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arXiv:2312.16660 · Journal of High Energy Physics

Unveiling chiral states in the XXZ chain: finite-size scaling probing symmetry-enriched c = 1 conformal field theories

Chenan Wei, V. V. Mkhitaryan, Tigran Sedrakyan

Published Jun 19, 2024• 6 citations
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Abstract

A bstract We study the low-energy properties of the one-dimensional spin-1/2 XXZ chain with time-reversal symmetry-breaking pseudo-scalar chiral interaction and propose a phase diagram for the model. In the integrable case of the isotropic Heisenberg model with the chiral interaction, we employ the thermodynamic Bethe ansatz to find “chiralization”, the response of the ground state versus the strength of the pseudo-scalar chiral interaction of a chiral Heisenberg chain. Unlike the magnetization case, the chirality of the ground state remains zero until the transition point corresponding to critical coupling α c = 2 J/π with J being the antiferromagnetic spin-exchange interaction. The central-charge c = 1 conformal field theories (CFTs) describe the two phases with zero and finite chirality. We show for this particular case and conjecture more generally for similar phase transitions that the difference between two emergent CFTs with identical central charges lies in the symmetry of their ground state (lightest weight) primary fields, i.e., the two phases are symmetry-enriched CFTs. At finite but small temperatures, the non-chiral Heisenberg phase acquires a finite chirality that scales with the temperature quadratically. We show that the finite-size effect around the transition point probes the transition.

DOI
10.1007/jhep06(2024)125

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A bstract We study the low-energy properties of the one-dimensional spin-1/2 XXZ chain with time-reversal symmetry-breaking pseudo-scalar chiral interaction and propose a phase diagram for the model. In the integrable case of the isotropic Heisenberg model with the chiral interaction, we employ the thermodynamic Bethe ansatz to find “chiralization”, the response of the ground state versus the strength of the pseudo-scalar chiral interaction of a chiral Heisenberg chain. Unlike the magnetization case, the chirality of the ground state remains zero until the transition point corresponding to critical coupling α c = 2 J/π with J being the antiferromagnetic spin-exchange interaction. The central-charge c = 1 conformal field theories (CFTs) describe the two phases with zero and finite chirality. We show for this particular case and conjecture more generally for similar phase transitions that the difference between two emergent CFTs with identical central charges lies in the symmetry of their ground state (lightest weight) primary fields, i.e., the two phases are symmetry-enriched CFTs. At finite but small temperatures, the non-chiral Heisenberg phase acquires a finite chirality that scales with the temperature quadratically. We show that the finite-size effect around the transition point probes the transition.
We study the zero-temperature properties of a spin-1/21/21/2 Heisenberg chain with an additional three-site chiral term of strength ggg. The model is known to be integrable and previous work using Bethe ansatz has identified a phase transition from the usual critical state for g<gc=2/πg< g_c = 2/πg<gc​=2/π to a chiral phase for g>gcg>g_cg>gc​. Here we combine analytical and computational approaches to demonstrate that the transition point comprises an unusual Lifshitz transition with dynamical critical exponent z=3z=3z=3, and the chiral phase is a partially spin-polarized phase which spontaneously breaks SU(2) symmetry. We derive a strongly interacting chiral boson field theory for the critical point, and show that even its small-momentum, low-energy response is non-trivial. We also determine the universal properties of the chiral phase, and specifically show that it has a simultaneous quadratically dispersing ferromagnetic "magnon" mode coexisting with the power-law singularities and linearly dispersing excitations typical of a Luttinger liquid. Our conclusions are validated using numerics based on matrix product state methods and exact diagonalization.
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We revisit the phase diagram of the dimerized XXZ spin-12\frac{1}{2}21​ chain with nearest-neighbor couplings which was studied numerically by Mondal et al. [Phys. Rev. B 106, L201106 (2022)]. The model has isotropic XYXYXY couplings which have a uniform value and ZZZZZZ couplings which have a dimerized form, with strengths Ja{J}_{a}Ja​ and Jb{J}_{b}Jb​ on alternate bonds. We find a rich phase diagram in the region of positive Ja,Jb{J}_{a},\phantom{\rule{0.28em}{0ex}}{J}_{b}Ja​,Jb​. We provide a detailed understanding of the different phases and associated quantum phase transitions using a combination of mean-field theory, low-energy effective Hamiltonians, renormalization group calculations employing the technique of bosonization, and numerical calculations using the density-matrix renormalization group (DMRG) method. The phase diagram consists of two Ising paramagnetic phases called IPM0{\mathrm{IPM}}_{0}IPM0​ and ${\mathrm{IPM}}_{\ensuremath{\pi}}$, and a phase with Ising Neel (IN) order; all these phases are gapped. The phases IPM0{\mathrm{IPM}}_{0}IPM0​ and ${\mathrm{IPM}}_{\ensuremath{\pi}}$ are separated by a gapless phase transition line $0\ensuremath{\le}{J}_{a}={J}_{b}\ensuremath{\le}1$---the well-known Tomonaga Luttinger liquid captured by a conformal field theory (CFT) with central charge c=1c=1c=1. In contrast, there are two phase transition lines separating IPM0{\mathrm{IPM}}_{0}IPM0​ or the ${\mathrm{IPM}}_{\ensuremath{\pi}}$ from IN that are described by c=12c=\frac{1}{2}c=21​ CFTs and correspond to quantum Ising transitions. In fact, the c=1c=1c=1 line bifurcates into the two c=12c=\frac{1}{2}c=21​ lines at the point Ja=Jb=1{J}_{a}={J}_{b}=1Ja​=Jb​=1; the shape of the bifurcation is found analytically using RG calculations. A symmetry analysis shows that IPM0{\mathrm{IPM}}_{0}IPM0​ is a topologically trivial phase while ${\mathrm{IPM}}_{\ensuremath{\pi}}$ is a time-reversal symmetry-protected topological phase (SPT) with spin-12\frac{1}{2}21​ states at the two ends of an open system. A gap opens in the low-energy spectrum whenever one moves away from one of the phase transition lines; the scaling of the gap with the distance from a transition line is found analytically using the RG method. The numerical results obtained by the DMRG method are in good agreement with the analytical results. Further, we show that useful insights into the nature of the phase diagram of the above Hamiltonian, particularly the SPT as well as the phase transitions, is obtained via Kramers-Wannier duality to a deformed quantum Ashkin-Teller model. Finally, we propose experimental platforms for testing our results.
Phases and phase transitions in a dimerized spin-12\mathbf{\frac{1}{2}}21​ XXZ chain
We revisit the phase diagram of the dimerized XXZ spin-12\frac{1}{2}21​ chain with nearest-neighbor couplings which was studied numerically in Phys. Rev. B 106, L201106 (2022). The model has isotropic XYXYXY couplings which have a uniform value and ZZZZZZ couplings which have a dimerized form, with strengths JaJ_aJa​ and JbJ_bJb​ on alternate bonds. We find a rich phase diagram in the region of positive Ja, JbJ_a, ~J_bJa​, Jb​. We provide a detailed understanding of the different phases and associated quantum phase transitions using a combination of mean-field theory, low-energy effective Hamiltonians, renormalization group calculations employing the technique of bosonization, and numerical calculations using the density-matrix renormalization group (DMRG) method. The phase diagram consists of two Ising paramagnetic phases called IPM0_00​ and IPMπ_ππ​, and a phase with Ising Neel order called IN; all these phases are gapped. The phases IPM0_00​ and IPMπ_ππ​ are separated by a gapless phase transition line given by 0≤Ja=Jb≤10 \le J_a = J_b \le 10≤Ja​=Jb​≤1 which is described by a conformal field theory with central charge c=1c=1c=1. There are two gapless phase transition lines separating IPM0_00​ from IN and IPMπ_ππ​ from IN; these are described by conformal field theories with c=12c=\frac{1}{2}c=21​ corresponding to quantum Ising transitions. The c=1c=1c=1 line bifurcates into the two c=12c=\frac{1}{2}c=21​ lines at the point Ja=Jb=1J_a = J_b = 1Ja​=Jb​=1; the shape of the bifurcation is found analytically using RG calculations. A symmetry analysis shows that IPM0_00​ is a topologically trivial phase while IPMπ_ππ​ is a time-reversal symmetry-protected topological phase (SPT) with spin-12\frac{1}{2}21​ states at the two ends of an open system. The numerical results obtained by the DMRG method are in good agreement with the analytical results. Finally we propose experimental platforms for testing our results.
We study zero temperature properties of a system of two coupled quantum spin chains subject to fields explicitly breaking time reversal symmetry and parity. A suitable choice of the strength of these fields gives a model soluble by Bethe ansatz methods which allows us to determine the complete magnetic phase diagram of the system and the asymptotics of correlation functions from the finite size spectrum. The chiral properties of the system for both the integrable and the nonintegrable case are studied using numerical techniques.
The method of de Vega and Woynarovitch (1985) is used to calculate finite-size corrections to the ground-state energy in different sectors for the XXZ Heisenberg chain. Finite-size scaling amplitudes and correction-to-scaling exponents in the critical region are derived. Using conformal invariance, a scaling dimension x=( pi - gamma )/2 pi is extracted corresponding to the electric field operator in the 8-vertex model: this confirms a conjecture of Baxter and Kelland (1974). Finite-size scaling properties near the Kosterlitz-Thouless critical point Delta =-1 are discussed.