The generic Mott transition in the sine-Gordon model through an embedded worm algorithm

TL;DR

This work targets the generic Mott transition in one-dimensional quantum systems described by the tilted sine-Gordon model. It introduces the Smooth Worm (SmoWo) Monte Carlo algorithm, which combines worm updates with extended event-chain moves in a current-fluctuation representation to smoothly connect topological sectors and sample large systems. The study reveals a clear phase separation between a Luttinger liquid and a Mott insulator, providing precise critical properties (e.g., $z=2$, $\nu\approx0.5$, $\eta\approx3$) and identifying Hubbard-band structures within the MI. The SmoWo method significantly improves sampling efficiency from exponential to polynomial scaling, enabling detailed finite-size scaling analyses and broader applicability to other bosonized 1D models.

Abstract

The generic Mott transition in one-dimensional quantum systems can be described by the sine-Gordon model with a tilt via bosonization. Because the configuration space of the sine-Gordon model separates into distinct topological sectors, standard local Monte Carlo schemes are limited to very small system sizes. To overcome this limitation, we introduce the smooth worm (SmoWo) Monte Carlo algorithm which enlarges the configuration space to allow smooth transitions between topological sectors. The method combines worm updates with event-chain Monte Carlo moves. We explicitly prove its validity and quantify its performance. Thanks to the substantial acceleration achieved by the SmoWo algorithm, we are able to simulate large system sizes, providing a precise picture of the different phases and critical behaviour of the sine-Gordon model.

Figures (20)

• Figure 1: Top: In the bosonized picture, particles (or up spins) sitting above the density $\rho_0$ create three consecutive particles and are represented by kinks of height $-\pi$. An extra hole (or spin down) creates a $+\pi$-kink. Middle: With a background density of 1 particle every two sites, a particle (hole) can decay into two half-particles (half-holes) corresponding to two consecutive particles (holes). This creates kinks of amplitude $\pm \pi/2$. Bottom: Due to their larger entropy, half-excitations are expected to be much more frequent than full-excitations.
• Figure 2: Typical field configurations contributing to the path integral \ref{['eq:partition_function']} on a lattice of size $256 \times 256$. All kinks are of amplitude $\pi/2$. Counting the number of space and time kinks shows that figure $({\rm i})$ has $N_x=2$, $N_\tau=0$, $({\rm ii})$ has $N_x=2$, $N_\tau=-1$, and $({\rm iii})$ has $N_x=8$, $N_\tau=0$. Since kinks identify as the worldlines of half particles, $N_x$ counts the number of full particles, and $N_\tau$ encodes the average (imaginary-time) particle current $j(x,\tau)=\frac{1}{\pi}\partial_\tau \phi(x,\tau)$. The quantization of $N_\tau$ can then be seen as arising from the indistinguishability of quantum particles (kinks). Indeed, the trace in $Z={\rm Tr}\, e^{-\beta \hat{H}}$ ensures the particles return to their original position up to a permutation.
• Figure 3: Current field $\vec{J}$ obtained from the field depicted in Fig. \ref{['fig:field_examples']}$({\rm i })$ by using Eqs. (\ref{['eq:n_f_def']},\ref{['eq:current_def']}). The current traces out the (oriented) topographic lines of the field $\phi$. For the sake of clarity, we do not display the very small current loops which would otherwise cover up the picture.
• Figure 4: The fields $\phi_i$, $n_i$, $f_i$ are all defined on the 2D lattice drawn in dashed lines. Its sites are labelled by $i$ and its plaquettes by $p$. The current field $\vec{J}_p$ (in solid lines) introduced in \ref{['eq:current_def']} lives on the edges of the dual lattice.
• Figure 5: Top: A field configuration $\phi$ with a worm ranging from $p_t$ to $p_h$ and obtained for a system of size $L=\beta=256$. The red lines are artificial discontinuities of amplitude $\pi$ which are needed to represent the field $\phi$. Bottom: Current field $\vec{J}$ associated to $\phi$. It is divergenceless everywhere except at $p_h$ and $p_t$. For the sake of clarity, we do not display the very small current loops which would otherwise cover up the picture.