Using a recently proposed perturbative numerical renormalization-group algorithm, we explore the connection between quantum criticality and the emergence of Luttinger liquid physics in $t-J$ chains coupled by frustrated interactions. This study is built on an earlier finding that at the maximally frustrated point, the ground state of weakly-coupled Heisenberg chains is disordered, the transverse exchanges being irrelevant. This result is extended here to transverse couplings up to $J_{\perp}=0.6$ , and we argue that it may also be valid at the isotropic point. A finite size analysis of coupled Heisenberg chains in the vicinity of the maximally frustrated point confirms that the transverse spin-spin correlations decay exponentially while the longitudinal ones revert to those of decoupled chains. We find that this behavior persists upon moderate hole doping $x \alt 0.75$. For larger doping, the frustration becomes inactive and the quantum critical point is suppressed.Using a recently proposed perturbative numerical renormalization-group algorithm, we explore the connection between quantum criticality and the emergence of Luttinger liquid physics in $t-J$ chains coupled by frustrated interactions. This study is built on an earlier finding that at the maximally frustrated point, the ground state of weakly-coupled Heisenberg chains is disordered, the transverse exchanges being irrelevant. This result is extended here to transverse couplings up to $J_{\perp}=0.6$, and we argue that it may also be valid at the isotropic point. A finite size analysis of coupled Heisenberg chains in the vicinity of the maximally frustrated point confirms that the transverse spin-spin correlations decay exponentially while the longitudinal ones revert to those of decoupled chains. We find that this behavior persists upon moderate hole doping $x \alt 0.75$. For larger doping, the frustration becomes inactive and the quantum critical point is suppressed.

arXiv:cond-mat/0504306Apr 13, 20050 citations

Néel and disordered phases of coupled Heisenberg chains withS= ½ toS= 4

S. Moukouri

We use the two-step density-matrix renormalization group method to study the effects of frustration in Heisenberg models for S = ½ to 4 in a two-dimensional anisotropic lattice. We find that as for S = ½ studied previously, the system is made up of nearly disconnected chains at the maximally frustrated point, J d / J ⊥ = 0.5, i.e., the transverse spin–spin correlations decay exponentially. This leads to the following consequences: (i) all half-integer spins systems are gapless, behaving like a sliding Luttinger liquid as for S = ½; (ii) for integer spins, there is an intermediate disordered phase with a spin gap, with the width of the disordered state roughly proportional to the 1D Haldane gap.We use the two-step density-matrix renormalization group method to study the effects of frustration in Heisenberg models for S = ½ to 4 in a two-dimensional anisotropic lattice. We find that as for S = ½ studied previously, the system is made up of nearly disconnected chains at the maximally frustrated point, J d / J ⊥ = 0.5, i.e., the transverse spin–spin correlations decay exponentially. This leads to the following consequences: (i) all half-integer spins systems are gapless, behaving like a sliding Luttinger liquid as for S = ½; (ii) for integer spins, there is an intermediate disordered phase with a spin gap, with the width of the disordered state roughly proportional to the 1D Haldane gap.

arXiv:cond-mat/0510552Journal of Statistical Mechanics Theory and ExperimentFeb 6, 200611 citations

Entanglement-entropy analysis of critical and topological quantum phases in a frustrated spin-1/2 Heisenberg ladder

Azadeh Ghannadan, Hana Vargová, Jozef Genzor +2 more

We investigate the ground-state phase diagram of a frustrated spin-1/2 Heisenberg ladder in the transverse magnetic field with an anisotropic inter-rung exchange coupling $α$ . This bond-anisotropic parameter continuously interpolates among the Ising-like limit $α=0$ , the isotropic point $α=1$ , and the XY-dominated regime $α\gg 1$ . Using density-matrix renormalization group calculations within a matrix-product-state framework, we analyze bipartite entanglement, magnetization, and spin correlations to characterize the emergent quantum phases. We identify six distinct ground-state phases, including rung-singlet, Haldane-like, Tomonaga-Luttinger liquid, canted Ising-ordered, XY-polarized, and ordered ferromagnetic states. While magnetization and local correlations provide a first insight into phase classifications, the finite-size scaling of the entanglement measures offers a more sensitive and unified diagnostics of both gapped and gapless regimes. In the gapless regime, we extract the central charge $c \simeq 1$ , confirming Tomonaga-Luttinger liquid behavior, while at the transition between the canted Ising and ferromagnetic phases, we observe $c \simeq 1/2$ , consistent with the Ising universality. Finally, we find that both the Haldane-like behavior and the extended critical Tomonaga-Luttinger liquid regime are strongly confined to the vicinity of the isotropic point $α= 1$ .We investigate the ground-state phase diagram of a frustrated spin-1/2 Heisenberg ladder in the transverse magnetic field with an anisotropic inter-rung exchange coupling $α$. This bond-anisotropic parameter continuously interpolates among the Ising-like limit $α=0$, the isotropic point $α=1$, and the XY-dominated regime $α\gg 1$. Using density-matrix renormalization group calculations within a matrix-product-state framework, we analyze bipartite entanglement, magnetization, and spin correlations to characterize the emergent quantum phases. We identify six distinct ground-state phases, including rung-singlet, Haldane-like, Tomonaga-Luttinger liquid, canted Ising-ordered, XY-polarized, and ordered ferromagnetic states. While magnetization and local correlations provide a first insight into phase classifications, the finite-size scaling of the entanglement measures offers a more sensitive and unified diagnostics of both gapped and gapless regimes. In the gapless regime, we extract the central charge $c \simeq 1$, confirming Tomonaga-Luttinger liquid behavior, while at the transition between the canted Ising and ferromagnetic phases, we observe $c \simeq 1/2$, consistent with the Ising universality. Finally, we find that both the Haldane-like behavior and the extended critical Tomonaga-Luttinger liquid regime are strongly confined to the vicinity of the isotropic point $α= 1$.

quant-pharXiv:2606.21242Jun 22, 20260 citations

Inhomogeneous superconducting phases in the frustrated Kondo-Heisenberg chain

A. Dobry, Akbar Jaefari, Eduardo Fradkin

We use bosonization and renormalization group methods to determine the ground-state phase diagram of a one-dimensional frustrated Kondo-Heisenberg system consisting of a one-dimensional spin- $\frac{1}{2}$ Luttinger liquid coupled by a Kondo exchange interaction ${J}_{K}$ to a frustrated quantum antiferromagnetic Heisenberg chain, with a nearest-neighbor exchange coupling ${J}_{1}$ and a next-nearest-neighbor (frustrating) exchange interaction ${J}_{2}$ . We analyze the interplay of quantum frustration in the antiferromagnetic chain with the Kondo exchange coupling ${J}_{K}$ with the Luttinger liquid. We discuss the structure of the phase diagram of this system as a function of the ratios ${J}_{K}/{J}_{1}$ , ${J}_{2}/{J}_{1}$ , and of the parameters of the Luttinger liquid. In particular, we discuss in detail the regimes in which a pair-density-wave state may be realized and its relation with the spin correlations in the frustrated antiferromagnetic chain.We use bosonization and renormalization group methods to determine the ground-state phase diagram of a one-dimensional frustrated Kondo-Heisenberg system consisting of a one-dimensional spin-$\frac{1}{2}$ Luttinger liquid coupled by a Kondo exchange interaction ${J}_{K}$ to a frustrated quantum antiferromagnetic Heisenberg chain, with a nearest-neighbor exchange coupling ${J}_{1}$ and a next-nearest-neighbor (frustrating) exchange interaction ${J}_{2}$. We analyze the interplay of quantum frustration in the antiferromagnetic chain with the Kondo exchange coupling ${J}_{K}$ with the Luttinger liquid. We discuss the structure of the phase diagram of this system as a function of the ratios ${J}_{K}/{J}_{1}$, ${J}_{2}/{J}_{1}$, and of the parameters of the Luttinger liquid. In particular, we discuss in detail the regimes in which a pair-density-wave state may be realized and its relation with the spin correlations in the frustrated antiferromagnetic chain.

arXiv:1304.5256Physical Review BJun 7, 20138 citations

Sliding Luttinger liquid phases

Ranjan Mukhopadhyay, C. L. Kane, T. C. Lubensky

We study systems of coupled spin-gapped and gapless Luttinger liquids. First, we establish the existence of a sliding Luttinger liquid phase for a system of weakly coupled parallel quantum wires, with and without disorder. It is shown that the coupling can stabilize a Luttinger liquid phase in the presence of disorder. We then extend our analysis to a system of crossed Luttinger liquids and establish the stability of a non-Fermi-liquid state: the crossed sliding Luttinger liquid phase. In this phase the system exhibits a finite-temperature, long-wavelength, isotropic electric conductivity that diverges as a power law in temperature T as $\stackrel{\ensuremath{\rightarrow}}{T}0.$ This two-dimensional system has many properties of a true isotropic Luttinger liquid, though at zero temperature it becomes anisotropic. An extension of this model to a three-dimensional stack exhibits a much higher in-plane conductivity than the conductivity in a perpendicular direction.We study systems of coupled spin-gapped and gapless Luttinger liquids. First, we establish the existence of a sliding Luttinger liquid phase for a system of weakly coupled parallel quantum wires, with and without disorder. It is shown that the coupling can stabilize a Luttinger liquid phase in the presence of disorder. We then extend our analysis to a system of crossed Luttinger liquids and establish the stability of a non-Fermi-liquid state: the crossed sliding Luttinger liquid phase. In this phase the system exhibits a finite-temperature, long-wavelength, isotropic electric conductivity that diverges as a power law in temperature T as $\stackrel{\ensuremath{\rightarrow}}{T}0.$ This two-dimensional system has many properties of a true isotropic Luttinger liquid, though at zero temperature it becomes anisotropic. An extension of this model to a three-dimensional stack exhibits a much higher in-plane conductivity than the conductivity in a perpendicular direction.

arXiv:cond-mat/0102163Physical review. B, Condensed matterJul 9, 2001131 citations

Stability of algebraic spin liquids coupled to quantum phonons

Francesco Ferrari, Josef Willsher, Urban F. P. Seifert +2 more

Algebraic spin liquids are quantum disordered phases of insulating magnets that exhibit fractionalized gapless excitations and power-law correlations. Quantum spin liquids in this category include the experimentally established one-dimensional (1D) Luttinger liquid, as well as the U(1) Dirac spin liquid (DSL) that has been a focus of recent candidate materials searches. Most notably, several exchange-frustrated Heisenberg materials on the triangular lattice have shown evidence of the U(1) DSL. In this work, we measure the algebraic correlations of spin-singlet excitations in the J 1 − J 2 antiferromagnetic Heisenberg model on the triangular lattice, prompting a detailed investigation of this model’s stability under spin-lattice coupling with dynamical phonons using variational Monte Carlo. As seen before in 1D spin chains, we observe a low-temperature transition from a U(1) DSL to valence bond order and predict the parameter regime where the model realizes a stable DSL ground state. To achieve this, we employ a series of finite-size scaling Ansätze inspired by the low-energy DSL’s conformal description in terms of quantum electrodynamics and show that emergent monopole operators drive the instability. We compare the physics of this transition to the 1D Luttinger liquid throughout our analysis. We derive the regime of stability against spin-Peierls ordering and argue that the DSL ground state might still be achievable in candidate materials, despite its tendency to valence bond solid ordering.Algebraic spin liquids are quantum disordered phases of insulating magnets that exhibit fractionalized gapless excitations and power-law correlations. Quantum spin liquids in this category include the experimentally established one-dimensional (1D) Luttinger liquid, as well as the U(1) Dirac spin liquid (DSL) that has been a focus of recent candidate materials searches. Most notably, several exchange-frustrated Heisenberg materials on the triangular lattice have shown evidence of the U(1) DSL. In this work, we measure the algebraic correlations of spin-singlet excitations in the J 1 − J 2 antiferromagnetic Heisenberg model on the triangular lattice, prompting a detailed investigation of this model’s stability under spin-lattice coupling with dynamical phonons using variational Monte Carlo. As seen before in 1D spin chains, we observe a low-temperature transition from a U(1) DSL to valence bond order and predict the parameter regime where the model realizes a stable DSL ground state. To achieve this, we employ a series of finite-size scaling Ansätze inspired by the low-energy DSL’s conformal description in terms of quantum electrodynamics and show that emergent monopole operators drive the instability. We compare the physics of this transition to the 1D Luttinger liquid throughout our analysis. We derive the regime of stability against spin-Peierls ordering and argue that the DSL ground state might still be achievable in candidate materials, despite its tendency to valence bond solid ordering.

arXiv:2410.16376Physical Review ResearchNov 16, 20252 citations

Using a recently proposed perturbative numerical renormalization-group algorithm, we explore the connection between quantum criticality and the emergence of Luttinger liquid physics in

t-J

chains coupled by frustrated interactions. This study is built on an earlier finding that at the maximally frustrated point, the ground state of weakly-coupled Heisenberg chains is disordered, the transverse exchanges being irrelevant. This result is extended here to transverse couplings up to

J_{\perp}=0.6

, and we argue that it may also be valid at the isotropic point. A finite size analysis of coupled Heisenberg chains in the vicinity of the maximally frustrated point confirms that the transverse spin-spin correlations decay exponentially while the longitudinal ones revert to those of decoupled chains. We find that this behavior persists upon moderate hole doping $x \alt 0.75$. For larger doping, the frustration becomes inactive and the quantum critical point is suppressed.

We investigate the ground-state phase diagram of a frustrated spin-1/2 Heisenberg ladder in the transverse magnetic field with an anisotropic inter-rung exchange coupling

α

. This bond-anisotropic parameter continuously interpolates among the Ising-like limit

α=0

, the isotropic point

α=1

, and the XY-dominated regime

α\gg 1

. Using density-matrix renormalization group calculations within a matrix-product-state framework, we analyze bipartite entanglement, magnetization, and spin correlations to characterize the emergent quantum phases. We identify six distinct ground-state phases, including rung-singlet, Haldane-like, Tomonaga-Luttinger liquid, canted Ising-ordered, XY-polarized, and ordered ferromagnetic states. While magnetization and local correlations provide a first insight into phase classifications, the finite-size scaling of the entanglement measures offers a more sensitive and unified diagnostics of both gapped and gapless regimes. In the gapless regime, we extract the central charge

c \simeq 1

, confirming Tomonaga-Luttinger liquid behavior, while at the transition between the canted Ising and ferromagnetic phases, we observe

c \simeq 1/2

, consistent with the Ising universality. Finally, we find that both the Haldane-like behavior and the extended critical Tomonaga-Luttinger liquid regime are strongly confined to the vicinity of the isotropic point

α= 1

We use bosonization and renormalization group methods to determine the ground-state phase diagram of a one-dimensional frustrated Kondo-Heisenberg system consisting of a one-dimensional spin-

\frac{1}{2}

Luttinger liquid coupled by a Kondo exchange interaction

{J}_{K}

to a frustrated quantum antiferromagnetic Heisenberg chain, with a nearest-neighbor exchange coupling

{J}_{1}

and a next-nearest-neighbor (frustrating) exchange interaction

{J}_{2}

. We analyze the interplay of quantum frustration in the antiferromagnetic chain with the Kondo exchange coupling

{J}_{K}

with the Luttinger liquid. We discuss the structure of the phase diagram of this system as a function of the ratios

{J}_{K}/{J}_{1}

{J}_{2}/{J}_{1}

, and of the parameters of the Luttinger liquid. In particular, we discuss in detail the regimes in which a pair-density-wave state may be realized and its relation with the spin correlations in the frustrated antiferromagnetic chain.