Emergence and spontaneous breaking of approximate O(4) symmetry at a weakly first-order deconfined phase transition

We investigate approximate emergent nonabelian symmetry in a class of weakly first order `deconfined' phase transitions using Monte Carlo simulations and a renormalization group analysis. We study a transition in a 3D classical loop model that is analogous to a deconfined 2+1D quantum phase transition in a magnet with reduced lattice symmetry. The transition is between the N\'eel phase and a twofold degenerate valence bond solid (lattice-symmetry-breaking) phase. The combined order parameter at the transition is effectively a four-component superspin. It has been argued that in some weakly first order `pseudocritical' deconfined phase transitions, the renormalization group flow can take the system very close to the ordered fixed point of the symmetric $O(N)$ sigma model, where $N$ is the total number of `soft' order parameter components, despite the fact that $O(N)$ is not a microscopic symmetry. This yields a first order transition with unconventional phenomenology. We argue that this occurs in the present model, with $N=4$. This means that there is a regime of lengthscales in which the transition resembles a `spin-flop' transition in the ordered $O(4)$ sigma model. We give numerical evidence for (i) the first order nature of the transition, (ii) the emergence of $O(4)$ symmetry to an accurate approximation, and (iii) the existence of a regime in which the emergent $O(4)$ is `spontaneously broken', with distinctive features in the order parameter probability distribution. These results may be relevant for other models studied in the literature, including 2+1D QED with two flavours, the `easy-plane' deconfined critical point, and the N\'eel--VBS transition on the rectangular lattice.Comment: 16 pages. v2: updated to journal versio

Topological Paramagnetism in Frustrated Spin-One Mott Insulators

Time reversal protected three dimensional (3D) topological paramagnets are magnetic analogs of the celebrated 3D topological insulators. Such paramagnets have a bulk gap, no exotic bulk excitations, but non-trivial surface states protected by symmetry. We propose that frustrated spin-1 quantum magnets are a natural setting for realising such states in 3D. We describe a physical picture of the ground state wavefunction for such a spin-1 topological paramagnet in terms of loops of fluctuating Haldane chains with non-trivial linking phases. We illustrate some aspects of such loop gases with simple exactly solvable models. We also show how 3D topological paramagnets can be very naturally accessed within a slave particle description of a spin-1 magnet. Specifically we construct slave particle mean field states which are naturally driven into the topological paramagnet upon including fluctuations. We propose bulk projected wave functions for the topological paramagnet based on this slave particle description. An alternate slave particle construction leads to a stable U(1) quantum spin liquid from which a topological paramagnet may be accessed by condensing the emergent magnetic monopole excitation of the spin liquid.Comment: 16 pages, 5 figure

Valence Bonds in Random Quantum Magnets: Theory and Application to YbMgGaO4

We analyze the effect of quenched disorder on spin-1/2 quantum magnets in which magnetic frustration promotes the formation of local singlets. Our results include a theory for 2d valence-bond solids subject to weak bond randomness, as well as extensions to stronger disorder regimes where we make connections with quantum spin liquids. We find, on various lattices, that the destruction of a valence-bond solid phase by weak quenched disorder leads inevitably to the nucleation of topological defects carrying spin-1/2 moments. This renormalizes the lattice into a strongly random spin network with interesting low-energy excitations. Similarly when short-ranged valence bonds would be pinned by stronger disorder, we find that this putative glass is unstable to defects that carry spin-1/2 magnetic moments, and whose residual interactions decide the ultimate low energy fate. Motivated by these results we conjecture Lieb-Schultz-Mattis-like restrictions on ground states for disordered magnets with spin-1/2 per statistical unit cell. These conjectures are supported by an argument for 1d spin chains. We apply insights from this study to the phenomenology of YbMgGaO$_4$, a recently discovered triangular lattice spin-1/2 insulator which was proposed to be a quantum spin liquid. We instead explore a description based on the present theory. Experimental signatures, including unusual specific heat, thermal conductivity, and dynamical structure factor, and their behavior in a magnetic field, are predicted from the theory, and compare favorably with existing measurements on YbMgGaO$_4$ and related materials.Comment: v2: Stylistic revisions to improve clarity. 22 pages, 8 figures, 2 tables main text; 13 pages, 3 figures appendice

Topological Constraints in Directed Polymer Melts

Polymers in a melt may be subject to topological constraints, as in the example of unlinked polymer rings. How to do statistical mechanics in the presence of such constraints remains a fundamental open problem. We study the effect of topological constraints on a melt of directed polymers, using simulations of a simple quasi-2D model. We find that fixing the global topology of the melt to be trivial changes the polymer conformations drastically. Polymers of length $L$ wander in the transverse direction only by a distance of order $(\ln L)^\zeta$ with $\zeta \simeq 1.5$. This is strongly suppressed in comparison with the Brownian $L^{1/2}$ scaling which holds in the absence of the topological constraint. It is also much smaller than the predictions of standard heuristic approaches - in particular the $L^{1/4}$ of a mean-field-like `array of obstacles' model - so our results present a sharp challenge to theory. Dynamics are also strongly affected by the constraints, and a tagged monomer in an infinite system performs logarithmically slow subdiffusion in the transverse direction. To cast light on the suppression of the strands' wandering, we analyse the topological complexity of subregions of the melt: the complexity is also logarithmically small, and is related to the wandering by a power law. We comment on insights the results give for 3D melts, directed and non-directed.Comment: 4 pages + appendices, 11 figures. Published versio

Measurement-Induced Phase Transitions in the Dynamics of Entanglement

We define dynamical universality classes for many-body systems whose unitary evolution is punctuated by projective measurements. In cases where such measurements occur randomly at a finite rate $p$ for each degree of freedom, we show that the system has two dynamical phases: `entangling' and `disentangling'. The former occurs for $p$ smaller than a critical rate $p_c$, and is characterized by volume-law entanglement in the steady-state and `ballistic' entanglement growth after a quench. By contrast, for $p > p_c$ the system can sustain only area-law entanglement. At $p = p_c$ the steady state is scale-invariant and, in 1+1D, the entanglement grows logarithmically after a quench. To obtain a simple heuristic picture for the entangling-disentangling transition, we first construct a toy model that describes the zeroth R\'{e}nyi entropy in discrete time. We solve this model exactly by mapping it to an optimization problem in classical percolation. The generic entangling-disentangling transition can be diagnosed using the von Neumann entropy and higher R\'{e}nyi entropies, and it shares many qualitative features with the toy problem. We study the generic transition numerically in quantum spin chains, and show that the phenomenology of the two phases is similar to that of the toy model, but with distinct `quantum' critical exponents, which we calculate numerically in $1+1$D. We examine two different cases for the unitary dynamics: Floquet dynamics for a nonintegrable Ising model, and random circuit dynamics. We obtain compatible universal properties in each case, indicating that the entangling-disentangling phase transition is generic for projectively measured many-body systems. We discuss the significance of this transition for numerical calculations of quantum observables in many-body systems.Comment: 17+4 pages, 16 figures; updated discussion and results for mutual information; graphics error fixe

Operator Spreading in Random Unitary Circuits

Random quantum circuits yield minimally structured models for chaotic quantum dynamics, able to capture for example universal properties of entanglement growth. We provide exact results and coarse-grained models for the spreading of operators by quantum circuits made of Haar-random unitaries. We study both 1+1D and higher dimensions, and argue that the coarse-grained pictures carry over to operator spreading in generic many-body systems. In 1+1D, we demonstrate that the out-of-time-order correlator (OTOC) satisfies a biased diffusion equation, which gives exact results for the spatial profile of the OTOC, and the butterfly speed $v_{B}$. We find that in 1+1D the `front' of the OTOC broadens diffusively, with a width scaling in time as $t^{1/2}$. We address fluctuations in the OTOC between different realizations of the random circuit, arguing that they are negligible in comparison to the broadening of the front. Turning to higher D, we show that the averaged OTOC can be understood exactly via a remarkable correspondence with a classical droplet growth problem. This implies that the width of the front of the averaged OTOC scales as $t^{1/3}$ in 2+1D and $t^{0.24}$ in 3+1D (KPZ exponents). We support our analytic argument with simulations in 2+1D. We point out that, in a lattice model, the late time shape of the spreading operator is in general not spherical. However when full spatial rotational symmetry is present in 2+1D, our mapping implies an exact asymptotic form for the OTOC in terms of the Tracy-Widom distribution. For an alternative perspective on the OTOC in 1+1D, we map it to the partition function of an Ising-like model. As a result of special structure arising from unitarity, this partition function reduces to a random walk calculation which can be performed exactly. We also use this mapping to give exact results for entanglement growth in 1+1D circuits.Comment: 29 pages, 16 figures. v2: new appendix on 'mean field

Dynamics of entanglement and transport in 1D systems with quenched randomness

Quenched randomness can have a dramatic effect on the dynamics of isolated 1D quantum many-body systems, even for systems that thermalize. This is because transport, entanglement, and operator spreading can be hindered by `Griffiths' rare regions which locally resemble the many-body-localized phase and thus act as weak links. We propose coarse-grained models for entanglement growth and for the spreading of quantum operators in the presence of such weak links. We also examine entanglement growth across a single weak link numerically. We show that these weak links have a stronger effect on entanglement growth than previously assumed: entanglement growth is sub-ballistic whenever such weak links have a power-law probability distribution at low couplings, i.e. throughout the entire thermal Griffiths phase. We argue that the probability distribution of the entanglement entropy across a cut can be understood from a simple picture in terms of a classical surface growth model. Surprisingly, the four length scales associated with (i) production of entanglement, (ii) spreading of conserved quantities, (iii) spreading of operators, and (iv) the width of the `front' of a spreading operator, are characterized by dynamical exponents that in general are all distinct. Our numerical analysis of entanglement growth between weakly coupled systems may be of independent interest.Comment: 17 pages, 16 figure

Quantum Entanglement Growth Under Random Unitary Dynamics

Characterizing how entanglement grows with time in a many-body system, for example after a quantum quench, is a key problem in non-equilibrium quantum physics. We study this problem for the case of random unitary dynamics, representing either Hamiltonian evolution with time--dependent noise or evolution by a random quantum circuit. Our results reveal a universal structure behind noisy entanglement growth, and also provide simple new heuristics for the `entanglement tsunami' in Hamiltonian systems without noise. In 1D, we show that noise causes the entanglement entropy across a cut to grow according to the celebrated Kardar--Parisi--Zhang (KPZ) equation. The mean entanglement grows linearly in time, while fluctuations grow like $(\text{time})^{1/3}$ and are spatially correlated over a distance $\propto (\text{time})^{2/3}$. We derive KPZ universal behaviour in three complementary ways, by mapping random entanglement growth to: (i) a stochastic model of a growing surface; (ii) a `minimal cut' picture, reminiscent of the Ryu--Takayanagi formula in holography; and (iii) a hydrodynamic problem involving the dynamical spreading of operators. We demonstrate KPZ universality in 1D numerically using simulations of random unitary circuits. Importantly, the leading order time dependence of the entropy is deterministic even in the presence of noise, allowing us to propose a simple `minimal cut' picture for the entanglement growth of generic Hamiltonians, even without noise, in arbitrary dimensionality. We clarify the meaning of the `velocity' of entanglement growth in the 1D `entanglement tsunami'. We show that in higher dimensions, noisy entanglement evolution maps to the well-studied problem of pinning of a membrane or domain wall by disorder