Spontaneous breaking of U(1) symmetry in coupled complex SYK models

Abstract

As shown in [1], two copies of the large N Majorana SYK model can produce spontaneous breaking of a Z2 symmetry when they are coupled by appropriate quartic terms. In this paper we similarly study two copies of the complex SYK model coupled by a quartic term preserving the U(1) × U(1) symmetry. We also present a tensor counterpart of this coupled model. When the coefficient α of the quartic term lies in a certain range, the coupled large N theory is nearly conformal. We calculate the scaling dimensions of fermion bilinear operators as functions of α. We show that the operator $$ {c}_{1i}^{\dagger }{c}_{2i} $$ c 1 i † c 2 i , which is charged under the axial U(1) symmetry, acquires a complex dimension outside of the line of fixed points. We derive the large N Dyson-Schwinger equations and show that, outside the fixed line, this U(1) symmetry is spontaneously broken at low temperatures because this operator acquires an expectation value. We support these findings by exact diagonalizations extrapolated to large N.

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“…Interestingly, such Schwarzian action matches the boundary action of the Jackiw-Teitelboim gravity in twodimensional nearly anti-de Sitter (AdS2) spacetime, and consequently, the SYK model is a concrete model with low-energy holographic description. Later, the model is generalized to study various problems in both high energy and condensed matter communities [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].…”

Instability of complex CFTs with operators in the principal series

“…Moreover, let the equation k σ (h, J) = 1 admit, for some fixed J and σ, a simple root of the form h = h ≡ d 2 + i r , with r ∈ R and different from zero. 16 As a comparison to previous literature, we notice that restricting to a1 = a2 and considering only fields with the same dimension ∆, only the even spin contributions survive, and we havê…”

“…Many examples of such merging of fixed points are known, see for example the list of references in [3]. In some instances, a special case of complex scaling dimension is found, one whose real part is equal to d/2, d being the space(-time) dimension; examples include non-supersymmetric orbifolds of N = 4 super Yang-Mills [4][5][6], gauge theories with matter in the Veneziano limit [7,8], and large-N theories dominated by melonic diagrams [9][10][11][12][13][14][15][16][17], or by fishnet diagrams [18][19][20][21][22][23]. The typical large-N mechanism leading to complex dimensions of such type is the following [6]: due to the large-N simplifications, the beta function…”

AdS3 gravity and the complex SYK models

“…1 An important ingredient in the JT/SYK or nAdS 2 /nCFT 1 correspondence is the emergence of conformal symmetry, and the pattern of symmetry breaking. In [23], a similar emergent symmetry and breaking pattern have been discussed for the complex SYK models, which involve Dirac fermions instead of Majorana [24][25][26][27][28]. The effective action of the complex SYK models takes the general form…”