arXiv:cond-mat/0312106 · Physical Review Letters

Flux Period, Spin Gap, and Pairing in the One-Dimensional t−J−J′ Model

Alexander Seidel, D. H. Lee

Published Jul 21, 2004• 9 citations

Abstract

Using the factorization of the wave function in the $t\mathrm{\text{\ensuremath{-}}}J\mathrm{\text{\ensuremath{-}}}{J}^{\ensuremath{'}}$ model at small exchange couplings, we demonstrate the connection between the existence of a spin gap and an $hc/2e$ flux periodicity of the ground state energy. We conjecture that all spin-gapped SU(2)-invariant Luttinger liquids have $hc/2e$ flux periodicity, and that this is connected to the fact that a gapped spin-$\frac{1}{2}$ chain always breaks translational symmetry by doubling the unit cell.

DOI: 10.1103/physrevlett.93.046401

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arXiv:cond-mat/0312106 · Physical Review Letters

Flux Period, Spin Gap, and Pairing in the One-Dimensional t−J−J′ Model

Alexander Seidel, D. H. Lee

Published Jul 21, 2004• 9 citations

arXiv HTML PDF

Abstract

Using the factorization of the wave function in the

t\mathrm{\text{\ensuremath{-}}}J\mathrm{\text{\ensuremath{-}}}{J}^{\ensuremath{'}}

model at small exchange couplings, we demonstrate the connection between the existence of a spin gap and an

hc/2e

flux periodicity of the ground state energy. We conjecture that all spin-gapped SU(2)-invariant Luttinger liquids have

hc/2e

flux periodicity, and that this is connected to the fact that a gapped spin-

\frac{1}{2}

chain always breaks translational symmetry by doubling the unit cell.

DOI: 10.1103/physrevlett.93.046401

Related papers

Ground-state phase diagram of the one-dimensional dimerized t−J model at quarter filling

Satoshi Nishimoto, Y. Ohta

The ground state of the one-dimensional dimerized $t\ensuremath{-}J$ model at quarter filling is studied by a Lanczos exact-diagonalization technique on small clusters. We calculate the charge gap, spin gap, binding energy, Tomonaga-Luttinger-liquid parameter, Drude weight, anomalous flux quantization, etc. We thereby show that the two types of dimerization, i.e., a dimerization of hopping integral and a dimerization of exchange interaction play a mutually competing role in controlling the ground state of the system and this leads to the emergence of various phases including the Mott insulating, Tomonaga-Luttinger-liquid, and spin-gap-liquid phases. The ground-state phase diagram of the model is given on the parameter space of the dimerizations.The ground state of the one-dimensional dimerized $t\ensuremath{-}J$ model at quarter filling is studied by a Lanczos exact-diagonalization technique on small clusters. We calculate the charge gap, spin gap, binding energy, Tomonaga-Luttinger-liquid parameter, Drude weight, anomalous flux quantization, etc. We thereby show that the two types of dimerization, i.e., a dimerization of hopping integral and a dimerization of exchange interaction play a mutually competing role in controlling the ground state of the system and this leads to the emergence of various phases including the Mott insulating, Tomonaga-Luttinger-liquid, and spin-gap-liquid phases. The ground-state phase diagram of the model is given on the parameter space of the dimerizations.

arXiv:cond-mat/9805280Physical review. B, Condensed matterFeb 15, 19998 citations

Ground-state phase diagram of the one-dimensional t - J model

Alexander Moreno, Alejandro Muramatsu, Salvatore R. Manmana

We examine the ground-state phase diagram of the $t$ - $J$ model in one dimension by means of the density matrix renormalization group. This model is characterized by a rich phase diagram as a function of the exchange interaction $J$ and the density $n$ , displaying Luttinger-liquid (LL) behavior both of repulsive and attractive (i.e., superconducting) natures, a spin-gap phase, and phase separation. The phase boundaries separating the repulsive from the attractive LL phase as $J$ is increased, and also the boundaries of the spin-gap region at low densities, and phase separation at even larger $J$ , are determined on the basis of correlation functions and energy gaps. In particular, we shed light on a contradiction between variational and renormalization-group (RG) results about the extent of the spin-gap phase, which results larger than the variational but smaller than the RG one. Furthermore, we show that the spin gap can reach a sizable value ( $~0.1t$ ) at low enough filling, such that preformed pairs should be observable at temperatures below these energy scales. No evidence for a phase with clustering of more than two particles is found on approaching phase separation.We examine the ground-state phase diagram of the $t$-$J$ model in one dimension by means of the density matrix renormalization group. This model is characterized by a rich phase diagram as a function of the exchange interaction $J$ and the density $n$, displaying Luttinger-liquid (LL) behavior both of repulsive and attractive (i.e., superconducting) natures, a spin-gap phase, and phase separation. The phase boundaries separating the repulsive from the attractive LL phase as $J$ is increased, and also the boundaries of the spin-gap region at low densities, and phase separation at even larger $J$, are determined on the basis of correlation functions and energy gaps. In particular, we shed light on a contradiction between variational and renormalization-group (RG) results about the extent of the spin-gap phase, which results larger than the variational but smaller than the RG one. Furthermore, we show that the spin gap can reach a sizable value ($~0.1t$) at low enough filling, such that preformed pairs should be observable at temperatures below these energy scales. No evidence for a phase with clustering of more than two particles is found on approaching phase separation.

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arXiv:cond-mat/0402663Physical Review BJan 14, 200527 citations

Singlet wave functions for thet-Jmodel

Didier Poilblanc

We investigate T=0 properties of the commensurate flux phases, a class of variational wave functions for the t-J model proposed recently by Anderson, Shastry, and Hristopoulos and which generalize the (1/2-flux state of Affleck and Marston away from half-filling. These states can be formally written in the quantum spin liquid form, the singlet bond amplitudes of which break the lattice translational symmetry. The supercell of the electromagnetic gauge (defining the fictitious flux) imposes the actual periodicities of the correlated wave function. Their order of commensurability with the lattice unit length is closely related to the hole density.We investigate T=0 properties of the commensurate flux phases, a class of variational wave functions for the t-J model proposed recently by Anderson, Shastry, and Hristopoulos and which generalize the (1/2-flux state of Affleck and Marston away from half-filling. These states can be formally written in the quantum spin liquid form, the singlet bond amplitudes of which break the lattice translational symmetry. The supercell of the electromagnetic gauge (defining the fictitious flux) imposes the actual periodicities of the correlated wave function. Their order of commensurability with the lattice unit length is closely related to the hole density.

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SU(2) formulation of the t−J model: Application to underdoped cuprates

Patrick A. Lee, Naoto Nagaosa, Tai Kai Ng +1 more

We develop a slave-boson theory for the $t\ensuremath{-}J$ model at finite doping that respects an SU(2) symmetry: a symmetry previously known to be important at half filling. The mean-field phase diagram is found to be consistent with the phases observed in the cuprate superconductors, which contain $d$ -wave superconductor, spin-gap, strange metal, and Fermi-liquid phases. The spin-gap phase is best understood as the staggered flux phase, which is nevertheless translationally invariant for physical quantities. The physical electron spectral function shows small Fermi segments at low doping that continuously evolve into the large Fermi surface at high-doping concentrations. The close relation between the SU(2) and the U(1) slave-boson theory is discussed. The low-energy effective theory for the low-lying fluctuations is derived and additional lying modes [which were overlooked in the U(1) theory] are identified.We develop a slave-boson theory for the $t\ensuremath{-}J$ model at finite doping that respects an SU(2) symmetry: a symmetry previously known to be important at half filling. The mean-field phase diagram is found to be consistent with the phases observed in the cuprate superconductors, which contain $d$-wave superconductor, spin-gap, strange metal, and Fermi-liquid phases. The spin-gap phase is best understood as the staggered flux phase, which is nevertheless translationally invariant for physical quantities. The physical electron spectral function shows small Fermi segments at low doping that continuously evolve into the large Fermi surface at high-doping concentrations. The close relation between the SU(2) and the U(1) slave-boson theory is discussed. The low-energy effective theory for the low-lying fluctuations is derived and additional lying modes [which were overlooked in the U(1) theory] are identified.

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Class of variational Ansätze for the spin-incoherent ground state of a Luttinger liquid coupled to a spin bath

Mohammad Soltanieh-ha, Adrian E. Feiguin

Interacting one-dimensional electron systems are generally referred to as ``Luttinger liquids,'' after the effective low-energy theory in which spin and charge behave as separate degrees of freedom with independent energy scales. The ``spin-incoherent Luttinger liquid'' describes a finite-temperature regime that is realized when the temperature is very small relative to the Fermi energy, but larger than the characteristic spin energy scale. Similar physics can take place in the ground state, when a Luttinger liquid is coupled to a spin bath, which effectively introduces a ``spin temperature'' through its entanglement with the spin degree of freedom. We show that the spin-incoherent state can be written as a factorized wave function, with a spin wave function that can be described within a valence bond formalism. This enables us to calculate exact expressions for the momentum distribution function and the entanglement entropy. This picture holds not only for two antiferromagnetically coupled $t\ensuremath{-}J$ chains, but also for the $t\ensuremath{-}J$-Kondo chain with strongly interacting conduction electrons. We argue that this theory is quite universal and may describe a family of problems that could be dubbed ``spin incoherent.''Interacting one-dimensional electron systems are generally referred to as ``Luttinger liquids,'' after the effective low-energy theory in which spin and charge behave as separate degrees of freedom with independent energy scales. The ``spin-incoherent Luttinger liquid'' describes a finite-temperature regime that is realized when the temperature is very small relative to the Fermi energy, but larger than the characteristic spin energy scale. Similar physics can take place in the ground state, when a Luttinger liquid is coupled to a spin bath, which effectively introduces a ``spin temperature'' through its entanglement with the spin degree of freedom. We show that the spin-incoherent state can be written as a factorized wave function, with a spin wave function that can be described within a valence bond formalism. This enables us to calculate exact expressions for the momentum distribution function and the entanglement entropy. This picture holds not only for two antiferromagnetically coupled $t\ensuremath{-}J$ chains, but also for the $t\ensuremath{-}J$-Kondo chain with strongly interacting conduction electrons. We argue that this theory is quite universal and may describe a family of problems that could be dubbed ``spin incoherent.''

arXiv:1211.0982Physical Review BNov 19, 20125 citations

Using the factorization of the wave function in the

t\mathrm{\text{\ensuremath{-}}}J\mathrm{\text{\ensuremath{-}}}{J}^{\ensuremath{'}}

model at small exchange couplings, we demonstrate the connection between the existence of a spin gap and an

hc/2e

flux periodicity of the ground state energy. We conjecture that all spin-gapped SU(2)-invariant Luttinger liquids have

hc/2e

flux periodicity, and that this is connected to the fact that a gapped spin-

\frac{1}{2}

chain always breaks translational symmetry by doubling the unit cell.

We examine the ground-state phase diagram of the

t

J

model in one dimension by means of the density matrix renormalization group. This model is characterized by a rich phase diagram as a function of the exchange interaction

J

and the density

n

, displaying Luttinger-liquid (LL) behavior both of repulsive and attractive (i.e., superconducting) natures, a spin-gap phase, and phase separation. The phase boundaries separating the repulsive from the attractive LL phase as

J

is increased, and also the boundaries of the spin-gap region at low densities, and phase separation at even larger

J

, are determined on the basis of correlation functions and energy gaps. In particular, we shed light on a contradiction between variational and renormalization-group (RG) results about the extent of the spin-gap phase, which results larger than the variational but smaller than the RG one. Furthermore, we show that the spin gap can reach a sizable value (

~0.1t

) at low enough filling, such that preformed pairs should be observable at temperatures below these energy scales. No evidence for a phase with clustering of more than two particles is found on approaching phase separation.

hc∕2e

, i.e., half a flux quantum, in the limit of large system size

L

. Finite size corrections are found to vanish exponentially in

L

. This behavior is contrasted to that of the spin gapless case, for both even and odd particle number. Generalizations to finite temperature are also discussed.

d

-wave superconductor, spin-gap, strange metal, and Fermi-liquid phases. The spin-gap phase is best understood as the staggered flux phase, which is nevertheless translationally invariant for physical quantities. The physical electron spectral function shows small Fermi segments at low doping that continuously evolve into the large Fermi surface at high-doping concentrations. The close relation between the SU(2) and the U(1) slave-boson theory is discussed. The low-energy effective theory for the low-lying fluctuations is derived and additional lying modes [which were overlooked in the U(1) theory] are identified.