Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2408.14474

Phases and phase transitions in a dimerized spin-$\mathbf{\frac{1}{2}}$ XXZ chain

Harsh Nigam, Ashirbad Padhan, Diptiman Sen, Tapan Mishra, Subhro Bhattacharjee

Published Aug 26, 2024• 0 citations
arXivHTMLPDF

Abstract

We revisit the phase diagram of the dimerized XXZ spin-$\frac{1}{2}$ chain with nearest-neighbor couplings which was studied numerically in Phys. Rev. B 106, L201106 (2022). The model has isotropic $XY$ couplings which have a uniform value and $ZZ$ couplings which have a dimerized form, with strengths $J_a$ and $J_b$ on alternate bonds. We find a rich phase diagram in the region of positive $J_a, ~J_b$. 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 IPM$_0$ and IPM$_π$, and a phase with Ising Neel order called IN; all these phases are gapped. The phases IPM$_0$ and IPM$_π$ are separated by a gapless phase transition line given by $0 \le J_a = J_b \le 1$ which is described by a conformal field theory with central charge $c=1$. There are two gapless phase transition lines separating IPM$_0$ from IN and IPM$_π$ from IN; these are described by conformal field theories with $c=\frac{1}{2}$ corresponding to quantum Ising transitions. The $c=1$ line bifurcates into the two $c=\frac{1}{2}$ lines at the point $J_a = J_b = 1$; the shape of the bifurcation is found analytically using RG calculations. A symmetry analysis shows that IPM$_0$ is a topologically trivial phase while IPM$_π$ is a time-reversal symmetry-protected topological phase (SPT) with spin-$\frac{1}{2}$ 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.

DOI
10.48550/arxiv.2408.14474

Related papers

Phases and phase transitions in a dimerized spin- 1 2 XXZ chain

Harsh Nigam, Ashirbad Padhan, Diptiman Sen +2 more

We revisit the phase diagram of the dimerized XXZ spin-$\frac{1}{2}$ chain with nearest-neighbor couplings which was studied numerically by Mondal et al. [Phys. Rev. B 106, L201106 (2022)]. The model has isotropic $XY$ couplings which have a uniform value and $ZZ$ couplings which have a dimerized form, with strengths ${J}_{a}$ and ${J}_{b}$ on alternate bonds. We find a rich phase diagram in the region of positive ${J}_{a},\phantom{\rule{0.28em}{0ex}}{J}_{b}$. 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 ${\mathrm{IPM}}_{0}$ and ${\mathrm{IPM}}_{\ensuremath{\pi}}$, and a phase with Ising Neel (IN) order; all these phases are gapped. The phases ${\mathrm{IPM}}_{0}$ 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 $c=1$. In contrast, there are two phase transition lines separating ${\mathrm{IPM}}_{0}$ or the ${\mathrm{IPM}}_{\ensuremath{\pi}}$ from IN that are described by $c=\frac{1}{2}$ CFTs and correspond to quantum Ising transitions. In fact, the $c=1$ line bifurcates into the two $c=\frac{1}{2}$ lines at the point ${J}_{a}={J}_{b}=1$; the shape of the bifurcation is found analytically using RG calculations. A symmetry analysis shows that ${\mathrm{IPM}}_{0}$ is a topologically trivial phase while ${\mathrm{IPM}}_{\ensuremath{\pi}}$ is a time-reversal symmetry-protected topological phase (SPT) with spin-$\frac{1}{2}$ 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.

Physical review. B./Physical review. B•May 16, 2025•4 citations

Quantum phase transitions beyond Landau-Ginzburg theory in one-dimensional space revisited

Christopher Mudry, Akira Furusaki, Takahiro Morimoto +1 more

The phase diagram of the quantum spin-$\frac{1}{2}$ antiferromagnetic ${J}_{1}^{\phantom{\rule{0.16em}{0ex}}}\text{\ensuremath{-}}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}$ XXZ chain was obtained by Haldane using bosonization techniques [Haldane, Phys. Rev. B 25, 4925 (1982); Phys. Rev. B 26, 5257 (1982)]. It supports three distinct phases for $0\ensuremath{\le}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}/{J}_{1}^{\phantom{\rule{0.16em}{0ex}}}<\frac{1}{2}$, i.e., a gapless algebraic spin-liquid phase, a gapped long-range ordered Neel phase, and a gapped long-range ordered dimer phase. Even though the Neel and dimer phases are not related hierarchically by a pattern of symmetry breaking, it was shown that they meet along a line of quantum critical points with a U(1) symmetry and central charge $c=1$. Here, we extend the analysis made by Haldane on the quantum spin-$\frac{1}{2}$ antiferromagnetic ${J}_{1}^{\phantom{\rule{0.16em}{0ex}}}\text{\ensuremath{-}}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}$ XYZ chain using both bosonization and numerical techniques. We show that there are three Neel phases and the dimer phase that are separated from each other by six planes of phase boundaries realizing Gaussian criticality when $0\ensuremath{\le}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}/{J}_{1}^{\phantom{\rule{0.16em}{0ex}}}<\frac{1}{2}$. We also show that each long-range ordered phase harbors topological point defects (domain walls) that are dual to those across the phase boundary in that a defect in one ordered phase locally binds the other type of order around its core. By using the bosonization approach, we identify the critical theory that describes simultaneous proliferation of these dual point defects, and show that it supports an emergent U(1) symmetry that originates from the discrete symmetries of the XYZ model. To confirm this numerically, we perform exact diagonalization and density-matrix renormalization-group calculations and show that the critical theory is characterized by the central charge $c=1$ with critical exponents that are consistent with those obtained from the bosonization approach. Furthermore, we generalize the field-theoretic description of direct continuous phase transition to higher dimensions, especially in $d=3$, by using a nonlinear sigma model (NLSM) with a topological term. In particular, we propose the $\ensuremath{\pi}$-flux phase on the cubic lattice with local quartic interactions as a platform for direct continuous phase transition and deconfined criticality. We discuss possible phase diagrams for the $\ensuremath{\pi}$-flux phase on the cubic lattice with these quartic interactions from the renormalization flow of NLSMs.

arXiv:1903.05646•Physical review. B./Physical review. B•May 29, 2019•65 citations

Symmetry-protected topological phases and transition in a frustrated spin- 12 XXZ chain

Hiroshi Ueda, Shigeki Onoda

A frustrated spin-$\frac{1}{2}$ XXZ zigzag chain relevant to ${\mathrm{Rb}}_{2}{\mathrm{Cu}}_{2}{\mathrm{Mo}}_{3}{\mathrm{O}}_{12}$ is revisited in the light of symmetry-protected topological (SPT) phases. Using a density-matrix renormalization group method for infinite systems, we identify projective representations for four distinct time-reversal invariant SPT phases; two parity-symmetric dimer phases near the Heisenberg and XX limits and two parity-broken vector-chiral (VC) dimer phases in between. A small bond alternation in the nearest-neighbor ferromagnetic exchange coupling induces a direct SPT transition between the two distinct VC dimer phases. It is also found numerically that two Berezinskii-Kosterlitz-Thouless transitions, which occur from the gapless to the two distinct gapped VC phases in the case of $\ensuremath{\delta}=0$, meet each other in the case of $\ensuremath{\delta}>0$ at a Gaussian criticality of the same Tomonaga-Luttinger parameter value as in the SU(2)-symmetric case.

arXiv:1406.5382•Physical Review B•Dec 15, 2014•13 citations

Vector-spin-chirality order in a dimerized frustrated spin- 12 chain

Hiroshi Ueda, Shigeki Onoda

A frustrated spin-$1/2$ XXZ chain model comprising a ferromagnetic nearest-neighbor coupling with the bond alternation, ${J}_{1}(1\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta})<0$, and an antiferromagnetic second-neighbor exchange coupling ${J}_{2}>0$ is studied at zero and weak magnetic fields by means of density-matrix renormalization-group calculations of order parameters, correlation functions, and the entanglement entropy, as well as an Abelian bosonization analysis. At zero magnetic field, the bond alternation $\ensuremath{\delta}>0$ suppresses the gapless phase characterized by a vector-chiral (VC) long-range order (LRO) and a quasi-LRO of an incommensurate spin spiral, whereas this phase occupies a large region in the space of ${J}_{1}/{J}_{2}$ and the easy-plane exchange anisotropy for $\ensuremath{\delta}=0$ [S. Furukawa et al., Phys. Rev. Lett. 105, 257205 (2010)]. Then, four gapped phases are found to appear as the exchange anisotropy varies from the SU(2)-symmetric case to the U(1)-symmetric case: the Haldane dimer (D${}_{+}$) phase with the same sign of the $x,y$- and $z$-component dimer order parameters, two VC dimer (VCD${}_{+}$/VCD${}_{\ensuremath{-}}$) phases with the sign of the $z$-component dimer order parameter being unaltered/reversed, and the even-parity dimer (D${}_{\ensuremath{-}}$) phase. At small magnetic fields, a field-induced ring-exchange interaction, which is proportional to a staggered scalar chirality and a magnetic flux penetrating the associated triangle, drives a transition from the D${}_{\ensuremath{-}}$ phase into a VC-Neel-dimer (VCND) phase, but not from the D${}_{+}$ phase. This VCND phase is stable up to the large magnetic field at which the Zeeman term closes the spin gap. A possible relevance to Rb${}_{2}$Cu${}_{2}$Mo${}_{3}$O${}_{12}$ is discussed.

arXiv:1308.5747•Physical Review B•Jan 10, 2014•11 citations

Ground-state phase diagram of a spin- 12 frustrated ferromagnetic XXZ chain: Haldane dimer phase and gapped/gapless chiral phases

Shunsuke Furukawa, Masahiro Sato, Shigeki Onoda +1 more

The ground-state phase diagram of a spin-$\frac{1}{2}$ XXZ chain with competing ferromagnetic nearest-neighbor (${J}_{1}<0$) and antiferromagnetic second-neighbor (${J}_{2}>0$) exchange couplings is studied by means of the infinite time evolving block decimation algorithm and effective field theories. For the SU(2)-symmetric (Heisenberg) case, we show that the nonmagnetic phase in the range $\ensuremath{-}4<{J}_{1}/{J}_{2}<0$ has a small but finite ferromagnetic dimer order. We argue that this spontaneous dimer order is associated with effective spin-1 degrees of freedom on dimerized bonds, which collectively form a valence bond solid state as in the spin-1 antiferromagnetic Heisenberg chain (the Haldane spin chain). We thus call this phase the Haldane dimer phase. With easy-plane anisotropy, the model exhibits a variety of phases including the vector chiral phase with gapless excitations and the even-parity dimer and N\'eel phases with gapped excitations, in addition to the Haldane dimer phase. Furthermore, we show the existence of gapped phases with coexisting orders in narrow regions that intervene between the gapless chiral phase and any one of Haldane dimer, even-parity dimer, and N\'eel phases. Possible implications for quasi-one-dimensional edge-sharing cuprates are discussed.

arXiv:1207.1059•Physical Review B•Sep 12, 2012•120 citations

Phase diagram of the interacting Majorana chain model

Armin Rahmani, Xiaoyu Zhu, Marcel Franz +1 more

The Hubbard and spinless fermion chains are paradigms of strongly correlated systems, very well understood using the Bethe ansatz, density matrix renormalization group (DMRG), and field theory/renormalization group (RG) methods. They have been applied to one-dimensional materials and have provided important insights for understanding higher-dimensional cases. Recently, an interacting fermion model has been introduced, with possible applications to topological materials. It has a single Majorana fermion operator on each lattice site and interactions with the shortest possible range that involve four sites. We present a thorough analysis of the phase diagram of this model in one dimension using field-theory/RG and DMRG methods. It includes a gapped supersymmetric region and a gapless phase with coexisting Luttinger liquid and Ising degrees of freedom. In addition to a first-order transition, three critical points occur: tricritical Ising, Lifshitz, and a generalization of the commensurate-incommensurate transition. We also survey various gapped phases of the system that arise when the translation symmetry is broken by dimerization and find both trivial and topological phases with 0, 1, and 2 Majorana zero modes bound to the edges of the chain with open boundary conditions.

arXiv:1505.03966•Physical Review B•Dec 14, 2015•117 citations
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 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.
The phase diagram of the quantum spin-12\frac{1}{2}21​ antiferromagnetic ${J}_{1}^{\phantom{\rule{0.16em}{0ex}}}\text{\ensuremath{-}}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}$ XXZ chain was obtained by Haldane using bosonization techniques [Haldane, Phys. Rev. B 25, 4925 (1982); Phys. Rev. B 26, 5257 (1982)]. It supports three distinct phases for $0\ensuremath{\le}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}/{J}_{1}^{\phantom{\rule{0.16em}{0ex}}}<\frac{1}{2}$, i.e., a gapless algebraic spin-liquid phase, a gapped long-range ordered Neel phase, and a gapped long-range ordered dimer phase. Even though the Neel and dimer phases are not related hierarchically by a pattern of symmetry breaking, it was shown that they meet along a line of quantum critical points with a U(1) symmetry and central charge c=1c=1c=1. Here, we extend the analysis made by Haldane on the quantum spin-12\frac{1}{2}21​ antiferromagnetic ${J}_{1}^{\phantom{\rule{0.16em}{0ex}}}\text{\ensuremath{-}}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}$ XYZ chain using both bosonization and numerical techniques. We show that there are three Neel phases and the dimer phase that are separated from each other by six planes of phase boundaries realizing Gaussian criticality when $0\ensuremath{\le}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}/{J}_{1}^{\phantom{\rule{0.16em}{0ex}}}<\frac{1}{2}$. We also show that each long-range ordered phase harbors topological point defects (domain walls) that are dual to those across the phase boundary in that a defect in one ordered phase locally binds the other type of order around its core. By using the bosonization approach, we identify the critical theory that describes simultaneous proliferation of these dual point defects, and show that it supports an emergent U(1) symmetry that originates from the discrete symmetries of the XYZ model. To confirm this numerically, we perform exact diagonalization and density-matrix renormalization-group calculations and show that the critical theory is characterized by the central charge c=1c=1c=1 with critical exponents that are consistent with those obtained from the bosonization approach. Furthermore, we generalize the field-theoretic description of direct continuous phase transition to higher dimensions, especially in d=3d=3d=3, by using a nonlinear sigma model (NLSM) with a topological term. In particular, we propose the $\ensuremath{\pi}$-flux phase on the cubic lattice with local quartic interactions as a platform for direct continuous phase transition and deconfined criticality. We discuss possible phase diagrams for the $\ensuremath{\pi}$-flux phase on the cubic lattice with these quartic interactions from the renormalization flow of NLSMs.
A frustrated spin-12\frac{1}{2}21​ XXZ zigzag chain relevant to Rb2Cu2Mo3O12{\mathrm{Rb}}_{2}{\mathrm{Cu}}_{2}{\mathrm{Mo}}_{3}{\mathrm{O}}_{12}Rb2​Cu2​Mo3​O12​ is revisited in the light of symmetry-protected topological (SPT) phases. Using a density-matrix renormalization group method for infinite systems, we identify projective representations for four distinct time-reversal invariant SPT phases; two parity-symmetric dimer phases near the Heisenberg and XX limits and two parity-broken vector-chiral (VC) dimer phases in between. A small bond alternation in the nearest-neighbor ferromagnetic exchange coupling induces a direct SPT transition between the two distinct VC dimer phases. It is also found numerically that two Berezinskii-Kosterlitz-Thouless transitions, which occur from the gapless to the two distinct gapped VC phases in the case of $\ensuremath{\delta}=0$, meet each other in the case of $\ensuremath{\delta}>0$ at a Gaussian criticality of the same Tomonaga-Luttinger parameter value as in the SU(2)-symmetric case.
A frustrated spin-1/21/21/2 XXZ chain model comprising a ferromagnetic nearest-neighbor coupling with the bond alternation, ${J}_{1}(1\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta})<0$, and an antiferromagnetic second-neighbor exchange coupling J2>0{J}_{2}>0J2​>0 is studied at zero and weak magnetic fields by means of density-matrix renormalization-group calculations of order parameters, correlation functions, and the entanglement entropy, as well as an Abelian bosonization analysis. At zero magnetic field, the bond alternation $\ensuremath{\delta}>0$ suppresses the gapless phase characterized by a vector-chiral (VC) long-range order (LRO) and a quasi-LRO of an incommensurate spin spiral, whereas this phase occupies a large region in the space of J1/J2{J}_{1}/{J}_{2}J1​/J2​ and the easy-plane exchange anisotropy for $\ensuremath{\delta}=0$ [S. Furukawa et al., Phys. Rev. Lett. 105, 257205 (2010)]. Then, four gapped phases are found to appear as the exchange anisotropy varies from the SU(2)-symmetric case to the U(1)-symmetric case: the Haldane dimer (D+{}_{+}+​) phase with the same sign of the x,yx,yx,y- and zzz-component dimer order parameters, two VC dimer (VCD+{}_{+}+​/VCD${}_{\ensuremath{-}}$) phases with the sign of the zzz-component dimer order parameter being unaltered/reversed, and the even-parity dimer (D${}_{\ensuremath{-}}$) phase. At small magnetic fields, a field-induced ring-exchange interaction, which is proportional to a staggered scalar chirality and a magnetic flux penetrating the associated triangle, drives a transition from the D${}_{\ensuremath{-}}$ phase into a VC-Neel-dimer (VCND) phase, but not from the D+{}_{+}+​ phase. This VCND phase is stable up to the large magnetic field at which the Zeeman term closes the spin gap. A possible relevance to Rb2{}_{2}2​Cu2{}_{2}2​Mo3{}_{3}3​O12{}_{12}12​ is discussed.
The ground-state phase diagram of a spin-12\frac{1}{2}21​ XXZ chain with competing ferromagnetic nearest-neighbor (J1<0{J}_{1}<0J1​<0) and antiferromagnetic second-neighbor (J2>0{J}_{2}>0J2​>0) exchange couplings is studied by means of the infinite time evolving block decimation algorithm and effective field theories. For the SU(2)-symmetric (Heisenberg) case, we show that the nonmagnetic phase in the range $\ensuremath{-}4<{J}_{1}/{J}_{2}<0$ has a small but finite ferromagnetic dimer order. We argue that this spontaneous dimer order is associated with effective spin-1 degrees of freedom on dimerized bonds, which collectively form a valence bond solid state as in the spin-1 antiferromagnetic Heisenberg chain (the Haldane spin chain). We thus call this phase the Haldane dimer phase. With easy-plane anisotropy, the model exhibits a variety of phases including the vector chiral phase with gapless excitations and the even-parity dimer and N\'eel phases with gapped excitations, in addition to the Haldane dimer phase. Furthermore, we show the existence of gapped phases with coexisting orders in narrow regions that intervene between the gapless chiral phase and any one of Haldane dimer, even-parity dimer, and N\'eel phases. Possible implications for quasi-one-dimensional edge-sharing cuprates are discussed.
The Hubbard and spinless fermion chains are paradigms of strongly correlated systems, very well understood using the Bethe ansatz, density matrix renormalization group (DMRG), and field theory/renormalization group (RG) methods. They have been applied to one-dimensional materials and have provided important insights for understanding higher-dimensional cases. Recently, an interacting fermion model has been introduced, with possible applications to topological materials. It has a single Majorana fermion operator on each lattice site and interactions with the shortest possible range that involve four sites. We present a thorough analysis of the phase diagram of this model in one dimension using field-theory/RG and DMRG methods. It includes a gapped supersymmetric region and a gapless phase with coexisting Luttinger liquid and Ising degrees of freedom. In addition to a first-order transition, three critical points occur: tricritical Ising, Lifshitz, and a generalization of the commensurate-incommensurate transition. We also survey various gapped phases of the system that arise when the translation symmetry is broken by dimerization and find both trivial and topological phases with 0, 1, and 2 Majorana zero modes bound to the edges of the chain with open boundary conditions.