🔬 condensed matter

Dissipation splits the Mott transition in one dimension

This paper demonstrates that local dissipation fundamentally alters the one-dimensional Mott transition by splitting it and inducing novel critical behavior with continuously varying exponents for bath exponents $s<3/2$ , a finding quantitatively supported by Monte Carlo simulations.

Original authors: Oscar Bouverot-Dupuis, Alberto Rosso, Laura Foini

Published 2026-07-02

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CC BY 4.0

Original authors: Oscar Bouverot-Dupuis, Alberto Rosso, Laura Foini

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a crowded dance floor where everyone is trying to move in perfect sync. In the world of quantum physics, this "dance floor" is a one-dimensional chain of particles (like electrons). Usually, these particles can either flow freely like a liquid (a Luttinger Liquid) or get stuck in a rigid, orderly pattern where they can't move at all (a Mott Insulator).

In a perfect, isolated world, switching from the flowing liquid to the stuck solid happens in one sudden jump. But this paper asks: What happens if the dance floor is noisy and the dancers are constantly being bumped by an invisible, jostling crowd?

Here is the story of what the researchers found, explained simply:

1. The Invisible Jostle (Dissipation)

The researchers imagined that every particle on the chain is connected to its own tiny, invisible "bath" of energy. Think of these baths as a crowd of people gently pushing and pulling the dancers.

The "Rhythm" of the Push: These pushes aren't random; they have a specific rhythm or "texture." The researchers call this texture $s$ .
- If the rhythm is "rough" (low $s$ ), the pushes are long-lasting and heavy.
- If the rhythm is "smooth" (high $s$ ), the pushes are quick and light.

2. The Big Discovery: A Middle Ground

The team found that if the "roughness" of the jostling is strong enough (specifically, if the rhythm parameter $s$ is less than 1.5), the transition from "flowing liquid" to "stuck solid" doesn't happen in one step. Instead, it splits into two steps, creating a mysterious middle phase called the Dissipative Phase (DP).

Think of it like this:

Step 1 (Liquid to Middle): The dancers stop flowing freely but haven't locked into a rigid grid yet. They are still moving around (the system is "compressible"), but they have lost their ability to glide smoothly across the floor (they have no "superfluid stiffness"). It's like a crowd that is still jostling but has lost its momentum to run.
Step 2 (Middle to Solid): As the jostling gets even stronger, the dancers finally freeze into their rigid, stuck positions.

3. The "Roughness" of the Path

To understand why this middle phase exists, the researchers looked at the path a single particle takes through time.

In a normal world: A particle's path is like a drunkard's walk—it wobbles and zig-zags randomly (like Brownian motion).
In this noisy world: The "jostling" acts like a stiffener.
- If the jostling is smooth (high $s$ ), the particle can still zig-zag freely. The transition is direct (Liquid $\to$ Solid).
- If the jostling is rough (low $s$ ), it forces the particle's path to become very straight and stiff. It can't wiggle anymore. This "straightening out" creates the new middle phase where the particles are stuck in a line but haven't fully frozen into a solid block yet.

4. The Two Scenarios

The paper maps out exactly what happens based on the "texture" of the noise ( $s$ ):

Scenario A: The Smooth Noise ( $s > 1.5$ )
The noise is too light to change the rules. The system behaves like a normal quantum system: it jumps straight from flowing liquid to stuck solid. The middle phase doesn't exist.
Scenario B: The Rough Noise ( $s < 1.5$ )
The noise is strong enough to break the rules.
- The Liquid $\to$ Middle transition: This happens like a classic "unzipping" of a zipper (a Berezinskii–Kosterlitz–Thouless transition).
- The Middle $\to$ Solid transition: This is a brand-new type of transition that has never been seen before in this context. The researchers calculated exactly how the particles behave here, finding that the "stiffness" of the transition changes continuously depending on how rough the noise is.

5. How They Knew It Was True

The researchers didn't just guess; they used two powerful tools:

Math: They simplified the complex quantum math into a model of a single "particle line" moving through time, allowing them to calculate the exact rules of this new phase.
Computer Simulations: They built massive virtual models of these particle chains and watched them evolve. The simulations confirmed that when they turned on the "rough" noise, the particles indeed got stuck in that strange, intermediate state before finally freezing.

Summary

In simple terms, this paper shows that noise can create new states of matter. If you shake a quantum system just right, you can trap it in a "limbo" state where it is neither a flowing liquid nor a frozen solid, but something entirely new. This happens because the noise changes the way particles wiggle through time, forcing them into a stiff, intermediate dance before they finally stop moving altogether.

Technical Summary: Dissipation Splits the Mott Transition in One Dimension

Problem Statement
The paper investigates how local dissipation modifies the one-dimensional (1D) Mott transition, a paradigmatic quantum phase transition separating a conducting Luttinger liquid (LL) from a Mott insulator (MI). While dissipation is traditionally viewed as a source of decoherence, coupling a quantum system to an environment can fundamentally alter its many-body properties. The specific challenge addressed is determining the fate of the LL-MI transition when the system is coupled to independent local baths at each lattice site, where the baths are coupled to the particle density. The nature of the bath is characterized by a low-energy spectral function $J(\omega) \sim |\omega|^s$ , where the exponent $s$ distinguishes sub-ohmic ( $s<1$ ), ohmic ( $s=1$ ), and super-ohmic ( $s>1$ ) regimes. Previous studies had suggested the existence of an intermediate dissipative phase (DP) but left the nature of the transition between this DP and the MI unexplored.

Methodology
The authors employ a multi-faceted approach combining analytical field theory, renormalization group (RG) analysis, and large-scale Monte Carlo simulations:

Microscopic to Field Theory Mapping: Starting from a 1D interacting fermionic model (equivalent to an XXZ spin chain via Jordan-Wigner transformation) coupled to harmonic oscillator baths, the authors integrate out the bath degrees of freedom exactly. Using bosonization, the system is mapped onto a $(1+1)$ -dimensional Euclidean field theory with long-range interactions in imaginary time. The resulting action contains competing terms: an umklapp term ( $g \cos 4\phi$ ) favoring the MI, a chemical potential term driving doping, and a non-local dissipative term arising from the bath coupling.
Perturbative Renormalization Group (RG): A perturbative RG analysis is performed in the canonical ensemble to study the flow of couplings ( $K, u, g, \alpha$ ). This analysis distinguishes between commensurate transitions (at half-filling, $\mu=0$ ) and incommensurate transitions (finite doping, $\mu \neq 0$ ).
Effective Critical Theory: To understand the non-perturbative DP-to-MI transition, the authors derive an effective $(0+1)$ -dimensional field theory. This theory describes the fluctuations of a single isolated kink (soliton) trajectory $X(\tau)$ in the limit of infinitesimal doping. The action for this trajectory includes a kinetic term and a non-local dissipative interaction.
Numerical Validation: State-of-the-art Monte Carlo simulations are performed on the discretized field theory. The authors analyze typical field configurations, compute observables such as compressibility ( $\kappa$ ), superfluid stiffness ( $\rho_s$ ), and doping ( $\delta\rho$ ), and perform finite-size scaling to extract critical exponents. The simulations validate both the perturbative RG predictions and the effective $(0+1)$ D theory.

Key Contributions and Results

Splitting of the Mott Transition: The central finding is that for bath exponents $s < 3/2$ $s < 3/2$ , the conventional direct LL-MI transition is replaced by a two-step process. An intermediate Dissipative Phase (DP) emerges between the LL and the MI.
- LL to DP: This transition is a Berezinskii–Kosterlitz–Thouless (BKT) transition. The critical Luttinger parameter is $K_c = (1-s)/2$ for $s<1$ and $K_c = 1/2$ for $s>1$ .
- DP to MI: This is a new type of commensurate-incommensurate (C/IC) transition. The DP is compressible and doped but possesses zero superfluid stiffness.
Universality Classes and Critical Exponents:
- For $s > 3/2$ , dissipation is irrelevant at the C/IC transition. The system reverts to the standard non-dissipative scenario with a direct LL-MI transition and standard critical exponents ( $\beta = \nu = 1/2, z=2$ ).
- For $1 < s < 3/2$ , the DP-MI transition is governed by a new universality class where critical exponents vary continuously with $s$ :
  $\beta = \nu = 1/z = s - 1$
  The roughness exponent of the kink trajectory is $\zeta = s-1$ .
- For $s < 1$ (sub-ohmic), the transition is governed by $\beta = \nu = 0$ and $z = \infty$ . The doping vanishes sharper than any power law, and the kink trajectory becomes completely flat (zero roughness).
Effective Theory Validation: The authors demonstrate that the critical properties of the DP-MI transition are governed by the fluctuations of a single kink worldline described by a $(0+1)$ D effective action. The roughness exponent $\zeta$ derived from this effective theory matches the exponents extracted from the full $(1+1)$ D Monte Carlo simulations, confirming that the transition is controlled by the "roughness" of the particle worldlines.
Phase Diagram: The paper provides a complete phase diagram in the plane of chemical potential ( $\mu$ ) and dissipation strength ( $\alpha$ ) for all $s$ , identifying the threshold $s_c = 3/2$ that separates the conventional Mott scenario from the dissipation-controlled scenario.

Significance
The paper claims to provide the first complete characterization of the dissipative Mott transition, resolving the nature of the previously unexplored DP-to-MI transition. By establishing that dissipation can qualitatively alter the nature of the Mott transition and generate novel critical behavior with continuously varying exponents, the work demonstrates that environmental coupling can stabilize new phases of matter (the DP) with no counterpart in isolated systems. The derivation of an effective $(0+1)$ D theory for the C/IC transition offers a powerful analytical tool for understanding non-equilibrium and dissipative quantum criticality in strongly correlated 1D systems. The results are quantitatively supported by Monte Carlo simulations, lending high confidence to the predicted universality classes and critical exponents.