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arXiv:cond-mat/0504692 · Progress of Theoretical Physics Supplement

Competing Orders and Non-Landau-Ginzburg-Wilson Criticality in (Bose) Mott Transitions

Leon Balents, Lorenz Bartosch, A. A. Burkov, Subir Sachdev A, K. Sengupta

Published Jan 1, 2005• 24 citations
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Abstract

This paper reviews a recent non-Landau-Ginzburg-Wilson (LGW) approach to superfluid to Mott insulator transitions in two dimensional bosonic lattice systems, using a dual vortex field theory [L. Balents, L. Bartosch, A. Burkov, S. Sachdev and K. Sengupta, Phys. Rev. B 71 (2005), 144508; Phys. Rev. B 71 (2005), 144509]. The physical interpretation of conventional LGW theory of quantum criticality is re-examined and similarities and differences with the vortex picture are discussed. The “unification” of various competing (insulating) orders, and the coincidence of these orders with the Mott transition are readily understood formulation. Some aspects of the recent theory of “deconfined” quantum criticality, which are to an extent subsumed approach, are discussed. A pedagogical presentation of the “nuts and bolts” of boson-vortex duality at the hamiltonian level is included, tailored to a condensed matter audience.

DOI
10.1143/ptps.160.314

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This paper reviews a recent non-Landau-Ginzburg-Wilson (LGW) approach to superfluid to Mott insulator transitions in two dimensional bosonic lattice systems, using a dual vortex field theory [L. Balents, L. Bartosch, A. Burkov, S. Sachdev and K. Sengupta, Phys. Rev. B 71 (2005), 144508; Phys. Rev. B 71 (2005), 144509]. The physical interpretation of conventional LGW theory of quantum criticality is re-examined and similarities and differences with the vortex picture are discussed. The “unification” of various competing (insulating) orders, and the coincidence of these orders with the Mott transition are readily understood formulation. Some aspects of the recent theory of “deconfined” quantum criticality, which are to an extent subsumed approach, are discussed. A pedagogical presentation of the “nuts and bolts” of boson-vortex duality at the hamiltonian level is included, tailored to a condensed matter audience.
We describe the localization transition of superfluids on two-dimensional lattices into commensurate Mott insulators with average particle density p∕qp∕qp∕q (p,qp,qp,q relatively prime integers) per lattice site. For bosons on the square lattice, we argue that the superfluid has at least qqq degenerate species of vortices which transform under a projective representation of the square-lattice space group (a PSG). The formation of a single-vortex condensate produces the Mott insulator, which is required by the PSG to have density wave order at wavelengths of q∕nq∕nq∕n lattice sites (nnn integer) along the principle axes; such a second-order transition is forbidden in the Landau-Ginzburg-Wilson frame-work. We also discuss the superfluid-insulator transition in the direct boson representation and find that an interpretation of the quantum criticality in terms of deconfined fractionalized bosons is only permitted at special values of qqq for which a permutative representation of the PSG exists. We argue [and demonstrate in detail in a companion paper: L. Balents et al., following paper, Phys. Rev. B 71, 144509 (2005)] that our results apply essentially unchanged to electronic systems with short-range pairing, with the PSG determined by the particle density of Cooper pairs. We also describe the effect of static impurities in the superfluid: the impurities locally break the degeneracy between the qqq vortex species, and this induces density-wave order near each vortex. We suggest that such a theory offers an appealing rationale for the local density-of-states modulations observed by Hoffman et al. [Science 295, 466 (2002)], in scanning tunneling microscopy (STM) studies of the vortex lattice of ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$ and allows a unified description of the nucleation of density-wave order in zero and finite magnetic fields. We note signatures of our theory that may be tested by future STM experiments.
We examine recent theoretical predictions for two-dimensional lattice bosonic/spin systems that superfluid-solid (SF-S) quantum phase transitions are often continuous and do not fit the canonical Ginzburg-Landau-Wilson (GLW) picture.We show that the GLW theory based on extended broken symmetry is consistent with existing numerical data, which report weak first-order transitions or problems with the finite-size scaling.This GLW theory excludes generic continuous SF-S transitions and quantum disordered phases.Remarkably, the phase diagram of the deconfined critical action itself features a weak first-order phase transition with spinon deconfinement, in accordance with the conventional theory.Problems with data collapse even in the regime of weak interactions between the spinons strongly suggest that the first-order deconfinement transition is the only possible outcome. * )Since the first study of the SF-VBS point within the J-current model, we have observed numerous first-order SF-VBS transitions for different sets of Hamiltonian parameters.
The Landau-Ginzburg-Wilson (LGW) paradigm is the backbone of the modern understanding of critical phenoma. It rests on the assumption that a continuous phase transition can be described solely in terms of a fluctuating order parameter. In this sandwich thesis, we analyze several quantum phase transitions in two-dimensional systems where this basic assumption is violated. Our analysis is structured in three parts: In the first part, two ordering transitions in metals are studied, where the presence of gapless fermionic modes invalidates the pure order parameter description. Instead, we apply a renormalization group approach which retains both fermionic and bosonic (order paramater) degrees of freedom, using the deviation from an upper critical dimension of d = 5/2 as a control parameter. We first apply this technique to describe the quantum phase transition between a normal metal and an inhomogeneous (FFLO) superconductor. Our analysis rigorously confirms the mean field expectation that this transition is continuous, and shows that interesting non-Fermi liquid physics can arise at the critical point, manifesting itself in unusual scaling of various observables. The second case study is the onset of incommensurate 2kF charge density wave order. While non-Fermi liquid features are less pronounced at this transition, we find a strong dynamical nesting of the Fermi surface, which stabilizes the density wave formation and results in a continuous transition, opposed to early theoretical claims. In the second part, we present a novel case study of “deconfined criticality” for a quantum magnet: Here, symmetries are broken on both sides of the phase transition, which would be first order within a conventional two-order-parameter LGW description. By contrast, the scenario of deconfined criticality predicts a continuous transition, driven by condensation of topological defects which carry quantum numbers of the opposite phase. Its paradigmatic application is the transition from a SU (2)-Neél state to a valence-bond solid (a singlet covering configuration that breaks spatial symmetries) on the square lattice. Here, we propose a novel extension and study a SU (3) antiferromagnet on a triangular lattice, which supports a transition between a magnetic, three-sublattice color-ordered phase and a trimerized SU (3) singlet phase. We provide a critical theory in terms of fractional bosonic degrees of freedom (and with a topological defect interpretation), and study its fixed point properties with functional renormalization group. This yields a critical fixed point in a suitable large-N -limit, implying a continuous transition in accordance with the deconfined criticality framework. In the third and final part, we study the “polaron”-problem of a single impurity coupled to a majority Fermi sea, which is particularly interesting when formation of a bound state (“molecule”) between impurity and majority is allowed for. Starting from the exactly solvable limit of infinite impurity mass, we present a controlled computation of impurity spectra (both single- and two-particle) for heavy impurities based on Feynman diagram techniques, with an eye for experiments on doped semiconductors and ultracold gases. Furthermore, we discuss various aspects of the “molecule-to-polaron” transition that occurs in these systems, which also defies a simple LGW description in the single-impurity limit.
We present the critical theory of a number of zero-temperature phase transitions of quantum antiferromagnets and interacting boson systems in two dimensions. The most important example is the transition of the S=1∕2S=1∕2S=1∕2 square lattice antiferromagnet between the N\'eel state (which breaks spin rotation invariance) and the paramagnetic valence bond solid (which preserves spin rotation invariance but breaks lattice symmetries). We show that these two states are separated by a second-order quantum phase transition. This conflicts with Landau-Ginzburg-Wilson theory, which predicts that such states with distinct broken symmetries are generically separated either by a first-order transition, or by a phase with co-existing orders. The critical theory of the second-order transition is not expressed in terms of the order parameters characterizing either state, but involves fractionalized degrees of freedom and an emergent, topological, global conservation law. A closely related theory describes the superfluid-insulator transition of bosons at half filling on a square lattice, in which the insulator has a bond density wave order. Similar considerations are shown to apply to transitions of antiferromagnets between the valence bond solid and the Z2{Z}_{2}Z2​ spin liquid: the critical theory has deconfined excitations interacting with an emergent U(1)\mathrm{U}(1)U(1) gauge force. We comment on the broader implications of our results for the study of quantum criticality in correlated electron systems.
The Mott critical point between a metal and a correlated insulator has usually been studied via density or spin density bosonic mode fluctuations according to the standard Ginzburg-Landau-Wilson phase transition paradigm. A moment's reflection leads to increasing doubts that such an approach should work as the transition is nonmagnetic, voiding the relevance of spin density modes. Charge density modes are irrelevelant since the long range Coulomb interaction leads to a large plasmon gap and their incompressibility. In solidarity with these doubts, recent measurements of the Mott critical point in low dimensional organic materials yield critical exponents in violent diasagreement with the bosonic mode criticality lore. We propose that fermionic fluctuations control the behavior of the Mott transition. The transition thus has an intrinsic quantum aspect despite being a finite temperature phase transition. We develop this hitherto unexplored physics, obtain experimental predictions and find agreement with one of the novel unexplained experimental exponents. We conclude that this Mott transition corresponds to a new universality class of finite temperature critical points that contains quantum effects and cannot be accounted for by conventional Ginzburg-Landau-Wilson wisdom.
Two-dimensional quantum systems with competing orders can feature a deconfined quantum critical point, yielding a continuous phase transition that is incompatible with the Landau-Ginzburg-Wilson scenario, predicting instead a first-order phase transition. This is caused by the LGW order parameter breaking up into new elementary excitations at the critical point. Canonical candidates for deconfined quantum criticality are quantum antiferromagnets with competing magnetic orders, captured by the easy-plane CP^{1} model. A delicate issue however is that numerics indicates the easy-plane CP^{1} antiferromagnet to exhibit a first-order transition. Here we show that an additional topological Chern-Simons term in the action changes this picture completely in several ways. We find that the topological easy-plane antiferromagnet undergoes a second-order transition with quantized critical exponents. Further, a particle-vortex duality naturally maps the partition function of the Chern-Simons easy-plane antiferromagnet into one of massless Dirac fermions.