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arXiv:2509.22226 · Journal of Statistical Mechanics Theory and Experiment

The quantum XY chain with boundary fields: finite-size gap and phase behavior

Aldo Coraggio, Andrea Pelissetto C

Published Feb 3, 2025• 1 citation
arXivHTMLPDF

Abstract

Abstract We present a detailed study of the finite-size one-dimensional quantum XY chain in a transverse field in the presence of boundary fields coupled with the order-parameter spin operator. We consider fields located at the chain boundaries that have the same strength and that are oppositely aligned. We derive exact expressions for the gap Δ as a function of the model parameters for large values of the chain length L. These results allow us to characterize the nature of the ordered phases of the model. We find a magnetic (M) phase ( Δ ∼ e − a L ), a magnetic-incommensurate (MI) phase ( Δ ∼ e − a L f MI ( L ) ), a kink (K) phase ( Δ ∼ L − 2 ) and a kink-incommensurate (KI) phase ( Δ ∼ L − 2 f KI ( L ) ); f MI ( L ) and f KI ( L ) are bounded oscillating functions of L. We also analyze the behavior along the phase boundaries. In particular, we characterize the universal crossover behavior across the K-KI phase boundary. On this boundary, the dynamic critical exponent is z = 4, i.e. Δ ∼ L − 4 for large values of L.

DOI
10.1088/1742-5468/ad8bb7

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Abstract We present a detailed study of the finite-size one-dimensional quantum XY chain in a transverse field in the presence of boundary fields coupled with the order-parameter spin operator. We consider fields located at the chain boundaries that have the same strength and that are oppositely aligned. We derive exact expressions for the gap Δ as a function of the model parameters for large values of the chain length L. These results allow us to characterize the nature of the ordered phases of the model. We find a magnetic (M) phase ( Δ ∼ e − a L ), a magnetic-incommensurate (MI) phase ( Δ ∼ e − a L f MI ( L ) ), a kink (K) phase ( Δ ∼ L − 2 ) and a kink-incommensurate (KI) phase ( Δ ∼ L − 2 f KI ( L ) ); f MI ( L ) and f KI ( L ) are bounded oscillating functions of L. We also analyze the behavior along the phase boundaries. In particular, we characterize the universal crossover behavior across the K-KI phase boundary. On this boundary, the dynamic critical exponent is z = 4, i.e. Δ ∼ L − 4 for large values of L.
We present a detailed study of the finite one-dimensional quantum Ising chain in a transverse field in the presence of boundary magnetic fields coupled with the order-parameter spin operator. We consider two magnetic fields located at the boundaries of the chain that have the same strength and that are aligned in the same or in the opposite direction. We derive analytic expressions for the gap in all phases for large values of the chain length L , as a function of the boundary field strength. We also investigate the behaviour of the chain in the quantum ferromagnetic phase for oppositely aligned fields, focusing on the magnet-to-kink transition that occurs at a finite value of the magnetic field strength. At this transition we compute analytically the finite-size crossover functions for the gap, the magnetisation profile, the two-point correlation function, and the density of fermionic modes. As the magnet-to-kink transition is equivalent to the wetting transition in two-dimensional classical Ising models, our results provide new analytic predictions for the finite-size behaviour of Ising systems in a strip geometry at this transition.
We systematically characterize a previously unidentified dynamical phase boundary alpha*(h/J) in the one-dimensional transverse-field Ising model with power-law decaying interactions ~1/r^alpha. Three main results are established: (i) the dominant oscillation frequency of the mean magnetization satisfies omega = Delta/2pi universally across all interaction ranges alpha tested; (ii) the finite-size scaling of the energy gap changes qualitatively at alpha*(h/J), separating exponential (long-range) from power-law (short-range effective) scaling; (iii) the boundary follows alpha*(h/J) = (c*J/h)^3 + b with c ~ e/2 and b ~ 5/3, and is characterized by the effective crossover condition Jeff(alpha*) ~ (h/J)/sqrt(3) + 1/2, suggesting the crossover is controlled by a balance between collective interaction strength and transverse field energy scale. Results are obtained via exact diagonalization (QuTiP, N=4-12) and symbolic regression (PySR).
We obtain the steady-state phase diagram of a transverse-field XY spin chain coupled at its ends to magnetic reservoirs held at different magnetic potentials. In the long-time limit, the magnetization bias across the system generates a current-carrying nonequilibrium steady state. We characterize the different nonequilibrium phases as functions of the chain's parameters and magnetic potentials, in terms of their correlation functions and entanglement content. The mixed-order transition, previously observed for the case of a transverse-field Ising chain, is established to emerge as a generic feature of a wider class of out-of-equilibrium problems. The critical exponents associated with this universality class are determined analytically. Results are also contrasted with those obtained in the limit of Markovian reservoirs. Our findings should prove helpful in establishing the properties of nonequilibrium phases and phase transitions of extended open quantum systems.
Abstract In this study, a spin-1/2 extended anisotropic XY chain has been introduced in which both time reversal and SU(2) symmetries are broken but Z 2 symmetry is preserved. Magnetic and topological phase diagrams in the parameter space have been drawn in the presence of transverse magnetic field. Entanglement measures like mutual information and quantum discord are also evaluated and it indicates that these transitions are second order in nature. Quantum phase transition is noted at zero magnetic field, as well as magnetic long range order is found to withstand magnetic field of any strength. Exact analytic results for spin-spin correlation functions have been obtained in terms of Jordan-Wigner fermionization. Existence of long range magnetic order has been investigated numerically by finding correlation functions as well as the Binder cumulant in the ground state. Dispersion relation, ground state energy, and energy gap are obtained analytically. In order to find the topologically nontrivial phase, sign of Pfaffian invariant and value of winding number have been evaluated. Both magnetic and topological phases are robust against the magnetic field and found to move coercively in the parameter space with the variation of its strength. Long range orders along two orthogonal directions and two different topological phases are found and their one-to-one correspondence has been found. Finally casting the spinless fermions onto Majorana fermions, properties of zero energy edge states are studied. Three different kinds of Majorana pairings are noted. In the trivial phase, next-nearest-neighbor Majorana pairing is found, whereas two different types of nearest-neighbor Majorana pairings are identified in the topological superconducting phase.
I study the universal finite-size scaling function for the lowest gap of the quantum Ising chain with a one-parameter family of ``defect'' boundary conditions, which includes periodic, open, and antiperiodic boundary conditions as special cases. The universal behavior can be described by the Majorana fermion field theory in 1+11+11+1 dimensions, with the mass proportional to the deviation from the critical point. Although the field theory appears to be symmetric with respect to the inversion of the mass (Kramers-Wannier duality), the actual gap is asymmetric, reflecting the spontaneous symmetry breaking in the ordered phase which leads to the two-fold ground-state degeneracy in the thermodynamic limit. The asymptotic ground-state degeneracy in the ordered phase is realized by (i) formation of a bound state at the defect (except for the periodic/antiperiodic boundary condition) and (ii) effective reversal of the fermion number parity in one of the sectors (except for the open boundary condition), resulting in a rather nontrivial crossover ``phase'' diagram in the space of the boundary condition (defect strength) and mass.
We calculate the entanglement and the universal boundary entropy (BE) in critical quantum spin chains, such as the transverse field Ising chain and the XXZ chain, with arbitrary direction of the boundary magnetic field (ADBMF). We determine the boundary universality class that an ADBMF induces. In particular, we show that the induced boundary conformal field theory depends on the point on the Bloch sphere where the boundary magnetic field directs. We show that the classification of the directions boils down to the simple fact that the boundary field breaks the bulk symmetry or does not. We present a procedure to estimate the universal BE, based on the finite-size corrections of the entanglement entropy, which apply to ADBMF. To calculate the universal BE in the XXZ chain, we use the density matrix renormalization group. The transverse field XY chain with ADBMF after Jordan-Wigner transformation is not a quadratic free fermion Hamiltonian. We map this model to a quadratic free fermion chain by introducing two extra ancillary spins coupled to the main chain at the boundaries, which makes the problem integrable. The eigenstates of the transverse field XY chain can be obtained by proper projection in the enlarged chain. Using this mapping, we are able to calculate the entanglement entropy of the transverse field XY chain using the usual correlation matrix technique up to relatively large sizes.