First order transitions are rounded in finite cells or in infinite strips of finite width (cylinders). In this work we have considered the rounding of the jump of the magnetization of an Ising system at a fixed temperature T below Tc when the external field h goes from a small positive, to a small negative value. This rounding is well-known in a finite cell (block geometry) but the situation is more complicated, and more interesting, in a cylinder. Indeed it is known from the work of Privman and Fisher that these rounding effects are related to the behaviour of the correlation length along the axis of the cylinder which increases exponentially with the cross-sectional area A of the strip. It then follows that if one studies, for a given magnetization M, the external field h, this field approaches exponentially with A its bulk limit, but the coefficient inside the exponential is different if M is smaller, larger than or equal to the bulk spontaneous magnetization M0. We present numerical data for the two-dimensional case (or equivalently for the one-dimensional quantum model in a transverse field) which are in satisfactory agreement with the theory.
Boundary conditions monitor the finite-size dependence of scaling functions for the Ising model. We study the low-temperature phase for the extremely anisotropic limit, or quantum version of the 2D classical Ising model, by means of combined exact results and large-size numerical calculations. The mass gap (inverse of correlation length) is the suitable order parameter for the finite system, and its finite-size behavior is studied as a function of variable boundary conditions. We find that the well-known exponential convergence to zero of the mass gap is only valid in a limited range of parameters; it strikingly changes into a power law for antiperiodic boundary conditions. We suggest that this puzzling phenomenon is associated with topological excitations.
We study finite-size effects at first-order quantum transitions (FOQTs). We show that the low-energy properties show a finite-size scaling (FSS) behavior, the relevant scaling variable being the ratio of the energy associated with the perturbation driving the transition and the finite-size energy gap at the FOQT point. The size dependence of the scaling variable is therefore essentially determined by the size dependence of the gap at the transition, which in turn depends on the boundary conditions. Our results have broad validity and, in particular, apply to any FOQT characterized by the degeneracy and crossing of the two lowest-energy states in the infinite-volume limit. In this case, a phenomenological two-level theory provides exact expressions for the scaling functions. Numerical results for the quantum Ising chain in transverse and parallel magnetic fields support the FSS Ansätzes.
The critical properties of an infinitely long Ising strip with finite width L joined periodically or antiperiodically are investigated by analyzing the distribution of partition function zeros. For periodic boundary condition, the the leading finite-size scaling of partition function zeros and its corrections are given. For antiperiodic boundary condition, the critical point of 2D Ising transition is one of the loci of the zeros, and the associated non-analyticity is identified as a first-order phase transition. The exact amount of the latent heat released by the transition is 4/L.
Various lattice geometries and boundary conditions are used to investigate valence-bond-solid (VBS) ordering in the ground state of an
S=1/2 square-lattice quantum spin model---the
J-
Q model, in which four- or six-spin interactions
Q are added to the standard Heisenberg exchange
J. Ground-state results for finite systems (with up to thousands of spins) are obtained using an unbiased projector quantum Monte Carlo method. It is found that great care has to be taken when extrapolating the order parameter to infinite lattice size, in particular, in cylinder geometry. Even though strong VBS order exists in two dimensions, and is established clearly with increasing system size on $L\ifmmode\times\else\texttimes\fi{}L$ lattices (or ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}$ lattices with a fixed aspect ratio
Lx/Ly of order 1), only short-range VBS correlations are observed on long cylinders (when ${L}_{x}\ensuremath{\rightarrow}\ensuremath{\infty}$ at fixed
Ly). The correlation length increases with the cylinder width, until long-range order sets in at a ``critical'' width. This width is very large even when the 2D order is relatively strong. For example, for a system in which the order parameter is
70 of the largest possible value,
Ly=8 is required for ordering. Extrapolations of the VBS order parameter based on correlation functions (the square of the order parameter) for small $L\ifmmode\times\else\texttimes\fi{}L$ lattices can also be misleading. For a
20-ordered system, results for
L up to $\ensuremath{\approx}$20 appear to extrapolate clearly to a vanishing order parameter, while for larger lattices the scaling behavior crosses over and extrapolates to a nonzero value (with exponentially small finite-size corrections). The VBS order parameter also exhibits interesting edge effects related the known emergent U(1) symmetry close to a ``deconfined'' critical point, which, if not considered properly, can lead to wrong conclusions for the thermodynamic limit. The observed finite-size behavior for small $L\ifmmode\times\else\texttimes\fi{}L$ lattices and long cylinders is very similar to that predicted for a
Z2 spin liquid. The results therefore raise concerns about recent numerical work claiming
Z2 spin-liquid ground states in 2D frustrated quantum spin systems, in particular, the Heisenberg model with nearest and next-nearest-neighbor couplings. Based on the results presented so far, a VBS state in this system cannot be ruled out.
Using thermodynamic fluctuation theory, we study the finite-size rounding of anomalies occurring at first-order phase transitions of the corresponding infinite system. Explicit expressions for thermodynamic functions are derived both for "symmetric transitions" (such as the jump of the spontaneous magnetization in the Ising model from
+Msp to $\ensuremath{-}{M}_{\mathrm{sp}}$ as the field changes from
0+ to ${0}^{\ensuremath{-}}$) as well as for asymmetric cases, but restricting attention to (hyper)cubic system shapes. As an explicit example for the usefulness of these considerations in Monte Carlo simulations where it may be a problem to (i) locate a phase transition and (ii) distinguish first-order from second-order transitions, we present numerical results for the two-dimensional nearest-neighbor Ising ferromagnet in a field, both below the critical temperature
Tc and at
Tc. The numerical results are found to be in very good agreement with the phenomenological theory and it is shown that one may extract the magnitudes of jumps occurring at first-order phase transitions in a well-defined and accurate way.