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arXiv:2512.07591

Critical Density-Wave Vestigial Phases of Commensurate Pair Density Wave

Gao, Chu-Tian, Zhou, Jing, Liu, Yu-Bo, Yang, Fan

Published Dec 8, 2025• 0 citations
arXivHTMLPDF

Abstract

The pair-density-wave (PDW) is an exotic pairing state hosting a spatially modulated pairing order parameter, which has attracted great interest. Due to its simultaneously breaking U(1)-gauge and translational symmetries, intriguing vestigial phases which restore only one broken symmetry can emerge at an intermediate temperature regime. Previously, investigations on the vestigial phases of PDW were mainly focused on incommensurate PDW. However, the experimentally observed PDW is usually commensurate, whose vestigial phases have not been systematically investigated. Here we study the vestigial phases of 2D commensurate PDW with $n$-times expanded unit vectors, hosting different numbers of wave vectors. Based on the Ginzburg-Landau theory, we get the low energy effective model Hamiltonian. Subsequent renormalization group (RG) and Monte-Carlo (MC) studies are conducted to obtain the phase diagram and spatial dependent correlation functions. Our RG and MC calculations consistently yield the following result. For $n\le 4$, besides the charge-4e/2e superconductivity, there exists the translational symmetry broken charge-density-wave (CDW) vetigial phase. Intriguingly, for $n\ge 5$, the restore of the translational symmetry with increasing temperature is realized through two successive Berezinskii-Kosterlitz-Thouless transitions. Such a two-step process leads into two critical vestigial phases, i.e. the critical-PDW and the critical-CDW phases, in which the discrete translational symmetry is quasily broken, leading into a power-law decaying density-density correlation even at 2D. Our work appeals for experimental verifications.

DOI
10.48550/arxiv.2512.07591

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Abstract The pair density wave (PDW) is an exotic pairing state hosting a spatially modulated pairing order parameter, which has attracted great interest. Due to its simultaneously breaking U (1)-gauge and translational symmetries, intriguing vestigial phases which restore only one broken symmetry can emerge at an intermediate temperature regime. Previously, investigations on the vestigial phases of PDW were mainly focused on incommensurate PDW. However, the experimentally observed PDW is usually commensurate, whose vestigial phases have not been systematically investigated. Here we study the vestigial phases of 2D commensurate PDW with n -times ( n ≥ 3) expanded unit vectors, hosting different numbers of wave vectors. Based on the Ginzburg–Landau theory, we get the low energy effective model Hamiltonian from typical PDW ground state configurations. Subsequent renormalization group (RG) and Monte-Carlo (MC) studies are conducted to obtain the phase diagram and spatial dependent correlation functions. Our RG and MC calculations consistently yield the following result. For nt ≤ 4 ( t = 1 for odd n and t = 1/2 for even n ), besides the charge-4 e /2 e superconductivity, there exists the translational symmetry broken charge-density-wave (CDW) vetigial phase. Intriguingly, for nt ≥ 5 (specifically, for odd n , n ≥ 5, and for even n , n ≥ 10), the restoration of the translational symmetry with increasing temperature is realized through two successive Berezinskii–Kosterlitz–Thouless transitions. Such a two-step process leads to two critical vestigial phases, i.e., the critical-PDW and the critical-CDW phases, in which the discrete translational symmetry is quasily broken, leading to a power-law decaying density–density correlation even at 2D. Our work appeals for experimental verifications.

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Pair density wave superconducting states and statistical mechanics of dimers

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The following thesis is divided in two main parts. Chapters 2, 3 and 4 are devoted to the study of the so called pair-density-wave (PDW) superconducting state and some of its connections to electronic liquid crystal (ELC) phases, its topological aspects in a one dimensional model and its appearance in a quasi-one dimensional system. On the other hand, chapter 5 is focused on the investigation of the classical statistical mechanics properties of dimers, in particular, the dimer model on the Aztec diamond graph and its relation with the octahedron equation. In chapter 2 we present a theory of superconducting states where the Cooper pairs have a nonzero center-of-mass momentum, inhomogeneous superconducting states known as a pair-density-waves (PDWs) states. We show that in a system of spin-1/2 fermions in two dimensions in an electronic nematic spin-triplet phase where rotational symmetry is broken in both real and spin space PDW phases arise naturally in a theory that can be analysed using controlled approximations. We show that several superfluid phases that may arise in this phase can be treated within a controlled BCS mean field theory, with the strength of the spin-triplet nematic order parameter playing the role of the small parameter of this theory. We find that in a spin-triplet nematic phase, in addition to a triplet p-wave and spin-singlet d-wave (or s depending on the nematic phase) uniform superconducting states, it is also possible to have a d-wave (or s) PDW superconductor. The PDW phases found here can be either unidirectional, bidirectional, or tridirectional depending on the spin-triplet nematic phase and which superconducting channel is dominant. In addition, a triple-helix state is found in a particular channel. We show that these PDW phases are present in the weak-coupling limit, in contrast to the usual Fulde-Ferrell-Larkin-Ovchinnikov phases, which require strong coupling physics in addition to a large magnetic field (and often both). In chapter 3 we show that the pair-density-wave (PDW) superconducting state emergent in extended Heisenberg-Hubbard models in two-leg ladders is topological in the presence of an Ising spin symmetry and supports a Majorana zero mode (MZM) at an open boundary and at a junction with a uniform d-wave one-dimensional superconductor. Similarly to a conventional finite-momentum paired state, the order parameter of the PDW state is a charge-2e field with finite momentum. However, the order parameter here is a quartic electron operator and conventional mean-field theory cannot be applied to study this state. We use bosonization to show that the 1D PDW state has a MZM at a boundary. This superconducting state is an exotic topological phase supporting Majorana fermions with finite-momentum pairing fields and charge-4e superconductivity. In chapter 4 we provide a quasi-one-dimensional model which can support a PDW state. The model consists of an array of strongly-interacting one-dimensional systems, where the one-dimensional systems are coupled to each other by local interactions.Within the interchain mean-field theory (MFT), we find several SC states from the model, including a conventional uniform SC state, PDW SC state, and a coexisting phase of the uniform SC and PDW states. In this quasi-1D regime we can treat the strong correlation physics essentially exactly using bosonization methods and the crossover to the 2D system by means of interchain MFT. The resulting critical temperatures of the SC phases generically exhibit a power-law scaling with the coupling constants of the array, instead of the essential singularity found in weak-coupling BCS-type theories. Electronic excitations with an open Fermi surface, which emerge from the electronic Luttinger liquid systems below their crossover temperature to the Fermi liquid, are then coupled to the SC order parameters via the proximity effect. From the Fermi surface thus coupled to the SC order parameters, we calculate the quasiparticle spectrum in detail. We show that the quasiparticle spectrum can be fully gapped or nodal in the uniform SC phase and also in the coexisting phase of the uniform SC and PDW parameters. In the pure PDW state, the excitation spectrum has a reconstructed Fermi surface in the form of Fermi pockets of Bogoliubov quasiparticles. In chapter 5 we study the octahedron relation (also known as the A∞ T-system), obeyed in particular by the partition function for dimer coverings of the Aztec Diamond graph. For a suitable class of doubly periodic initial conditions, we find exact solutions with a particularly simple factorized form. For these, we show that the density function that measures the average dimer occupation of a face of the Aztec graph, obeys a system of linear recursion relations with periodic coefficients. This allows us to explore the thermodynamic limit of the corresponding dimer models and to derive exact “arctic” curves separating the various phases of the system.

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In light of the new experimental and theoretical important developments in high-${T}_{c}$ superconductivity, we revisit the fermionic hot-spot model relevant to the phenomenology of the cuprates. We extend previous results by means of a complete two-loop order renormalization group (RG) framework. Here, we explicitly study the effect of the charge-density-wave (CDW) order parameter with a $d$-wave form factor with the experimentally observed modulation $(\ifmmode\pm\else\textpm\fi{}{Q}_{0},0)$ and $(0,\ifmmode\pm\else\textpm\fi{}{Q}_{0})$ at the infrared-stable nontrivial fixed point obtained previously for this model. Additionally, we proceed to investigate also the so-called pair-density-wave (PDW) order that was recently proposed in the literature as a possible candidate for the ``hidden'' order to describe the pseudogap phase observed in underdoped cuprates. We confirm that although the above two ordering tendencies are also found to be nearly degenerate both at one-loop and two-loop RG orders and linked by an emergent SU(2) pseudospin symmetry, they turn out to be subleading for weaker couplings in the present model to antiferromagnetism, $d$-wave bond-density wave (BDW) order with modulation along Brillouin zone diagonals $(\ifmmode\pm\else\textpm\fi{}{Q}_{0},\ifmmode\pm\else\textpm\fi{}{Q}_{0})$, and $d$-wave singlet superconductivity (SSC). However, as we increase the strength of the initial coupling towards moderate values, we do capture a tendency for the entangled PDW/CDW order to become leading compared to BDW/SSC in the model, which suggests that the former composite order might be indeed a viable concept to describe some cuprate superconductors at high temperatures in the underdoped regime, as has been recently alluded to by many authors in the literature.

arXiv:1503.00379•Physical Review B•Jul 31, 2015•45 citations
The pair-density-wave (PDW) is an exotic pairing state hosting a spatially modulated pairing order parameter, which has attracted great interest. Due to its simultaneously breaking U(1)-gauge and translational symmetries, intriguing vestigial phases which restore only one broken symmetry can emerge at an intermediate temperature regime. Previously, investigations on the vestigial phases of PDW were mainly focused on incommensurate PDW. However, the experimentally observed PDW is usually commensurate, whose vestigial phases have not been systematically investigated. Here we study the vestigial phases of 2D commensurate PDW with nnn-times expanded unit vectors, hosting different numbers of wave vectors. Based on the Ginzburg-Landau theory, we get the low energy effective model Hamiltonian. Subsequent renormalization group (RG) and Monte-Carlo (MC) studies are conducted to obtain the phase diagram and spatial dependent correlation functions. Our RG and MC calculations consistently yield the following result. For n≤4n\le 4n≤4, besides the charge-4e/2e superconductivity, there exists the translational symmetry broken charge-density-wave (CDW) vetigial phase. Intriguingly, for n≥5n\ge 5n≥5, the restore of the translational symmetry with increasing temperature is realized through two successive Berezinskii-Kosterlitz-Thouless transitions. Such a two-step process leads into two critical vestigial phases, i.e. the critical-PDW and the critical-CDW phases, in which the discrete translational symmetry is quasily broken, leading into a power-law decaying density-density correlation even at 2D. Our work appeals for experimental verifications.
Abstract The pair density wave (PDW) is an exotic pairing state hosting a spatially modulated pairing order parameter, which has attracted great interest. Due to its simultaneously breaking U (1)-gauge and translational symmetries, intriguing vestigial phases which restore only one broken symmetry can emerge at an intermediate temperature regime. Previously, investigations on the vestigial phases of PDW were mainly focused on incommensurate PDW. However, the experimentally observed PDW is usually commensurate, whose vestigial phases have not been systematically investigated. Here we study the vestigial phases of 2D commensurate PDW with n -times ( n ≥ 3) expanded unit vectors, hosting different numbers of wave vectors. Based on the Ginzburg–Landau theory, we get the low energy effective model Hamiltonian from typical PDW ground state configurations. Subsequent renormalization group (RG) and Monte-Carlo (MC) studies are conducted to obtain the phase diagram and spatial dependent correlation functions. Our RG and MC calculations consistently yield the following result. For nt ≤ 4 ( t = 1 for odd n and t = 1/2 for even n ), besides the charge-4 e /2 e superconductivity, there exists the translational symmetry broken charge-density-wave (CDW) vetigial phase. Intriguingly, for nt ≥ 5 (specifically, for odd n , n ≥ 5, and for even n , n ≥ 10), the restoration of the translational symmetry with increasing temperature is realized through two successive Berezinskii–Kosterlitz–Thouless transitions. Such a two-step process leads to two critical vestigial phases, i.e., the critical-PDW and the critical-CDW phases, in which the discrete translational symmetry is quasily broken, leading to a power-law decaying density–density correlation even at 2D. Our work appeals for experimental verifications.
Abstract Flat-band systems provide an ideal platform for studying strong electron correlations, in which multiple quasi-degenerate charge orders frequently coexist. Motivated by this context, we investigate the pair-density-wave (PDW) superconducting state in a quasi-one-dimensional Creutz lattice featuring two strictly flat bands. Using a renormalized mean-field approach for the strongly correlated t - J model, we obtain self-consistent solutions that demonstrate a PDW phase—characterized by finite-center-of-mass-momentum Cooper pairing—coexisting with charge density wave (CDW) and bond density wave (BDW) orders. Physically, this coexistence phenomenon originates from the perfect band nesting inherent to the flat-band system, which allows for changes in the particle momentum distributions without energy cost. Here, a pure PDW solution without CDW and BDW orders emerges in the underdoped regime when magnetic effects are neglected, while strong Hubbard correlations favor a pure CDW phase. We conclude that a robust PDW phase, accompanied by CDW and BDW orders, occupies the majority of the parameter space. Furthermore, both the PDW order parameter and its critical temperature scale linearly with the superexchange coupling strength—a hallmark of flat-band superconductivity. Overall, our results reveal that a PDW superconducting state can be realized on the Creutz lattice with strong correlations.
We show using bosonization methods that extended Hubbard-Heisenberg models on two types of two leg ladders (without flux and with flux $\ensuremath{\pi}$ per plaquette) have commensurate pair-density-wave (PDW) phases. In the case of the conventional (flux-less) ladder, the PDW arises when certain filling fractions for which commensurability conditions are met. For the flux $\ensuremath{\pi}$ ladder, the PDW phase is generally present. The PDW phase is characterized by a finite spin gap and a superconducting order parameter with a finite (commensurate in this case) wave vector and power-law superconducting correlations. In this phase, the uniform superconducting order parameter, the 2kF2{k}_{F}2kF​ charge-density-wave (CDW) order parameter and the spin-density-wave N\'eel order parameter exhibit short range (exponentially decaying) correlations. We discuss in detail the case in which the bonding band of the ladder is half filled for which the PDW phase appears even at weak coupling. The PDW phase is shown to be dual to a uniform superconducting (SC) phase with quasi-long-range order. By making use of bosonization and the renormalization group, we determine the phase diagram of the spin-gapped regime and study the quantum phase transition. The phase boundary between PDW and the uniform SC ordered phases is found to be in the Ising universality class. We generalize the analysis to the case of other commensurate fillings of the bonding band, where we find higher-order commensurate PDW states for which we determine the form of the effective bosonized field theory and discuss the phase diagram. We compare our results with recent findings in the Kondo-Heisenberg chain. We show that the formation of PDW order in the ladder embodies the notion of intertwined orders.
Pair-density-wave (PDW) superconductivity is a state in which the superconducting order parameter modulates at a finite wavevector. Using large-scale density matrix renormalization group, we study the doped altermagnetic Mott insulator in the checkerboard ttt-JJJ model, where altermagnetic exchange anisotropy is encoded microscopically through anisotropic ferromagnetic next-nearest-neighbor exchange. By mapping the ground-state phase diagram as a function of doping and altermagnetic anisotropy, mainly on six-leg cylinders, we identify a transition from a uniform ddd-wave superconducting regime with charge modulation to a PDW regime coexisting with stripe order. In the PDW regime, we report an unconventional wave-vector locking QPDW≈2QStripe\mathbf Q_{\mathrm{PDW}}\approx 2\mathbf Q_{\mathrm{Stripe}}QPDW​≈2QStripe​ along the cylinder direction, in contrast to the conventional relation. Pair correlations reveal a two-scale structure, consisting of short-distance local ddd-wave pairing and long-distance finite-momentum PDW correlations. A symmetry-based Ginzburg--Landau analysis is presented for the observed locking. Our results identify altermagnetism as a strong-coupling, microscopically grounded route to finite-momentum superconductivity in doped Mott insulators.
The following thesis is divided in two main parts. Chapters 2, 3 and 4 are devoted to the study of the so called pair-density-wave (PDW) superconducting state and some of its connections to electronic liquid crystal (ELC) phases, its topological aspects in a one dimensional model and its appearance in a quasi-one dimensional system. On the other hand, chapter 5 is focused on the investigation of the classical statistical mechanics properties of dimers, in particular, the dimer model on the Aztec diamond graph and its relation with the octahedron equation. In chapter 2 we present a theory of superconducting states where the Cooper pairs have a nonzero center-of-mass momentum, inhomogeneous superconducting states known as a pair-density-waves (PDWs) states. We show that in a system of spin-1/2 fermions in two dimensions in an electronic nematic spin-triplet phase where rotational symmetry is broken in both real and spin space PDW phases arise naturally in a theory that can be analysed using controlled approximations. We show that several superfluid phases that may arise in this phase can be treated within a controlled BCS mean field theory, with the strength of the spin-triplet nematic order parameter playing the role of the small parameter of this theory. We find that in a spin-triplet nematic phase, in addition to a triplet p-wave and spin-singlet d-wave (or s depending on the nematic phase) uniform superconducting states, it is also possible to have a d-wave (or s) PDW superconductor. The PDW phases found here can be either unidirectional, bidirectional, or tridirectional depending on the spin-triplet nematic phase and which superconducting channel is dominant. In addition, a triple-helix state is found in a particular channel. We show that these PDW phases are present in the weak-coupling limit, in contrast to the usual Fulde-Ferrell-Larkin-Ovchinnikov phases, which require strong coupling physics in addition to a large magnetic field (and often both). In chapter 3 we show that the pair-density-wave (PDW) superconducting state emergent in extended Heisenberg-Hubbard models in two-leg ladders is topological in the presence of an Ising spin symmetry and supports a Majorana zero mode (MZM) at an open boundary and at a junction with a uniform d-wave one-dimensional superconductor. Similarly to a conventional finite-momentum paired state, the order parameter of the PDW state is a charge-2e field with finite momentum. However, the order parameter here is a quartic electron operator and conventional mean-field theory cannot be applied to study this state. We use bosonization to show that the 1D PDW state has a MZM at a boundary. This superconducting state is an exotic topological phase supporting Majorana fermions with finite-momentum pairing fields and charge-4e superconductivity. In chapter 4 we provide a quasi-one-dimensional model which can support a PDW state. The model consists of an array of strongly-interacting one-dimensional systems, where the one-dimensional systems are coupled to each other by local interactions.Within the interchain mean-field theory (MFT), we find several SC states from the model, including a conventional uniform SC state, PDW SC state, and a coexisting phase of the uniform SC and PDW states. In this quasi-1D regime we can treat the strong correlation physics essentially exactly using bosonization methods and the crossover to the 2D system by means of interchain MFT. The resulting critical temperatures of the SC phases generically exhibit a power-law scaling with the coupling constants of the array, instead of the essential singularity found in weak-coupling BCS-type theories. Electronic excitations with an open Fermi surface, which emerge from the electronic Luttinger liquid systems below their crossover temperature to the Fermi liquid, are then coupled to the SC order parameters via the proximity effect. From the Fermi surface thus coupled to the SC order parameters, we calculate the quasiparticle spectrum in detail. We show that the quasiparticle spectrum can be fully gapped or nodal in the uniform SC phase and also in the coexisting phase of the uniform SC and PDW parameters. In the pure PDW state, the excitation spectrum has a reconstructed Fermi surface in the form of Fermi pockets of Bogoliubov quasiparticles. In chapter 5 we study the octahedron relation (also known as the A∞ T-system), obeyed in particular by the partition function for dimer coverings of the Aztec Diamond graph. For a suitable class of doubly periodic initial conditions, we find exact solutions with a particularly simple factorized form. For these, we show that the density function that measures the average dimer occupation of a face of the Aztec graph, obeys a system of linear recursion relations with periodic coefficients. This allows us to explore the thermodynamic limit of the corresponding dimer models and to derive exact “arctic” curves separating the various phases of the system.
In light of the new experimental and theoretical important developments in high-Tc{T}_{c}Tc​ superconductivity, we revisit the fermionic hot-spot model relevant to the phenomenology of the cuprates. We extend previous results by means of a complete two-loop order renormalization group (RG) framework. Here, we explicitly study the effect of the charge-density-wave (CDW) order parameter with a ddd-wave form factor with the experimentally observed modulation $(\ifmmode\pm\else\textpm\fi{}{Q}_{0},0)$ and $(0,\ifmmode\pm\else\textpm\fi{}{Q}_{0})$ at the infrared-stable nontrivial fixed point obtained previously for this model. Additionally, we proceed to investigate also the so-called pair-density-wave (PDW) order that was recently proposed in the literature as a possible candidate for the ``hidden'' order to describe the pseudogap phase observed in underdoped cuprates. We confirm that although the above two ordering tendencies are also found to be nearly degenerate both at one-loop and two-loop RG orders and linked by an emergent SU(2) pseudospin symmetry, they turn out to be subleading for weaker couplings in the present model to antiferromagnetism, ddd-wave bond-density wave (BDW) order with modulation along Brillouin zone diagonals $(\ifmmode\pm\else\textpm\fi{}{Q}_{0},\ifmmode\pm\else\textpm\fi{}{Q}_{0})$, and ddd-wave singlet superconductivity (SSC). However, as we increase the strength of the initial coupling towards moderate values, we do capture a tendency for the entangled PDW/CDW order to become leading compared to BDW/SSC in the model, which suggests that the former composite order might be indeed a viable concept to describe some cuprate superconductors at high temperatures in the underdoped regime, as has been recently alluded to by many authors in the literature.