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Finite-temperature fluid–insulator transition of strongly interacting 1D disordered bosons Vincent P. Michala, Igor L. Aleinerb, Boris L. Altshulerb,c,1, and Georgy V. Shlyapnikova,c,d,e a Laboratoire de Physique Théorique et Modèles Statistiques, CNRS, Université Paris-Sud and Université Paris-Saclay, 91405 Orsay, France; bPhysics Department, Columbia University, New York, NY 10027; cWuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China; d Van der Waals-Zeeman Institute, University of Amsterdam, 1098 XH Amsterdam, The Netherlands; and eRussian Quantum Center, Moscow Region 143025, Russia
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 → fluid transition, and, for large interactions, there is a reentrance to the insulator regime. It is feasible to experimentally verify these predictions by tuning the interaction strength with the use of Feshbach or confinement-induced resonances, for example, in 7Li or 39K. many-body localization
| disordered bosons | ultracold atomic gases
D
espite 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 T = 0 for the Luttinger liquid parameter K < 3=2, even an arbitrary weak short-range disorder leads to localization. Up to now, the finite temperature behavior of 1D disordered bosons was well understood only for weak interactions. Here, we
www.pnas.org/cgi/doi/10.1073/pnas.1606908113
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 → fluid transition when the interaction strength increases from zero to a certain critical value, and then a reentrance to the insulator phase at sufficiently strong interactions. 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. 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. 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. 1
To whom correspondence should be addressed. Email: [email protected].
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 1073/pnas.1606908113/-/DCSupplemental.
PNAS | Published online July 19, 2016 | E4455–E4459
PHYSICS
Contributed by Boris L. Altshuler, May 2, 2016 (sent for review December 24, 2015; reviewed by Thierry Giamarchi and Igor Lerner)
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 UðxÞ is present, the Hamiltonian reads HB =
N X X Z2 − ∂2xj + U xj + g δ xj − xk , 2m j=1 1≤j 0 is the coupling constant, and m is the atom mass. The interaction strength is characterized by the dimensionless parameter mg γ= 2 , Z n
N X Z2 − ∂2xj + U xj + VF , 2m j=1
[3]
with the interaction operator (37) Z4 VF = 2 m g
X
∂xj − ∂xk δ xj − xk ∂xj − ∂xk ,
and the eigenfunctions of HF (Eq. 3) coincide with the bosonic eigenfunctions of HB (Eq. 1) when the coordinates xj are ordered xj < xj+1, and differ by a sign upon exchange of the coordinates (SI Materials and Methods). Strongly repulsive bosons map onto weakly attractive spinless fermions with the Fermi momentum kF = πn and Fermi energy EF = Z2 k2F =2m. In the degenerate regime, the fermionic dimensionless coupling constant can be estimated as the ratio of a typical interaction energy (Eq. 4) per particle to the Fermi energy, Z4 k3F m2 g 1 ≈ − 1. EF γ
[5]
Having in mind the comparison between strongly and weakly interacting bosons, we use below the temperature of quantum degeneracy Td =
π 2 Z2 n2 , 2m
which coincides with EF of effective fermions in the case of strongly interacting bosons. One-Dimensional Disorder and Single-Particle Localization Without loss of generality, we can represent the static disorder by a Gaussian random potential UðxÞ with zero mean, variance U02, and correlation length σ, hUðxÞi = 0;
hUðxÞUðx′Þi = U02 f ðjx − x′j=σÞ,
where f ðzÞ → 1 for z → 0, f ðzÞ → 0 for z → ∞, and the symbol h. . .i denotes averaging over the realizations of the disorder. Below, E4456 | www.pnas.org/cgi/doi/10.1073/pnas.1606908113
2 4 1=3 mσ U0 U0 . Z2
Derivation of this equation, which was made in ref. 21, is described in SI Materials and Methods in more detail. For a weak disorder, «p Td, the single-particle localization length at characteristic energy « ≈ Td, given by (41–43) ζð«Þ ≈
Z« 3=2
m1=2 «p
[6]
,
greatly exceeds the mean interparticle distance n−1. This ensures the presence of a small parameter, D=
«p Td
3=2 1.
[7]
Disorder Renormalization and Many-Body LDT In the limit of γ → ∞, the fermions are free (30), and an arbitrary weak disorder localizes them irrespective of their energy, leading to an insulator state at any temperature. A weak interaction between the fermions (and thus a strong but finite interaction between the bosons) causes many-body delocalization, i.e., destroys the insulator at the critical temperature (see equation 42 in ref. 7 and SI Materials and Methods) Tc =
[4]
1≤j 7.9). In this sense, our Eq. 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 ðγ − γ 0 Þ D1=2. Close to the zerotemperature transition, this condition is violated, and one has to use the coupled renormalization group equations for the interaction and disorder. For γ < γ 0, this leads to the Berezinskii−Kosterlitz−Thouless (BKT) criticality with temperature (SI Materials and Methods) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Tc ≈ Td exp −16π c21 D − c22 ðγ − γ 0 Þ2 ,
Dγ . ln γ
PHYSICS
~ ðq, k1 , k2 Þ = − Z ðk1 − k2 + 2qÞðk1 − k2 Þ. V m2 g
Tc ≈ Td
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where the operator aðkÞ annihilates a fermion with momentum k, and
Fig. 1. Phase diagram in terms of the disorder strength D and temperature at a fixed dimensionless coupling constant γ. In A, γ 1. The yellow part of the curve corresponds to Eq. 24, the blue part corresponds to Eq. 25, and the red part corresponds to Eq. 26. In B, γ > γ 0 ≈ 8, and the blue part of the curve corresponds to Eq. 18, whereas the red part corresponds to Eq. 22.
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δF ðkÞ = −arctan
Z2 k k = −arctan . 2mg 2γn
[23]
A
For T < Td, the fermions are degenerate, and k ≈ kF ≈ n. The phase shift δF ðkÞ ≈ −arctanðγ −1 Þ is thus small as long as γ 1, i.e., the fermions interact weakly. In the nondegenerate case, T Td, theffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
momentum k is of the order of the thermal momentum mT=Z2, and, at very high temperatures, the fermions interact strongly, δF ≈ 1. However, at T K Tc (Eq. 22), the phase shift is still small, δF ≤ D1=3 =γ 2=3 1, and the many-body LDT single-particle picture leading to Eq. 22 is valid. In Fig. 1, we show the phase diagram in terms of the amplitude of the disorder (D) and temperature at a fixed interaction strength (γ). As expected, the critical disorder for the fluid−insulator transition vanishes as T → 0 if γ > 8, whereas, for γ < 8, it remains finite. The phase diagram in terms of dimensionless temperature T=Td and interaction strength γ at a fixed weak disorder (D 1) is displayed in Fig. 2. It combines the results of the present paper for strong coupling of the bosons, γ 1, with the weak coupling (γ 1) results of ref. 21. We clearly see that, in a weak disorder (D 1) at any finite temperature, one has two insulator−fluid transitions as γ increases from small to large values. The first insulator-to-fluid transition occurs when the interaction between the bosons is weak (γ 1). The results of ref. 21 for this case can be written in terms of the critical coupling γ c1 of the transition as γ c1 ≈ D2=3 ; γ c1 ≈ DTd =T;
T D1=3 Td , D1=3 Td T Td ,
γ c1 ≈ DðTd =TÞ1=2 ;
T Td .
[24] [25] [26]
The reentrance to the insulator takes place at strong interactions (γ 1), pffiffiffiffi [27] γ c2 = γ 0 − c D; T → 0, γ c2 ≈ D−1 ðT=Td Þ3=2 ;
T Td .
[28]
For finite temperatures T Td, the critical coupling is determined by a transcendental equation following from Eq. 18. The T → 0
Fig. 2. Fluid−insulator transition of finite temperature repulsive bosons in a weak disorder, D 1 (see text).
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B
Fig. 3. Phase diagram in terms of the disorder strength D and dimensionless coupling constant γ at a fixed temperature. In A, T = 0, described by Eq. 24 for γ 1 and by Eq. 27 for γ close to γ 0 ≈ 8. In B, T Td , described by Eq. 26 for γ 1 and by Eq. 28 for γ 1. The dashed parts of the curves show the expected behavior at intermediate coupling.
asymptotics corresponds to the 1+1 BKT quantum transition (26). In Fig. 3, we show the dependence of the critical disorder on γ for T → 0 and for T Td. One clearly sees the presence of two insulator−fluid transitions when increasing γ from small to large values at a fixed temperature. The difference γ c2 − γ c1 between the two critical interaction constants should decrease as the disorder (D) increases. The two insulating regimes at T = 0 are believed to merge (γ c1 = γ c2) at a certain critical value of the disorder, Dc ≈ 1 (40). 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 7Li 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).
Michal et al.
Michal et al.
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26. Giamarchi T, Schulz HJ (1988) Anderson localization and interactions in onedimensional metals. Phys Rev B Condens Matter 37(1):325–340. 27. Gertsenshtein ME, Vasil’ev VB (1959) Waveguides with random inhomogeneities and Brownian motion in the Lobachevsky plane. Theory Probab Appl 4(4):391–398. 28. Mott NF, Twose WD (1961) The theory of impurity conduction. Adv Phys 10(38): 107–163. 29. Berezinskii VL (1973) Kinetika kvantovoi chastitsy v odnomernom sluchainom potentsiale [Kinetics of a quantum particle in a one-dimensional random potential]. ZhETF 65(3):1251–1266. Russian. 30. Girardeau M (1960) Relationship between systems of impenetrable bosons and fermions in one dimension. J Math Phys 1(6):516–523. 31. Gorkov LP, Dzyaloshinski IE (1973) Conductivity of quasi-one-dimensional crystals in the metallic phase. JETP Lett 18(11):401–403. 32. Matveev KA, Yue D, Glazman LI (1993) Tunneling in one-dimensional non-Luttinger electron liquid. Phys Rev Lett 71(20):3351–3354. 33. Yue D, Matveev KA, Glazman LI (1994) Conduction of a weakly interacting onedimensional electron gas through a single barrier. Phys Rev B 49(3):1966–1975. 34. Giamarchi T (2003) Disordered systems. Quantum Physics in One Dimension (Oxford Univ Press, Oxford), pp 270–302. 35. Lieb EH, Liniger W (1963) Exact analysis of an interacting Bose gas. I. The general solution and the ground state. Phys Rev 130(4):1605–1616. 36. Cheon T, Shigehara T (1999) Fermion-boson duality of one-dimensional quantum particles with generalized contact interactions. Phys Rev Lett 82(12):2536–2539. 37. Grosse H, Langmann E, Paufler C (2004) Exact solution of a 1D quantum many-body system with momentum dependent interactions. J Phys Math Gen 37(16):4579–4592. 38. Sen D (1999) Perturbation theory for singular potentials in quantum mechanics. Int J Mod Phys A 14(11):1789–1807. 39. Sen D (2003) The fermionic limit of the δ-function Bose gas: A pseudopotential approach. J Phys A 36(27):7517. 40. Cazalilla MA, Citro R, Giamarchi T, Orignac E, Rigol M (2011) One dimensional bosons: From condensed matter systems to ultracold gases. Rev Mod Phys 83(4):1405–1466. 41. Lifshitz IM (1965) Energy spectrum structure and quantum states of disordered condensed systems. Sov Phys Usp 7(4):549–573. 42. Halperin BI, Lax M (1966) Impurity-band tails in the high-density limit. I. Minimum counting methods. Phys Rev 148(2):722–740. 43. Zittartz J, Langer JS (1966) Theory of bound states in a random potential. Phys Rev 148(2):741–747. 44. Brand J, Cherny AY (2005) Dynamic structure factor of the one-dimensional Bose gas near the Tonks-Girardeau limit. Phys Rev A 72(3):033619. 45. Pollack SE, et al. (2009) Extreme tunability of interactions in a 7Li Bose-Einstein condensate. Phys Rev Lett 102(9):090402. 46. Khaykovich L, et al. (2002) Formation of a matter-wave bright soliton. Science 296(5571):1290–1293. 47. Nguyen JHV, Dyke P, Luo D, Malomed BA, Hulet RG (2014) Collisions of matter-wave solitons. Nat Phys 10(12):918–922. 48. Haller E, et al. (2009) Realization of an excited, strongly correlated quantum gas phase. Science 325(5945):1224–1227. 49. Landau LD, Lifshitz EM (1977) Quantum Mechanics (Non-relativistic Theory), Course of Theoretical Physics (Elsevier, New York), 3rd Ed, Vol 3. 50. Haldane FDM (1981) Effective harmonic-fluid approach to low-energy properties of one-dimensional quantum fluids. Phys Rev Lett 47(25):1840–1843.
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Finite-temperature fluid–insulator transition of strongly interacting 1D disordered bosons Vincent P. Michala, Igor L. Aleinerb, Boris L. Altshulerb,c,1, and Georgy V. Shlyapnikova,c,d,e a Laboratoire de Physique Théorique et Modèles Statistiques, CNRS, Université Paris-Sud and Université Paris-Saclay, 91405 Orsay, France; bPhysics Department, Columbia University, New York, NY 10027; cWuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China; d Van der Waals-Zeeman Institute, University of Amsterdam, 1098 XH Amsterdam, The Netherlands; and eRussian Quantum Center, Moscow Region 143025, Russia
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 → fluid transition, and, for large interactions, there is a reentrance to the insulator regime. It is feasible to experimentally verify these predictions by tuning the interaction strength with the use of Feshbach or confinement-induced resonances, for example, in 7Li or 39K. many-body localization
| disordered bosons | ultracold atomic gases
D
espite 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 T = 0 for the Luttinger liquid parameter K < 3=2, even an arbitrary weak short-range disorder leads to localization. Up to now, the finite temperature behavior of 1D disordered bosons was well understood only for weak interactions. Here, we
www.pnas.org/cgi/doi/10.1073/pnas.1606908113
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 → fluid transition when the interaction strength increases from zero to a certain critical value, and then a reentrance to the insulator phase at sufficiently strong interactions. 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. 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. 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. 1
To whom correspondence should be addressed. Email: [email protected].
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 1073/pnas.1606908113/-/DCSupplemental.
PNAS | Published online July 19, 2016 | E4455–E4459
PHYSICS
Contributed by Boris L. Altshuler, May 2, 2016 (sent for review December 24, 2015; reviewed by Thierry Giamarchi and Igor Lerner)
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 UðxÞ is present, the Hamiltonian reads HB =
N X X Z2 − ∂2xj + U xj + g δ xj − xk , 2m j=1 1≤j 0 is the coupling constant, and m is the atom mass. The interaction strength is characterized by the dimensionless parameter mg γ= 2 , Z n
N X Z2 − ∂2xj + U xj + VF , 2m j=1
[3]
with the interaction operator (37) Z4 VF = 2 m g
X
∂xj − ∂xk δ xj − xk ∂xj − ∂xk ,
and the eigenfunctions of HF (Eq. 3) coincide with the bosonic eigenfunctions of HB (Eq. 1) when the coordinates xj are ordered xj < xj+1, and differ by a sign upon exchange of the coordinates (SI Materials and Methods). Strongly repulsive bosons map onto weakly attractive spinless fermions with the Fermi momentum kF = πn and Fermi energy EF = Z2 k2F =2m. In the degenerate regime, the fermionic dimensionless coupling constant can be estimated as the ratio of a typical interaction energy (Eq. 4) per particle to the Fermi energy, Z4 k3F m2 g 1 ≈ − 1. EF γ
[5]
Having in mind the comparison between strongly and weakly interacting bosons, we use below the temperature of quantum degeneracy Td =
π 2 Z2 n2 , 2m
which coincides with EF of effective fermions in the case of strongly interacting bosons. One-Dimensional Disorder and Single-Particle Localization Without loss of generality, we can represent the static disorder by a Gaussian random potential UðxÞ with zero mean, variance U02, and correlation length σ, hUðxÞi = 0;
hUðxÞUðx′Þi = U02 f ðjx − x′j=σÞ,
where f ðzÞ → 1 for z → 0, f ðzÞ → 0 for z → ∞, and the symbol h. . .i denotes averaging over the realizations of the disorder. Below, E4456 | www.pnas.org/cgi/doi/10.1073/pnas.1606908113
2 4 1=3 mσ U0 U0 . Z2
Derivation of this equation, which was made in ref. 21, is described in SI Materials and Methods in more detail. For a weak disorder, «p Td, the single-particle localization length at characteristic energy « ≈ Td, given by (41–43) ζð«Þ ≈
Z« 3=2
m1=2 «p
[6]
,
greatly exceeds the mean interparticle distance n−1. This ensures the presence of a small parameter, D=
«p Td
3=2 1.
[7]
Disorder Renormalization and Many-Body LDT In the limit of γ → ∞, the fermions are free (30), and an arbitrary weak disorder localizes them irrespective of their energy, leading to an insulator state at any temperature. A weak interaction between the fermions (and thus a strong but finite interaction between the bosons) causes many-body delocalization, i.e., destroys the insulator at the critical temperature (see equation 42 in ref. 7 and SI Materials and Methods) Tc =
[4]
1≤j 7.9). In this sense, our Eq. 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 ðγ − γ 0 Þ D1=2. Close to the zerotemperature transition, this condition is violated, and one has to use the coupled renormalization group equations for the interaction and disorder. For γ < γ 0, this leads to the Berezinskii−Kosterlitz−Thouless (BKT) criticality with temperature (SI Materials and Methods) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Tc ≈ Td exp −16π c21 D − c22 ðγ − γ 0 Þ2 ,
Dγ . ln γ
PHYSICS
~ ðq, k1 , k2 Þ = − Z ðk1 − k2 + 2qÞðk1 − k2 Þ. V m2 g
Tc ≈ Td
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where the operator aðkÞ annihilates a fermion with momentum k, and
Fig. 1. Phase diagram in terms of the disorder strength D and temperature at a fixed dimensionless coupling constant γ. In A, γ 1. The yellow part of the curve corresponds to Eq. 24, the blue part corresponds to Eq. 25, and the red part corresponds to Eq. 26. In B, γ > γ 0 ≈ 8, and the blue part of the curve corresponds to Eq. 18, whereas the red part corresponds to Eq. 22.
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δF ðkÞ = −arctan
Z2 k k = −arctan . 2mg 2γn
[23]
A
For T < Td, the fermions are degenerate, and k ≈ kF ≈ n. The phase shift δF ðkÞ ≈ −arctanðγ −1 Þ is thus small as long as γ 1, i.e., the fermions interact weakly. In the nondegenerate case, T Td, theffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
momentum k is of the order of the thermal momentum mT=Z2, and, at very high temperatures, the fermions interact strongly, δF ≈ 1. However, at T K Tc (Eq. 22), the phase shift is still small, δF ≤ D1=3 =γ 2=3 1, and the many-body LDT single-particle picture leading to Eq. 22 is valid. In Fig. 1, we show the phase diagram in terms of the amplitude of the disorder (D) and temperature at a fixed interaction strength (γ). As expected, the critical disorder for the fluid−insulator transition vanishes as T → 0 if γ > 8, whereas, for γ < 8, it remains finite. The phase diagram in terms of dimensionless temperature T=Td and interaction strength γ at a fixed weak disorder (D 1) is displayed in Fig. 2. It combines the results of the present paper for strong coupling of the bosons, γ 1, with the weak coupling (γ 1) results of ref. 21. We clearly see that, in a weak disorder (D 1) at any finite temperature, one has two insulator−fluid transitions as γ increases from small to large values. The first insulator-to-fluid transition occurs when the interaction between the bosons is weak (γ 1). The results of ref. 21 for this case can be written in terms of the critical coupling γ c1 of the transition as γ c1 ≈ D2=3 ; γ c1 ≈ DTd =T;
T D1=3 Td , D1=3 Td T Td ,
γ c1 ≈ DðTd =TÞ1=2 ;
T Td .
[24] [25] [26]
The reentrance to the insulator takes place at strong interactions (γ 1), pffiffiffiffi [27] γ c2 = γ 0 − c D; T → 0, γ c2 ≈ D−1 ðT=Td Þ3=2 ;
T Td .
[28]
For finite temperatures T Td, the critical coupling is determined by a transcendental equation following from Eq. 18. The T → 0
Fig. 2. Fluid−insulator transition of finite temperature repulsive bosons in a weak disorder, D 1 (see text).
E4458 | www.pnas.org/cgi/doi/10.1073/pnas.1606908113
B
Fig. 3. Phase diagram in terms of the disorder strength D and dimensionless coupling constant γ at a fixed temperature. In A, T = 0, described by Eq. 24 for γ 1 and by Eq. 27 for γ close to γ 0 ≈ 8. In B, T Td , described by Eq. 26 for γ 1 and by Eq. 28 for γ 1. The dashed parts of the curves show the expected behavior at intermediate coupling.
asymptotics corresponds to the 1+1 BKT quantum transition (26). In Fig. 3, we show the dependence of the critical disorder on γ for T → 0 and for T Td. One clearly sees the presence of two insulator−fluid transitions when increasing γ from small to large values at a fixed temperature. The difference γ c2 − γ c1 between the two critical interaction constants should decrease as the disorder (D) increases. The two insulating regimes at T = 0 are believed to merge (γ c1 = γ c2) at a certain critical value of the disorder, Dc ≈ 1 (40). 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 7Li 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).
Michal et al.
Michal et al.
PNAS | Published online July 19, 2016 | E4459
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