c) $\beta =-0.2$ and (d) $\beta =-0.3$. We see that the dimerization first rises continuously as we cross ${\alpha }_{\delta }^l\left(\beta \right)$, before acquring a cusp at the C-IC transition $({\alpha }_{dL}$ line), and then decreases to zero in the incommensurate dimer phase. The transition line ${\alpha }_{\delta }\left(\beta \right)$ between the incommensurate dimer phase and the NNN-AKLT phase is defined when dimerization $\delta (\alpha ,\beta )$ becomes vanishingly small in Eq. (4).

Figure 20

The dependence of the gap $\Delta$, on the number of kept states $M$ in each relevant phase: (a) dimer, (b) Haldane, and (c) NNN-AKLT.

Figure 21

The ground-state energy $E_{gs}$, for various different number of kept states $M$ for (a) $\beta =-0.2$ and (b) $\beta =0.2$ as a function of $\alpha$. In contrast to the gap (Fig. 20), the $M$ dependence of $E_{gs}$ is much weaker. (c) The most significant $M$ dependence of the ground-state energy is observed in the NNN-AKLT phase.

Figure 22

The variational matrix product state (MPS) wave function is constructed using three states, AKLT, NNN-AKLT, and dimer, which can be represented individually as MPSs, see Eq. (15). Parameters $A,B,C$ can be chosen to interpolate between the three states, which can each be recovered by a special choice of parameters shown in the figure above.

Figure 23

Approximate dispersion curves $\varepsilon \left(k\right)$ calculated for representative choices of couplings $\alpha ,\beta$ within the single mode or “crackion” approach. The dotted line (blue) is evaluated for $\alpha =0,\beta =0$, is shown for reference and captures the physics deep within the AKLT phase; the dashed-dotted (orange) line corresponds to $\alpha >0$ and approximates the excitations in the NNN-AKLT phase; the dashed (purple) line corresponds to $\beta <0$, the dimer phase, and the solid (red) line corresponds to $\beta >0$. We observe a shift in the minimum of the dispersion curve away from the AKLT phase, in which the minimum occurs at $k=\pi$ as we tune $\alpha ,\beta$, see also Fig. 18.

Figure 24

Values of $A,B,C$ (blue circles, red squares, yellow diamonds, respectively) shown as next-nearest-neighbor coupling $\alpha$ and biquadratic coupling $\beta$ are tuned across the pure Heisenberg coupling point $\alpha =0,\beta =0$ deep inside the Haldane phase. (a) For a fixed $\beta =0$ we show $A,B,C$ as a function of $\alpha$. (b) For a fixed $\alpha =0$, we show $A,B,C$ as a function of $\beta$—the point $\beta =1/3$ corresponds to the AKLT point for which the AKLT MPS state is the exact ground state (corresponding to $A=B=1/3,C=0$. Note that, within our approximation, the AKLT state also coincidentally describes the point $\alpha =0.374,\beta =0$ shown in (a). Moreover, we observe that the parameters $A,B,C$ are “close” to the AKLT values for a large portion of the phase diagram (with the exception of regions deep in the NNN-AKLT phase $\alpha \gtrsim 0.8)$.