Thermal disruption of a Luttinger liquid Danyel Cavazos-Cavazos1 Ruwan Senaratne1 Aashish Kafle1 and Randall G. Hulet1 emailrandyrice.edu 1Department of Physics and Astronomy Rice University Houston Texas 77005 USA

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Thermal disruption of a Luttinger liquid

Danyel Cavazos-Cavazos1, Ruwan Senaratne1, Aashish Kaﬂe1, and Randall G. Hulet1(email:randy@rice.edu)

1Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA

The Tomonaga-Luttinger liquid (TLL) theory

describes the low-energy excitations of strongly

correlated one-dimensional (1D) fermions. In

the past years, a number of studies have pro-

vided a detailed understanding of this universality

class. More recently, theoretical investigations that

go beyond the standard low-temperature, linear-

response, TLL regime have been developed. While

these provide a basis for understanding the dynam-

ics of the spin-incoherent Luttinger liquid, there

are few experimental investigations in this regime.

Here we report the observation of a thermally-

induced, spin-incoherent Luttinger liquid in a 6Li

atomic Fermi gas conﬁned to 1D. We use Bragg

spectroscopy to measure the suppression of spin-

charge separation and the decay of correlations as

the temperature is increased. Our results probe

the crossover between the coherent and incoher-

ent regimes of the Luttinger liquid, and elucidate

the roles of the charge and the spin degrees of

freedom in this regime.

Studies of strongly interacting atomic gases in 1D,

aided by exactly solvable models1–6, have provided re-

markable insight into the physics of highly-correlated

quantum many-body systems in regimes that are in-

creasingly accessible to experiment7–14. The low-energy

properties of spin-1/2 fermions in 1D are well under-

stood in terms of the TLL theory15–19, which features

low-energy collective spin- and charge-density waves

(SDWs/CDWs). These sound waves propagate with

diﬀerent velocities, thus resulting in a spin-charge sepa-

ration. At its core, the TLL universality class is charac-

terized by collective excitations that are coherent and

linearly dispersing. Several regimes, however, have

been found to extend beyond this spin-charge separa-

tion paradigm, allowing access to new classes of un-

conventional Luttinger liquids where the coherence of

the excitations is disrupted20–22. Higher-order eﬀects

such as band-curvature and back-scattering, for exam-

ple, produce a nonlinear Luttinger liquid23, for which the

linearity of the dispersion is disrupted. Spin polariza-

tion is expected to control a quantum phase transition,

at which the TLL turns quantum critical and all ther-

modynamic quantities exhibit universal scaling22. Al-

lowing for anisotropic coupling between 1D chains of

fermions could realize the sliding Luttinger liquid (SLL)

phase24. Topological materials such as single- and bi-

layer graphene25 provide access to phases such as chiral

Luttinger liquids26 (χLL), which host excitation modes

with a preferred sense of propagation.

Finite temperature represents another pathway for

disrupting the correlations in a TLL (Fig. 1a). In the

low temperature (T) limit, the thermal energy kBTis

the lowest energy scale and both the charge- and spin-

density waves propagate coherently in accordance with

the standard TLL theory, thus deﬁning the spin-coherent

(SC) regime. As Tis increased, thermal ﬂuctuations

disrupt the coherence in the spin sector ﬁrst, and the

system enters the spin-incoherent (SI) Luttinger liquid

regime27. In the SI regime, spin-spin correlations are

expected to exhibit a rapid exponential decay while the

density-density correlations retain a slower algebraic de-

cay, leading to correlations that are independent from

the spin sector28. The SI regime has been investigated

theoretically with the Bethe ansatz28, 29 and a bosonized

path integral approach31–33 to describe both fermions34

and bosons35. Recent studies have also identiﬁed den-

sity correlations28, 34, 36 that distinguish the SC and the

SI regimes. Experimental evidence for the SI regime,

however, remains scarce. Studies of quasi-1D solid-

state materials using angle-resolved photoemission spec-

troscopy37, 38 have suggested that signatures of the SI

regime arise for small electron densities39. The con-

trol and tunability aﬀorded by ultracold gases, on the

other hand, facilitate systematic study of Luttinger liq-

uid physics10–12, 14.

Here, we explore the crossover between a SC Lut-

tinger liquid and the SI regime in a pseudo-spin-1/2 gas

of 6Li atoms loaded into an array of 1D waveguides. We

use Bragg spectroscopy to show the suppression of spin-

charge separation and the systematic loss of coherence

with increasing T. Surprisingly, signatures of the spin

degree of freedom persist even for T > TF, where TFis

the Fermi temperature.

Spin-charge separation results in a separation of en-

ergy scales for the spin and the charge sectors of the TLL

arXiv:2210.06306v2 [cond-mat.quant-gas] 2 Jun 2023

Hamiltonian. These are given by Es,c =ℏnvs,c, where vs

and vcare the propagation speeds of each mode, and n

is the 1D density27. The speed of the SDW is less than

that of the CDW1, and thus Es< Ecin the SC regime.

In the SI regime, where Es< kBT < Ec, the spin

conﬁgurations are mixed, even though the charge corre-

lations remain unaﬀected. Consequently, it is expected

that the SDW no longer propagates in the SI regime,

whereas the CDW continues to propagate33. For suf-

ﬁciently high T, such that kBT > Ec, the coherence

in both sectors is expected to vanish, corresponding to

the thermal regime. The interplay between T, inter-

action strength, and waveguide occupancy Ndeﬁnes

an energy hierarchy for the Luttinger liquid, as shown

schematically in Fig. 1b. These regimes are expected to

Fig. 1 |Energy hierarchy of a Luttinger liq-

uid.a, Schematic diagram showing the energy regimes

of a Luttinger liquid in the spin-coherent (SC), spin-

incoherent (SI) and thermal regimes, illustrating the ef-

fect of decoherence of the spin and charge correlations.

b, Crossover hierarchy of a quasi-1D atomic Fermi gas

(see Methods). The incoherent regimes can be reached

either by increasing the scattering length a, increasing

the temperature T, or by reducing the number of atoms

per tube, N. Dashed lines correspond to the boundaries

between the diﬀerent regimes, deﬁned by Esand Ec,

which are functions of aand N. The solid line illus-

trates a trajectory corresponding to constant Nand a.

As Tis increased, the system ﬁrst loses its spin coher-

ence for Es< kBT < Ec, and at a suﬃciently high T,

such that Es< Ec< kBT, all coherence in the system

is lost.

be separated by smooth crossovers, and the transition

between them remains a subject of active research27.

Our methods for preparing and probing quantum de-

generate, pseudo-spin-1/2 Fermi gases of 6Li atoms,

and characterizing them by Bragg spectroscopy12, 14, 40

are described in the Methods. The pseudo-spin-1/2 sys-

tem consists of a balanced spin mixture of the lowest and

third-to-lowest hyperﬁne ground states of 6Li, which we

label as |1⟩and |3⟩. The interactions depend on the

s-wave scattering length, a, which is ﬁxed to be 500 a0,

where a0is the Bohr radius, by using the |1⟩-|3⟩mag-

netic Feshbach resonance located at 690 G41. We found

that 500 a0is the largest value of aachievable without

incurring an unacceptably large atom loss arising from

3-body recombination12. We vary Tby modifying the

duration and depth of the evaporative cooling trajec-

tory ﬁrst in a crossed-beam dipole trap and then in a

3D harmonic trap. Following evaporation, the atoms are

loaded into a 3D optical lattice, and then into a 2D opti-

cal lattice with a depth of 15 Er, where Er=kB×1.4 µK

is the recoil energy due to a lattice photon of wavelength

Fig. 2 |Symmetric and antisymmetric excitations

via Bragg spectroscopy. Partial energy-level diagram

of 6Li showing relevant transitions and laser detunings

of the Bragg pulses. We generate a symmetric light

shift by symmetrically detuning the frequency of the

Bragg beams far (∆σ= 11 GHz) from the 2S →2P

resonance. For an antisymmetric excitation, the Bragg

beams are detuned by ∆α=±80 MHz from the 2S →

3P resonance frequencies.

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ThermaldisruptionofaLuttingerliquidDanyelCavazos-Cavazos1,RuwanSenaratne1,AashishKafle1,andRandallG.Hulet1(email:randy@rice.edu)1DepartmentofPhysicsandAstronomy,RiceUniversity,Houston,Texas77005,USATheTomonaga-Luttingerliquid(TLL)theorydescribesthelow-energyexcitationsofstronglycorrelatedone-dimensi...

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Thermal disruption of a Luttinger liquid Danyel Cavazos-Cavazos1 Ruwan Senaratne1 Aashish Kafle1 and Randall G. Hulet1 emailrandyrice.edu 1Department of Physics and Astronomy Rice University Houston Texas 77005 USA

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