Localization induced by spatially uncorrelated subohmic baths in one dimension

Saptarshi Majumdar, Laura Foini, Thierry Giamarchi, and Alberto Rosso

Phys. Rev. B 108, 205138 – Published 17 November 2023

Abstract

We study an incommensurate XXZ spin chain coupled to a collection of local harmonic baths. At zero temperature, by varying the strength of the coupling to the bath the chain undergoes a quantum phase transition between a Luttinger liquid phase and a spin-density wave (SDW). Our results are consistent with the Berezinskii-Kosterlitz-Thouless (BKT) transition. As opposed to the standard mechanism, the SDW emerges in the absence of the opening of a gap, but it is due to “fractional excitations” induced by the bath. We also show, by computing the DC conductivity, that the system is insulating in the presence of a subohmic bath. We interpret this phenomenon as localization induced by the bath à la Caldeira and Leggett.

Received 18 July 2023
Revised 4 October 2023
Accepted 31 October 2023

DOI:https://doi.org/10.1103/PhysRevB.108.205138

Physics Subject Headings (PhySH)

Bosonization Dissipative dynamics Spin density waves

Langevin algorithm Luttinger liquid model Quantum spin chains

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Saptarshi Majumdar ¹, Laura Foini², Thierry Giamarchi ³, and Alberto Rosso¹

¹Université Paris Saclay, CNRS, LPTMS, 91405, Orsay, France
²IPhT, CNRS, CEA, Université Paris Saclay, 91191 Gif-sur-Yvette, France
³Department of Quantum Matter Physics, University of Geneva, 24 Quai Ernest-Ansermet, CH-1211 Geneva, Switzerland

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Vol. 108, Iss. 20 — 15 November 2023

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Images

Figure 1
Schematic diagram of a one-dimensional quantum XXZ spin chain coupled with local dissipative baths. $J_{x y}$ denotes the hopping energy, and $J_{z}$ is the interaction between the two nearest-neighbor spins. The baths are characterized by their spectral function $J (Ω) \sim α Ω^{s}$ . At zero temperature, the baths induce an SDW phase with periodicity $π / q_{F}$ , where $q_{F}$ is the Fermi momentum related to the magnetization of the chain (see text).
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Figure 2
$F (ω_{n})$ for ohmic ( $s = 1$ , left) and subohmic ( $s = 0.5$ , right) bath obtained by numerical solution of Eq. (14) (with $β = 1024$ and $α = 5$ ). In the dissipative phase, $F (ω_{n})$ behaves as $0.301 | ω_{n} |$ (purple square points) for ohmic ( $K = 0.15$ ) and $0.415 \sqrt{| ω_{n} |}$ for (purple circular points) subohmic bath ( $K = 0.3$ ). In the LL phase, $F (ω_{n}) = 0.06 ω_{n}^{2}$ for the ohmic bath (black square points) and $F (ω_{n}) = 0.054 ω_{n}^{2}$ for the subohmic bath ( $K = 1$ ) (black circular points).
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Figure 3
The parameters $η$ and $ν$ obtained from the numerical solution of Eq. (14) (with $β = 1024$ and $α = 5$ ). For $η$ (top row), we use $α^{'}$ and $Λ$ from Eq. (11) as fitting parameters. For the ohmic case (purple square), $α^{'} = 10.096, Λ = 1.963$ , and for the subohmic case (purple circle), $α^{'} = 8.29, Λ = 3.29$ . For the plot of $ν$ (bottom row), the fitting parameters are $τ_{c}$ and $Λ$ from Eq. (12). For the ohmic case (black square), $τ_{c} = 1.68, Λ = 0.272$ , and for the subohmic case (black circle), $τ_{c} = 1.241, Λ = 0.415$ .
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Figure 4
Calculation of different quantities for $K = 1$ that characterizes the LL ( $α = 2$ , top row) and dissipative phases ( $α = 6$ , bottom row). Red points correspond to $L = β = 128$ , and blue points correspond to $L = β = 320$ . Green points correspond to $ω_{n} C (ω_{n})$ calculated from numerically solving the self-consistent variational Eq. (7). (left) $π χ$ remains unrenormalized for all values of $α$ and $q$ . (middle) For $α = 2, ω_{n} C (ω_{n})$ saturates to $K_{r} / 2 = 0.46$ for small $ω$ , whereas $ω_{n}^{0.25} C (ω_{n})$ saturates to $1 / \sqrt{α_{r}} = 1.072$ for $α = 6$ . The variational solution saturates to $K_{r, var} / 2 = 0.486$ for $α = 2$ and fails to correctly predict the dissipative phase for $α = 6$ . (right) For $α = 2, 〈 cos 2 ϕ 〉$ decays as a power law with the exponent $K_{r} = 0.915$ . However, it saturates to a constant $c_{1}$ algebraically for $α = 6$ . The fit for the order parameter in the dissipative phase gives $c_{1} = 0.096, c_{2} = 0.112$ , and $c_{3} = 3.37$ . In the LL phase, $〈 cos (2 ϕ) 〉$ calculated with the variational method decays with $K_{r, var} = 0.973$ .
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Figure 5
Renormalized value of different parameters of the action for $K = 1, s = 0.5$ . $K_{r} / u_{r}$ (red square) remains constant and equal to 1 for all values of $α$ . $K_{r}$ (purple circles calculated from $〈 cos 2 ϕ 〉$ and blue triangle calculated from $C (ω_{n})$ ) decreases from $K = 1$ to $K_{c} = 0.75$ as $α$ increases and approaches $α_{c}$ . $α_{r}$ (green circle) becomes relevant in the dissipative phase and increases as a function of $α$ . This behavior of the parameters helps us locate the critical region $α_{c} \in (3, 4)$ .
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