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:cond-mat/9806251 · Physical review. B, Condensed matter

Diagonal ladders: A class of models for strongly coupled electron systems

Germán Sierra, Miguel Angel Martin-Delgado, Steven R. White, Douglas J. Scalapino, J. Dukelsky

Published Mar 15, 1999• 24 citations
arXivHTMLPDF

Abstract

We introduce a class of models defined on ladders with a diagonal structure generated by ${n}_{p}$ plaquettes. The case ${n}_{p}=1$ corresponds to the necklace ladder and has remarkable properties that are studied using density matrix renormalization-group and recurrent variational ansatzes. The antiferromagnetic Heisenberg (AFH) model on this ladder is equivalent to the alternating spin-1/spin-$\frac{1}{2}$ AFH chain, which is known to have a ferromagnetic ground state (GS). For doping 1/3 the GS is a fully doped (1,1) stripe with the holes located mostly along the principal diagonal while the minor diagonals are occupied by spin singlets. This state can be seen as a Mott insulator of localized Cooper pairs on the plaquettes. A physical picture of our results is provided by a ${t}_{p}\ensuremath{-}{J}_{p}\ensuremath{-}{t}_{d}$ model of plaquettes coupled diagonally with a hopping parameter ${t}_{d}.$ In the limit ${t}_{d}\ensuremath{\rightarrow}\ensuremath{\infty}$ we recover the original $t\ensuremath{-}J$ model on the necklace ladder while for a weak hopping parameter the model is easily solvable. The GS in the strong hopping regime is essentially an ``on link'' Gutzwiller projection of the weak hopping GS. We generalize the ${t}_{p}\ensuremath{-}{J}_{p}\ensuremath{-}{t}_{d}$ model to diagonal ladders with ${n}_{p}>1$ and the two-dimensional square lattice. We use in our construction concepts familiar in statistical mechanics such as medial graphs and Bratelli diagrams.

DOI
10.1103/physrevb.59.7973

Related papers

Low-energy properties of spin-1/2 two-leg antiferromagnetic Heisenberg ladders with ferromagnetic diagonal coupling

Ningsheng Zhu, Xiaoqun Wang, Changfeng Chen

Using the density matrix renormalization group method, we have obtained the ground-state phase diagram of a spin-1/2 two-leg antiferromagnetic Heisenberg ladder with a ferromagnetic diagonal interaction ${J}_{\ifmmode\times\else\texttimes\fi{}}.$ The boundary between the two phases found in the system, the Haldane phase and the singlet phase, lies in the frustrated area of the parameter space, i.e., both the rung coupling ${J}_{\ensuremath{\perp}}$ and the diagonal coupling ${J}_{\ifmmode\times\else\texttimes\fi{}}$ are ferromagnetic. The spin gap is found to be insensitive to the change of the ferromagnetic diagonal coupling strength when the ratio of the rung coupling constant ${J}_{\ensuremath{\perp}}$ to the leg coupling constant ${J}_{\ensuremath{\Vert}}$ is 0.5, in agreement with the analysis of recent experimental spin susceptibility data.

Physical review. B, Condensed matter•Dec 11, 2000•14 citations

Weak-coupling phase diagrams of bond-aligned and diagonal, doped Hubbard ladders

Shirit Baruch, Dror Orgad

We study, using a perturbative renormalization group technique, the phase diagrams of bond-aligned and diagonal Hubbard ladders defined as sections of a square lattice with nearest-neighbor and next-nearest-neighbor hopping. We find that for not too large hole doping and small next-nearest-neighbor hopping, the bond-aligned systems exhibit a fully spin-gapped phase while the diagonal systems remain gapless. Increasing the next-nearest-neighbor hopping typically leads to a decrease of the gap in the bond-aligned ladders, and to a transition into a gapped phase in the diagonal ladders. Embedding the ladders in an antiferromagnetic environment can lead to a reduction in the extent of the gapped phases. These findings may suggest a relation between the orientation of hole-rich stripes and superconductivity as observed in ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}\mathrm{Cu}{\mathrm{O}}_{4}$.

arXiv:cond-mat/0410037•Physical Review B•May 10, 2005•2 citations

Hole and pair structures in the t-J model

Steven R. White, Douglas J. Scalapino

Using numerical results from density matrix renormalization group (DMRG) calculations for the t-J model, on systems as large as 10\ifmmode\times\else\texttimes\fi{}7, we examine the structure of the one and two hole ground states in ladder systems and in two dimensional clusters. A simple theoretical framework is used to explain why holes bind in pairs in two-dimensional antiferromagnets. For the case J/t=0.5, which we have studied, the hole pairs reside predominantly on a 2\ifmmode\times\else\texttimes\fi{}2 core plaquette with the probability that the holes are on diagonal sites greater than nearest-neighbor sites. There is a strong singlet bond connecting the spins on the two remaining sites of the plaquette. We find that a general characteristic of dynamic holes in an antiferromagnet is the presence of frustrating antiferromagnetic bonds connecting next-nearest-neighbor sites across the holes. Pairs of holes bind in order to share the frustrating bonds. At low doping, in addition to hole pairs, there are two additional low-energy structures which spontaneously form on certain finite systems. The first is an undoped L\ifmmode\times\else\texttimes\fi{}2 spin-liquid region, or ladder. The second is a hole moving along a one dimensional chain of sites. At higher doping we expect that hole pairing is always favored.

arXiv:cond-mat/9605143•Physical review. B, Condensed matter•Mar 1, 1997•165 citations

Ground State ofS=1 Heisenberg Ladders

Munehisa Matsumoto, Synge Todo, Chitoshi Yasuda +1 more

The ground state of S=1 antiferromagnetic Heisenberg (AFH) two-leg ladder has been shown to be gapful, in a whole range from the weak rung coupling limit to the strong rung coupling limit. The well-known valence-bond solid (VBS) picture for single AFH chains does not hold for this disordered ground state. We generalize the VBS picture to the plaquette-singlet solid (PSS) one to obtain a good description of the S=1 ladder. The validity of the PSS picture for this system is also supported by the finiteness of a newly-introduced hidden order parameter analogous to the string order parameter. We observe that there is also no critical point in the ground state of ladders with in-phase leg-bond alternation.

Progress of Theoretical Physics Supplement•Jan 1, 2002•1 citation

Spin-1/2 string correlations and singlet-triplet gaps of frustrated ladders with ferromagnetic (F) legs and alternate F and AF rungs

Chatterjee, Monalisa, Manoranjan Kumar, Z. G. Soos

The frustrated ladder with alternate ferromagnetic(F) exchange $-J_F$ and AF exchange $J_A$ to first neighbors and F exchange $-J_L$ to second neighbors is studied by exact diagonalization (ED) and density matrix renormalization group (DMRG) calculations in systems of $2N$ spins-1/2 with periodic boundary conditions. The ground state is a singlet $(S = 0)$ and the singlet-triplet gap $\varepsilon_T$ is finite for the exchanges considered. Spin-1/2 string correlation functions $g_1(N)$ and $g_2(N)$ are defined for an even number $N$ of consecutive spins in systems with two spins per unit cell; the ladder has string order $g_2(\infty)> 0$ and $g_1(\infty) = 0$. The minimum $N^*$ of $g_2(N)$ is related to the range of ground-state spin correlations. Convergence to $g_2(\infty)$ is from below, and $g_1(N)$ decreases exponentially for $N \geq N^*$. Singlet valence bond (VB) diagrams account for the size dependencies. The frustrated ladder at special values of $J_F$, $J_L$ and $J_A$ reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and the $J_1-J_2$ model, among others. Numerical analysis of ladders matches previous results for spin-1 gaps or string correlation functions and extends them to spin-1/2 systems. The nondegenerate singlet ground state of ladder is a bond-order wave, a Kekulé VB diagrams at $J_L = J_F/2 \leq J_A$, that is reversed on interchanging $-J_F$ and $J_A$. Inversion symmetry is spontaneously broken in the dimer phase of the $J_1-J_2$ model where the Kekulé diagrams are the doubly degenerate ground states at $J_2/J_1 = 1/2$.

arXiv:2403.05238•Mar 8, 2024•0 citations

Bond-Alternating Antiferromagnetic S = 1/2 Heisenberg Ladder with Ferromagnetic Diagonal Coupling

Li An-Kang, LU Jun-Zhe, Ma Lei

Using the density matrix renormalization group method, we determine the phase diagram of a frustrated bond-alternating S = 1/2 Heisenberg ladder with ferro-antiferromagnetic couplings at zero temperature. With the interactions between spins along the rungs set, we identify three spin-gapped phases (the Haldane phase, the singlet phase and the dimer phase) in the whole parameter range. The analysis of our data shows that two-leg spin bond-alternating ladders have a rich phase diagram if both rung and diagonal couplings are taken into account.

Chinese Physics Letters•Dec 1, 2009•0 citations
We introduce a class of models defined on ladders with a diagonal structure generated by np{n}_{p}np​ plaquettes. The case np=1{n}_{p}=1np​=1 corresponds to the necklace ladder and has remarkable properties that are studied using density matrix renormalization-group and recurrent variational ansatzes. The antiferromagnetic Heisenberg (AFH) model on this ladder is equivalent to the alternating spin-1/spin-12\frac{1}{2}21​ AFH chain, which is known to have a ferromagnetic ground state (GS). For doping 1/3 the GS is a fully doped (1,1) stripe with the holes located mostly along the principal diagonal while the minor diagonals are occupied by spin singlets. This state can be seen as a Mott insulator of localized Cooper pairs on the plaquettes. A physical picture of our results is provided by a tp\ensuremath−Jp\ensuremath−td{t}_{p}\ensuremath{-}{J}_{p}\ensuremath{-}{t}_{d}tp​\ensuremath−Jp​\ensuremath−td​ model of plaquettes coupled diagonally with a hopping parameter td.{t}_{d}.td​. In the limit td\ensuremath→\ensuremath∞{t}_{d}\ensuremath{\rightarrow}\ensuremath{\infty}td​\ensuremath→\ensuremath∞ we recover the original t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model on the necklace ladder while for a weak hopping parameter the model is easily solvable. The GS in the strong hopping regime is essentially an ``on link'' Gutzwiller projection of the weak hopping GS. We generalize the tp\ensuremath−Jp\ensuremath−td{t}_{p}\ensuremath{-}{J}_{p}\ensuremath{-}{t}_{d}tp​\ensuremath−Jp​\ensuremath−td​ model to diagonal ladders with np>1{n}_{p}>1np​>1 and the two-dimensional square lattice. We use in our construction concepts familiar in statistical mechanics such as medial graphs and Bratelli diagrams.
Using the density matrix renormalization group method, we have obtained the ground-state phase diagram of a spin-1/2 two-leg antiferromagnetic Heisenberg ladder with a ferromagnetic diagonal interaction ${J}_{\ifmmode\times\else\texttimes\fi{}}.$ The boundary between the two phases found in the system, the Haldane phase and the singlet phase, lies in the frustrated area of the parameter space, i.e., both the rung coupling ${J}_{\ensuremath{\perp}}$ and the diagonal coupling ${J}_{\ifmmode\times\else\texttimes\fi{}}$ are ferromagnetic. The spin gap is found to be insensitive to the change of the ferromagnetic diagonal coupling strength when the ratio of the rung coupling constant ${J}_{\ensuremath{\perp}}$ to the leg coupling constant ${J}_{\ensuremath{\Vert}}$ is 0.5, in agreement with the analysis of recent experimental spin susceptibility data.
We study, using a perturbative renormalization group technique, the phase diagrams of bond-aligned and diagonal Hubbard ladders defined as sections of a square lattice with nearest-neighbor and next-nearest-neighbor hopping. We find that for not too large hole doping and small next-nearest-neighbor hopping, the bond-aligned systems exhibit a fully spin-gapped phase while the diagonal systems remain gapless. Increasing the next-nearest-neighbor hopping typically leads to a decrease of the gap in the bond-aligned ladders, and to a transition into a gapped phase in the diagonal ladders. Embedding the ladders in an antiferromagnetic environment can lead to a reduction in the extent of the gapped phases. These findings may suggest a relation between the orientation of hole-rich stripes and superconductivity as observed in ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}\mathrm{Cu}{\mathrm{O}}_{4}$.
Using numerical results from density matrix renormalization group (DMRG) calculations for the t-J model, on systems as large as 10\ifmmode\times\else\texttimes\fi{}7, we examine the structure of the one and two hole ground states in ladder systems and in two dimensional clusters. A simple theoretical framework is used to explain why holes bind in pairs in two-dimensional antiferromagnets. For the case J/t=0.5, which we have studied, the hole pairs reside predominantly on a 2\ifmmode\times\else\texttimes\fi{}2 core plaquette with the probability that the holes are on diagonal sites greater than nearest-neighbor sites. There is a strong singlet bond connecting the spins on the two remaining sites of the plaquette. We find that a general characteristic of dynamic holes in an antiferromagnet is the presence of frustrating antiferromagnetic bonds connecting next-nearest-neighbor sites across the holes. Pairs of holes bind in order to share the frustrating bonds. At low doping, in addition to hole pairs, there are two additional low-energy structures which spontaneously form on certain finite systems. The first is an undoped L\ifmmode\times\else\texttimes\fi{}2 spin-liquid region, or ladder. The second is a hole moving along a one dimensional chain of sites. At higher doping we expect that hole pairing is always favored.
The ground state of S=1 antiferromagnetic Heisenberg (AFH) two-leg ladder has been shown to be gapful, in a whole range from the weak rung coupling limit to the strong rung coupling limit. The well-known valence-bond solid (VBS) picture for single AFH chains does not hold for this disordered ground state. We generalize the VBS picture to the plaquette-singlet solid (PSS) one to obtain a good description of the S=1 ladder. The validity of the PSS picture for this system is also supported by the finiteness of a newly-introduced hidden order parameter analogous to the string order parameter. We observe that there is also no critical point in the ground state of ladders with in-phase leg-bond alternation.
The frustrated ladder with alternate ferromagnetic(F) exchange −JF-J_F−JF​ and AF exchange JAJ_AJA​ to first neighbors and F exchange −JL-J_L−JL​ to second neighbors is studied by exact diagonalization (ED) and density matrix renormalization group (DMRG) calculations in systems of 2N2N2N spins-1/2 with periodic boundary conditions. The ground state is a singlet (S=0)(S = 0)(S=0) and the singlet-triplet gap εT\varepsilon_TεT​ is finite for the exchanges considered. Spin-1/2 string correlation functions g1(N)g_1(N)g1​(N) and g2(N)g_2(N)g2​(N) are defined for an even number NNN of consecutive spins in systems with two spins per unit cell; the ladder has string order g2(∞)>0g_2(\infty)> 0g2​(∞)>0 and g1(∞)=0g_1(\infty) = 0g1​(∞)=0. The minimum N∗N^*N∗ of g2(N)g_2(N)g2​(N) is related to the range of ground-state spin correlations. Convergence to g2(∞)g_2(\infty)g2​(∞) is from below, and g1(N)g_1(N)g1​(N) decreases exponentially for N≥N∗N \geq N^*N≥N∗. Singlet valence bond (VB) diagrams account for the size dependencies. The frustrated ladder at special values of JFJ_FJF​, JLJ_LJL​ and JAJ_AJA​ reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and the J1−J2J_1-J_2J1​−J2​ model, among others. Numerical analysis of ladders matches previous results for spin-1 gaps or string correlation functions and extends them to spin-1/2 systems. The nondegenerate singlet ground state of ladder is a bond-order wave, a Kekulé VB diagrams at JL=JF/2≤JAJ_L = J_F/2 \leq J_AJL​=JF​/2≤JA​, that is reversed on interchanging −JF-J_F−JF​ and JAJ_AJA​. Inversion symmetry is spontaneously broken in the dimer phase of the J1−J2J_1-J_2J1​−J2​ model where the Kekulé diagrams are the doubly degenerate ground states at J2/J1=1/2J_2/J_1 = 1/2J2​/J1​=1/2.
Using the density matrix renormalization group method, we determine the phase diagram of a frustrated bond-alternating S = 1/2 Heisenberg ladder with ferro-antiferromagnetic couplings at zero temperature. With the interactions between spins along the rungs set, we identify three spin-gapped phases (the Haldane phase, the singlet phase and the dimer phase) in the whole parameter range. The analysis of our data shows that two-leg spin bond-alternating ladders have a rich phase diagram if both rung and diagonal couplings are taken into account.