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arXiv:1912.12315 · Physical review. B./Physical review. B

Theory of deconfined pseudocriticality

Ruochen Ma, Chong Wang

Published Jul 21, 2020• 92 citations
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Abstract

It has been proposed that the deconfined criticality in $(2+1)d$---the quantum phase transition between a N\'eel antiferromagnet and a valence-bond solid (VBS)---may actually be pseudocritical, in the sense that it is a weakly first-order transition with a generically long correlation length. The underlying field theory of the transition would be a slightly complex (nonunitary) fixed point as a result of fixed points annihilation. This proposal was motivated by existing numerical results from large scale Monte Carlo simulations as well as a conformal bootstrap. However, an actual theory of such a complex fixed point, incorporating key features of the transition such as the emergent SO(5) symmetry, is so far absent. Here we propose a Wess-Zumino-Witten (WZW) nonlinear sigma model with level $k=1$, defined in $2+\ensuremath{\epsilon}$ dimensions, with target space ${S}^{3+\ensuremath{\epsilon}}$ and global symmetry $\text{SO}(4+\ensuremath{\epsilon})$. This gives a formal interpolation between the deconfined criticality at $d=3$ and the $\text{SU}{(2)}_{1}$ WZW theory at $d=2$ describing the spin-$1/2$ Heisenberg chain. The theory can be formally controlled, at least to leading order, in terms of the inverse of the WZW level $1/k$. We show that at leading order there is a fixed point annihilation at ${d}^{*}\ensuremath{\approx}2.77$, with complex fixed points above this dimension including the physical $d=3$ case. The pseudocritical properties such as correlation length, scaling dimensions, and the drifts of scaling dimensions as the system size increases, calculated crudely to leading order, are qualitatively consistent with existing numerics.

DOI
10.1103/physrevb.102.020407

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It has been proposed that the deconfined criticality in (2+1)d(2+1)d(2+1)d---the quantum phase transition between a N'eel antiferromagnet and a valence-bond solid (VBS)---may actually be pseudocritical, in the sense that it is a weakly first-order transition with a generically long correlation length. The underlying field theory of the transition would be a slightly complex (nonunitary) fixed point as a result of fixed points annihilation. This proposal was motivated by existing numerical results from large scale Monte Carlo simulations as well as a conformal bootstrap. However, an actual theory of such a complex fixed point, incorporating key features of the transition such as the emergent SO(5) symmetry, is so far absent. Here we propose a Wess-Zumino-Witten (WZW) nonlinear sigma model with level k=1k=1k=1, defined in 2+\ensuremathϵ2+\ensuremath{\epsilon}2+\ensuremathϵ dimensions, with target space S3+\ensuremathϵ{S}^{3+\ensuremath{\epsilon}}S3+\ensuremathϵ and global symmetry SO(4+\ensuremathϵ)\text{SO}(4+\ensuremath{\epsilon})SO(4+\ensuremathϵ). This gives a formal interpolation between the deconfined criticality at d=3d=3d=3 and the SU(2)1\text{SU}{(2)}_{1}SU(2)1​ WZW theory at d=2d=2d=2 describing the spin-1/21/21/2 Heisenberg chain. The theory can be formally controlled, at least to leading order, in terms of the inverse of the WZW level 1/k1/k1/k. We show that at leading order there is a fixed point annihilation at d∗\ensuremath≈2.77{d}^{*}\ensuremath{\approx}2.77d∗\ensuremath≈2.77, with complex fixed points above this dimension including the physical d=3d=3d=3 case. The pseudocritical properties such as correlation length, scaling dimensions, and the drifts of scaling dimensions as the system size increases, calculated crudely to leading order, are qualitatively consistent with existing numerics.
We suggest the possibility that the two-dimensional SU(2)k\mathrm{SU}{(2)}_{k}SU(2)k​ Wess-Zumino-Witten (WZW) theory, which has global SO(4)\mathrm{SO}(4)SO(4) symmetry, can be continued to $2+\ensuremath{\epsilon}$ dimensions by enlarging the symmetry to $\mathrm{SO}(4+\ensuremath{\epsilon})$. This is motivated by the three-dimensional sigma model with SO(5)\mathrm{SO}(5)SO(5) symmetry and a WZW term, which is relevant to deconfined criticality. If such a continuation exists, the structure of the renormalization group flows at small $\ensuremath{\epsilon}$ may be fixed by assuming analyticity in $\ensuremath{\epsilon}$. This leads to the conjecture that the WZW fixed point annihilates with a new, unstable fixed point at a critical dimensionality dc>2{d}_{c}>2dc​>2. We suggest that dc<3{d}_{c}<3dc​<3 for all kkk, and we compute dc{d}_{c}dc​ in the limit of large kkk. The flows support the conjecture that the deconfined phase transition in SU(2)\mathrm{SU}(2)SU(2) magnets is a ``pseudocritical'' point with approximate SO(5)\mathrm{SO}(5)SO(5), controlled by a fixed point slightly outside the physical parameter space.
Abstract Berry phase interference arguments that underlie the theory of deconfined quantum criticality (DQC) for S = 1 / 2 antiferromagnets have also been invoked to allow for continuous transitions in S = 1 magnets including a Néel to (columnar) valence bond solid (cVBS) transition. We provide a microscopic model realization of this transition on the square lattice consisting of Heisenberg exchange ( J H ) and biquadratic exchange ( J B ) that favor a Néel phase, and a designed Q -term ( Q B ) interaction which favors a cVBS through large-scale quantum Monte Carlo (QMC) simulations. For J H = 0 , this model is equivalent to the SU(3) JQ model with a Néel-cVBS transition that has been argued to be DQC through QMC. Upon turning on J H which brings down the symmetry to SU(2), we find multiple signatures—a single critical point, high quality collapse of correlation ratios and order parameters, ‘U(1)-symmetric’ cVBS histograms and lack of double-peak in order parameter histograms for largest sizes studied near the critical point—that are highly suggestive of a continuous transition scenario. However, Binder analysis finds negative dips that grow sub-extensively that we interpret as these transitions rather being pseudocritical. This, along with recent results on spin- 1 2 models, suggests that deconfined pseudocriticality is the more generic scenario.
We resolve the nature of the quantum phase transition between antiferromagnetic and valence-bond solid ground states in square-lattice spin-1/2 magnets - the putative deconfined quantum-critical point. Using quantum Monte Carlo simulations of J-Q models, in which Heisenberg exchange J competes with multi-spin interactions Qn formed by n singlet projectors, we demonstrate first-order discontinuities increasing with n. For n=2 and n=3, the first-order signatures are very weak and we extract universal scaling dimensions of a proximate multicritical point. Their values coincide closely with known bounds for an SO(5) symmetric conformal field theory (CFT) containing a relevant SO(5) singlet operator that destabilizes the generic deconfined critical point. We characterize the emergent SO(5) symmetry by its scaling dimensions of the leading irrelevant perturbations. Using the J-Q2-Q6 model, we tune the transition from weak to stronger first-order and find scaling of the coexisting order parameters with a large exponent beta ~ 0.85, in agreement with the relevant scaling dimensions of the multicritical point. The exponent that had previously been identified with the correlation length does not have a corresponding CFT level, and we explain it as pseudo-critical length scale of bubbles of one phase inside the other that are compressed in size by an SO(5)-violating irrelevant repulsive interaction between the two order parameters. The bubble scale reflects a hierarchy of excitation gaps generated by the SO(5) perturbation in the neighborhood of the first-order transition, akin to a Higgs mechanism. We argue that the multicritical point is the tip of a gapless spin liquid phase in frustrated Heisenberg models into which the J-Q models can be deformed, but the spin liquid, and perhaps the multicritical point itself, exist outside the regime amenable to sign-free simulations.
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Square-lattice J-Q models provide a sign-problem-free setting for probing the quantum phase transition between Néel antiferromagnet and columnar valence-bond solid. We analyze this transition through the scaling of Lee--Yang zeros, computed within stochastic series expansion quantum Monte Carlo by reweighting configurations sampled near criticality in the presence of complex source fields. Benchmark studies of the dimerized Heisenberg model and the checkerboard J-Q model validate the method, yielding stable O(3)\mathrm{O}(3)O(3) critical scaling in the former and clear spacetime-volume scaling in the latter, as expected for a first-order transition. Applying the same analysis to the J-Q models, we find a pronounced and systematic drift of the leading-zero scaling with increasing system size, consistent with an extended pseudocritical regime. The Lee--Yang scaling implies an effective scaling dimension ΔϕΔ_ϕΔϕ​ of the SO(5)\mathrm{SO}(5)SO(5) order-parameter field that decreases with size and is consistent with vanishing in the thermodynamic limit. Such behavior lies below the scalar unitarity bound of any unitary relativistic conformal field theory in 2+12{+}12+1 dimensions and enforces inverse spacetime-volume scaling of the zeros, the hallmark of a first-order transition. These results support a weakly first-order interpretation of the Néel--VBS transition and establish the finite-size Lee--Yang zeros as a sensitive, symmetry-resolved diagnostic of pseudocriticality and transition order in the J-Q family.