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We investigate a quantum dynamical phase transition induced by the competition between local unitary evolution and dissipation in a qubit chain with a strong, on-site $\mathbb{Z}_2$ symmetry. While the steady-state of this evolution is always maximally-mixed, we show that the dynamical behavior of certain non-local observables on the approach to this steady-state is dictated by a quantum Ising model with a $\textit{complex}$ transverse-field (cTFIM). We investigate these observables analytically, uncovering a dynamical phase transition as the relative rate of unitary evolution and dissipation is tuned. We show that the weak-dissipation limit corresponds to a cTFIM with a large magnitude of the imaginary transverse-field, for which the many-body "ground-state" (with smallest real eigenvalue) is gapless, exhibiting quasi-long-range correlations of the local magnetization with a continuously-varying exponent. Correspondingly, the dynamics of the non-local observables show oscillatory behavior with an amplitude decaying exponentially in time. The strong-dissipation limit corresponds to a gapped ferromagnetic phase of the cTFIM, and non-local observables show exponential decay on the approach to equilibrium. This transition in (1+1)-dimensions has a peculiar, "two-sided" nature appearing as either first- or second-order depending on the phase from which the transition is approached, an analytic result which is corroborated by numerical studies. In higher dimensions, we present a field-theoretic understanding of the first-order nature of this transition, when approaching from the ferromagnetic phase of the cTFIM, though the nature of the phase with large imaginary transverse-field remains to be understood., Comment: 15 + 14 pages, 8 figures

We demonstrate that, starting with a simple fermion wave function, the steady mixed state of the evolution of a class of Lindbladians, and the ensemble created by strong local measurement of fermion density without post-selection can be mapped to the "Gutzwiller projected" wave functions in the doubled Hilbert space -- the representation of the density matrix through the Choi-Jamiolkowski isomorphism. A Gutzwiller projection is a broadly used approach of constructing spin liquid states. For example, if one starts with a gapless free Dirac fermion pure quantum state, the constructed mixed state corresponds to an algebraic spin liquid in the doubled Hilbert space. We also predict that for some initial fermion wave function, the mixed state created following the procedure described above is expected to have a spontaneous "strong-to-weak" U(1) symmetry breaking, which corresponds to the emergence of superconductivity in the doubled Hilbert space. We also design the experimental protocol to construct the desired physics of mixed states., Comment: 11 pages, 2 figures, more results and new authors added

Measurements can detect errors in a decohered quantum memory allowing active error correction to increase the memory time. Previous understanding of this mechanism has focused on evaluating the performance of error correction algorithms based on measurement results. In this work, we instead intrinsically characterize the information dynamics in a quantum memory under repeated measurements, using coherent information and relative entropy. We consider the dynamics of a $d$-dimensional stabilizer code subject to Pauli errors and noisy stabilizer measurements and develop a $(d+1)$-dimensional statistical mechanics model for the information-theoretic diagnostics. Our model is dual to the model previously obtained for the optimal decoding algorithm, and the potential decoding transition in the quantum memory again manifests as a thermal phase transition in the statistical mechanics model. We explicitly derive the model and study the phase transition in information encoding in three examples: surface codes, repetition codes, and the XZZX code., Comment: 27 pages, 9 figures

Measurements allow efficient preparation of interesting quantum many-body states with long-range entanglement, conditioned on additional transformations based on measurement outcomes. Here, we demonstrate that the so-called conformal quantum critical points (CQCP) can be obtained by performing general single-site measurements in an appropriate basis on the cluster states in $d\geq2$. The equal-time correlators of the said states are described by correlation functions of certain $d$-dimensional classical models at finite temperatures and feature spatial conformal invariance. This establishes an exact correspondence between the measurement-prepared critical states and conformal field theories of a range of critical spin models, including familiar Ising models and gauge theories. Furthermore, by mapping the long-range entanglement structure of measured quantum states into the correlations of the corresponding thermal spin model, we rigorously establish the stability condition of the long-range entanglement in the measurement-prepared quantum states deviating from the ideal setting. Most importantly, we describe protocols to decode the resulting quantum phases and transitions without post-selection, thus transferring the exponential measurement complexity to a polynomial classical computation. Therefore, our findings suggest a novel mechanism in which a quantum critical wavefunction emerges, providing new practical ways to study quantum phases and conformal quantum critical points., Comment: 37 pages, 11 figures

We introduce an exact mapping of Clifford (stabilizer) random tensor networks (RTNs) and monitored quantum circuits, onto a statistical mechanics model. With Haar unitaries, the fundamental degrees of freedom ('spins') are permutations because all operators commuting with the action of the unitaries on a tensor product arise from permutations of the tensor factors ('Schur-Weyl duality'). For unitaries restricted to the smaller Clifford group, the set of commuting operators, the 'commutant', forming the new 'spin' degrees of freedom, will be larger. We use the recent full characterization of this commutant by Gross et al., Comm. Math. Phys. 385, 1325 (2021), to construct the Clifford statistical mechanics models for on-site Hilbert space dimensions which are powers of a prime number $p$. We show that the Boltzmann weights are invariant under a symmetry group involving orthogonal matrices with entries in the finite number field ${\bf F}_p$. This implies that the symmetry group, and consequently all universal properties of entanglement transitions in Clifford circuits and RTNs will in general depend on, and only on the prime $p$. We show that Clifford monitored circuits with on-site Hilbert space dimension $d=p^M$ are described by percolation in the limits $d \to \infty$ at (a) $p=$ fixed but $M\to \infty$, and at (b) $M= 1$ but $p \to \infty$. In the limit (a) we calculate the effective central charge, and in the limit (b) we derive the following universal minimal cut entanglement entropy $S_A =(\sqrt{3}/\pi)\ln p \ln L_A$ for $d=p$ large at the transition. We verify those predictions numerically, and present extensive numerical results for critical exponents at the transition in monitored Clifford circuits for prime number on-site Hilbert space dimension $d=p$ for a variety of different values of $p$, and find that they approach percolation values at large $p$., Comment: 23 pages, 5 figures. Abstract shortened to meet arxiv requirements, see pdf for full abstract

We propose entanglement negativity as a fine-grained probe of measurement-induced criticality. We motivate this proposal in stabilizer states, where for two disjoint subregions, comparing their "mutual negativity" and their mutual information leads to a precise distinction between bipartite and multipartite entanglement. In a measurement-only stabilizer circuit that maps exactly to two-dimensional critical percolation, we show that the mutual information and the mutual negativity are governed by boundary conformal fields of different scaling dimensions at long distances. We then consider a class of "hybrid" circuit models obtained by perturbing the measurement-only circuit with unitary gates of progressive levels of complexity. While other critical exponents vary appreciably for different choices of unitary gate ensembles at their respective critical points, the mutual negativity has scaling dimension 3 across remarkably many of the hybrid circuits, which is notably different from that in percolation. We contrast our results with limiting cases where a geometrical minimal-cut picture is available.

We establish the emergence of a conformal field theory (CFT) in a (1+1)-dimensional hybrid quantum circuit right at the measurement-driven entanglement transition by revealing space-time conformal covariance of entanglement entropies and mutual information for various subregions at different circuit depths. While the evolution takes place in real time, the spacetime manifold of the circuit appears to host a Euclidean field theory with imaginary time. Throughout the paper we investigate Clifford circuits with several different boundary conditions by injecting physical qubits at the spatial and/or temporal boundaries, all giving consistent characterizations of the underlying "Clifford CFT." We emphasize (super)universal results that are consequences solely of the conformal invariance and do not depend crucially on the precise nature of the CFT. Among these are the infinite entangling speed as a consequence of measurement-induced quantum nonlocality and the critical purification dynamics of a mixed initial state., Comment: 31 pages, 16 figures. v2: updated references and acknowledgements. v3: new Appendix on "localizable entanglement". Published version

We introduce a procedure to systematically search for a local unitary transformation that maps a wavefunction with a non-trivial sign structure into a positive-real form. The transformation is parametrized as a quantum circuit compiled into a set of one and two qubit gates. We design a cost function that maximizes the average sign of the output state and removes its complex phases. The optimization of the gates is performed through automatic differentiation algorithms, widely used in the machine learning community. We provide numerical evidence for significant improvements in the average sign for a two-leg triangular Heisenberg ladder with next-to-nearest neighbour and ring-exchange interactions. This model exhibits phases where the sign structure can be removed by simple local one-qubit unitaries, but also an exotic Bose-metal phase whose sign structure induces "Bose surfaces" with a fermionic character and a higher entanglement that requires deeper circuits., Comment: 9 pages, 5 figures

In this paper we continue to explore "hybrid" quantum circuit models in one-dimension with both unitary and measurement gates, focussing on the entanglement properties of wavefunction trajectories at long times, in the steady state. We simulate a large class of Clifford circuits, including models with or without randomness in the unitary gates, and with or without randomness in the locations of measurement gates, using stabilizer techniques to access the long time dynamics of systems up to 512 qubits. In all models we find a volume law entangled phase for low measurement rates, which exhibits a sub-dominant logarithmic behavior in the entanglement entropy, $S_A = {\alpha} \ln |A| + s|A|$, with sub-system size $|A|$. With increasing measurement rate the volume law phase is unstable to a disentangled area law phase, passing through a single entanglement transition at a critical rate of measurement. At criticality we find a purely logarithmic entanglement entropy, $S_A = {\alpha}(p_c) \ln|A|$, a power law decay and conformal symmetry of the mutual information, with exponential decay off criticality. Various spin-spin correlation functions also show slow decay at criticality. Critical exponents are consistent across all models, indicative of a single universality class. These results suggest the existence of an effective underlying statistical mechanical model for the entanglement transition. Beyond Clifford circuit models, numerical simulations of up to 20 qubits give consistent results., Comment: 30 pages, 28 figures. Published version, w/ updated discussion, acknowledgement, and references

We introduce and explore a one-dimensional "hybrid" quantum circuit model consisting of both unitary gates and projective measurements. While the unitary gates are drawn from a random distribution and act uniformly in the circuit, the measurements are made at random positions and times throughout the system. By varying the measurement rate we can tune between the volume law entangled phase for the random unitary circuit model (no measurements) and a "quantum Zeno phase" where strong measurements suppress the entanglement growth to saturate in an area-law. Extensive numerical simulations of the quantum trajectories of the many-particle wavefunctions (exploiting Clifford circuitry to access systems up to 512 qubits) provide evidence for a stable "weak measurement phase" that exhibits volume-law entanglement entropy, with a coefficient decreasing with increasing measurement rate. We also present evidence for a novel continuous quantum dynamical phase transition between the "weak measurement phase" and the "quantum Zeno phase", driven by a competition between the entangling tendencies of unitary evolution and the disentangling tendencies of projective measurements. Detailed steady-state and dynamic critical properties of this novel quantum entanglement transition are accessed., Comment: 10 pages, 6 figures. Updated references & minor changes

We investigate the implications of integrability for the existence of quantum disentangled liquid (QDL) states in the half-filled one-dimensional Hubbard model. We argue that there exist finite energy-density eigenstates that exhibit QDL behaviour in the sense of J. Stat. Mech. P10010 (2014). These states are atypical in the sense that their entropy density is smaller than that of thermal states at the same energy density. Furthermore, we show that thermal states in a particular temperature window exhibit a weaker form of the QDL property, in agreement with recent results obtained by strong-coupling expansion methods in arXiv:1611.02075. This article is part of the themed issue `Breakdown of ergodicity in quantum systems: from solids to synthetics matter'., Comment: 11 pages, 3 figures. arXiv admin note: text overlap with arXiv:1611.02075

We investigate the existence of quantum disentangled liquid (QDL) states in the half-filled Hubbard model on bipartite lattices. In the one dimensional case we employ a combination of integrability and strong coupling expansion methods to argue that there are indeed finite energy-density eigenstates that exhibit QDL behaviour in the sense of J. Stat. Mech. P10010 (2014). The states exhibiting the QDL property are atypical in the sense that while their entropy density is non-zero, it is smaller than that of thermal states at the same energy density. We argue that for U >> t these latter thermal states exhibit a weaker form of the QDL property, which carries over to the higher dimensional case., Comment: 11 pages, 3 figures, accepted version

We study the possible breakdown of quantum thermalization in a model of itinerant electrons on a one-dimensional chain without disorder, with both spin and charge degrees of freedom. The eigenstates of this model exhibit peculiar properties in the entanglement entropy, the apparent scaling of which is modified from a "volume law" to an "area law" after performing a partial, site-wise measurement on the system. These properties and others suggest that this model realizes a new, non-thermal phase of matter, known as a quantum disentangled liquid (QDL). The putative existence of this phase has striking implications for the foundations of quantum statistical mechanics., Comment: As accepted to PRB

We provide an effective description of a particle-hole symmetric state of electrons in a half-filled Landau level, starting from the traditional approach pioneered by Halperin, Lee and Read. Specifically, we study a system consisting of alternating quasi-one-dimensional strips of composite Fermi liquid (CFL) and composite hole liquid (CHL), both of which break particle-hole symmetry. When the CFL and CHL strips are identical in size, the resulting state is manifestly invariant under the combined action of a particle-hole transformation with respect to a single Landau level (which interchanges the CFL and CHL) and translation by one unit, equal to the strip width, in the direction transverse to the strips. At distances long compared to the strip width, we demonstrate that the system is described by a Dirac fermion coupled to an emergent gauge field, with an anti-unitary particle-hole symmetry, as recently proposed by Son., Comment: 27 pages, 4 figures

The half-filled Landau level is widely believed to be described by the Halperin-Lee-Read theory of the composite Fermi liquid (CFL). In this paper, we develop a theory for the particle-hole conjugate of the CFL, the Anti-CFL, which we argue to be a distinct phase of matter as compared with the CFL. The Anti-CFL provides a possible explanation of a recent experiment [Kamburov et. al., Phys. Rev. Lett. 113, 196801 (2014)] demonstrating that the density of composite fermions in GaAs quantum wells corresponds to the electron density when the filling fraction $\nu < 1/2$ and to the hole density when $\nu > 1/2$. We introduce a local field theory for the CFL and Anti-CFL in the presence of a boundary, which we use to study CFL - Insulator - CFL junctions, and the interface between the Anti-CFL and CFL. We show that the CFL - Anti-CFL interface allows partially fused boundary phases in which "composite electrons" can directly tunnel into "composite holes," providing a non-trivial example of transmutation between topologically distinct quasiparticles. We discuss several observable consequences of the Anti-CFL, including a predicted resistivity jump at a first order transition between uniform CFL and Anti-CFL phases. We also present a theory of a continuous quantum phase transition between the CFL and Anti-CFL. We conclude that particle-hole symmetry requires a modified view of the half-filled Landau level, in the presence of strong electron-electron interactions and weak disorder, as a critical point between the CFL and the Anti-CFL., Comment: 29 pages, 8 figures. v2 includes additional discussion of Landau level mixing (Sec. VII D) and paired composite fermion states

Many body quantum eigenstates of generic Hamiltonians at finite energy density typically satisfy "volume law" of entanglement entropy: the von Neumann entanglement entropy and the Renyi entropies for a subregion scale in proportion to its volume. Here we provide a connection between the volume law and the sign structure of eigenstates. In particular, we ask the question: can a positive wavefunction support a volume law entanglement? Remarkably, we find that a typical random positive wavefunction, exhibits a constant law for Renyi entanglement entropies $S_n$ for $n>1$, despite arbitrary large amplitude fluctuations. We also provide evidence that the modulus of the finite energy density eigenstates of generic local Hamiltonians show similar behavior., Comment: 6 pages, 3 figures

More than half a century after first being proposed by Sir Nevill Mott, the deceptively simple question of whether the interaction-driven electronic metal-insulator transition may be continuous remains enigmatic. Recent experiments on two-dimensional materials suggest that when the insulator is a quantum spin liquid, lack of magnetic long-range order on the insulating side may cause the transition to be continuous, or only very weakly first order. Motivated by this, we study a half-filled extended Hubbard model on a triangular lattice strip geometry. We argue, through use of large-scale numerical simulations and analytical bosonization, that this model harbors a continuous (Kosterlitz-Thouless-like) quantum phase transition between a metal and a gapless spin liquid characterized by a spinon Fermi surface, i.e., a "spinon metal." These results may provide a rare insight into the development of Mott criticality in strongly interacting two-dimensional materials and represent one of the first numerical demonstrations of a Mott insulating quantum spin liquid phase in a genuinely electronic microscopic model., Comment: 18 pages, 9 figures

We study the spin-1/2 Heisenberg model on the square lattice with first- and second-neighbor antiferromagnetic interactions J1 and J2, which possesses a nonmagnetic region that has been debated for many years and might realize the interesting Z2 spin liquid. We use the density matrix renormalization group approach with explicit implementation of SU(2) spin rotation symmetry and study the model accurately on open cylinders with different boundary conditions. With increasing J2, we find a Neel phase, a plaquette valence-bond (PVB) phase with a finite spin gap, and a possible spin liquid in a small region of J2 between these two phases. From the finite-size scaling of the magnetic order parameter, we estimate that the Neel order vanishes at J2/J1~0.44. For 0.5

We study point-contact tunneling in the integer quantum Hall state of bosons. This symmetry-protected topological state has electrical Hall conductivity equal to $2 e^2/h$ and vanishing thermal Hall conductivity. In contrast to the integer quantum Hall state of fermions, a point contact can have a dramatic effect on the low energy physics. In the absence of disorder, a point contact generically leads to a partially-split Hall bar geometry. We describe the resulting intermediate fixed point via the two-terminal electrical (Hall) conductance of the edge modes. Disorder along the edge, however, both restores the universality of the two-terminal conductance and helps preserve the integrity of the Hall bar within the relevant parameter regime., Comment: 12 pages, 5 figures; v.2: typos fixed and clarified some arguments

Non-Abelian anyons promise to reveal spectacular features of quantum mechanics that could ultimately provide the foundation for a decoherence-free quantum computer. A key breakthrough in the pursuit of these exotic particles originated from Read and Green's observation that the Moore-Read quantum Hall state and a (relatively simple) two-dimensional p+ip superconductor both support so-called Ising non-Abelian anyons. Here we establish a similar correspondence between the Z_3 Read-Rezayi quantum Hall state and a novel two-dimensional superconductor in which charge-2e Cooper pairs are built from fractionalized quasiparticles. In particular, both phases harbor Fibonacci anyons that---unlike Ising anyons---allow for universal topological quantum computation solely through braiding. Using a variant of Teo and Kane's construction of non-Abelian phases from weakly coupled chains, we provide a blueprint for such a superconductor using Abelian quantum Hall states interlaced with an array of superconducting islands. Fibonacci anyons appear as neutral deconfined particles that lead to a two-fold ground-state degeneracy on a torus. In contrast to a p+ip superconductor, vortices do not yield additional particle types yet depending on non-universal energetics can serve as a trap for Fibonacci anyons. These results imply that one can, in principle, combine well-understood and widely available phases of matter to realize non-Abelian anyons with universal braid statistics. Numerous future directions are discussed, including speculations on alternative realizations with fewer experimental requirements., Comment: 37 pages. v2 fixed typos and clarified some issue

We propose and explore a new finite temperature phase of translationally invariant multi-component liquids which we call a "Quantum Disentangled Liquid" (QDL) phase. We contemplate the possibility that in fluids consisting of two (or more) species of indistinguishable quantum particles with a large mass ratio, the light particles might "localize" on the heavy particles. We give a precise, formal definition of this Quantum Disentangled Liquid phase in terms of the finite energy density many-particle wavefunctions. While the heavy particles are fully thermalized, for a typical fixed configuration of the heavy particles, the entanglement entropy of the light particles satisfies an area law; this implies that the light particles have not thermalized. Thus, in a QDL phase, thermal equilibration is incomplete, and the canonical assumptions of statistical mechanics are not fully operative. We explore the possibility of QDL in water, with the light proton degrees of freedom becoming "localized" on the oxygen ions. We do not presently know whether a local, generic Hamiltonian can have eigenstates of the QDL form, and if it can not, then the non-thermal behavior discussed here will exist as an interesting crossover phenomena at time scales that diverge as the ratio of the mass of the heavy to the light species also diverges., Comment: 14 pages

We use density matrix renormalization group (DMRG) algorithm to study the phase diagram of the spin-1/2 Heisenberg model on honeycomb lattice with first (J1) and second (J2) neighbor antiferromagnetic interactions, where a Z2 spin liquid region has been proposed. By implementing SU(2) symmetry in the DMRG code, we are able to obtain accurate results for long cylinders with width slightly over 15 lattice spacings and torus up to the size N=2*6*6. With increasing J2, we find a Neel phase with vanishing spin gap and a plaquette valence-bond (PVB) phase with non-zero spin gap. By extrapolating the square of the staggered magnetic moment ms^2 on finite-size cylinders to thermodynamic limit, we find the Neel order vanishing at J2/J1~0.22. For 0.25

A 3d electron topological insulator (ETI) is a phase of matter protected by particle-number conservation and time-reversal symmetry. It was previously believed that the surface of an ETI must be gapless unless one of these symmetries is broken. A well-known symmetry-preserving, gapless surface termination of an ETI supports an odd number of Dirac cones. In this paper we deduce a symmetry-respecting, gapped surface termination of an ETI, which carries an intrinsic 2d topological order, Moore-Read x anti-semion. The Moore-Read sector supports non-Abelian charge 1/4 anyons, while the Abelian, anti-semion sector is electrically neutral. Time-reversal symmetry is implemented in this surface phase in a highly non-trivial way. Moreover, it is impossible to realize this phase strictly in 2d, simultaneously preserving its implementation of both the particle number and time-reversal symmetries. A 1d edge on the ETI surface between the topologically-ordered phase and the topologically trivial time-reversal-broken phase with a Hall conductivity sigma_H = 1/2, carries a right-moving neutral Majorana mode, a right-moving bosonic charge mode and a left-moving bosonic neutral mode. The topologically-ordered phase is separated from the surface superconductor by a direct second order phase transition in the XY* universality class, which is driven by the condensation of a charge 1/2 boson, when approached from the topologically-ordered side, and proliferation of a flux 4\pi-vortex, when approached from the superconducting side. In addition, we prove that time-reversal invariant (interacting) electron insulators with no intrinsic topological order and electromagnetic response characterized by a \theta-angle, \theta = \pi, do not exist if the electrons transform as Kramers singlets under time-reversal., Comment: 35 pages

It is well-known that one signature of the three-dimensional electron topological insulator is the Witten effect: if the system is coupled to a compact electromagnetic gauge field, a monopole in the bulk acquires a half-odd-integer polarization charge. In the present work, we propose a corresponding phenomenon for the topological insulator of bosons in 3d protected by particle number conservation and time-reversal symmetry. We claim that although a monopole inside a topological insulator of bosons can remain electrically neutral, its statistics are transmuted from bosonic to fermionic. We demonstrate that this ``statistical Witten effect" directly implies that if the surface of the topological insulator is neither gapless, nor spontaneously breaks the symmetry, it necessarily supports an intrinsic two-dimensional topological order. Moreover, the surface properties cannot be fully realized in a purely 2d system. We also confirm that the surface phases of the bosonic topological insulator proposed by Vishwanath and Senthil (arXiv:1209.3058) provide a consistent termination of a bulk exhibiting the statistical Witten effect. In a companion paper, we will provide an explicit field-theoretic, lattice-regularized, construction of the 3d topological insulator of bosons, employing a parton decomposition and subsequent condensation of parton-monopole composites., Comment: 27 pages, 1 figure

We calculate the electron spectral function for a spin-liquid with a spinon Fermi surface and a Dirac spin-liquid. Calculations are based upon the slave-rotor mean-field theory. We consider the effect of gauge fluctuations using a simple model and find the behavior is not strongly modified. The results, distinct from conventional Mott insulator or band theory predictions, suggest that measuring the spectral function e.g. via ARPES could help in the experimental verification and characterization of spin liquids., Comment: 7 pages, 7 figures

Developing a theoretical framework for conducting electronic fluids qualitatively distinct from those described by Landau's Fermi-liquid theory is of central importance to many outstanding problems in condensed matter physics. One such problem is that, above the transition temperature and near optimal doping, high-transition-temperature copper-oxide superconductors exhibit `strange metal' behaviour that is inconsistent with being a traditional Landau Fermi liquid. Indeed, a microscopic theory of a strange-metal quantum phase could shed new light on the interesting low-temperature behaviour in the pseudogap regime and on the d-wave superconductor itself. Here we present a theory for a specific example of a strange metal---the 'd-wave metal'. Using variational wavefunctions, gauge theoretic arguments, and ultimately large-scale density matrix renormalization group calculations, we show that this remarkable quantum phase is the ground state of a reasonable microscopic Hamiltonian---the usual t-J model with electron kinetic energy $t$ and two-spin exchange $J$ supplemented with a frustrated electron `ring-exchange' term, which we here examine extensively on the square lattice two-leg ladder. These findings constitute an explicit theoretical example of a genuine non-Fermi-liquid metal existing as the ground state of a realistic model., Comment: 22 pages, 12 figures: 6 pages, 7 figures of main text + 16 pages, 5 figures of Supplementary Information; this is approximately the version published in Nature, minus various subedits in the main text

We establish compelling evidence for the existence of new quasi-one-dimensional descendants of the d-wave Bose liquid (DBL), an exotic two-dimensional quantum phase of uncondensed itinerant bosons characterized by surfaces of gapless excitations in momentum space [O. I. Motrunich and M. P. A. Fisher, Phys. Rev. B {\bf 75}, 235116 (2007)]. In particular, motivated by a strong-coupling analysis of the gauge theory for the DBL, we study a model of hard-core bosons moving on the $N$-leg square ladder with frustrating four-site ring exchange. Here, we focus on four- and three-leg systems where we have identified two novel phases: a compressible gapless Bose metal on the four-leg ladder and an incompressible gapless Mott insulator on the three-leg ladder. The former is conducting along the ladder and has five gapless modes, one more than the number of legs. This represents a significant step forward in establishing the potential stability of the DBL in two dimensions. The latter, on the other hand, is a fundamentally quasi-one-dimensional phase that is insulating along the ladder but has two gapless modes and incommensurate power law transverse density-density correlations. In both cases, we can understand the nature of the phase using slave-particle-inspired variational wave functions consisting of a product of two distinct Slater determinants, the properties of which compare impressively well to a density matrix renormalization group solution of the model Hamiltonian. Stability arguments are made in favor of both quantum phases by accessing the universal low-energy physics with a bosonization analysis of the appropriate quasi-1D gauge theory. We will briefly discuss the potential relevance of these findings to high-temperature superconductors, cold atomic gases, and frustrated quantum magnets., Comment: 33 pages, 16 figures; this is the print version, only very minor changes from v1

Motivated by recent experiments on material Ba3NiSb2O9, we propose two novel spin liquid phases (A and B) for spin-1 systems on a triangular lattice. At the mean field level, both spin liquid phases have gapless fermionic spinon excitations with quadratic band touching, thus in both phases the spin susceptibility and C_v/T saturate to a constant at zero temperature, which are consistent with the experimental results on Ba3NiSb2O9. On the lattice scale, these spin liquid phases have Sp(4) ~ SO(5) gauge fluctuation; while in the long wavelength limit this Sp(4) gauge symmetry is broken down to U(1)xZ_2 in type A spin liquid phase, and broken down to Z_4 in type B phase. We also demonstrate that the $A$ phase is the parent state of the ferro-quadrupole state, nematic state, and the noncollinear spin density wave state., Comment: 5 pages, 1 figure

We show that long-ranged superconducting order is not necessary to guarantee the existence of Majorana fermion zero modes at the ends of a quantum wire. We formulate a concrete model which applies, for instance, to a semiconducting quantum wire with strong spin-orbit coupling and Zeeman splitting coupled to a wire with algebraically-decaying superconducting fluctuations. We solve this model by bosonization and show that it supports Majorana fermion zero modes. We argue that a large class of models will also show the same phenomenon. We discuss the implications for experiments on spin-orbit coupled nanowires coated with superconducting film and for LaAlO3/SrTiO3 interfaces., Comment: 14 pages. Figures added and a discussion of the effects of quantum phase slips. References Added. Fourth author added

Among the broad spectrum of systems predicted to exhibit topological superconductivity and Majorana fermions, one-dimensional wires with strong spin-orbit coupling provide one of the most promising experimental candidates. Here we investigate the fate of the topological superconducting phase in such wires when repulsive interactions are present. Using a combination of Density Matrix Renormalization Group, bosonization, and Hartree-Fock techniques, we demonstrate that while interactions degrade the bulk gap -consistent with recent results of Gangadharaiah et al.- they also greatly expand the parameter range over which the topological phase arises. In particular, we show that with interactions this phase can be accessed over a broader chemical potential window, thereby leading to greater immunity against disorder-induced chemical potential fluctuations in the wire. We also suggest that in certain wires strong interactions may allow Majorana fermions to be generated without requiring a magnetic field., Comment: 14 pages, 11 figures

We show that semiconductor nanowires coupled to an s-wave superconductor provide a playground to study effects of interactions between different topological superconducting phases supporting Majorana zero-energy modes. We consider quasi-one dimensional system where the topological phases emerge from different transverse subbands in the nanowire. In a certain parameter space, we show that there is a multicritical point in the phase diagram where the low-energy theory is equivalent to the one describing two coupled Majorana chains. We study effect of interactions as well as symmetry-breaking perturbations on the topological phase diagram in the vicinity of this multicritical point. Our results shed light on the stability of the topological phase around the multicritical point and have important implications for the experiments on Majorana nanowires., Comment: 8 pages, 2 figures; final version to appear in PRB

We study a spin-1/2 system with Heisenberg plus ring exchanges on a four-leg triangular ladder using the density matrix renormalization group and Gutzwiller variational wave functions. Near an isotropic lattice regime, for moderate to large ring exchanges we find a spin Bose-metal phase with a spinon Fermi sea consisting of three partially filled bands. Going away from the triangular towards the square lattice regime, we find a staggered dimer phase with dimers in the transverse direction, while for small ring exchanges the system is in a featureless rung phase. We also discuss parent states and a possible phase diagram in two dimensions., Comment: 4 pages, 5 figures, v3 is the print version

We present evidence for an exotic gapless insulating phase of hard-core bosons on multi-leg ladders with a density commensurate with the number of legs. In particular, we study in detail a model of bosons moving with direct hopping and frustrating ring exchange on a 3-leg ladder at $\nu=1/3$ filling. For sufficiently large ring exchange, the system is insulating along the ladder but has two gapless modes and power law transverse density correlations at incommensurate wave vectors. We propose a determinantal wave function for this phase and find excellent comparison between variational Monte Carlo and density matrix renormalization group calculations on the model Hamiltonian, thus providing strong evidence for the existence of this exotic phase. Finally, we discuss extensions of our results to other $N$-leg systems and to $N$-layer two-dimensional structures., Comment: 5 pages, 4 figures; v3 is the print version; supplemental material attached

Fermions in two-dimensions (2D) when subject to anisotropic spin-dependent hopping can potentially give rise to unusual paired states in {\it unpolarized} mixtures that can behave as non-Fermi liquids. One possibility is a fully paired state with a gap for fermion excitations in which the Cooper pairs remain uncondensed. Such a "Cooper-pair Bose-metal" phase would be expected to have a singular Bose-surface in momentum space. As demonstrated in the context of 2D bosons hopping with a frustrating ring-exchange interaction, an analogous Bose-metal phase has a set of quasi-1D descendent states when put on a ladder geometry. Here we present a density matrix renormalization group (DMRG) study of the attractive Hubbard model with spin-dependent hopping on a two-leg ladder geometry. In our setup, one spin species moves preferentially along the leg direction, while the other does so along the rung direction. We find compelling evidence for the existence of a novel Cooper-pair Bose-metal phase in a region of the phase diagram at intermediate coupling. We further explore the phase diagram of this model as a function of hopping anisotropy, density, and interaction strength, finding a conventional superfluid phase, as well as a phase of paired Cooper pairs with d-wave symmetry, similar to the one found in models of hard-core bosons with ring-exchange. We argue that simulating this model with cold Fermi gases on spin dependent optical lattices is a promising direction for realizing exotic quantum states., Comment: 10 pages, 12 figures

Composite fermions have played a seminal role in understanding the quantum Hall effect, particularly the formation of a compressible `composite Fermi liquid' (CFL) at filling factor nu = 1/2. Here we suggest that in multi-layer systems interlayer Coulomb repulsion can similarly generate `metallic' behavior of composite fermions between layers, even if the electrons remain insulating. Specifically, we propose that a quantum Hall bilayer with nu = 1/2 per layer at intermediate layer separation may host such an interlayer coherent CFL, driven by exciton condensation of composite fermions. This phase has a number of remarkable properties: the presence of `bonding' and `antibonding' composite Fermi seas, compressible behavior with respect to symmetric currents, and fractional quantum Hall behavior in the counterflow channel. Quantum oscillations associated with the Fermi seas give rise to a new series of incompressible states at fillings nu = p/[2(p \pm 1)] per layer (p an integer), which is a bilayer analogue of the Jain sequence., Comment: 4 pages, 3 figures

We propose a model for realizing exotic paired states in cold atomic Fermi gases. By using a {\it spin dependent} optical lattice it is possible to engineer spatially anisotropic Fermi surfaces for each hyperfine species, that are rotated 90 degrees with respect to one another. We consider a balanced population of the fermions with an attractive interaction. We explore the BCS mean field phase diagram as a function of the anisotropy, density, and interaction strength, and find the existence of an unusual paired superfluid state with coexisting pockets of momentum space with gapless unpaired carriers. This state is a relative of the Sarma or breached pair states in polarized mixtures, but in our case the Fermi gas is unpolarized. We also propose the possible existence of an exotic paired "Cooper-pair Bose-Metal" (CPBM) phase, which has a gap for single fermion excitations but gapless and uncondensed "Cooper pair" excitations residing on a "Bose-surface" in momentum space., Comment: 4 pages, 3 figs

Recent experiments on triangular lattice organic Mott insulators have found evidence for a 2D spin liquid in proximity to the metal-insulator transition. A Gutzwiller wavefunction study of the triangular lattice Heisenberg model with appropriate four-spin ring exchanges has found that the projected spinon Fermi sea state has a low variational energy. This wavefunction, together with a slave particle gauge theory, suggests that such spin liquid possesses spin correlations that are singular along surfaces in momentum space ("Bose surfaces"). Signatures of this state, which we refer to as a "Spin Bose-Metal" (SBM), are expected to be manifest in quasi-1D ladder systems: The discrete transverse momenta cut through the 2D Bose surface leading to a distinct pattern of 1D gapless modes. Here we search for a quasi-1D descendant of the triangular lattice SBM state by exploring the Heisenberg plus ring model on a two-leg strip (zigzag chain). Using DMRG, variational wavefunctions, and a Bosonization analysis, we map out the full phase diagram. Without ring exchange the model is equivalent to the J_1 - J_2 Heisenberg chain, and we find the expected Bethe-chain and dimerized phases. Remarkably, moderate ring exchange reveals a new gapless phase over a large swath of the phase diagram. Spin and dimer correlations possess particular singular wavevectors and allow us to identify this phase as the hoped for quasi-1D descendant SBM state. We derive a low energy theory and find three gapless modes and one Luttinger parameter controlling all power laws. Potential instabilities out of the zigzag SBM give rise to other interesting phases such as a period-3 VBS or a period-4 Chirality order, which we discover in the DMRG; we also find an interesting SBM state with partial ferromagnetism., Comment: 30 pages, 23 figures

Characterizing and accessing quantum phases of itinerant bosons or fermions in two dimensions (2D) that exhibit singular structure along surfaces in momentum space but have no quasi-particle description remains as a central challenge in the field of strongly correlated physics. Fortuitously, signatures of such 2D strongly correlated phases are expected to be manifest in quasi-one-dimensional "$N$-leg ladder" systems. The ladder discretization of the transverse momentum cuts through the 2D surface, leading to a quasi-1D descendant state with a set of low-energy modes whose number grows with the number of legs and whose momenta are inherited from the 2D surfaces. These multi-mode quasi-1D liquids constitute a new and previously unanticipated class of quantum states interesting in their own right. But more importantly they carry a distinctive quasi-1D "fingerprint" of the parent 2D quantum fluid state. This can be exploited to access the 2D phases from controlled numerical and analytical studies in quasi-1D models. The preliminary successes and future prospects in this endeavor will be briefly summarized., Comment: Proceedings of the 24th Solvay conference on physics - "Quantum theory of condensed matter", Brussels, 11 - 13 October 2008

One of the earliest proposed phase transitions beyond the Landau-Ginzburg-Wilson paradigm is the quantum critical point separating an antiferromagnet and a valence-bond-solid on a square lattice. The low energy description of this transition is believed to be given by the 2+1 dimensional CP(1) model -- a theory of bosonic spinons coupled to an abelian gauge field. Monopole defects of the gauge field play a prominent role in the physics of this phase transition. In the present paper, we use the state-operator correspondence of conformal field theory in conjunction with the 1/N expansion to study monopole operators at the critical fixed point of the CP(N-1) model. This elegant method reproduces the result for monopole scaling dimension obtained through a direct calculation by Murthy and Sachdev. The technical simplicity of our approach makes it the method of choice when dealing with monopole operators in a conformal field theory., Comment: 14 pages, 1 figure

Developing a theoretical framework to access quantum phases of itinerant bosons or fermions in two dimensions (2D) that exhibit singular structure along surfaces in momentum space but have no quasi-particle description remains as a central challenge in the field of strongly correlated physics. In this paper we propose that distinctive signatures of such 2D strongly correlated phases will be manifest in quasi-one-dimensional "N-leg ladder" systems. Characteristic of each parent 2D quantum liquid would be a precise pattern of 1D gapless modes on the N-leg ladder. These signatures could be potentially exploited to approach the 2D phases from controlled numerical and analytical studies in quasi-1D. As a first step we explore itinerant boson models with a frustrating ring exchange interaction on the 2-leg ladder, searching for signatures of the recently proposed two-dimensional d-wave correlated Bose liquid (DBL) phase. A combination of exact diagonalization, density matrix renormalization group, variational Monte Carlo, and bosonization analysis of a quasi-1D gauge theory, all provide compelling evidence for the existence of a new strong-coupling phase of bosons on the 2-leg ladder which can be understood as a descendant of the two-dimensional DBL. We suggest several generalizations to quantum spin and electron Hamiltonians on ladders which could likewise reveal fingerprints of such 2D non-Fermi liquid phases., Comment: 22 pages, 15 figures

We correct an error in our paper Phys. Rev. B 72, 104404 (2005) [cond-mat/0502215]. We show that a particular fermion bilinear is not related to the other ``competing orders'' of the algebraic spin liquid, and does not possess their slowly decaying correlations. For the square lattice staggered flux spin liquid (equivalently, d-wave RVB state), this observable corresponds to the uniform spin chirality., Comment: 1.25 pages

Using the hierarchy picture of the fractional quantum Hall effect, we study the the ground state periodicity of a finite size quantum Hall droplet in a quantum Hall fluid of a different filling factor. The droplet edge charge is periodically modulated with flux through the droplet and will lead to a periodic variation in the conductance of a nearby point contact, such as occurs in some quantum Hall interferometers. Our model is consistent with experiment and predicts that superperiods can be observed in geometries where no interfering trajectories occur. The model may also provide an experimentally feasible method of detecting elusive neutral modes and otherwise obtaining information about the microscopic edge structure in fractional quantum Hall states., Comment: 4 pages, 4 figures

We develop a description of a new quantum liquid phase of interacting bosons in 2d which possesses relative D-wave two-body correlations and which we call a D-wave Bose Liquid (DBL). The DBL has no broken symmetries, supports gapless boson excitations residing on "Bose surfaces" in momentum space, and exhibits power law correlations with continuously variable exponents. While the DBL can be constructed for bosons in the 2d continuum, the state only respects the point group symmetries of the square lattice. On the lattice the DBL respects all symmetries and does not require a particular filling. But lattice effects allow a second distinct phase, a quasi-local variant which we call a D-wave Local Bose Liquid (DLBL). Remarkably, the DLBL has short-range boson correlations and hence no Bose surfaces, despite sharing gapless excitations and other critical signatures with the DBL. Moreover, both phases are metals with a resistance that vanishes as a power of the temperature. We establish these results by constructing a class of many-particle wavefunctions for the DBL, which are time reversal invariant analogs of Laughlin's quantum Hall wavefunction for bosons at $\nu=1/2$. A gauge theory formulation leads to a simple mean field theory, and an N-flavor generalization enables incorporation of gauge field fluctuations to deduce the properties of the DBL/DLBL; various equal time correlation functions are in qualitative accord with the properties inferred from the wavefunctions. We also identify a promising Hamiltonian which might manifest the DBL or DLBL, and perform a variational study comparing to other competing phases. We suggest how the DBL wavefunction can be generalized to describe an itinerant non-Fermi liquid phase of electrons on the square lattice with a no double occupancy constraint, a D-wave metal phase., Comment: 33 pages, 17 figures

There is growing evidence from both experiment and numerical studies that low half-odd integer quantum spins on a kagome lattice with predominant antiferromagnetic near neighbor interactions do not order magnetically or break lattice symmetries even at temperatures much lower than the exchange interaction strength. Moreover, there appear to be a plethora of low energy excitations, predominantly singlets but also spin carrying, which suggest that the putative underlying quantum spin liquid is a gapless ``critical spin liquid'' rather than a gapped spin liquid with topological order. Here, we develop an effective field theory approach for the spin-1/2 Heisenberg model with easy-plane anisotropy on the kagome lattice. By employing a vortex duality transformation, followed by a fermionization and flux-smearing, we obtain access to a gapless yet stable critical spin liquid phase, which is described by (2+1)-dimensional quantum electrodynamics (QED$_3$) with an emergent $\mathrm{SU}(8)$ flavor symmetry. The specific heat, thermal conductivity, and dynamical structure factor are extracted from the effective field theory, and contrasted with other theoretical approaches to the kagome antiferromagnet., Comment: 14 pages, 8 figures

When cold atoms are trapped in a square or cubic optical lattice, it should be possible to pump the atoms into excited $p-$level orbitals within each well. Following earlier work, we explore the metastable equilibrium that can be established before the atoms decay into the $s-$wave orbital ground state. We will discuss the situation with integer number of bosons on every site, and consider the strong correlation "insulating" regime. By employing a spin-wave analysis together with a new duality transformation, we establish the existence and stability of a novel gapless "critical phase", which we refer to as a "bond algebraic liquid". The gapless nature of this phase is stabilized due to the emergence of symmetries which lead to a quasi-one dimensional behavior. Within the algebraic liquid phase, both bond operators and particle flavor occupation number operators have correlations which decay algebraically in space and time. Upon varying parameters, the algebraic bond liquid can be unstable to either a Mott insulator phase which spontaneously breaks lattice symmetries, or a $\mathbb{Z}_2$ phase. The possibility of detecting the algebraic liquid phase in cold atom experiments is addressed. Although the momentum distribution function is insufficient to distinguish the algebraic bond liquid from other phases, the density correlation function can in principle be used to detect this new phase of matter., Comment: 15 pages, 10 figures

Although magnetically ordered at low temperatures, the spin-1/2 triangular antiferromagnet Cs_2CuCl_4 exhibits remarkable spin dynamics that strongly suggest proximity to a spin liquid phase. Here we address the question of whether a proximate spin liquid may also occur in an applied magnetic field, leaving a similar imprint on the dynamical spin correlations of this material. Specifically, we explore a spatially anisotropic Heisenberg spin-1/2 triangular antiferromagnet at 1/3 magnetization from a dual vortex perspective, and indeed find a new ``critical'' spin liquid phase described by QED3 with an emergent SU(6) symmetry. A number of nontrivial predictions are given for the dynamical spin structure factor in this ``algebraic vortex liquid'' phase, which can be tested experimentally via inelastic neutron scattering. We also discuss how the well-studied ``up-up-down'' magnetization plateaus can be captured within our approach, and further predict the existence of a stable gapless solid phase in a weakly ordered up-up-down state. Finally, we predict several anomalous ``roton'' minima in the excitation spectrum in the regime of lattice anisotropy where the canted Neel state appears., Comment: 5 pages, 2 figures; expanded intro & discussion of theory; minor correction to structure factor

We explore spin-1/2 triangular antiferromagnets with both easy-plane and lattice exchange anisotropies by employing a dual vortex mapping followed by a fermionization of the vortices. Over a broad range of exchange anisotropy, this approach leads naturally to a ``critical'' spin liquid--the algebraic vortex liquid--which appears to be distinct from other known spin liquids. We present a detailed characterization of this state, which is described in terms of non-compact QED3 with an emergent SU(4) symmetry. Descendant phases of the algebraic vortex liquid are also explored, which include the Kalmeyer-Laughlin spin liquid, a variety of magnetically ordered states such as the well known coplanar spiral state, and supersolids. In the range of exchange anisotropy where the ``square lattice'' Neel ground state arises, we demonstrate that anomalous ``roton'' minima in the excitation spectrum recently reported in series expansions can be accounted for within our approach., Comment: 20 pages, 10 figures

We present a pedagogical overview of recent theoretical work on unconventional quantum phases and quantum phase transitions in condensed matter systems. Strong correlations between electrons can lead to a breakdown of two traditional paradigms of solid state physics: Landau's theories of Fermi liquids and phase transitions. We discuss two resulting "exotic" states of matter: topological and critical spin liquids. These two quantum phases do not display any long-range order even at zero temperature. In each case, we show how a gauge theory description is useful to describe the new concepts of topological order, fractionalization and deconfinement of excitations which can be present in such spin liquids. We make brief connections, when possible, to experiments in which the corresponding physics can be probed. Finally, we review recent work on deconfined quantum critical points. The tone of these lecture notes is expository: focus is on gaining a physical picture and understanding, with technical details kept to a minimum., Comment: 22 pages, 15 figures; Notes of the Lectures at the International Summer School on Fundamental Problems in Statistical Physics XI, September 2005, Leuven, Belgium; High-resolution version available at http://w3-phystheo.ups-tlse.fr/~alet/leuven.html

A number of examples have demonstrated the failure of the Landau-Ginzburg-Wilson(LGW) paradigm in describing the competing phases and phase transitions of two dimensional quantum magnets. In this paper we argue that such magnets possess field theoretic descriptions in terms of their slow fluctuating orders provided certain topological terms are included in the action. These topological terms may thus be viewed as what goes wrong within the conventional LGW thinking. The field theoretic descriptions we develop are possible alternates to the popular gauge theories of such non-LGW behavior. Examples that are studied include weakly coupled quasi-one dimensional spin chains, deconfined critical points in fully two dimensional magnets, and two component massless $QED_3$. A prominent role is played by an anisotropic O(4) non-linear sigma model in three space-time dimensions with a topological theta term. Some properties of this model are discussed. We suggest that similar sigma model descriptions might exist for fermionic algebraic spin liquid phases., Comment: 11 pages, 1 figure

Motivated by inelastic neutron scattering data on Cs_2CuCl_4, we explore spin-1/2 triangular lattice antiferromagnets with both spatial and easy-plane exchange anisotropies, the latter due to an observed Dzyaloshinskii-Moriya interaction. Exploiting a duality mapping followed by a fermionization of the dual vortex degrees of freedom, we find a novel "critical" spin-liquid phase described in terms of Dirac fermions with an emergent global SU(4) symmetry minimally coupled to a non-compact U(1) gauge field. This ``algebraic vortex liquid" supports gapless spin excitations and universal power-law correlations in the dynamical spin structure factor which are consistent with those observed in Cs_2CuCl_4. We suggest future neutron scattering experiments that should help distinguish between the algebraic vortex liquid and other spin liquids and quantum critical points previously proposed in the context of Cs_2CuCl_4., Comment: 4 pages, 3 figures; minor revisions, momenta in Fig. 2 corrected