Unwrapping phase fluctuations in one dimension

Correlation functions in one-dimensional complex scalar field theory provide a toy model for phase fluctuations, sign problems, and signal-to-noise problems in lattice field theory. Phase unwrapping techniques from signal processing are applied to lattice field theory in order to map compact random phases to noncompact random variables that can be numerically sampled without sign or signal-to-noise problems. A cumulant expansion can be used to reconstruct average correlation functions from moments of unwrapped phases, but points where the field magnitude fluctuates close to zero lead to ambiguities in the definition of the unwrapped phase and significant noise at higher orders in the cumulant expansion. Phase unwrapping algorithms that average fluctuations over physical length scales improve, but do not completely resolve, these issues in one dimension. Similar issues are seen in other applications of phase unwrapping, where they are found to be more tractable in higher dimensions.Comment: 14 pages, 7 figures. arXiv admin note: text overlap with arXiv:1806.0183

Phase transitions in one dimension and less

Phase transitions can occur in one-dimensional classical statistical mechanics at non-zero temperature when the number of components N of the spin is infinite. We show how to solve such magnets in one dimension for any N, and how the phase transition develops at N = infinity. We discuss SU(N) and Sp(N) magnets, where the transition is second-order. In the new high-temperature phase, the correlation length is zero. We also show that for the SU(N) magnet on exactly three sites with periodic boundary conditions, the transition becomes first order.Comment: 16 pages, 1 figur

Scattering phase shifts in quasi-one-dimension

Scattering of an electron in quasi-one dimensional quantum wires have many unusual features, not found in one, two or three dimensions. In this work we analyze the scattering phase shifts due to an impurity in a multi-channel quantum wire with special emphasis on negative slopes in the scattering phase shift versus incident energy curves and the Wigner delay time. Although at first sight, the large number of scattering matrix elements show phase shifts of different character and nature, it is possible to see some pattern and understand these features. The behavior of scattering phase shifts in one-dimension can be seen as a special case of these features observed in quasi-one-dimensions. The negative slopes can occur at any arbitrary energy and Friedel sum rule is completely violated in quasi-one-dimension at any arbitrary energy and any arbitrary regime. This is in contrast to one, two or three dimensions where such negative slopes and violation of Friedel sum rule happen only at low energy where the incident electron feels the potential very strongly (i.e., there is a very well defined regime, the WKB regime, where FSR works very well). There are some novel behavior of scattering phase shifts at the critical energies where $S$-matrix changes dimension.Comment: Minor corrections mad

Phase Transitions in one-dimensional nonequilibrium systems

The phenomenon of phase transitions in one-dimensional systems is discussed. Equilibrium systems are reviewed and some properties of an energy function which may allow phase transitions and phase ordering in one dimension are identified. We then give an overview of the one-dimensional phase transitions which we have been studied in nonequilibrium systems. A particularly simple model, the zero-range process, for which the steady state is know exactly as a product measure, is discussed in some detail. Generalisations of the model, for which a product measure still holds, are also discussed. We analyse in detail a condensation phase transition in the model and show how conditions under which it may occur may be related to the existence of an effective long-range energy function. Although the zero-range process is not well known within the physics community, several nonequilibrium models have been proposed that are examples of a zero-range process, or closely related to it, and we review these applications here.Comment: latex, 28 pages, review article; references update

The Deconfinement Phase Transition in One-Flavour QCD

We present a study of the deconfinement phase transition of one-flavour QCD, using the multiboson algorithm. The mass of the Wilson fermions relevant for this study is moderately large and the non-hermitian multiboson method is a superior simulation algorithm. Finite size scaling is studied on lattices of size $8^3\times 4$, $12^3\times 4$ and $16^3\times 4$. The behaviours of the peak of the Polyakov loop susceptibility, the deconfinement ratio and the distribution of the norm of the Polyakov loop are all characteristic of a first-order phase transition for heavy quarks. As the quark mass decreases, the first-order transition gets weaker and turns into a crossover. To investigate finite size scaling on larger spatial lattices we use an effective action in the same universality class as QCD. This effective action is constructed by replacing the fermionic determinant with the Polyakov loop identified as the most relevant Z(3) symmetry breaking term. Higher-order effects are incorporated in an effective Z(3)-breaking field, $h$, which couples to the Polyakov loop. Finite size scaling determines the value of $h$ where the first order transition ends. Our analysis at the end - point, $h_{ep}$, indicates that the effective model and thus QCD is consistent with the universality class of the three dimensional Ising model. Matching the field strength at the end point, $h_{ep}$, to the $\kappa$ values used in the dynamical quark simulations we estimate the end point, $\kappa_{ep}$, of the first-order phase transition. We find $\kappa_{ep}\sim 0.08$ which corresponds to a quark mass of about 1.4 GeV .Comment: LaTex, 25 pages, 18 figure

Haldane phase in one-dimensional topological Kondo insulators

We investigate the groundstate properties of a recently proposed model for a topological Kondo insulator in one dimension (i.e., the $p$-wave Kondo-Heisenberg lattice model) by means of the Density Matrix Renormalization Group method. The non-standard Kondo interaction in this model is different from the usual (i.e., local) Kondo interaction in that the localized spins couple to the "$p$-wave" spin density of conduction electrons, inducing a topologically non-trivial insulating groundstate. Based on the analysis of the charge- and spin-excitation gaps, the string order parameter, and the spin profile in the groundstate, we show that, at half-filling and low energies, the system is in the Haldane phase and hosts topologically protected spin-1/2 end-states. Beyond its intrinsic interest as a useful "toy-model" to understand the effects of strong correlations on topological insulators, we show that the $p$-wave Kondo-Heisenberg model can be implemented in $p-$band optical lattices loaded with ultra-cold Fermi gases.Comment: 8 pages, 4 figures, 1 appendi

Topological phase in one-dimensional interacting fermion system

We study a one-dimensional interacting topological model by means of exact diagonalization method. The topological properties are firstly examined with the existence of the edge states at half-filling. We find that the topological phases are not only robust to small repulsive interactions but also are stabilized by small attractive interactions, and also finite repulsive interaction can drive a topological non-trivial phase into a trivial one while the attractive interaction can drive a trivial phase into a non-trivial one. Next we calculate the Berry phase and parity of the bulk system and find that they are equivalent in characterizing the topological phases. With them we obtain the critical interaction strengths and construct part of the phase diagram in the parameters space. Finally we discuss the effective Hamiltonian at large-U limit and provide additional understanding of the numerical results. Our these results could be realized experimentally using cold atoms trapped in the 1D optical lattice.Comment: 7 pages, 5 figures; revised version, references added, Accepted for publication in Physical Review