Unified phase diagram of antiferromagnetic SU($N$) spin ladders

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 $\mathrm{SU}\left(N\right)$ 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 $2\le N\le 6$, we show that the phase diagrams as a function of the interchain (rung) to intrachain (leg) coupling ratio, $J_{\perp }/J_{\parallel }$, strongly differ for even versus odd $N$. For even $N=4$ and 6, we demonstrate that the phase diagram consists of a single valence bond crystal (VBC) with a spatial period of $N/2$ rungs. For odd $N=3$ 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 $N$ 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 $N$, albeit of a different nature for $N=3$ as compared to $N=5$. For $N=3$, we find a novel gapless intermediate phase with $J_{\perp }$-dependent incommensurate spatial fluctuations in a sizeable region of the phase diagram. For $N=5$, 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 $\mathrm{SU}\left(N\right)$ symmetry.

Figure 1

Phase diagrams of two-leg $\mathrm{SU}\left(N\right)$ antiferromagnetic ladders for $N\le 6$ vs $J_{\perp }$, setting $J_{\parallel }=1$. At $J_{\perp }=0$, all ladders are critical [two decoupled Sutherland models (SM), with total central charge $c=2\times (N-1)$]. For finite $J_{\perp }>0$, for even $N$, a single symmetry-broken $4k_F$ valence-bond crystal (VBC) phase with an $N$-site plaquette is found, which for $N=2$ reduces to a trivial nondegenerate rung singlet phase (note that for $N\ge 8$ and large $J_{\perp }$, a critical phase with $c=N-1$ is expected even for even $N$). For odd $N$, in stark contrast, we find three phases. For small $J_{\perp }$, a $2k_F$-VBC, for large $J_{\perp }$, a critical Luttinger liquid (LL) with central charge $c=N-1$, and a newly uncovered intermediate phase. This intermediate phase for $N=3$ ($J_{\perp }\in [\sim 1,1.562]$) is incommensurate, and for $N=5$ ($J_{\perp }\in [0.581,0.596]$) is reminiscent of topological phases.

Figure 2

$\mathrm{SU}\left(N\right)$ DMRG results for the nearest-neighbor spin correlations ${\langle S·S\rangle }_{\parallel }$ for fixed bonds along the legs as a function of $J_{\perp }$ for $N=4$ (left panel) and $N=6$ (right panel). In both cases, we witness the appearance of two different sets of values due to the spontaneous formation of $2\times 2$ plaquettes for $N=4$ and $3\times 2$ plaquettes for $N=6$, as depicted in the lower right insets based on actual bond energies. Panel (a) shows dimerization (and finite-size tetramerization for $J_{\perp }\lesssim 0.5$) in uniform $\mathrm{SU}\left(4\right)$ ladders of length $L=16$, 32, 240, together with an extrapolation $L\to \infty$ (thick lines) using open boundaries. By grouping data at fixed bond position $xmodN$, two branches emerge for larger $J_{\perp }$. Tetramerization manifests itself by the additional splitting of branches towards smaller $J_{\perp }$. It vanishes in the thermodynamic limit. The inset shows the dimerization strength $\mathrm{\Delta }$ of the extrapolated data ($L\to \infty$), where “tetra #$n$” refers to the dimerization relative to the individual tetramerized branches $n=1,2$. The analysis for the $\mathrm{SU}\left(6\right)$ ladder in (b) is analogous to (a), yet takes data from a single DMRG scan [cf. Appendix pp1] that is linear in $J_{\perp }$ over the range shown, using $L=300$. By grouping data with respect to $xmod6$, again two branches emerge, but here due to trimerization (see inset for data on an isotropic ladder for $J_{\perp }=1$) with no indication for hexamerization.

Figure 3

$\mathrm{SU}\left(3\right)$ Heisenberg ladder—ED results showing the local rung parity for small ladders with OBC. $J_{\perp }$ varies from 1.35 to 1.60 in steps of 0.01 for (a) to (c) and in steps of 0.05 for (d). The color code groups the $J_{\perp }$ values with similar profiles $\langle n_{\mathrm{S}}\left(x\right)\rangle$ [the $J_{\perp }$ values where these profiles switch are summarized in Fig. 7]. The number $\left(m\right)$ indicates the number of significant local maxima, and coincides with the number of ground-state level crossings encountered, starting with $m=0$ when coming from large $J_{\perp }$ (see main text).

Figure 4

$\mathrm{SU}\left(3\right)$ Heisenberg ladder: (a) Bond energies ${\langle S·S\rangle }_{\parallel }$ vs $J_{\perp }$ along the legs in the center of uniform systems (the analysis is completely analogous to Fig. 2 except that data is grouped with respect to $xmod3$; see legend). Within the incommensurate phase $J_{\perp }\in [\sim 1,1.562]$ (shaded area) the bond energies vs $J_{\perp }$ become irregular. Inset visualizes the bond energies in the center of a uniform $L=300$ ladder at $J_{\perp }=1$ (in the VBC phase). (b) Bond strengths ${\langle S·S\rangle }_{\perp }$ along the rungs [identical analysis as in (a) otherwise]. Inset shows a polynomial extrapolation of $J_{\perp }^{c1}\left(L\right)$ in the thermodynamic limit, suggesting $J_{\perp }^{c1}(L\to \infty )\sim 1$. (c) Norm-squared (arb. units) of the Fourier transform of ${\langle S·S\rangle }_{\perp }$ along uniforms ladders vs $J_{\perp }$. The peaks related to the incommensurate wave vector $q\left(J_{\perp }\right)$ are tracked by red dots [see also Fig. 8]. Additional maxima appear at $k=\pm 2\pi /3$ (horizontal black dashed lines) that indicate the superimposed three-rung periodicity. The vertical dashed line marks $J_{\perp }^{c2}$. (d) Ground-state parity $P\left(L\right)$ vs ladder length $L=3p$ with integer $p$, superimposed with scaled nearest-neighbor ${\langle S·S\rangle }_{\parallel }\left(x\right)$ data along the legs (blue) and rungs (orange), all at fixed $J_{\perp }=1.4$. Further details are provided in Fig. 10 in Appendix pp3.

Figure 5

Incommensurate profiles in the $\mathrm{SU}\left(3\right)$ Heisenberg ladder at length $L=60$, using $\langle n_{\mathrm{S}}\rangle ={\langle S·S\rangle }_{\perp }+\frac{2}{3}$: (a)–(g) show different snapshots across the incommensurate regime, with (a) already in the commensurate small-$J_{\perp }$ regime [note that $J_{\perp }^{c1}(L=60)\gtrsim 1.20$; cf. Fig. 4], whereas (g) is just outside $J_{\perp }^{c2}$. Small red arrows indicate “domain walls” (upper crossings amongst light-colored lines), which is useful for $J_{\perp }$ closer to $J_{\perp }^{c1}$. Conversely, small blue arrows mark local maxima in the plain data sequence (blue), which is useful for larger $J_{\perp }$. Every panel also specifies the integrated density $\langle N_{\mathrm{S}}\rangle$ of symmetric rung multiplets. (h) summarizes $\langle N_{\mathrm{S}}\rangle$ vs $J_{\perp }$ over the full incommensurate range (shaded area) down to $J_{\perp }=0$.

Figure 6

$\mathrm{SU}\left(5\right)$ Heisenberg ladder: (a) similar analysis as in Fig. 2, yet grouping data with respect to $xmod5$ (see legend). This results in three branches for $J_{\perp }<0.581\equiv J_{\perp }^{c1}$ due to pentamerization, as shown in the lower left inset for $J_{\perp }=0.5$. From our numerical data, we conclude that this gapped phase persists down to $J_{\perp }\to 0^{+}$. The lower right inset shows a zoom into the extremely narrow intermediate phase for $J_{\perp }\in [0.581,0.596]$. (b) Nearest-neighbor ${\langle S·S\rangle }_{\perp }^{\mathrm{NN}}$ rung data at the open left boundary of the system. By color coding the ranges in $J_{\perp }$ as with the $x$ axis in the lower right inset of (a), the intermediate phase is linked to the sharp emergence of an edge feature at the open boundary of the ladder. (c) Analysis of the entanglement flow for the same window in $J_{\perp }$ as the right inset in (a). These entanglement spectra were computed for fixed $J_{\perp }$ in the center of uniform length $L=120$ ladders Lines with the same color belong to the same $\mathrm{SU}\left(5\right)$ symmetry sector, using a compact Dynkin labeling scheme in the legend (see Appendix pp1). The results are also consistent with DMRG scans in $J_{\perp }$ (e.g., see Fig. 11 in Appendix pp4).

Figure 7

Locations of the changes in the ground-state parity obtained from ED results on $\mathrm{SU}\left(3\right)$ Heisenberg $2\times L$ ladder as a function of $1/L$. The number $\left(N_{\mathrm{S}}\right)$ indicates the number of ground-state level crossings encountered coming from large $J_{\perp }$ (see Fig. 3 in the main text). There are $L/3$ level crossings distributed over a significant region of $J_{\perp }$, compatible with the extension of the intermediate phase found in DMRG.

Figure 8

Incommensurate wave length in uniform SU(3) Heisenberg ladders of length $L=60$ with open boundaries in the incommensurate regime $J_{\perp }\in [J_{\perp }^{c1},J_{\perp }^{c2}]\equiv [\sim 1,1.562]$ [data taken from Fig. 4 in the main paper, where red points trace the incommensurate maxima in Fourier space]. The inset suggests a scaling for the incommensurate wave vector that is $\left(\frac{2\pi }{3}-q\right)\propto {|J_{\perp }-J_{\perp }^{c1}|}^{\sim 3}$ close to $J_{\perp }^{c1}$, and $q\propto \sqrt{|J_{\perp }-J_{\perp }^{c2}|}$ close to $J_{\perp }^{c2}$.

Figure 9

Analysis of excited states across the lower boundary $J_{\perp }^{c1}$ of the incommensurate phase for the $\mathrm{SU}\left(3\right)$ Heisenberg ladder. (a) shows, for $J_{\perp }=0.75$, a small but finite energy gap when extrapolating to the thermodynamic limit $1/L\to 0$. P(b) at $J_{\perp }=1$ suggests the onset of criticality, whereas for (c) at $J_{\perp }=1.2,$ due to the presence of incommensurate correlations, other states take over the ground state for sufficiently large systems (see crossing of the lowest blue line, which served as energy reference). For the simulations here, several lowest system eigenstates are targeted simultaneously within a single DMRG run for uniform systems up to length $L=96$ using open boundary conditions. The colors encode symmetry sectors, which shows that two multiplets were included in the $q=\left(00\right)\equiv \mathbf{1}$ sector (singlets; blue data points) as well as two multiplets in the adjoint representation $q=\left(11\right)\equiv \mathbf{8}$ (orange data points). Therefore while only including 4 multiplets total, this DMRG simulations actually targeted the lowest 18 states. The insets of (a) and (b) show the maximum discarded weight within DMRG in the last sweep while simultaneously targeting the low-energy multiplets above. For all ladders, at least 1024 multiplets ($>51000$ states) were kept. For $L\le 69$, at least 2048 multiplets ($>111000$ states) were kept. The gray vertical lines indicate the region that was used to fit the data (crosses) to the thermodynamic limit $L\to \infty$ using a plain quadratic polynomial fit (solid lines).

Figure 10

Analysis of the incommensurate phase in the $\mathrm{SU}\left(3\right)$ Heisenberg ladder for $J_{\perp }=1.4$. (a) shows the block parity $P_L\left(x\right)\in [-1,1]$ of the ground state [cf. Eq. (C1)], where the data is vertically offset by the integers $L/3$ for different system sizes $L=15,18,...,120$. The incommensurate wave is highlighted by simply also connecting only every third data point (light colors). The total rung parity, i.e., the last data point $P_L\left(L\right)$ of each curve determines whether the full ground state is symmetric (blue) or antisymmetric (orange) under rung exchange. (b) compares the block parity $P_L\left(x\right)$ for a single system of fixed length $L=96$ [data horizontally offset by $dx=17.5$; data also marked in darker colors in (a)] to the full parity $P\left(L\right)$ for systems vs varying length $L$ (dots) [this corresponds to the last data points $P_L\left(L\right)$ of each curve in (a)]. The resulting incommensurate period of $P_L\left(x\right)$ clearly matches the period seen in $P\left(L\right)$. For comparison, (c) shows the incommensurate behavior seen in the $\langle S·S\rangle$ data [similar to (b), again for $L=96$, and data horizontally offset]. The incommensurate wavelength is the same up to a factor of 2 (see text). (d) shows the Fourier transform of the data in (a). In order to focus on the incommensurate behavior, this is the norm-squared of the FFT data when including rungs $x=i,i+3,i+6,...$, subsequently averaged over $i=1,2,3$. The incommensurate wavelength $\lambda$ is obtained by tracking the peaks in the FFT data (red dots). Finite-size effects lead to diminishing jumps in $\lambda$ as $L$ increases. For the standard deviation, $\delta \lambda$ indicated in the panel, the data was taken in the range $L\in [80,120]$

Figure 11

DMRG scan across the narrow intermediate phase of the $\mathrm{SU}\left(5\right)$ ladder [$J_{\perp }\in [0.57,0.62]$ tuned linearly along a length $L=1200$ ladder with open boundaries, keeping up to $D^{*}\le 2048$ multiplets ($D\le 711,159$ states)]. (a) shows the block entanglement entropy $S$ along the system. The intermediate phase $J_{\perp }\in [J_{\perp }^{c1},J_{\perp }^{c2}]=[0.581,0.596]$ is marked by vertical dashed lines [similar in (c) and (d)]. The data points are also connected by a line of light matching color, with the effect of shading data with significant period-5 variations for $J_{\perp }\lesssim J_{\perp }^{c2}$. (b1) shows the growth of the entanglement entropy $S$ vs $1/D$ at specific cuts $x$ along the ladder. The cuts correspond to specific values of $J_{\perp }$ [color matched with the vertical markers in (a)]. The range $J_{\perp }<J_{\perp }^{c1}$ is well converged (blue), whereas the range $J_{\perp }>J_{\perp }^{c1}$ still shows (minor) growth of the entanglement entropy for the largest $D$ employed [see remainder of data in (b), and also the increase in (a) from the last forward ($\gg$; blue dots) to the last backward sweep ($\ll$; red dots)]. Panel (b2) shows the largest discarded DMRG weight $\delta {\rho }_{\mathrm{disc}}$ over the entire ladder vs $1/D$, together with the convergence $\delta e_0$ of the overall “ground-state” energy $e_0\equiv E_0/L$. (b3) and (4) show the entanglement spectrum in the gapped and the intermediate phase at $J_{\perp }=0.575$ and 0.590, respectively. The number on top of each level indicates the multiplet degeneracy. (c) shows the $\langle S·S\rangle$ nearest-neighbor correlations where the shading again indicates period-5 variations of the corresponding data in matching color. A zoom into these variations around $J_{\perp }\simeq 0.576$ [vertical solid gray marker] is shown in the inset. (d) shows the flow of entanglement spectra along the ladder, i.e., vs $J_{\perp }$, where different colors represent different symmetry sectors (see legend). The lines connect data from every fifth bond, where bonds corresponding to a block size of a multiple of 5 are shown in strong colors. The inset shows a semilogarithm plot of the same data, yet relative to the lowest level for each bond.

Figure 12

Uniform $\mathrm{SU}\left(5\right)$ ladder in the middle of the intermediate phase at $J_{\perp }=0.590$. The $\mathrm{SU}(N=5)$ DMRG simulation was performed on a ladder of length $L=120$ with open boundary conditions keeping up to $D{}^{*}=4096$ multiplets (about $D\le 1.5$ million states). (a) and (b) analyze the block parity $P_L\left(x\right)$. Up to an alternating behavior, and within finite-size variations [cf. Fig. 10], a period of $\lambda \sim 2N=10$ rungs is observed. This period is also marked via vertical dotted lines in (b). (c) and (d) show the nearest-neighbor $\langle S·S\rangle$ correlation along rungs and legs, respectively. (d) shows convergence within DMRG by adding data lines for intermediate sweeps at $D^{*}$ as specified with the legend to the right, together with an extrapolation to $D^{*}\to \infty$. This demonstrates that the last sweep at $D^{*}=4096$ is already well converged. (e) shows the entanglement spectra along the ladder where each line connects data from every bond, while combining data for all ladder positions $x$ [therefore, e.g., only iterations where $x$ includes an integer multiple of 5 rungs support the scalar (0000) symmetry sector]. Colors themselves indicate individual symmetry sectors as indicated with the legend. Again as with Fig. 11, the number $n_Y$ of boxes in the corresponding Young tableau also identifies the rung position $x$ modulo $N$ along the ladder, i.e., $\mathrm{mod}(n_Y,N)=\mathrm{mod}(2x,N)$, resulting in 5 data sets with respect to rung position. (f) shows block parity vs energy in the entanglement spectrum (see text) at fixed bond position $x=60$ in the system center based on $-{log}_{10}\left(\rho \right)$. The inset shows a zoom around $ES\sim 8$, which demonstrates that a systematic pairing of multiplets with opposite rung parity occurs all the way down to eigenvalues of the block density matrix well below ${10}^{-8}$.

Figure 13

Analysis of singlet gap for uniform $\mathrm{SU}\left(5\right)$ ladders up to 230 rungs at $J_{\perp }=0.590$, i.e., in the middle of the intermediate phase. From left to right, respectively, the simulations targeted a total of $N_{\psi }=1,2,4,8$ low-energy states in the singlet sector (0000). The panels in the left column, (a) and (b), show entanglement spectra in the center of the ladder for the ground state only. In contrast to Fig. 11, (a) is derived from a uniform system here. The data as in (a) was computed for system sizes up to $L=160$ and plotted vs $1/L$ in (b). The remainder of the panels show DMRG results that simultaneously targeted multiple states. (c)–(g) show ground-state convergence, with the maximum discarded weight vs $1/L$ shown as inset to each panel. (d)–(h) show the energy of the excited states relative to the ground state vs $1/L$, together with a quadratic extrapolation towards the thermodynamic limit, $L\to \infty$.

Figure 14

Central charge for $\mathrm{SU}\left(3\right)$ ladder in the large $J_{\perp }$ phase, actually having $J_{\perp }=1.6$ just above $J_{\perp }^{c2}=1.562$, based on the scaling of the block entanglement entropy using periodic boundary conditions (see Refs. [83, 84] and also text). For a fixed number of $D\to \infty$ kept states ($D^{*}$ multiplets), a quadratic fit for the range $x>x_0=-0.25$ (see vertical gray marker), e.g., results in the blue line for the last DMRG sweep performed. Its slope at $x=0$ yields the central charge $c=2.008$ as indicated with the legend. The left dotted marker indicates $(1/3)ln(L/\pi )$. Therefore with four data points between this marker and the one at $x_0$, the overwhelming part of the ladder (22 rungs out of 30 rungs) falls into $x>x_0$. The upper inset analyses the convergence of the central charge vs average discarded weight $\delta \rho$ for given DMRG sweep at fixed $D$. The data was obtained by evaluating the slope of $S\left(x\right)$ as shown in the main panel for fixed $D$ at $x=x_0$ and $x=0$ (gray and black data, respectively). The red asterisk corresponds to the extrapolated value $D\to \infty$ as specified with the legend in the main panel. The lower inset shows discarded weight $\delta \rho$ vs number $D$ of kept states.

Figure 15

Central charge for $\mathrm{SU}\left(5\right)$ ladder in the large $J_{\perp }$ phase for $L=30$, with an analysis of the entanglement entropy that is completely analogous to the case of $\mathrm{SU}\left(3\right)$ in Fig. 14. In the extrapolated limit, $D\to \infty$ and by evaluating the slope at $x=0$, this yields a central charge of $c=3.994$ (see legend and also red asterisk in upper inset), in excellent agreement with the expected value of $c=4$.