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# Finite-temperature fluid–insulator transition of strongly interacting 1D disordered bosons

Contributed by Boris L. Altshuler, May 2, 2016 (sent for review December 24, 2015; reviewed by Thierry Giamarchi and Igor Lerner)

## Significance

One-dimensional bosons in disorder provide a perfect system for studying a generic phenomenon of many-body localization–delocalization transition. After the observation of single-particle Anderson localization in dilute clouds of bosonic atoms, the obvious direction of research is to describe the effects of repulsion between the bosons. Theoretical studies of 1D interacting bosons have a long history. In the case of strong enough repulsion and zero temperature, the problem was solved by Giamarchi and Schulz in 1988, and, more recently, the limit of weak repulsion was analyzed. However, the full picture of finite temperature and arbitrary interaction strength has remained an open problem. In this paper, we develop such a theory and establish predictions that can be confronted to experiment.

## Abstract

We consider the many-body localization–delocalization transition for strongly interacting one-dimensional disordered bosons and construct the full picture of finite temperature behavior of this system. This picture shows two insulator–fluid transitions at any finite temperature when varying the interaction strength. At weak interactions, an increase in the interaction strength leads to insulator ^{7}Li or ^{39}K.

Despite intensive studies during several decades, Anderson localization of quantum particles in disorder (1) remains one of the most active directions of research in condensed matter physics (2). A subtle question is how the interaction between particles affects localization. The conventional theory of quantum charge transport in solids aims to describe disordered systems coupled to an outside bath, e.g., phonons. Phonon-assisted hopping (3) causes a finite, albeit small, conductivity in the insulating (localized) state at any temperature. The interaction between electrons was believed to only modify it (4) rather than to lead to any transport in the absence of phonons. In a metallic state far from the localization transition, both the coupling to phonons and the interaction between particles can be analyzed by means of perturbation theory (5), which fails in the vicinity of the Anderson transition.

The progress in developing solid-state coherent quantum devices and in understanding neutral atom quantum gases brought up systems with dramatically reduced coupling to the bath. The destruction of localization by many-body effects in disordered quantum systems decoupled from a bath was first discussed in ref. 6. Further studies of such systems (7) have led to the concept of many-body localization. It became clear that interacting quantum particles can undergo the localization−delocalization transition (LDT) transition from the insulator to fluid state.

Effect of the interactions on the LDT recently became crucial for understanding the physics of ultracold neutral atoms in the presence of disorder. After the first experiments on the observation of Anderson localization in expanding dilute clouds of bosonic atoms (8, 9), research on quantum gases in disorder grew rapidly (10⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–22). Many-body LDT for disordered weakly interacting 1D bosons has been discussed at both zero (23⇓–25) and finite temperatures (21). In the latter case, many-body LDT manifests itself as a nonconventional insulator−normal fluid phase transition, the transport properties being singular at the transition point. In the fluid phase, mass transport is possible, whereas, in the insulator phase, it is completely blocked, although the temperature *T* is finite. The fluid−insulator transition for strong interactions has been discussed by Giamarchi and Schulz (26), who predicted that, at

Up to now, the finite temperature behavior of 1D disordered bosons was well understood only for weak interactions. Here, we extend this understanding to the general case of strong and moderate interactions. At any finite temperature, we show the presence of two insulator−fluid transitions: the insulator

The physical picture can be interpreted as follows. Localization of all single-particle quantum eigenstates in one dimension by an arbitrary weak disorder (27⇓–29) implies the insulating phase in the absence of interaction between the bosons. In ref. 21, it was demonstrated that arbitrary weak interactions are unable to destroy the insulator: The boson density gets fragmented into lakes with irrelevant tunneling between them. The tunneling becomes relevant and drives the system into a fluid state at a critical interaction strength determined by the disorder.

On the other hand, it is well known that bosons with an infinitely strong repulsion are equivalent to free fermions (30) and, hence, they are localized by an arbitrary weak disorder. Accordingly, there should be a second critical interaction strength, above which one should expect an insulating state. The physics of the second transition at strong coupling is analogous to the fluid−insulator transition of spinless fermionic atoms as in ref. 7. In one dimension, it is also important to account for the renormalization of the disorder by fermion−fermion interaction (31⇓⇓–34) (see *Disorder Renormalization and Many-Body LDT*). To describe this transition using the 1D boson−fermion duality, one should determine an effective interaction between the fermions when the boson−boson interaction is strong but finite.

## Model and Boson–Fermion Duality

The system of *N* 1D bosonic atoms repulsively interacting with each other via a short-range potential is well described by the Lieb−Liniger model (35). When a generic one-body potential *m* is the atom mass. The interaction strength is characterized by the dimensionless parameter*L* the system length. In the regime of strong interactions, we have **1**) maps onto spinless (spin-polarized) fermions interacting with each other via an odd-wave momentum-dependent attractive interaction (36⇓⇓–40) (*SI Materials and Methods*). More precisely, the fermions are governed by the Hamiltonian**3**) coincide with the bosonic eigenfunctions of **1**) when the coordinates *SI Materials and Methods*). Strongly repulsive bosons map onto weakly attractive spinless fermions with the Fermi momentum **4**) per particle to the Fermi energy,

## SI Materials and Methods

We detail here five points of the paper. We begin with the description of the disorder potential and the length and energy scales of the 1D localized states in the absence of interaction, then we explain the fermion/boson duality in one dimension in the presence of an external potential. Furthermore, we characterize the finite-temperature fluid−insulator transition of weakly interacting spinless fermions in the degenerate and dilute regimes. In the fourth part, we detail the renormalization of the disorder strength due to weak interfermion interaction. Finally, we point out the relation between our results and the Giamarchi−Schulz renormalization group formulation of the weakly disordered Luttinger liquid.

### Disorder Potential Length and Energy Scales.

We consider a Gaussian random potential *σ* and the amplitude *One-Dimensional Disorder and Single-Particle Localization*). The energy dependence of the localization length is known (41⇓–43),*ζ*, which is made of fragments of size *σ*, where the value of the potential is ∼*U*_{0}. The number of such fragments is *ζ* gives the potential energy contribution of a typical fragment, *ζ* and taking into account Eq. **S1**, one obtains the length and energy scales of the low-energy bound state**S2** directly leads to the condition stated in *One-Dimensional Disorder and Single-Particle Localization*,

### Boson/Fermion Correspondence in One Dimension.

Here we extend the arguments of ref. 37 to the case of an external (random) potential. We demonstrate that, although the model is not exactly solvable, the 1D fermion−boson duality remains valid.

Consider the Schrödinger equation for the *N*-fermion wavefunction **4**, which can be rewritten as

For noncoinciding coordinates, Eq. **S8** reduces to**S8** over an infinitesimally small interval **S9** and **S10** and taking into account the identity **S13** can be written as

Integrating Eq. **S13** over *x* in an infinitesimally small interval **S15** takes the form

Note that Eqs. **S11** and **S16** could equally be obtained for the Hamiltonian of Eq. **1**,*N*-body bosonic wavefunctions. As a consequence, the many-body spectra of the models (Eqs. **S8** and **S17**) are identical, and the eigenfunctions are equal in magnitude and only differ by a sign upon exchange of coordinates.

### Conductor–Insulator Transition of Disordered Weakly Interacting 1D Spinless Fermions with Gradient-Dependent Interaction.

In the case where all single-particle eigenstates are localized (as in one dimension), there is a critical temperature **S10**, writes**S19**) evaluate to*α*,*γ* and *β*,*δ* are neighbors in energy (7, 21), we obtain

In the quantum degenerate regime, *n* is the average density of particles), the two-body processes responsible for the fluid−insulator transition occur at the energy scale *λ* is of the order of the inverse of the Lieb−Liniger parameter,*ζ* writes**S18**) with two-body matrix elements (Eq. **S23**) leads to the critical temperature in the quantum degenerate regime (7),*C* is a model-dependent numerical constant of the order 1. Taking into account the duality (Eq. **S24**) and the level spacing renormalization due to interparticle interaction (described in *Renormalization of the Impurity Scattering Amplitude*), we obtain Eq. **18**.

In the dilute high-temperature case *T* gives the relevant energy scale of the states participating in the interaction processes. The critical temperature in this regime is given by (21)*ε*,**S1**. In Eq. **S27**, the quantity **S1** and **S28**–**S30** into Eq. **S27**, we obtain Eq. **22**.

### Renormalization of the Impurity Scattering Amplitude.

In this section, we describe the renormalization of the disorder strength by the interaction and derive the resulting Born approximation backscattering rate in the case of many impurities. The renormalization of a single impurity transmission amplitude *t* due to weak interaction between spinless fermions was computed by Matveev et al. and Yue et al. (32, 33). The result is a temperature dependence of the transmission coefficient

In quantum mechanics (49), a particle of energy *σ* and amplitude **[S7]**. On the other hand, the backscattering rate caused by many impurities with concentration **S33**, we get**S35** does not depend on the properties of the single impurity potential, and therefore it is not affected by the renormalization (Eq. **S31**). As a consequence, we can write the renormalized backscattering rate as

### Giamarchi−Schulz Renormalization Group for the Disordered Bosons.

We now turn to the Luttinger liquid formulation of the disordered 1D bosons and point out the relation between its predictions and our theory. The renormalization group approach of the disordered Luttinger liquid of bosons was developed by Giamarchi and Schulz (26), and it describes the zero-temperature algebraic superfluid to Bose glass transition at moderate interactions and small disorder. The influence of disorder is taken into account by the backscattering field *K* is the Luttinger parameter, and *K* and *a* is a short-distance cutoff that we set equal to **S42** and our strong coupling critical temperature in the degenerate regime given in the main text. Indeed, given the development of the Luttinger parameter with respect to *K*, and our result extends Eq. **S42** to the whole temperature regime

On the other hand, when *K* and **S42** and the dependence **19**.

## One-Dimensional Disorder and Single-Particle Localization

Without loss of generality, we can represent the static disorder by a Gaussian random potential *σ*,*SI Materials and Methods* in more detail.

For a weak disorder,

## Disorder Renormalization and Many-Body LDT

In the limit of *SI Materials and Methods*)*ζ*. For a weak disorder in the 1D case, *τ* being the transport time. Thus, we have

It is known that the interaction between 1D fermions renormalizes the disorder (see refs. 31⇓⇓–34),*SI Materials and Methods*),

The dimensionless fermionic coupling constant is defined (32, 33) as**4**) for transferred momentum *q*. In the secondly quantized form, the potential (Eq. **4**) writes*k*, and**15** into Eq. **12** yields the fermionic coupling constant

Thus, according to Eqs. **10** and **9**, the renormalized impurity backscattering rate and the effective level spacing decrease with temperature due to interfermion interaction (Eq. **4**). Using Eqs. **7**, **11**, and **16**, we find**17** into Eq. **9** and using Eq. **8**, we obtain a critical temperature in the quantum degenerate regime,**18** indicates that **18** connects with the zero temperature result of ref. 26.

In the derivation of Eq. **18**, we took into account the renormalization of the disorder due to interaction of the effective spinless fermions with Friedel oscillations and neglected the renormalization of the interaction by the disorder. It is known (26) that this approximation works as long as *SI Materials and Methods*)*γ* becomes larger than **18**.

The derivation of Eq. **19** is similar to that for the localization length scale **19**, except for the coefficients

In the limit of extremely strong coupling, *γ* in Eq. **18** becomes equal to 1, which gives**5** into Eq. **20** transforms it to the result of ref. 7 for

For a nondegenerate gas where **6**. According to Eq. **14**, we have *γ* corresponding to a crossover from the classical to quantum degenerate regime is obtained from Eq. **22** setting

It should be noted that, at temperatures of the order of **22**), the fermions remain weakly interacting. The scattering phase shift **4**) is*k* is of the order of the thermal momentum **22**), the phase shift is still small, **22** is valid.

In Fig. 1, we show the phase diagram in terms of the amplitude of the disorder (*γ*). As expected, the critical disorder for the fluid−insulator transition vanishes as

The phase diagram in terms of dimensionless temperature *γ* at a fixed weak disorder (*γ* increases from small to large values. The first insulator-to-fluid transition occurs when the interaction between the bosons is weak (**18**. The *γ* for *γ* from small to large values at a fixed temperature.

The difference

It is feasible to verify experimentally the full picture of the finite temperature behavior of 1D disordered bosons constructed in the present paper. A suitable candidate would be the gas of ^{7}Li atoms where the coupling constant *g* can be varied by Feshbach resonance from very small to very large values (45), and the 1D regime has already been achieved (46, 47). The regime of strong interactions can be reached, in particular, by using a confinement-induced resonance as in the cesium experiments (48), and the disorder can be introduced by using optical speckles like in the first experiments on the observation of Anderson localization (8). The insulator−fluid transition can be identified in expansion experiments (see the discussion in ref. 21), or by analyzing the momentum distribution like in recent experiments for bosons in 1D quasiperiodic potentials (16, 17).

## Acknowledgments

We are grateful to M. B. Zvonarev and A. I. Gudyma for fruitful discussions and acknowledge support from IFRAF and from the Dutch Foundation FOM. The research leading to these results received funding from the European Research Council under European Community’s Seventh Framework Programme (FR7/2007-2013 Grant Agreement 341197).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: bla{at}phys.columbia.edu.

Author contributions: V.P.M., I.L.A., B.L.A., and G.V.S. designed research, performed research, and wrote the paper.

Reviewers: T.G., University of Geneva; and I.L., University of Birmingham.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1606908113/-/DCSupplemental.

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