Localisation Phase Diagram in 2D KT Systems

• The localisation phase diagram is a map of 2D systems that distinguishes quasi-long-range (algebraic) order from localized (exponential decay) regimes via vortex binding and unbinding.
• It highlights critical features such as an essential singularity in the correlation length and a universal jump in stiffness or conductance observed in quantum magnets and topological insulators.
• Finite-size scaling methods and renormalization group techniques provide actionable strategies for extracting phase boundaries and understanding dynamic localisation phenomena.

A localisation phase diagram characterizes the different regimes of spatial coherence and transport in 2D physical systems, particularly in the context of the Kosterlitz–Thouless (KT) universality. Such diagrams delineate phases with algebraic (quasi-long-range) order, exponential decay (localized or disordered), and the critical boundary—typically defined by an essential singularity in correlation length and a universal jump in a stiffness or conductance. The classic realisation is in 2D quantum magnets, superconductors, superfluids, and topological insulators, but the KT paradigm extends to Bose gases, photonic lattices, and disordered solids, as well as certain quantum gauge field theories.

1. Vortex Unbinding and Criticality in 2D XY-Type Systems

The archetype for localisation phase diagrams in KT physics is the 2D classical XY model, Bose superfluids, and analogous systems. At low temperature (or high stiffness), vortex–antivortex pairs remain bound, yielding algebraic correlations: $$G(r) = \langle e^{i[\theta(r)-\theta(0)]} \rangle \sim r^{-\eta(T)}$$ with $\eta(T)$ increasing with temperature/stiffness but bounded above by the universal value $\eta_c=1/4$ at the KT transition. Above $T_c$, unbound vortices proliferate, and correlations are exponentially screened: $G(r) \sim \exp(-r/\xi(T))$ The phase diagram typically exhibits a precise boundary, where the correlation length diverges with an essential singularity: $$\xi(T) \sim \xi_0 \exp\left[\frac{b}{\sqrt{|T/T_c - 1|}}\right]$$ and the superfluid stiffness drops discontinuously at the transition: $\rho_s(T_c^-) = \frac{2}{\pi} T_c$ The transition line separates algebraic (delocalized, quasi-ordered) and exponential (localized, disordered) regimes (2207.13748, Begun et al., 2024).

2. Localisation Phenomena in Topological Insulators and Disordered Systems

In 2D topological insulators, notably the quantum spin Hall phase, disorder induces a KT-type metal–insulator transition. The Thouless conductance $g$ is size-independent below a critical disorder strength $W_c$, signaling critically delocalized electron states. The associated beta function,

$\beta = \frac{d \ln g}{d \ln M}$

pins to zero in the "metallic" regime, and the localisation length diverges as

$$\xi(W) \propto \exp\left[\frac{\alpha}{\sqrt{W - W_c}}\right] \quad (W > W_c)$$

The microscopic mechanism is vortex–antivortex unbinding in local current loops, and the transition persists even if time-reversal symmetry is broken (Xu et al., 2012). The phase diagram thus separates metallic (critical KT) and insulating phases via the KT essential singularity.

3. Phase Diagram Extraction from Monte Carlo and Renormalization Group

In both classical and quantum simulations, the localisation phase boundary is mapped by finite-size scaling of stiffness, correlation length, and critical conductance:

• Helicity modulus crossing: $\rho_s(T)=2T/\pi$ at $T_c$ (Banerjee, 2023, Bietenholz et al., 2013)
• Heat capacity: Non-divergent, broad peaks decreasing with system size near $T_c$
• Binder cumulant: Merging plateaus signal KT-like criticality, opposed to Ising-type crossings (Baek et al., 2011)
• Critical scaling: $T_c(L) = T_c(\infty) + {A}/{(\ln L)^2}$, and exponential divergence of $\xi(t)$ above $T_c$ In disordered films, spatial coherence shrinks onto a ramified percolating network at $T_c$, which can be directly imaged by local probes (Erez et al., 2013, Erez et al., 2010).

Summary Table: Localisation Phase Boundaries in Prototypical KT Systems

Physical System	Order Parameter	Critical Boundary
2D XY model	Helicity modulus ($\rho_s$)	$\rho_s(T_c^-)=2T_c/\pi$, $\xi\propto\exp(b/|T-T_c|^{1/2})$
Topological insulator (QSH)	Thouless conductance ($g$)	$\xi(W)\propto\exp[\alpha/\sqrt{W-W_c}]$, $\beta=0$ for $g>g_c$
Bose superfluid	Superfluid density ($\rho_s$)	$\rho_s(T_c^-)=2T_c/\pi$, $\eta(T_c)=1/4$
Disordered superconductor	Edge–edge correlation	Percolation threshold $p_c=1/2$, vortex and bond percolation duality

4. Nonequilibrium and Dynamical Localisation Phase Diagrams

The KT transition can separate dynamical steady states with distinct localisation properties. In quench-driven Bose gases, the long-time coherence function $g^{(1)}(r)$ undergoes an algebraic-to-exponential crossover, defining the dynamic localisation boundary (Scoquart et al., 2022, Mathey et al., 2011). Real-time RG flow equations,

$$dy/d\ell = 2 (1-1/\tau) y\quad d\tau/d\ell = (64\pi^2 \alpha/\tau) y^2$$

track the evolution of correlation exponents; crossing $\eta=1/4$ signals dynamic vortex unbinding and loss of delocalization (Mathey et al., 2011). Similar transitions manifest in driven open quantum systems, with modified exponent bounds due to nonequilibrium noise (Dagvadorj et al., 2014).

5. Quantum KT Transitions and Localisation in Field Theories

Quantum generalizations involve transitions at $T=0$ driven by coupling constants rather than temperature. In compact U(1) gauge theories with diverging dielectric constant, the "quantum localisation" transition is induced by monopole (vortex) unbinding, with

$$\xi \sim \exp\left[b/|g-g_c|^{1/2}\right],\quad z\rightarrow\infty$$

and algebraic-to-exponential boundary in two-point correlations. The localisation phase diagram in these models is defined analogously by RG flow and a critical coupling, not temperature (Diamantini et al., 8 Oct 2025, Jensen et al., 2010).

6. Extensions: Anisotropy, Lattice Effects, and Symmetry Constraints

Variations in ground-state symmetry and anisotropy yield alternative localisation phase boundaries. For instance, anisotropic XY models or dipolar stripe systems feature direction-dependent stiffness tensors and phase boundaries set by geometric mean stiffness (Bombin et al., 2019, Lin et al., 2010), sometimes with multiple KT transitions involving half-vortex–half-dislocations. Ground-state symmetry on different lattices (e.g., honeycomb with six-fold symmetry) maps localisation boundaries onto KT universality, contrasted with Ising-type boundaries in fourfold symmetric systems (Baek et al., 2011).

7. Experimental and Numerical Mapping of Localisation Phase Diagrams

The boundaries are established by direct measurement and simulation:

• Interferometric measurement of $g^{(1)}(r)$ or fringe visibility in ultracold atomic or photonic gases
• Scaling of stiffness, correlation length, vortex unbinding, and conductivity in 2D superconductors (magnetic field tuning provides crossover from KT to BCS-like behaviour) (Sharma et al., 2024).
• Numerical RG, cluster algorithms, and finite-size extrapolations In tuneable systems, weak external fields or disorder can continuously tune localisation boundaries, revealing critical behaviour, crossover regimes, and essential singularities.

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A localisation phase diagram, when constructed for 2D systems with KT-like topological transitions, is fundamentally a map of phase-coherence, critical exponents, vortex binding/unbinding, and their dependence on microscopic couplings (temperature, disorder, interaction, field strength). It provides a unifying quantitative paradigm for distinguishing quasi-ordered (delocalized), localized, and critical phases across magnets, fluids, insulators, and correlated electron systems (2207.13748, Begun et al., 2024, Xu et al., 2012, Situ et al., 2013, Mathey et al., 2011, Erez et al., 2010, Erez et al., 2013, Sharma et al., 2024, Diamantini et al., 8 Oct 2025).