Bath-induced phase transition in a Luttinger liquid

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

Phys. Rev. B 107, 165113 – Published 7 April 2023

Abstract

We study an XXZ spin chain, where each spin is coupled to an independent ohmic bath of harmonic oscillators at zero temperature. Using bosonization and numerical techniques, we show the existence of two phases separated by a Kosterlitz-Thouless transition. At low coupling with the bath, the chain remains in a Luttinger liquid (LL) phase with a reduced but finite spin stiffness, while above a critical coupling, the system is in a dissipative phase characterized by a vanishing spin stiffness. We argue that the transport properties are also inhibited: The LL is a perfect conductor, while the dissipative phase displays finite resistivity. Our results show that the effect of the bath can be interpreted as annealed disorder-inducing signatures of localization.

Received 21 October 2022
Revised 9 March 2023
Accepted 21 March 2023

DOI:https://doi.org/10.1103/PhysRevB.107.165113

Physics Subject Headings (PhySH)

Dissipative dynamics

1-dimensional spin chains

Luttinger liquid model Spin chains

Condensed Matter, Materials & Applied PhysicsStatistical Physics & Thermodynamics

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

^*saptarshi.majumdar@universite-paris-saclay.fr

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Vol. 107, Iss. 16 — 15 April 2023

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Figure 1
Schematic representation of the microscopic system: A one-dimensional XXZ spin chain (blue color) with each spin coupled to its individual dissipative bath (red color). The baths are described by a collection of simple harmonic oscillators kept at zero temperature. The parameter $α$ is a measure of the coupling strength between the bath and the associated spin.
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Figure 2
Calculation of different quantities for $K = 0.75$ that characterizes Luttinger liquid (LL; $α = 1$ , top row) and dissipative phase ( $α = 8$ , bottom row). Blue and red points correspond to $L = β = 384$ and 128, respectively. The average is performed over 6000 to 12 000 configurations. (left) Due to symmetry, $π χ = K_{r} / u_{r}$ is equal to $K / u = 0.75$ for all values of $α$ and all length scales. (middle) For $α = 1, ω_{n} C (ω_{n})$ saturates to $K_{r} / 2 = 0.349$ as $ω_{n} \to 0$ ; whereas for $α = 8, \sqrt{ω_{n}} C (ω_{n})$ saturates to ${[K_{r} π / (8 α_{r} u_{r})]}^{1 / 2} = 0.392$ . The other fitting constants are $a_{1} = 0.2493$ and $a_{2} = 3.572$ . (right) For $α = 1, 〈 cos (ϕ) 〉$ decays as a power law, which allows us to extract $K_{r} = 0.694$ , consistent with the fit of $ω_{n} C (ω_{n})$ . For $α = 8$ , it saturates to a constant, as predicted by the variational ansatz (the fit gives $c_{1} = 0.62, c_{2} = 0.603$ , and $c_{3} = - 0.531$ ).
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Figure 3
Behavior of different renormalized parameters as a function of dissipative coupling $α$ . $K_{r} / u_{r}$ (red square points) remains constant and equal to $K / u = 0.75$ for all values of $α$ . $K_{r}$ , extracted from $〈 cos (ϕ) 〉$ analysis (purple circular points) agrees with the one from the $C (ω_{n})$ analysis (blue triangular points). It approaches $K_{c} = 0.5$ as $α$ reaches the critical point. The parameter $α_{r}$ (green circular points) starts to be defined in the dissipative phase and increases rapidly with increase in $α$ . This behavior of the parameter allows us to locate the different phases: For $α < 3$ , the system is in Luttinger liquid (LL) phase, whereas for $α > 4$ , the system is in dissipative phase. The phase transition takes place for $α \in (3, 4)$ .
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Figure 4
Calculation of different quantities for $K = 0.55$ that characterizes Luttinger liquid (LL; $α = 0.05$ , top row) and dissipative phase ( $α = 10$ , bottom row). Blue and red points correspond to $L = β = 384$ and 128, respectively. (left) Due to symmetry, $π χ = K_{r} / u_{r}$ is equal to $K / u = 0.55$ for all values of $α$ and all length scales. (middle) For $α = 0.05, ω_{n} C (ω_{n})$ saturates to $K_{r} / 2 = 0.273$ as $ω_{n} \to 0$ ; whereas for $α = 10, \sqrt{ω_{n}} C (ω_{n})$ saturates to ${[K_{r} π / (8 α_{r} u_{r})]}^{1 / 2} = 0.156$ . The other fitting constants are $a_{1} = 16.61$ and $a_{2} = 571.4$ . (right) For $α = 0.05, 〈 cos (ϕ) 〉$ decays as a power law, which allows us to extract $K_{r} = 0.546$ , consistent with the fit of $ω_{n} C (ω_{n})$ . For $α = 10$ , it saturates to a constant, as predicted by the variational ansatz (the fit gives $c_{1} = 0.788, c_{2} = 0.215$ , and $c_{3} = 0.012$ ). (bottom row) Behavior of different renormalized parameters as a function of dissipative coupling $α$ . $K_{r} / u_{r}$ (red square points) remains constant and equal to $K / u = 0.55$ for all values of $α$ . $K_{r}$ , extracted from $〈 cos (ϕ) 〉$ analysis (purple circular points) agrees with the one from the $C (ω)$ analysis (blue triangular points). It approaches $K_{c} = 0.5$ as $α$ reaches the critical point. The parameter $α_{r}$ (green circular points) starts to be defined in the dissipative phase and increases rapidly with increase in $α$ . This behavior of the parameter allows us to locate the different phases: For $α < 0.25$ , the system is in LL phase, whereas for $α > 2$ , the system is in dissipative phase. The phase transition takes place for $α \in (0.25, 2)$ . We believe $α = 0.25$ to be in the critical region as $K_{r}$ extracted from $〈 cos (ϕ) 〉$ is very close to $K_{c} = 0.5, α_{r}$ is very small, and $ω_{n} C (ω_{n})$ saturates for a long range of $ω_{n}$ but then starts decreasing.
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