Emergent dimerization and localization in disordered quantum chains

We uncover a novel mechanism for inducing a gapful phase in interacting many-body quantum chains. The mechanism is nonperturbative, being triggered only in the presence of both strong interactions and strong aperiodic (disordered) modulation. In the context of the critical antiferromagnetic spin-1/2 XXZ chain, we identify an emerging dimerization, which removes the system from criticality and stabilizes the novel phase. This mechanism is shown to be quite general in strongly interacting quantum chains in the presence of strongly modulated quasiperiodic disorder, which is, surprisingly, perturbatively irrelevant. Finally, we also characterize the associated quantum phase transition via the corresponding critical exponents and thermodynamic properties.

Figure 1

Schematic phase diagram of the XXZ spin-$1/2$ chain Eq. (1) in the presence of perturbatively irrelevant deterministic aperiodicity. The anisotropy parameter $\mathrm{\Delta }$ parameterizes the strength of the interactions while the modulation $r$ parameterizes the strength of the aperiodicity. For the aperiodic sequence defined in Eq. (2), $r_c\approx 0.13$ and ${\mathrm{\Delta }}^{*}\approx 0.69$.

Figure 2

The upper lattice shows the leftmost portion of the aperiodic sequence of bonds in Eq. (2). Dashed blue lines represent weak ($J^{\left(a\right)}$) couplings and solid red lines represent strong ($J^{\left(b\right)}$) couplings. The middle lattice is obtained after the first two renormalization steps (consisting in decimating the strong bonds $J^{\left(b\right)}$ and all remaining weak bonds $J^{\left(a\right)}$). The lower lattice shows the effective chain produced in the latest stages of the SDRG method in the Heisenberg limit $\mathrm{\Delta }=1$. Notice the alternating pattern of strong (red/short/thick/solid) and weak (green/long/thin/broken) effective couplings, revealing an emergent dimerization. Circles and polygons represent real spins and low-energy effective spin-1/2 degrees of freedom (effective spins), respectively. In the latter case, the number of sides (3, 7, or 9) is the number of real spins within. The numbers indicate the original position of the central real spin.

Figure 3

Main plot: Logarithm of the conductance as a function of the number of effective spins $\ell$ of the effective tight-binding Hamiltonian (8) describing the lowest-energy many-body excitation band of the Heisenberg chain with couplings following the sequence in Eq. (2) in the strong-modulation limit. The blue dashed curve is a fit given by $-9.20{\ell }^{\psi }$, with $\psi =1/2$. The chemical potential corresponds to the band center. (Inset) Average inverse participation as a function of the effective system size. The green dotted line is proportional to $1/\ell$.

Figure 4

Linear-logarithmic plot the energy gap as a function of the coupling modulation $r$ for the Heisenberg chain with couplings following the sequence in Eq. (2) for various chain lengths $L$. The inset shows the position of the relative minimum in the curves for large $L$, using also the intermediate values $L=2226$ and 4582 (not shown for the sake of clarity).

Figure 5

Susceptibility $\chi$ as a function of temperature $T$ for the Heisenberg chain with couplings following the sequence in Eq. (2), for a coupling ratio $J^{\left(a\right)}/J^{\left(b\right)}=1/10$. The results were obtained by using the SSE QMC algorithm, with open chains containing $N\in \left\{86,178,366,754\right\}$ spins. The inset shows that the product $\chi T$ follows $exp\left(-\mathrm{\Delta }E/T\right)$, with a size-dependent energy gap $\mathrm{\Delta }E$, which approaches $\approx 5\times {10}^{-4}{J}_b$ as $N\to \infty$. Temperature is measured in units of $J^{\left(b\right)}/k_B$. The oscillations in $d\chi /dT$ for temperatures between $T\simeq 10$ and $\simeq {10}^{-3}$ reflect the energy scales associated with the formation of “spin molecules,” as predicted by the SDRG scheme (see main text).

Figure 6

Same data as in Fig. 4 with the energy gap rescaled by ${\left(1-r\right)}^2$. The inset shows the data collapse obtained by the finite-size scaling hypothesis in Eq. (15) using $r>r_c=0.135, z=1$, and $\nu =2$.

Figure 7

Comparison between geometric fluctuations induced by the Rudin-Shapiro sequence (A3) as calculated for all lengths (solid line) and only for the natural lengths of the sequence (circles). The red dashed line is proportional to $N^{1/2}$. Notice the existence of both stronger and weaker fluctuations in the neighborhood of the natural lengths.

Figure 8

Comparison between geometric fluctuations induced by the Fibonacci sequence (A8) as calculated for all lengths (solid line) and only for the natural lengths of the sequence (circles). The red dashed red line is proportional to $N^0$. Notice that the stronger fluctuations scale at most logarithmically with $N$ (blue dot-dashed line, with $c_1$ and $c_2$ constants of order 1).

Figure 9

Comparison between geometric fluctuations induced by the sequence in Eq. (2) as calculated for all lengths (solid line) and only for the natural lengths of the sequence (circles). The red dashed curve is proportional to $N^{-1}$. Notice that the stronger fluctuations scale as $N^0$ for large $N$.

Figure 10

Fourier spectrum of the Rudin-Shapiro sequence. The inset shows $Y\left(Q\right)={\sum }_{q=\pi }^Q{y}^2\left(q\right)$, which behaves as ${\left|Q-\pi \right|}^{2\alpha +1}$ if $y\left(Q\right)\sim {\left|Q-\pi \right|}^{\alpha }$.

Figure 11

Fourier spectrum of the 6-3 sequence. The inset shows $Y\left(Q\right)={\sum }_{q=\pi }^Q{y}^2\left(q\right)$, which behaves as ${\left|Q-\pi \right|}^{2\alpha +1}$ if $y\left(Q\right)\sim {\left|Q-\pi \right|}^{\alpha }$.

Figure 12

Fourier spectrum of the sequence in Eq. (A11). The inset shows the finite-size scaling behavior of the average value of the Fourier weights in a region of width $\mathrm{\Delta }{Q}^{*}={10}^{-3}\pi$ around $Q=\pi$.

Figure 13

Relation between the strengths $J$ and the lengths $l$ of the weak effective bonds corresponding to the low-energy effective chain when couplings follow the aperiodic sequence in Eq. (2). The continuous line is a fit using Eq. (C1). The coupling ratio corresponds to $r=J_a/J_b=1/10$, and the strengths are given in units of $J_b$. For this coupling ratio, the strength of the strong effective bonds, as predicted by the SDRG approach, is $\approx 1.5\times {10}^{-3}$, with a length of ten lattice parameters.