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arXiv:1311.3493 · Physical Review B

Asymptotically exact scenario of strong-disorder criticality in one-dimensional superfluids

Lode Pollet, Nikolay V. Prokof'ev, Boris V. Svistunov

Published Feb 14, 2014• 25 citations
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Abstract

We present a controlled rare-weak-link theory of the superfluid-to-Bose/Mott glass transition in one-dimensional disordered systems. The transition has Kosterlitz-Thouless critical properties but may occur at an arbitrary large value of the Luttinger parameter $K$. The hydrodynamic description is valid under the correlation radius and defines criticality via mutual renormalization of the strength of microscopic weak links and superfluid stiffness. The link strength renormalizes along the lines of Kane and Fisher [Phys. Rev. Lett. 68, 1220 (1992)], while the renormalization of superfluid stiffness follows the lines of classical-field flow. The hallmark of the theory is the relation ${K}^{(c)}=1/\ensuremath{\zeta}$ between the critical value of the Luttinger parameter at macroscopic scales and the microscopic (irrenormalizable) exponent $\ensuremath{\zeta}$ describing the scaling $\ensuremath{\propto}1/{N}^{1\ensuremath{-}\ensuremath{\zeta}}$ for the strength of the weakest link among the $N\ensuremath{\gg}L$ disorder realizations in a system of fixed mesoscopic size $L$.

DOI
10.1103/physrevb.89.054204

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•Apr 12, 2021•0 citations
We present a controlled rare-weak-link theory of the superfluid-to-Bose/Mott glass transition in one-dimensional disordered systems. The transition has Kosterlitz-Thouless critical properties but may occur at an arbitrary large value of the Luttinger parameter KKK. The hydrodynamic description is valid under the correlation radius and defines criticality via mutual renormalization of the strength of microscopic weak links and superfluid stiffness. The link strength renormalizes along the lines of Kane and Fisher [Phys. Rev. Lett. 68, 1220 (1992)], while the renormalization of superfluid stiffness follows the lines of classical-field flow. The hallmark of the theory is the relation K(c)=1/\ensuremathζ{K}^{(c)}=1/\ensuremath{\zeta}K(c)=1/\ensuremathζ between the critical value of the Luttinger parameter at macroscopic scales and the microscopic (irrenormalizable) exponent \ensuremathζ\ensuremath{\zeta}\ensuremathζ describing the scaling \ensuremath∝1/N1\ensuremath−\ensuremathζ\ensuremath{\propto}1/{N}^{1\ensuremath{-}\ensuremath{\zeta}}\ensuremath∝1/N1\ensuremath−\ensuremathζ for the strength of the weakest link among the N\ensuremath≫LN\ensuremath{\gg}LN\ensuremath≫L disorder realizations in a system of fixed mesoscopic size LLL.
We present numerical evidence from Monte Carlo simulations that the superfluid-insulator quantum phase transition of interacting bosons subject to strong disorder in one dimension is controlled by the strong-randomness critical point. At this critical point the distribution of superfluid stiffness over disorder realizations develops a power-law tail reflecting a universal distribution of weak links. The Luttinger parameter on the other hand does not take on a universal value at this critical point, in marked contrast to the known Berezinskii-Kosterlitz-Thouless-like superfluid-insulator transition in weakly disordered systems. We develop a finite-size scaling procedure which allows us to directly compare the numerical results from systems of linear size up to 1024 sites with theoretical predictions obtained by Altman et al. [Phys. Rev. Lett. 93, 150402 (2004)] using a strong disorder renormalization group approach. The data shows good agreement with the scaling expected at the strong-randomness critical point.
Using large-scale simulations based on matrix product state and quantum Monte Carlo techniques, we study the superfluid to Bose glass transition for one-dimensional attractive hard-core bosons at zero temperature, across the full regime from weak to strong disorder. As a function of interaction and disorder strength, we identify a Berezinskii-Kosterlitz-Thouless critical line with two different regimes. At small attraction where critical disorder is weak compared to the bandwidth, the critical Luttinger parameter Kc{K}_{c}Kc​ takes its universal Giamarchi-Schulz value Kc=3/2{K}_{c}=3/2Kc​=3/2. Conversely, a nonuniversal Kc>3/2{K}_{c}>3/2Kc​>3/2 emerges for stronger attraction where weak-link physics is relevant. In this strong-disorder regime, the transition is characterized by self-similar power-law-distributed weak links with a continuously varying characteristic exponent $\ensuremath{\alpha}$.
We show that in the regime when strong disorder is more relevant than field quantization, the superfluid-to-Bose-glass criticality of one-dimensional bosons is preceded by the prolonged logarithmically slow classical-field renormalization flow of the superfluid stiffness at mesoscopic scales. With the system compressibility remaining constant, the quantum nature of the system manifests itself only in the renormalization of dilute weak links. On the insulating side, the flow ultimately reaches a value of the Luttinger parameter at which the instanton--anti-instanton pairs start to proliferate, in accordance with the universal quantum scenario. This happens first at astronomical system sizes because of the suppressed instanton fugacity. We illustrate our result by first-principles simulations.
We study the superfluid-insulator transition in a one-dimensional system of interacting bosons, modeled as a disordered Josephson array, using a strong-randomness real-space renormalization-group technique. Unlike perturbative methods, this approach does not suffer from run-away flows and allows us to study the complete phase diagram. We show that the superfluid-insulator transition is always Kosterlitz-Thouless like in the way that length and time scales diverge at the critical point. Interestingly however, we find that the transition at strong disorder occurs at a nonuniversal value of the Luttinger parameter, which depends on the disorder strength. This result places the transition in a universality class different from the weak disorder transition first analyzed by Giamarchi and Schulz [Europhys. Lett. 3, 1287 (1987)]. While the details of the disorder potential are unimportant at the critical point, the type of disorder does influence the properties of the insulating phases. We find three classes of insulators which arise for different classes of disorder potential. For disorder only in the charging energies and Josephson coupling constants, at integer filling we find an incompressible but gapless Mott-glass phase. If both integer and half-integer filling factors are allowed then the corresponding phase is a random-singlet insulator, which has a divergent compressibility. Finally in a generic disorder potential the insulator is a Bose glass with a finite compressibility.
We study the superfluid--Bose-glass transition in a one-dimensional lattice boson model with power-law decaying hopping amplitude ${t}_{i\ensuremath{-}j}\ensuremath{\sim}1/{|i\ensuremath{-}j|}^{\ensuremath{\alpha}}$, using bosonization and the nonperturbative functional renormalization group (FRG). When $\ensuremath{\alpha}$ is smaller than a critical value ${\ensuremath{\alpha}}_{c}<3$, the U(1) symmetry is spontaneously broken, which leads to a density mode with nonlinear dispersion and dynamical exponent $z=(\ensuremath{\alpha}\ensuremath{-}1)/2$; the superfluid phase is then stable for sufficiently weak disorder, contrary to the case of short-range hopping where disorder is a relevant perturbation when the Luttinger parameter is smaller than 3/23/23/2. In the presence of disorder, however, long-range hopping has no effect in the infrared limit and the FRG flow eventually becomes similar to that of a boson system with short-range hopping. This implies that the superfluid phase, when stable, exhibits a density mode with linear dispersion (z=1z=1z=1) and the superfluid--Bose-glass transition remains in the Berezinskii-Kosterlitz-Thouless universality class, while the Bose-glass fixed point is insensitive to long-range hopping. We compare our findings with a recent numerical study.
In this dissertation, we study the superfluid-insulator quantum phase transition in disordered boson systems. Recently, there has been considerable controversy over the validity of the scaling relations of the superfluid--Bose-glass quantum phase transition in three dimensions. Results from experimental and numerical studies on disordered quantum magnets contradict the scaling relations and the associated conventional scaling hypothesis for the singular part of the free energy. We determine various critical exponents of the superfluid--Bose-glass quantum phase transition in three-dimensional disordered Bose-Hubbard model through extensive Monte Carlo simulations. Our numerical study shows the previous studies on disordered quantum magnets were performed outside the quantum critical region, and our results for the critical exponents are in perfect agreement with previous theoretical predictions. We next move to study theoretically and numerically the superfluid-insulator quantum phase transition in one-dimensional disordered boson systems. While the superfluid-insulator transition in the weak disorder limit is well understood through the perturbative renormalization group study by Giamarchi and Schulz, transitions in the strong disorder regime are beyond the reach of this method. This problem was recently attacked by Altman et al. with the real space renormalization group method. They reached the conclusion that the superfluid-insulator transition in the strong disorder regime can be explained by the Coulomb blockade physics of weak links. However, their method is actually uncontrolled. Taking account of the crucial role of the hydrodynamic renormalization of weak links, finally, Pollet et al. put forward an asymptotically exact renormalization group theory of the superfluid-insulator transition. Based on this theory, we are able to provide an accurate description of the interplay between the well-known Giamarchi-Schulz criticality and the new weak-link criticality. A significant part of the ground-state phase diagram of one-dimensional disordered Bose-Hubbard model at unit filling is also determined numerically. In particular, we established the position of the multicritical point beyond which the new weak-link criticality takes over of the Giamarchi–Schulz criticality.