Stiefel Liquids: Possible Non-Lagrangian Quantum Criticality from Intertwined Orders

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Stiefel Liquids: Possible Non-Lagrangian Quantum Criticality from Intertwined Orders

Liujun Zou, Yin-Chen He, and Chong Wang

Phys. Rev. X 11, 031043 – Published 25 August 2021

Abstract

We propose a new type of quantum liquids, dubbed Stiefel liquids, based on ( $2 + 1$ )-dimensional nonlinear sigma models on target space $S O (N) / S O (4)$ , supplemented with Wess-Zumino-Witten terms. We argue that the Stiefel liquids form a class of critical quantum liquids with extraordinary properties, such as large emergent symmetries, a cascade structure, and nontrivial quantum anomalies. We show that the well-known deconfined quantum critical point and $U (1)$ Dirac spin liquid are unified as two special examples of Stiefel liquids, $N = 5$ and $N = 6$ , respectively. Furthermore, we conjecture that Stiefel liquids with $N > 6$ are non-Lagrangian, in the sense that under renormalization group they flow to infrared (conformally invariant) fixed points that cannot be described by any renormalizable continuum Lagrangian. Such non-Lagrangian states are beyond the paradigm of parton gauge mean-field theory familiar in the study of exotic quantum liquids in condensed matter physics. The intrinsic absence of (conventional or partonlike) mean-field construction also means that, within the traditional approaches, will be difficult to decide whether a non-Lagrangian state can actually emerge from a specific UV system (such as a lattice spin system). For this purpose we hypothesize that a quantum state is emergible from a lattice system if its quantum anomalies match with the constraints from the (generalized) Lieb-Schultz-Mattis theorems. Based on this hypothesis, we find that some of the non-Lagrangian Stiefel liquids can indeed be realized in frustrated quantum spin systems, for example, on triangular or kagome lattice, through the intertwinement between noncoplanar magnetic orders and valence-bond-solid orders.

Received 31 January 2021
Revised 10 June 2021
Accepted 30 June 2021

DOI:https://doi.org/10.1103/PhysRevX.11.031043

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Anomalies Conformal field theory Gauge theories Quantum criticality Quantum spin liquid

Condensed Matter, Materials & Applied PhysicsParticles & FieldsStatistical Physics & Thermodynamics

Authors & Affiliations

Liujun Zou , Yin-Chen He , and Chong Wang

Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5

Popular Summary

To date, most, if not all, known quantum states relevant to the studies of quantum matter can be described by the framework of renormalizable quantum field theories, that is, quantum field theories constructed of particles that are weakly interacting when they are close to each other. An important question is, are there quantum states that cannot be captured by this framework? Here, we suggest there are.

Specifically, we propose an infinite family of novel quantum states. These states, we argue, are “quantum critical,” meaning that they possess intricate structures of quantum entanglement so that they can serve as parent states of many other states—perturbing them results in other states. Perhaps what is most remarkable is that these novel states appear to be non-Lagrangian: They cannot be described by renormalizable quantum field theories but only by nonrenormalizable ones. Furthermore, we argue that these states can arise in experimentally relevant systems because of the competition between intertwined symmetry-breaking orders.

This work raises the interesting possibility that non-Lagrangian quantum states can emerge in realistic quantum matter, challenging the established framework to understand these states, and opening a whole new landscape of quantum states to be explored.

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Vol. 11, Iss. 3 — July - September 2021

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Figure 1
The Stiefel liquid out of intertwined orders in quantum magnets. (a) $(N = 5, k = 1)$ SL is the widely studied deconfined phase transition, which can arise from the intertwinement of collinear magnetic order (e.g., Néel state) and valence bond solid. (b) $(N = 6, k = 1)$ SL is the widely studied $U (1)$ Dirac spin liquid, which can arise from the intertwinement of noncollinear magnetic order (but coplanar) and valence bond solid. (c) $(N = 7, k = 1)$ SL is a new critical quantum liquid, which can arise from the intertwinement of noncoplanar magnetic order and valence bond solid. On triangular lattice the noncoplanar order is known as the tetrahedral order. On kagome lattice the noncoplanar order is known as the cuboctahedral order, in which the magnetizations on the three sublattices are $S_{A} = - Q_{3} \cos (n π) - Q_{1} \cos [(m + n) π]$ , $S_{B} = Q_{3} \cos (n π) - Q_{2} \cos (m π)$ , $S_{C} = Q_{2} \cos (m π) + Q_{1} \cos [(m + n) π]$ , with $Q_{1, 2, 3}$ being orthogonal to each other.
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Figure 2
The cascade structure of Stiefel liquids.
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Figure 3
Fixed point structure and schematic phase diagram of ${SL}^{(N, k)}$ . In (a), different fixed point structures for different values of $α$ are shown. There is always an attractive fixed point, represented by the red circle, which corresponds to the symmetry-broken state. The structure of the other fixed points depends on the relation between $α$ and $α_{c} \sim O (1)$ . The precise value of $α_{c}$ depends on $β_{0}$ and $C$ . If $α < α_{c}$ , there are two other fixed points, a repulsive one represented by the blue circle, corresponding to an order-disorder transition, and an attractive one, represented by the yellow circle, corresponding to a stable critical quantum liquid, i.e., the SL. As $α$ increases and approaches $α_{c}$ , the blue and yellow fixed points approach each other and collide when $α = α_{c}$ . When $α > α_{c}$ , the original blue and yellow fixed points disappear (become complex fixed points [25, 86]). We would like to take $α$ below $α_{c}$ . In (b), a schematic phase diagram of ${SL}^{(N, k)}$ is shown. For the $(N, k)$ in the critical regime, it is possible to tune the parameters of the system to yield a critical quantum liquid, while in the symmetry-breaking regime this is not possible. The yellow star represents $(N, k) = (6, 1)$ , i.e., the $U (1)$ DSL, and the black star represents $(N, k) = (5, 1)$ , the DQCP. The precise boundary between the critical and symmetry-breaking regimes is currently undetermined.
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Figure 4
Square lattice and the relevant symmetries. Each filled red circle represents an odd number of spin- $1 / 2$ ’s. The $C_{2}$ rotation is around a site that hosts the spins, and the dashed line is the reflection axis of $R_{y}$ .
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Figure 5
Triangular lattice and the relevant symmetries. Each filled red circle represents an odd number of spin- $1 / 2$ ’s. The $C_{6}$ rotation is around a site that host the spins, and the dashed line is the reflection axis of $R_{y}$ .
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Figure 6
Kagome lattice and the relevant symmetries. Each filled red circle represents an odd number of spin- $1 / 2$ ’s. In this case the rotation center of $C_{6}$ does not host any spin. Again, the dashed line is the reflection axis of $R_{y}$ .
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