Nonclassical Rotational Inertia Research Articles

Abnormal quantum moment of inertia and structural properties of electrons in 2D and 3D quantum dots: an ab initio path-integral Monte Carlo study

We present extensive new direct path-integral Monte Carlo results for electrons in quantum dots in two and three dimensions. This allows us to investigate the nonclassical rotational inertia (NCRI) of the system, and we find an abnormal negative quantum moment of inertia (2014 Phys. Rev. Lett. 112 235301) under some conditions. In addition, we study the structural properties by computing a re-normalized, angular-resolved center-two particle correlation function. Remarkably, we find no connection between the spatial structure and the NCRI, since the former can be nearly identical for Fermi- and Bose-statistics for parameters where the superfluid fraction is diverging towards negative infinity.

Topological order in Mott insulators

In this commemorative issue for Phil Anderson, I reproduce parts of a joint unpublished paper that first got me thinking about topological order in Mott insulators. With Phil, we explored aspects of the momentum distribution function and the non-classical rotational inertia in the Bose Hubbard model close to the quantum phase transition. My interest on this question continued, and later in the article I take up the question of topological order in the Kitaev Mott insulator on the honeycomb lattice with bond-dependent interactions in a finite magnetic field. Kitaev obtained the exact solution for zero field that demonstrated an exotic Z2 quantum spin liquid phase with topological order. In a finite magnetic field, we discovered a new intermediate U(1) gapless quantum spin liquid phase, sandwiched between the known gapped Kitaev spin liquid and a polarized phase, that harbors fractionalized neutral fermionic excitations, dubbed spinons, that remarkably form a Fermi surface in a Mott insulator.

Commensurate-incommensurate supersolid ground state of a spin-orbit-coupled Bose-Einstein condensate in one-dimensional optical lattices

Supersolid, a fascinating quantum state of matter, features novel phenomena such as the non-classical rotational inertia and transport anomalies. It is a long standing issue of the coexistence of superfluidity and broken translational symmetry in condensed matter physics. By recent experimental advances to create tunable synthetic spin-orbit coupling in ultracold gases, such highly controllable atomic systems would provide new possibilities to access supersolidity with no counterpart in solids. Here we report that the combination of anharmonicity of trapping potential and spin-orbit coupling will provide a new paradigm to achieve supersolids. By means of imaginary time evolution of the Gross-Pitaevskii equation, we demonstrate that a supersolid state can be found when considering a trapped Rashba-type spin-orbit coupled bosonic atoms loaded in a one-dimensional optical lattice. Furthermore, a skyrmion-anti-skyrmion lattice is associated with the appearance of such supersoildity, indicating the topological nontrivial properties of our proposed supersolids.

Path-integral Monte Carlo simulations of quantum dipole systems in traps: Superfluidity, quantum statistics, and structural properties

We present extensive \textit{ab initio} path integral Monte Carlo (PIMC) simulations of two-dimensional quantum dipole systems in a harmonic confinement, taking into account both Bose- and Fermi-statistics. This allows us to study the nonclassical rotational inertia, which can lead to a negative superfluid fraction in the case of fermions [Phys. Rev. Lett. \textbf{112}, 235301 (2014)]. Moreover, we study in detail the structural characteristics of such systems, and are able to clearly resolve the impact of quantum statistics on density profiles and the respective shell structure. Further, we present results for a more advanced center-two particle correlation function [Phys. Rev. E \textbf{91}, 043104 (2015)], which allows to detect differences between Fermi- and Bose-systems that do not manifest in other observables like the density. Overall, we find that bosonic systems sensitively react to even small values of the dipole--dipole coupling strength, whereas such a weak interaction is effectively masked for fermions by the Pauli exclusion principle. In addition, the abnormal superfluid fraction for fermions is not reflected by the structural properties of the system, which are equal to the bosonic case even though the moments of inertia diverge from each other. Lastly, we have demonstrated that fermionic PIMC simulations of quantum dipole systems are feasible despite the notorious fermion sign problem, which opens up new avenues for future investigations in this field.

Perturbative Approach to Superfluidity under Nonuniform Potential

A perturbative way to investigate superfluid properties of various systems under nonuniform potential is presented. We derive the perturbation expansion of the superfluid fraction, which indicates how liquid exhibits nonclassical rotational inertia, in terms of the strength of nonuniform potential and find that the coefficient of the leading term reflects the density fluctuation of the system. Our formulation does not assume anything about Bose-Einstein condensation and thus is applicable to wide variety of systems. Superfluid properties of some examples including (non-)interacting Bose systems, especially Bose gas in the mean field limit, (non-)interacting Fermi sytems, Tomonaga-Luttinger liquid and spinless chiral $p$-wave superfluid are investigated.

Upper limit of supersolidity in solid helium

The resonant period drop observed at low temperatures in torsional oscillators containing solid helium had been interpreted as a signature of a supersolid. However, it was found that the shear modulus increase found in solid helium at the same low temperature could also decrease the resonant period of the torsional oscillator. We report the results of a study in two different torsional oscillators that were specially designed to minimize the shear modulus effect and maximize any possible supersolid response. We were able to place an upper limit on the nonclassical rotational inertia or supersolid fraction of $4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$. Moreover, we have repeated an earlier experiment on the hcp $^{3}\mathrm{He}$ solid, which shows similar low-temperature stiffening to that in hcp $^{4}\mathrm{He}$. We found that the small drop of the resonant period measured in the hcp $^{3}\mathrm{He}$ sample is comparable in size to that found in the hcp $^{4}\mathrm{He}$ samples. These results strongly suggest that the resonant period drop reported in most torsional oscillator studies in the last decade is primarily a consequence of the shear modulus stiffening effect.

Drude Weight, Meissner Weight, Rotational Inertia of Bosonic Superfluids: How Are They Distinguished?

The Drude weight, the quantity which distinguishes metals from insulators, is proportional to the second derivative of the ground state energy with respect to a flux at zero flux. The same expression also appears in the definition of the Meissner weight, the quantity which indicates superconductivity, as well as in the definition of non-classical rotational inertia of bosonic superfluids. It is shown that the difference between these quantities depends on the interpretation of the average momentum term, which can be understood as the expectation value of the total momentum (Drude weight), the sum of the expectation values of single momenta (rotational inertia of a superfluid), or the sum over expectation values of momentum pairs (Meissner weight). This distinction appears naturally when the current from which the particular transport quantity is derived is cast in terms of shift operators.

Dislocation model for the TO-period anomaly

The dislocation-vibration model can explain both the anomaly in the period of torsional-oscillator (TO) containing solid 4He, i.e. the so-called nonclassical rotational inertia (NCRI), and the anomaly in the shear modulus of solid 4He on the same basis. This dislocation model is extended to understand the amplitude dependence and hysteresis of the TO period by introducing an exponential distribution function for the network pinning lengths of dislocation segments. Pinning and unpinning of dislocations by 3He impurity atoms depending on the history of temperature and stress play an important role. An actual distribution of the network pinning lengths is determined from the experimental data.

Absence of Supersolidity in Solid Helium in Porous Vycor Glass

In 2004, Kim and Chan carried out torsional oscillator measurements of solid helium confined in porous Vycor glass and found an abrupt drop in the resonant period below 200 mK. The period drop was interpreted as probable experimental evidence of nonclassical rotational inertia. This experiment sparked considerable activities in the studies of superfluidity in solid helium. More recent ultrasound and torsional oscillator studies, however, found evidence that shear modulus stiffening is responsible for at least a fraction of the period drop found in bulk solid helium samples. The experimental configuration of Kim and Chan makes it unavoidable to have a small amount of bulk solid inside the torsion cell containing the Vycor disk. We report here the results of a new helium in Vycor experiment with a design that is completely free from any bulk solid shear modulus stiffening effect. We found no measurable period drop that can be attributed to nonclassical rotational inertia.

Anomalous Response of 4He Confined in Nanoporous Media to Torsional Oscillation

The existence of “Non-Classical Rotational Inertia (NCRI)” in solid 4He below 0.2 K has been controversial and interpreted by a number of different theories. We report on torsional oscillator measurements for 4He in a nanoporous Gelsil glass, which has a network of nanopores with 3.5 nm in diameter. In addition to the usual “low-T NCRI” with an onset temperature 0.15 K, we find a larger decrease in rotational moment of inertia in a broad range of temperature from 0.2 to 1.9 K. This “high-T inertial anomaly” is accompanied with multiple dissipation peaks, but has no dependence on torsional oscillation velocity unlike the low-T NCRI. Since the high-T anomaly is observed also in confined liquid states, it originates in amorphous solid 4He layer near the pore wall. Our result shows that different types of supersolid—like phenomena, i.e. inertial anomalies, can coexist in a single 4He sample, even with genuine superfluidity of liquid 4He.

On the Superfluid Fraction and the Hydrodynamics of Supersolids

This review largely considers the author’s extensions of two foundation works in supersolids: Andreev and Lifshitz’s hydrodynamics, and Leggett’s Non-Classical Rotational Inertia (NCRI) with estimates of the T=0 upper limit. Relative to the case of a perfect lattice, the Andreev and Lifshitz equations contain an additional degree of freedom, which permits a liquid-like internal pressure P that is distinct from the lattice stress (or ‘solid-like pressure’). This is particularly relevant to 4He, which requires an applied pressure P a to solidify; use of a Maxwell relation relating P and strain yields that \(P\sim P_{a}^{2}\); we estimate that near the melting pressure the liquid-like pressure in equilibrium is about 1/4 of the solid-like pressure. This new freedom also permits vacancy diffusion, which we have studied for both ordinary solids and supersolids. In both cases, for the vacancy diffusion mode the liquid-like pressure and the lattice stress cancel. Further, since at T=0 the supersolid fraction f s is less than unity and the excitation part of the normal fraction is zero, we argue that there must be an additional source of “normal” mass, to which we attribute a velocity that in principle is distinct from the lattice velocity associated with elasticity. Relative to NCRI we have made numerous estimates of the upper limit for the superfluid fraction f s ; we find f s values on the order of 0.2 for realistic models of the atomic density. Correlation effects in the solid cause the superfluid velocity \(\vec{v}_{s}\) of one particle to depend on correlations with the positions of other particles, and this leads to a more complex theory for the flow pattern and for the upper limit on f s .

Quantum crystals in a trapped Rydberg-dressed Bose-Einstein condensate

Spontaneously crystalline ground states, called quantum crystals, of a trapped Rydberg-dressed Bose-Einstein condensate are numerically investigated. As a result described by a mean-field order parameter, such states simultaneously possess crystalline and superfluid properties. A hexagonal droplet lattice is observed in a quasi-two-dimensional system when dressing interaction is sufficiently strong. Onset of these states is characterized by a drastic drop of the non-classical rotational inertia proposed by Leggett [Phys. Rev. Lett. 25, 1543 (1970)]. In addition, an AB stacking bilayer lattice can also be attained. Due to an anisotropic interaction possibly induced by an external electric field, transition from a hexagonal to a nearly square droplet lattice is also observed.

Colloquium: Supersolids: What and where are they?

The ongoing experimental and theoretical effort aimed at understanding nonclassical rotational inertia in solid helium has sparked renewed interest in the supersolid phase of matter, its microscopic origin and character, and its experimental detection. The purpose of this Colloquium is to provide a general theoretical framework for the phenomenon of supersolidity, review some of the experimental evidence for solid $^{4}\mathrm{He}$, and discuss its possible interpretation in terms of physical effects underlain by extended defects (such as dislocations). Quantitative support to our theoretical scenarios by means of first-principle numerical simulations is provided. Alternate avenues for the observation of the supersolid phase, not involving helium but rather assemblies of ultracold atoms, are also discussed.

Measurements of the Dielectric Constant of Solid Helium at Low Temperatures

A careful measurement of the dielectric constant of solid 4He was made to search for signatures associated with the reported onset of nonclassical rotational inertia and shear modulus stiffening. The samples studied include zero-pressure liquid, solid-liquid mixtures on the melting-curve and solid helium. It was found that the dielectric constant of solid 4He decreases smoothly with temperature below 300 mK, showing no measurable anomalies down to 50 mK. The new measurements do not show the anomalous upturn effect that was reported in our previous measurements.

A Microscopic Description of Vacancies in Solid 4He

The changes that vacancies produce in the properties of hcp solid 4He are studied by means of quantum Monte Carlo methods. Our results show that the introduction of vacancies produces significant changes in the behavior of solid 4He, even when the vacancy concentration is very small. We show that there is an onset temperature where the properties of incommensurate 4He change significantly. Below this temperature, we observe the emergence of off-diagonal long range order and a complete spatial delocalization of the vacancies. This temperature is quite close to the temperature where non-classical rotational inertia has been experimentally observed. Finally, we report results on the influence of vacancies in the elastic properties of hcp 4He at zero temperature.

Onset Temperature of Bose-Einstein Condensation in Incommensurate SolidHe4

The temperature dependence of the one-body density matrix in (4)He crystals presenting vacancies is computed with path integral Monte Carlo simulations. The main purpose of this study is to estimate the onset temperature T(0) of Bose-Einstein condensation in these systems. We see that T(0) depends on the vacancy concentration X(v) of the simulated system, but not following the law T(0) ~ X(v)(2/3) obtained assuming noninteracting vacancies. For the lowest X(v) we study, that is X(v)= 1/256, we get T(0) = 0.15 ± 0.05 K, close to the temperatures at which a finite fraction of nonclassical rotational inertia is experimentally observed. Below T(0), vacancies do not act as classical point defects becoming completely delocalized entities.

Probing phase coherence in solid helium using torsional oscillators of different path lengths

Long-range phase coherence is a critical signature of macroscopic quantum phenomena. To date, nonclassical rotational inertia (NCRI) of solid helium has been reported only in samples with physical dimension of at most 5 cm. We have investigated solid helium in longer path-length torsional oscillators. Samples of length ranging from 6 to 100 cm were grown inside toroids and in self-connected long capillaries. NCRI of $4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ and $3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ were found in cells with path length of 6 and 9 cm. In cells with path length of 30 and 100 cm, NCRI, if it exists, is less than $7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ and $4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$, respectively.

Phenomenological Ginzburg-Landau theory for supersolidity

A Ginzburg-Landau theory is proposed in which the supersolid state is viewed as a system displaying features of an ordinary solid and of a superfluid. The theory shows that the superfluid part is responsible for a nonclassical rotational inertia (NCRI) behavior, but the ordinary part (the lattice) is responsible for elastic behaviors usually seen in solids. Moreover, the superfluid part contributes to an excess of heat capacity near the supersolid--ordinary solid transition. The theory provides a coherent picture, at least at the macroscopic scale, of supersolidity that reconciles (NCRI) and the heat-capacity measurements. The parameters of the Ginzburg-Landau free energy are estimated using experimental data, hence a healing length of the order of $100\phantom{\rule{0.16em}{0ex}}\mathrm{nm}$ and a critical speed of the order of $0.1\phantom{\rule{0.16em}{0ex}}\mathrm{m}/\mathrm{s}$ are predicted, both results consistent with recent studies by Kubota and co-workers.

Theory of superfluidity and drag force in the one-dimensional Bose gas

The one-dimensional Bose gas is an unusual superfluid. In contrast to higher spatial dimensions, the existence of non-classical rotational inertia is not directly linked to the dissipationless motion of infinitesimal impurities. Recently, experimental tests with ultracold atoms have begun and quantitative predictions for the drag force experienced by moving obstacles have become available. This topical review discusses the drag force obtained from linear response theory in relation to Landau's criterion of superfluidity. Based upon improved analytical and numerical understanding of the dynamical structure factor, results for different obstacle potentials are obtained, including single impurities, optical lattices and random potentials generated from speckle patterns. The dynamical breakdown of superfluidity in random potentials is discussed in relation to Anderson localization and the predicted superfluid-insulator transition in these systems.

Josephson effects in one-dimensional supersolids

We demonstrate that superflow past an obstacle is possible in a solid phase in the one-dimensional Gross-Pitaevskii equation with a finite-range two-body interaction. The phenomenon we find is analogous to the dc Josephson effect in superconductors, and we deduce the ``Josephson relation'' between the current and phase difference of the condensates separated by the obstacle. We also discuss persistent currents and nonclassical rotational inertia in annular containers with a penetrable potential barrier. The phase diagram in the plane of the current and the interaction strength is given. Our result provides a simple theoretical example of supersolidity in the presence of an obstacle.